Research ArticleStability and Bogdanov-Takens Bifurcation of an SIS EpidemicModel with Saturated Treatment Function

Yanju Xiao1 Weipeng Zhang1 Guifeng Deng2 and Zhehua Liu1

1School of Mathematics and Statistics Northeast Normal University Changchun Jilin 130024 China2School of Mathematics and Information Shanghai Lixin University of Commerce Shanghai 201620 China

Correspondence should be addressed to Weipeng Zhang wpzhang808163com

Received 11 July 2014 Revised 18 October 2014 Accepted 28 October 2014

Academic Editor Shifei Ding

Copyright copy 2015 Yanju Xiao et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper introduces the global dynamics of an SIS model with bilinear incidence rate and saturated treatment function Thetreatment function is a continuous and differential function which shows the effect of delayed treatment when the rate of treatmentis lower and the number of infected individuals is getting larger Sufficient conditions for the existence and global asymptotic stabilityof the disease-free and endemic equilibria are given in this paper The first Lyapunov coefficient is computed to determine varioustypes of Hopf bifurcation such as subcritical or supercritical By some complex algebra the Bogdanov-Takens normal form andthe three types of bifurcation curves are derived Finally mathematical analysis and numerical simulations are given to support ourtheoretical results

1 Introduction

In the mathematical modeling of epidemic transmissionthere are several factors that substantially affect the dynamicalbehavior of the models Incidence rate functions are seen asa major factor in producing the rich dynamics of epidemicmodels in many literatures (see [1ndash25]) In most classicalmodels of epidemics the incidence rate is taken to be massaction incidence with bilinear interaction that is 120573119868119878 where120573 is the probability of transmission per contact and 119878 and 119868represent the number of susceptible and infected individualsrespectively Epidemic models with such bilinear incidencerates usually have at most one endemic equilibrium andthen the diseases will be eradicated if the basic reproductionnumber is less than one and will persist otherwise Besidesthere are also many other types of incidence rate func-tions such as nonlinear incidence rate standard incidencerate and saturated incidence rate Recently there are manystudies that have demonstrated the nonlinear incidence ratewhich is one of the key factors that induce periodic oscil-lations in epidemic models (see [1 2 9 10 15]) Moreover

Liu et al [12] introduced a nonlinear incidence rate of theform

119891 (119868) 119878 =120573119868119901119878

1 + 120572119868119902 (1)

where 120573119868119901 means the infection force of the disease and1(1+120572119868119902)measures the inhibition effect from the behavioralchange of the susceptible individuals when the number ofinfective individuals increases So we can see that the bilinearincidence rate 120573119878119868 is a special case of (1) with 119901 = 1and 120572 = 0 or 119902 = 0 Furthermore Wang and Ruan in[16] studied the global dynamics of an SIRS model withthe incidence function 119891(119868) = 1205731198682(1 + 1205721198682) that is 119901 =119902 = 2 they also showed that the SIRS epidemic modelundergoes a Bogdanov-Takens bifurcation that is saddle-node Hopf and homoclinic bifurcations To have a betterunderstanding of the dynamics of the system Tang et al [17]calculated higher order Lyapunov values of the weak focusand reduced the system to a universal unfolding form for acusp of codimension 3 and they gave the bifurcation surfaces

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 745732 14 pageshttpdxdoiorg1011552015745732

2 Mathematical Problems in Engineering

and displayed all limit cycles and monoclinic loops of orderup to 2 Wei and Cui explored an SIS epidemic model withstandard incidence rate in [19] They took the incidence rateform 119891(119868)119878 = 120573119868119878(119868 + 119878) and showed the dynamics andbackward bifurcation of the SIS epidemic model

Recently in order to prevent and control the spread ofthe infectious diseases such as measles tuberculosis andflu many mathematicians (see [7 11 16 20 23 25ndash29])have begun to investigate the role of treatment functions inepidemiological models In some classical epidemic modelsthe treatment function is an important method to decreasethe spread of the epidemiological diseases Generally speak-ing the treatment function of the infective individuals isassumed to be proportional to the number of the infectiveindividuals But every community should have a maximalcapacity for the treatment of a disease and the resources fortreatment should be very largeTherefore it is very importantto adopt a suitable treatment function In [16] Wang andRuan introduced a constant treatment function of diseases inan SIR model that is

119879 (119868) = 119903 119868 gt 0

0 119868 = 0(2)

This means that they use the maximal treatment capacityto cure infective individuals so that the disease can beeradicatedThey also found that themodel undergoes saddle-node bifurcation Hopf bifurcation and Bogdanov-Takensbifurcation Further a piecewise linear treatment functionwas considered in [20] that is

119879 (119868) = 119896119868 0 le 119868 le 1198680

119898 119868 gt 1198680(3)

where 119898 = 1198961198680 and 119896 and 1198680 are positive constants Thismeans that the treatment rate is proportional to the numberof the infective individuals when the capacity of treatment hasnot been reached otherwise it takes the maximal capacity oftreatment 1198961198680 By considering the above treatment functionWang [20] found that a backward bifurcation takes place in anSIR epidemicmodel In [30] J C Eckalbar andW L Eckalbarconstructed an SIR epidemic model with a quadratic treat-ment function that is119879(119868) = max119903119868minus1198921198682 0 119903 119892 gt 0Theyfound that the system has as many as four equilibria withpossible bistability backward bifurcations and limit cycles

Recently saturated treatment function has been widelyapplied inmany epidemicmodels In particular in paper [23]Zhang and Liu took a continuous and differentiable saturatedtreatment function 119879(119868) = 119903119868(1 + 120572119868) where 119903 gt 0 120572 ge 0119903 stands for the cure rate and 120572 measures the extent of theeffect of the infected being delayed for treatment We cansee that the treatment function 119879(119868) sim 119903119868 when 119868 is smallenoughwhereas119879(119868) sim 119903120572when 119868 is large enough It ismorerealistic and it has the convenience of being continuous anddifferential compared to the previous ones Furthermore theauthors in [23] found that 1198770 = 1 is a critical threshold fordisease eradication when this delayed effect for treatment isweak and a backward bifurcation will take place when thiseffect is strong So it is really important to adequately stress

the interesting connection recently established between thechoice of saturated treatment functions in epidemic modelsand the occurrence of backward bifurcation in the relatedsystem dynamics In fact recently saturated-type treatmentfunctions have been indicated as responsible for the occur-rence of backward bifurcations for SIR [20 23] for SIS [1921] and for SEIR [28 29] models supporting the generalcircumstance that saturated-type treatments can be one ofthe causes of backward bifurcations in epidemic models Inparticular in [29] such connection has also been shown andvalidated in a specific concrete disease-control setting

In the real world some infectious diseases do not conferimmunity Such infections do not have a recovered state andindividuals become susceptible again after infection Thistype of disease can be modeled by the SIS type So the SISepidemic model has been adopted by many mathematicians(see [8 11 18 19 21 25]) And SIS models are appropriatefor some bacterial agent diseases such as meningitis plagueand venereal diseases and for protozoan agent diseases suchas malaria and sleeping sickness

Motivated by the above points we will consider the fol-lowing SIS model with bilinear incidence rate and saturatedtreatment function

119889119878

119889119905= 119860 minus 119889119878 minus 120582119878119868 + 120576119868 +

119903119868

1 + 120572119868

119889119868

119889119905= 120582119878119868 minus (119889 + 120576 + 120583) 119868 minus

119903119868

1 + 120572119868

(4)

where 119878 and 119868 denote the numbers of susceptible and infectiveindividuals respectively Positive constant 119860 is the recruit-ment rate of the population Positive constant 119889 is the naturedeath rate of population The bilinear incidence rate is 120582119878119868where 120582 is positive Positive constant 120576 is the natural recoveryrate of infective individuals Positive constant 120583 is the disease-related death rate The saturated treatment function ℎ(119868) ≜119903119868(1 + 120572119868) where 119903 is positive and 120572 is nonnegative

This paper focuses on the detailed dynamics analysisof the model (4) The local stability of these equilibria isinvestigated which enables us to classify the types of modelequilibria (eg attractor saddle or repeller) We show thatthe system has backward bifurcation and Bogdanov-Takensbifurcation (ie Hopf bifurcation saddle-node bifurcationand homoclinic bifurcation) under some certain conditionsFinally the three bifurcation curves and the complicatedglobal bifurcation phase portraits are derived by applyingthe Bogdanov-Takens normal form and the correspond-ing parameters which satisfy the conditions that ensureBogdanov-Takens bifurcation exists

The organization of this paper is as follows In Section 2we study the existence and local stability of equilibria andbackward bifurcation In Section 3 we investigate the globalstability of the model In Section 4 we give the supercriticaland subcritical bifurcation under two different conditionsin system (4) In Section 5 we show that the system (4)undergoes Bogdanov-Takens bifurcation under some certainconditions In Section 6 some numerical simulations aredisplayed in detail We close with a discussion in Section 7 onour mathematical results and epidemiological implications

Mathematical Problems in Engineering 3

0 5 10 15 20 25 30 35 4002468

101214161820

S

I

E0

Figure 1 The disease-free equilibrium 1198640 is locally asymptoticallystable when 1198770 lt 1 with the parameter values 119860 = 2 119889 = 01119896 = 01 120583 = 001 120582 = 001 120572 = 1 119903 = 2 and 120576 = 01

2 Equilibria and Backward Bifurcation

21 Disease-Free Equilibrium Obviously system (4) has adisease-free equilibrium 1198640 = (119860119889 0) The Jacobian matrixof (4) at 1198640 is

119872(1198640) = (minus119889

minus120582119860

119889+ 120576 + 119903

0120582119860

119889minus (119889 + 120576 + 120583) minus 119903

) (5)

By using the next generation matrix of (4) we get the basicreproduction number 1198770 = 120582119860119889(119889 + 120576 + 119903 + 120583)119872(1198640) hasnegative eigenvalues if 120582119860119889 minus (119889 + 120576 + 120583) minus 119903 lt 0 Then wehave the following result

Theorem 1 The disease-free equilibrium 1198640 is locally asymp-totically stable when 1198770 lt 1 (see Figure 1) and is unstable when1198770 gt 1 (see Figure 2)

22 Endemic Equilibria An endemic equilibrium alwayssatisfies

119860 minus 119889119878 minus 120582119878119868 + 120576119868 +119903119868

1 + 120572119868= 0

120582119878119868 minus (119889 + 120576 + 120583) 119868 minus119903119868

1 + 120572119868= 0

(6)

In view of 119889119878 + (119889 + 120583)119868 = 119860 we get 119878 = (119860 minus (119889 + 120583)119868)119889 andsubstitute it into the second equation of (6) When 119868 = 0 weobtain

120582[119860 minus (119889 + 120583) 119868

119889] minus (119889 + 120576 + 120583) minus

119903

1 + 120572119868= 0 (7)

Then we have an equation of the form

1198861198682 + 119887119868 + 119888 = 0 (8)

0 50 100 150 200 25002468

101214161820

S

I

E0

Figure 2 The disease-free equilibrium is unstable when 1198770 gt 1with the parameter values 119860 = 15 119889 = 01 119896 = 01 120583 = 001120582 = 001 120572 = 1 119903 = 08 and 120576 = 01

with

119886 = 120572120582 (119889 + 120583)

119887 = 120582 (119889 + 120583) + 120572119889 (119889 + 120576 + 120583) minus 120572120582119860

119888 = 119889 (119889 + 120576 + 119903 + 120583) minus 120582119860

(9)

This equation may admit positive solution

1198681 =minus119887 minus radic1198872 minus 4119886119888

2119886 1198682 =

minus119887 + radic1198872 minus 4119886119888

2119886 (10)

Obviously if 1198770 = 1 then 119888 = 0 if 1198770 gt 1 then 119888 lt 0 andif 1198770 lt 1 then 119888 gt 0 From (8) it is obvious that we have thefollowing results

Theorem 2 The following results hold

(1198671) Let 120572 = 0 Equation (8) is a linear equation with aunique solution 119868 = minus119888119887 Then the system (4) has aunique endemic equilibrium when 1198770 gt 1 and has noendemic equilibrium when 1198770 le 1

(1198672) Let 120572 gt 0 If 119887 gt 0 system (4) has a unique endemicequilibrium when 1198770 gt 1 and no endemic equilibriumwhen 1198770 le 1

(1198673) Let 120572 gt 0 If 119887 lt 0 system (4) has a unique endemicequilibrium when 1198770 ge 1 no endemic equilibriumwhen 1198770 lt 119877lowast0 and two endemic equilibria 1198641 and 1198642when 119877lowast

0le 1198770 lt 1 When 1198770 = 119877lowast0 and 1198641 = 1198642 one

has 1198872 minus 4119886119888 = 0 which is equivalent to

1198770 =4119886120582119860

1198872 + 4119886120582119860

=41205822120572 (119889 + 120583)119860

[120582 (119889 + 120583) + 120572119889 (119889 + 120576 + 120583) minus 120572120582119860]2+ 41205822120572 (119889 + 120583)119860

(11)

4 Mathematical Problems in Engineering

181614121080604020

10

20

30

40

50

60

I

Rlowast

0

R0

Figure 3 The figure of infective sizes at equilibria versus 1198770 when120572 = 1 119889 = 01 120583 = 001 120582 = 005 119903 = 2 and 120576 = 01

We define the right-hand side of (11) as

119877lowast0≜

41205822120572 (119889 + 120583)119860

[120582 (119889 + 120583) + 120572119889 (119889 + 120576 + 120583) minus 120572120582119860]2+ 41205822120572 (119889 + 120583)119860

(12)

Therefore according to the qualitative approach recentlyproposed by [31] which is based on the analysis of theequilibria curve in the neighboring of the critical threshold1198770 = 1 we have the following theorem

Theorem 3 If 120572 gt 0 119887 lt 0 then system (4) has a backwardbifurcation at 1198770 = 1 (see Figure 3)

In order to verify the bifurcation curve (the graph of 119868as a function of 1198770) in Figure 3 we think of 119903 as a variablewith the other parameters as constant Through implicitdifferentiation of (8) with respect to 119903 we get

(2119886119868 + 119887)119889119868

119889119903= minus119889 lt 0 (13)

From (13) we know that the sign of 119889119868119889119903 is opposite to thatof 2119886119868 + 119887 And from the definition of 1198770 we know that 1198770decreases when 119903 increases It implies that the bifurcationcurve has positive slope at equilibrium values with 2119886119868+119887 gt 0and negative slope at equilibrium values with 2119886119868 + 119887 lt 0 Ifthere is no backward bifurcation at 1198770 = 1 then the uniqueendemic equilibrium for 1198770 gt 1 satisfies

2119886119868 + 119887 = radic1198872 minus 4119886119888 gt 0 (14)

and the bifurcation curve has positive slope at all pointswhere119868 gt 0 If there is a backward bifurcation at 1198770 = 1 then thereis an interval onwhich there are two endemic equilibria givenby

2119886119868 + 119887 = plusmnradic1198872 minus 4119886119888 (15)

The bifurcation curve has negative slope at the smaller oneandpositive slope at the larger oneThus the bifurcation curveis as shown in Figure 3

Under the conditions of Theorem 3 if a backward bifur-cation takes place we can see from Figure 3 that there is acritical value 119877lowast

0at the turning point In this case the disease

will not die out when1198770 lt 1 However the diseasewill die outwhen 1198770 lt 119877

lowast

0 Therefore the critical value 119877lowast

0can be taken

as a new threshold for the control of the diseaseIn the following we give an explicit criterion of a

backward bifurcation at 1198770 = 1For convenience we define

1205720 =120582 (119889 + 120583)

119889119903 (16)

Corollary 4 When 120572 gt 1205720 system (4) has a backward bifur-cation at 1198770 = 1

Proof When 1198770 le 1 hArr 119888 ge 0

120582119860 le 119889 (119889 + 120576 + 120583 + 119903) (17)

The condition 119887 lt 0 is equivalent to

120582 (119889 + 120583) + 120572119889 (119889 + 120576 + 120583) lt 120572120582119860 (18)

From (17) and (18) we get 120582(119889 + 120583) + 120572119889(119889 + 120576 + 120583) lt 120572119889(119889 +120576 + 119903 + 120583) which reduces to

120572 gt120582 (119889 + 120583)

119889119903≜ 1205720 (19)

It means that 120572 is big enough to lead a backward bifurcationwith two endemic equilibria when 1198770 lt 1 Therefore theproof is complete

Next we consider the local stability of the uniqueendemic equilibrium when 1198770 gt 1

Theorem 5 When 1198770 gt 1 and 0 le 120572 lt 120582119903 the uniqueendemic equilibrium 119864lowast is locally asymptotically stable

Proof Firstly from Theorem 2 we can know that system(4) has a unique endemic equilibrium 119864lowast when 1198770 gt 1Moreover the Jacobian matrix of system (4) is

119872 =(

minus119889 minus 120582119868 minus120582119878 + 120576 +119903

1 + 120572119868minus

119903120572119868

(1 + 120572119868)2

120582119868 120582119878 minus (119889 + 120576 + 120583) minus119903

1 + 120572119868+

119903120572119868

(1 + 120572119868)2

)

(20)

From the second equation of (6) we have

minus120582119878 + 120576 +119903

1 + 120572119868= minus119889 minus 120583 (21)

From (21) the Jacobian matrix119872 reduces to

119872 =(

minus119889 minus 120582119868 minus119889 minus 120583 minus119903120572119868

(1 + 120572119868)2

120582119868119903120572119868

(1 + 120572119868)2

) (22)

Mathematical Problems in Engineering 5

We obtain

det (119872) = 119868

(1 + 120572119868)2[120582 (119889 + 120583) (1 + 120572119868)

2 minus 119903120572119889] (23)

In fact there is 120582(119889 + 120583)(1 + 120572119868)2 gt 120582119889 Since 1198770 gt 1 and0 le 120572 lt 120582119903 we have

120582 (119889 + 120583) (1 + 120572119868)2 minus 119903120572119889 gt 120582119889 minus 119903120572119889 gt 0 (24)

So we get det(119872) gt 0 The trace of119872 is given by

tr (119872) = 1

(1 + 120572119868)2[minus (119889 + 120582119868) (1 + 120572119868)

2 + 119903120572119868] (25)

In the same way as the above calculation of (24) we have

minus (119889 + 120582119868) (1 + 120572119868)2 + 119903120572119868 lt minus120582119868 + 119903120572119868 lt 0 (26)

So we get tr(119872) lt 0 The proof is complete

Now we consider the case that there are two endemicequilibria 1198641 and 1198642 let 119872119894 be the Jacobian matrix at 119864119894119894 = 1 2

Theorem 6 The endemic equilibrium 1198641 is a saddle wheneverit exists

Proof Since 1198681 = (minus119887 minus radic1198872 minus 4119886119888)2119886 and Δ = 1198872 minus 4119886119888 wehave 1198681 = (minus119887 minus radicΔ)2119886 Thus

det (1198721)

=1198681

(1 + 1205721198681)2[120582 (119889 + 120583) (1 + 1205721198681)

2minus 119903120572119889]

≜1198681

(1 + 1205721198681)2120595 (1198681)

(27)

From (1198673) of Theorem 2 we can know that if 120572 gt 0 119887 lt 0and 119877lowast

0lt 1198770 lt 1 then 1198641 exists Hence we can get 120595(0) =

120582(119889 + 120583) minus 119903120572119889 lt 0 and 1205951015840(1198681) = 2120582(119889 + 120583)(1 + 1205721198681)120572 gt 0 Itfollows that there exists a unique 119868lowast gt 0 such that

120595 (1198681) = 0 when 1198681 = 119868lowast

120595 (1198681) lt 0 when 0 lt 1198681 lt 119868lowast

120595 (1198681) gt 0 when 1198681 gt 119868lowast

(28)

where

119868lowast = radic119903119889

120572120582 (119889 + 120583)minus1

120572 (29)

On the other hand

1198681 = minus119887

2119886minusradicΔ

2119886

=120572120582119860 minus 120582 (119889 + 120583) minus 120572119889 (119889 + 120576 + 120583)

2120572120582 (119889 + 120583)minusradicΔ

2119886

= radic119903119889

120572120582 (119889 + 120583)minus1

120572+1

120572minus radic

119903119889

120572120582 (119889 + 120583)

+120572120582119860 minus 120582 (119889 + 120583) minus 120572119889 (119889 + 120576 + 120583)

2120572120582 (119889 + 120583)minusradicΔ

2119886

= 119868lowast + ([120582 (119889 + 120583) + 120572120582119860 minus 2radic119903119889120572120582 (119889 + 120583)

minus 120572119889 (119889 + 120576 + 120583)] minus radicΔ)times (2120572120582 (119889 + 120583))minus1

(30)

Δ = 120582 (119889 + 120583) minus [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]2

minus 4120572120582 (119889 + 120583) [119889 (119889 + 120576 + 119903 + 120583) minus 120582119860]

= 120582 (119889 + 120583) + [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

minus 2radic119903119889120572120582(119889 + 120583)2

minus 4119903119889120572120582 (119889 + 120583)

minus 4120582 (119889 + 120583) [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

+ 4radic119903119889120572120582 (119889 + 120583) 120582 (119889 + 120583) + [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

minus 4120572120582 (119889 + 120583) [minus120582119860 + 119889 (119889 + 120576 + 119903 + 120583)]

≜ 120582 (119889 + 120583) + [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

minus 2radic119903119889120572120582 (119889 + 120583)2

+ 119875

(31)

where

119875 = minus8119903119889120572120582 (119889 + 120583) + 4radic119903119889120572120582 (119889 + 120583)

times 120582 (119889 + 120583) + [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

(32)

In the following we will show that 119875 gt 0 Since 119877lowast0lt 1198770 we

have

4120572120582 (119889 + 120583)

[120582 (119889 + 120583) + 120572119889 (119889 + 120576 + 120583) minus 120572120582119860]2+ 41205721205822 (119889 + 120583)119860

lt1

119889 (119889 + 120576 + 119903 + 120583)

(33)

6 Mathematical Problems in Engineering

that is

[120582 (119889 + 120583) + 120572119889 (119889 + 120576 + 120583) minus 120572120582119860]2

gt 4120572120582 (119889 + 120583) 119889 (119889 + 120576 + 119903 + 120583) minus 41205721205822 (119889 + 120583)119860(34)

Therefore

120582 (119889 + 120583) + [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]2

= 120582 (119889 + 120583) minus [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]2

+ 4120582 (119889 + 120583) [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

gt 4120572120582 (119889 + 120583) 119889 (119889 + 120576 + 119903 + 120583) minus 41205721205822 (119889 + 120583)119860

+ 4120582 (119889 + 120583) [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

= 4119903119889120572120582 (119889 + 120583)

(35)

Obviously we have 119875 gt 0 From (30) one has 1198681 lt 119868lowast So

we get det(1198721) lt 0 Hence the endemic equilibrium 1198641 is asaddle The proof is complete

In order to explore the stability of the endemic equilib-rium 1198642 define

1198981 = (21198891205721198862 + 1205821198862 minus 1205761205721198862 minus 1198861198881205821205722)

minus 119887 (2120582120572119886 + 1198891205722119886 minus 1198871205821205722)

1198982 = 1198862119889 minus 119888 (2120582120572119886 + 1198891205722119886 minus 1198871205821205722)

(36)

Theorem 7 If 120578 gt 0 then endemic equilibrium 1198642 is locallyasymptotically stable if 120578 lt 0 then endemic equilibrium 1198642 isunstable where 120578 = 21198861198982 + 1198981(radic1198872 minus 4119886119888 minus 119887)

Proof Consider

det (1198722) =1198682

(1 + 1205721198682)2[120582 (119889 + 120583) (1 + 1205721198682)

2minus 120576120572119889]

=1198682

(1 + 1205721198682)2times 120595 (1198682)

(37)

By carrying out arguments similar to that of Theorem 6 wehave 1198682 gt 119868

lowast Therefore det(1198722) gt 0 In addition we have

tr (1198722) = minus(119889 + 1205821198682) (1 + 1205721198682)

2minus 1205761205721198682

(1 + 1205721198682)2

= minus120582120572211986832+ (2120582120572 + 1198891205722) 1198682

2+ (2119889120572 + 120582 minus 120576120572) 1198682 + 119889

(1 + 1205721198682)2

(38)

and then sgn(tr(1198722)) = minus sgn(119866(1198682)) where

119866 (119909) = 12058212057221199093 + (2120582120572 + 1198891205722) 1199092 + (2119889120572 + 120582 minus 120576120572) 119909 + 119889

(39)

I

0 50 100 1500

5

10

15

20

25

30

35

40

S

E0

E1

E2

Figure 4 One region of disease persistence and one region ofdisease extinction when 119860 = 10 120572 = 1 119889 = 01 120583 = 001 120582 = 001119903 = 01 and 120576 = 01

Using the expression of1198981 = (21198891205721198862 +1205821198862 minus 1205761205721198862 minus1198861198881205821205722) minus

119887(2120582120572119886+1198891205722119886minus1198871205821205722) and1198982 = 1198862119889minus119888(2120582120572119886+1198891205722119886minus1198871205821205722)

one has

119866 (1198682) = (1198861198682

2+ 1198871198682 + 119888) 1205930 +

11989811198682 + 11989821198862

(40)

where1205930 is a first degree polynomial of 1198682 Since 1198861198682

2+1198871198682+119888 =

0 sgn(tr(1198722)) = minus sgn(119866(1198682)) = minus sgn(11989811198682 + 1198982) From theexpression of 1198682 we have

sgn (11989811198682 + 1198982)

= sgn (21198861198982 + 1198981 (radic1198872 minus 4119886119888 minus 119887)) ≜ sgn (120578) (41)

Thus 1198642 is locally asymptotically stable if 120578 gt 0 and 1198642 isunstable if 120578 lt 0 The proof is complete

From the above discussion we have the following con-clusion If two endemic equilibria 1198641 and 1198642 exist the stablemanifolds of the saddle 1198641 split 119877

2

+into two regions The

disease is persistent in the upper region and dies out in thelower region (see Figure 4)

3 Global Analysis

Firstly we consider the global stability of the disease-freeequilibrium 1198640 Let 119873 = 119878 + 119868 be the total population sizeNow we note that the equation for total population is givenby 119889119873119889119905 = 119860 minus 119889119878 minus (119889 + 120583)119868 le 119860 minus 119889119873 It follows thatlim119905rarr+infin119873(119905) le 119860119889 Let

R = (119878 119868) isin 1198772

+ 119878 + 119868 le

119860

119889 (42)

which is positive invariant with respect to system (4)

Theorem 8 If 1198770 lt 119877lowast0 the disease-free equilibrium 1198640(119860119889 0) is globally asymptotically stable that is the disease diesout

Mathematical Problems in Engineering 7

Proof Suppose 1198770 lt 119877lowast0 From the (1198673) of Theorem 2 weknow that the model has no endemic equilibrium Fromthe corollary of Poincare-Bendixson theorem [32] we knowthat there is no periodic orbit in R as there is a disease-free equilibrium in R Since R is a bounded positivelyinvariant region and 1198640 is the only equilibrium in R thelocal stability of 1198640 implies that every solution initiating inR approaches 1198640 Thus the disease-free equilibrium 1198640 isglobally asymptotically stable The proof is complete

Now we analyze the global dynamics of the endemicequilibrium when 1198770 gt 1

Theorem 9 If 1198770 gt 1 and 0 le 120572 lt 120582119903 the system (4) has nolimit cycles

Proof We use Dulac theorem to exclude the limit cycle Let

119875 (119878 119868) = 119860 minus 119889119878 minus 120582119878119868 + 120576119868 +119903119868

1 + 120572119868

119876 (119878 119868) = 120582119878119868 minus (119889 + 120576 + 120583) 119868 minus119903119868

1 + 120572119868

(43)

and take the Dulac function

119863 =1

119868 (44)

According to 0 le 120572 lt 120582119903 we can get

120597 (119875119863)

120597119878+120597 (119876119863)

120597119868

= minus119889

119868minus 120582 +

120572119903

(1 + 120572119868)2

lt minus119889

119868minus 120582 +

120582

(1 + 120572119868)2

=1

119868 (1 + 120572119868)2minus119889 (1 + 120572119868)

2 minus 120582119868 [(1 + 120572119868)2 minus 1]

lt 0

(45)

Hence the system (4) has no limit cycles The proof iscomplete

Therefore we obtain the global result of the uniqueendemic equilibrium

Theorem 10 If 1198770 gt 1 and 0 le 120572 lt 120582119903 the unique endemicequilibrium 119864lowast is globally asymptotically stable (see Figure 5)

4 Hopf Bifurcation

In this section we study the Hopf bifurcation of system (4)From the above discussion we know that there is no closedorbit surrounding1198640 or1198641 because the 119878-axis is invariantwith

0 50 100 1500

50

100

150

S

I Elowast

Figure 5The unique endemic equilibrium 119864lowast is globally asymptot-ically stable when 119860 = 15 120572 = 001 119889 = 01 120583 = 001 120582 = 001119903 = 08 and 120576 = 01

respect to system (4) and 1198641 is always a saddle ThereforeHopf bifurcation can only occur at 1198642 Set

120590 = [minus120582120572 minus1198881 (1198882 + 21198884)

11988612]1198634

+ [minus 120582120572119886211minus 21205821205721198861111988612 + (1198882 minus

211988611119888111988612)

times (119886111198882 minus119886211119888111988612)

minus (21198862111198881

11988612minus 119886111198882 + 2119888411988611 minus 119888311988612)

times (1198884 minus11988611119888111988612)]1198632

+ 11988611 (119886211119888111988612minus 119886111198882 + 119888411988611 minus 119888311988612)

times (119886121198883 minus 2119886111198884 minus 119886111198882)

(46)

where 119886119894119895 (119894 119895 = 1 2) 119888119896 (119896 = 1 2 3 4) and 119863 are defined by(50) and (52)

Theorem 11 System (4) undergoes a Hopf bifurcation if 120578 = 0Moreover if 120590 lt 0 there is a family of stable period orbits ofsystem (4) as 120578 decreases from 0 that is a supercritical Hopfbifurcation occurs if 120590 gt 0 there is a family of unstable periodorbits of system (4) as 120578 increases from 0 that is a subcriticalHopf bifurcation occurs

Proof The proof of Theorem 7 shows that tr(1198722) = 0 if andonly if 120578 = 0 and det(1198722) gt 0 when 1198642 exists Therefore theeigenvalues of 1198722 are a pair of pure imaginary roots if andonly if 120578 = 0 The direct calculations show that

119889 (tr (1198722))119889120578

100381610038161003816100381610038161003816100381610038161003816120578=0= minus

1

21198863 (1 + 1205721198682)2lt 0 (47)

8 Mathematical Problems in Engineering

ByTheorem 342 in [24] 120578 = 0 is the Hopf bifurcation pointfor system (4)

To be concise in notations rescale (4) by 120591 = 119905(1 + 120572119868)For simplicity we still use 119905 instead of 120591 Then we obtain

119889119878

119889119905= (119860 minus 119889119878 minus 120582119878119868 + 120576119868) (1 + 120572119868) + 119903119868

119889119868

119889119905= 120582119878119868 (1 + 120572119868) minus (119889 + 120576 + 120583) 119868 (1 + 120572119868) minus 119903119868

(48)

Let 119909 = 119878 minus 1198782 and 119910 = 119868 minus 1198682 then (48) becomes

119889119909

119889119905= 11988611119909 + 11988612119910 + 1198881119910

2 + 1198882119909119910 minus 1205821205721199091199102

119889119910

119889119905= 11988621119909 + 11988622119910 + 1198883119909119910 + 1198884119910

2 + 1205821205721199091199102

(49)

where

11988611 = minus (119889 + 1205821198682) (1 + 1205721198682)

11988612 = (minus1205821198782 + 119903) (1 + 1205721198682)

+ 120572 (119860 minus 1198891198782 minus 12058211987821198682 + 1199031198682) + 120576

11988621 = 1205821198682 (1 + 1205721198682)

11988622 = [1205821198782 minus (119889 + 119903 + 120583)] (1 + 1205721198682)

+ 120572 [12058211987821198682 minus (119889 + 119903 + 120583) 1198682] minus 120576

1198881 = 120572 (minus1205821198782 + 119903)

1198882 = 120572 (minus119889 minus 1205821198682) minus 120582 (1 + 1205721198682)

1198883 = 120582 (1 + 1205721198682) + 1205821198682120572

1198884 = [1205821198782 minus (119889 + 119903 + 120583)] 120572

(50)

Let 119864lowast denote the origin of 119909-119910 plane Since 1198642 = (1198782 1198682)satisfies (6) we obtain

det (119872 (119864lowast))

= minus (119889 + 1205821198682) 1205721205761198682 + (119889 + 120583) (1 + 1205721198682

2)2

1205821198682 + 1205721205821205761198682

2

= 1198682 times 120595 (1198682)

(51)

From the proof of Theorem 7 it follows that 120595(1198682) is alwayspositive It is easy to verify that 11988611 + 11988622 = 0 if and only if120578 = 0 Set

119863 = radicdet (119872 (119864lowast)) (52)

and let 119906 = minus119909 and V = (11988611119863)119909+(11988612119863)119910 then the normalform of (48) for Hopf bifurcation reads

119889119906

119889119905= minus119863V + 119891 (119906 V)

119889V119889119905= 119863119906 + 119892 (119906 V)

(53)

where

119891 (119906 V) = (11988611119886121198882 minus119886211

119886212

1198881)1199062 minus11986321198881119886212

V2

+ (119863119888211988612minus 2119863119886111198881119886212

)119906V minus120582120572119886211

119886212

1199063

minus 211986311988611120582120572

119886212

1199062V minus1198632120582120572

119886212

119906V2

119892 (119906 V) =11988611119863(119886211

119886212

1198881 minus11988611119886121198882 +11988611119886121198884 minus 1198883)119906

2

+ (119863119886111198881119886212

+119863119888411988612) V2

+ (2119886211

119886212

1198881 minus11988611119886121198882 + 2

11988611119886121198884 minus 1198883)119906V

+120582120572119886211

11988612119863(1

11988612minus 1)1199063 +

21205821205721198861111988612

(1198861111988612minus 1)1199062V

+119863120582120572

11988612(1198861111988612minus 1)119906V2

(54)

Now we evaluate the first Lyapunov coefficient Γ of system(4) as follows

Γ =1

16[119891119906119906119906 + 119891119906VV + 119892119906119906V + 119892VVV] +

1

16119863

times [119891119906V (119891119906119906 + 119891VV) minus 119892119906V (119892119906119906 + 119892VV) minus 119891119906119906119892119906119906 + 119891VV119892VV]

(55)

where 119891119906V denotes (1205972119891120597119906120597V)(0 0) and so forth Then

Γ =1

81198862121198632[minus120582120572 minus

1198881 (1198882 + 21198884)

11988612]1198634

+ [minus 120582120572119886211minus 21205821205721198861111988612 + (1198882 minus

211988611119888111988612)

times (119886111198882 minus119886211119888111988612)

minus (21198862111198881

11988612minus 119886111198882 + 2119888411988611 minus 119888311988612)

times (1198884 minus11988611119888111988612)]1198632

+ 11988611 (119886211119888111988612minus 119886111198882 + 119888411988611 minus 119888311988612)

times (119886121198883 minus 2119886111198884 minus 119886111198882)

=120590

81198862121198632

(56)

Mathematical Problems in Engineering 9

Obviously the sign of Γ is determined by 120590 ByTheorem 342and (3411) in [24] the rest of the claims in Theorem 11 arevalid The proof is complete

5 Bogdanov-Takens Bifurcations

The purpose of this section is to study the Bogdanov-Takensbifurcation of (4) Now we assume the following

(1198601) 1198872 minus 4119886119888 = 0

(1198602) 21198861198892 + 119887120582120583 = 0

Then (6) admits a unique positive equilibrium 119864lowast = (119878lowast 119868lowast)where

119868lowast = minus119887

2119886 119878lowast =

119860 minus (119889 + 120583) 119868lowast

119889 (57)

The Jacobian matrix of system (4) at 119864lowast is

119872lowast = (

minus119889 minus 120582119868lowast minus120582119878lowast + 120576 +119903

(1 + 120572119868lowast)2

120582119868lowast 120582119878lowast minus (119889 + 120576 + 120583) minus119903

(1 + 120572119868lowast)2)

(58)

By (58) we have

det (119872lowast) = 119868lowast

(1 + 120572119868lowast)2[(119889 + 120583) 120582 (1 + 120572119868lowast)

2minus 119903119889120572]

= 0

(59)

because of

(1 + 120572119868lowast)2=41198862 minus 4119886120572119887 + 12057221198872

41198862

=41198862 minus 4119886120572119887 + 12057224119886119888

41198862

=119886 minus 120572119887 + 1205722119888

119886

=1205722119903119889

119886

(60)

Furthermore (1198602) implies that

tr (119872lowast) = 0 (61)

Thus (1198601) and (1198602) imply that the Jacobian matrix has azero eigenvalue withmultiplicity 2This suggests that (4)mayadmit a Bogdanov-Takens singularity

Theorem 12 Suppose that (1198601) (1198602) and 21198871 + 1198874 = 0 holdThen the endemic equilibrium 119864lowast = (119878lowast 119868lowast) of (4) is a cuspof codimension 2 that is it is a Bogdanov-Takens singularityHere 1198871 and 1198874 are defined by (65)

Proof Using the transformation of 119909 = 119868minus 119868lowast and 119910 = 119878minus 119878lowastsystem (4) becomes

119889119909

119889119905= 1198861119909 + 1198862119910 + 120582119909119910 + 11988611119909

2 minus 1198751 (119909)

119889119910

119889119905= minus11988621

1198862119909 minus 1198861119910 minus 120582119909119910 minus 11988611119909

2 + 1198751 (119909)

(62)

where 1198751(119909) is a smooth function of 119909 at least of the thirdorder and

1198861 = 120582119878lowast minus (119889 + 120576 + 120583) minus

119903

(1 + 120572119868lowast)2gt 0

1198862 = 120582119868lowast gt 0

11988611 =119903120572

(1 + 120572119868lowast)3gt 0

(63)

Set119883 = 119909 119884 = 1198861119909 + 1198862119910 Then (62) is transformed into

119889119883

119889119905= 119884 + 1198871119883

2 + 1198872119883119884 + 1198761 (119883)

119889119884

119889119905= 1198873119883

2 + 1198874119883119884 + 1198762 (119883)

(64)

where 119876119894(119883) are smooth functions of 119883 at least of the thirdorder and

1198871 = 11988611 minus12058211988611198862

1198872 =120582

1198862

1198873 = 119886111988611 minus12058211988621

1198862+ 1198861120582 minus 119886211988611

1198874 =12058211988611198862minus 120582

(65)

In order to obtain the canonical normal form we perform thetransformation of variables by

119906 = 119883 minus119887221198832 V = 119884 + 1198871119883

2 (66)

Then we obtain

119889119906

119889119905= V + 1198771 (119906)

119889V119889119905= 11988731199062 + (21198871 + 1198874) 119906V + 1198772 (119906)

(67)

where 119877119894(119906) are smooth functions of 119906 at least of the thirdorder Note that 1198873 lt 0 and 21198871 + 1198874 = 0 It follows from [33ndash35] that (4) admits a Bogdanov-Takens bifurcation

In the following wewill find the versal unfolding in termsof the original parameters in (4) In this way we will know theapproximate saddle-node Hopf and homoclinic bifurcation

10 Mathematical Problems in Engineering

curvesWe choose119860 and 119889 as bifurcation parameters Fix 120582 =1205820 120576 = 1205760 119903 = 1199030 120572 = 1205720 and 120583 = 1205830 Let 119860 = 1198600 + 1205791 and119889 = 1198890 + 1205792 where 1205791 and 1205792 are parameters which vary in asmall neighborhood of the origin

Suppose that 119860 = 1198600 119889 = 1198890 120582 = 1205820 120576 = 1205760 119903 = 1199030 120572 =1205720 and 120583 = 1205830 satisfy (1198601) and (1198602) Consider the followingsystem

119889119868

119889119905= 1205820119878119868 minus (1198890 + 1205792 + 1205760 + 1205830) 119868 minus

1199030119868

1 + 1205720119868

119889119878

119889119905= 1198600 + 1205791 minus (1198890 + 1205792) 119878 minus 1205820119878119868 + 1205760119868 +

1199030119868

1 + 1205720119868

(68)

By the transformations of 119909 = 119868 minus 119868lowast and 119910 = 119878 minus 119878lowast system(68) becomes

119889119909

119889119905= minus 1205792119868

lowast + 1198881119909 + 1198882119910 + 119888111199092 + 1205820119909119910 minus 1199081 (119909)

119889119910

119889119905= (1205791 minus 1205792119878

lowast) + 1198883119909 + 1198884119910 minus 119888111199092 minus 1205820119909119910 + 1199081 (119909)

(69)

where 1199081(119909) is a smooth function of 119909 at least of the thirdorder and

1198881 = 1205820119878lowast minus (1198890 + 1205792 + 1205760 + 1205830) minus

1199030

(1 + 1205720119868lowast)2

1198882 = 1205820119868lowast

1198883 = minus 1205820119878lowast + 1205760 +

1199030

(1 + 1205720119868lowast)2

1198884 = minus (1198890 + 1205792) minus 1205820119868lowast

11988811 =11990301205720

(1 + 1205720119868lowast)3

(70)

Making the change of variables 119883 = 119909 119884 = minus1205792119868lowast + 1198881119909 +

1198882119910 + 119888111199092 + 1205820119909119910 minus 1199081(119909) and rewriting 119883 119884 as 119909 and 119910

respectively we have

119889119909

119889119905= 119910

119889119910

119889119905= 1198900+ 1198901119909 + 1198902119910 + 11989011119909

2+ 11989012119909119910 + 119890221199102+ 1199082 (119909 119910 120579)

(71)

where 120579 = (1205791 1205792) 1199082(119909 119910 120579) is a smooth function of 119909 1199101205791 and 1205792 at least of the third order and

1198900 = 1198882 (1205791 minus 1205792119878lowast) + 11988841205792119868

lowast

1198901 = 1205820 (1205791 minus 1205792119878lowast) + 11988821198883 minus 11988811198884 minus 12058201205792119868

lowast

1198902 = 1198881 + 1198884 + 12058201205792119868lowast

1198882

11989011 = 12058201198883 minus 119888411988811 minus 119888111198882 + 11988811205820

11989012 = minus 1205820 + 211988811 + 1205820minus11988811198882 minus 12058201205792119868

lowast

11988822

11989022 =12058201198882

(72)

Next introduce a new time variable 120591 by 119889119905 = (1 minus(12058201198882)119909)119889120591 Rewriting 120591 as 119905 we obtain

119889119909

119889119905= 119910(1 minus

12058201198882119909)

119889119910

119889119905= (1 minus

12058201198882119909)

times (1198900 + 1198901119909 + 1198902119910 + 119890111199092

+ 11989012119909119910 + 119890221199102 + 1199082 (119909 119910 120579))

(73)

Let119883 = 119909 119884 = 119910(1minus(12058201198882)119909) and rename119883 and119884 as 119909 and119910 we have

119889119909

119889119905= 119910

119889119910

119889119905= 1198900 + 1198921119909 + 1198902119910 + 11989211119909

2 + 11989212119909119910 + 1199083 (119909 119910 120579)

(74)

where 120579 = (1205791 1205792) 1199083(119909 119910 120579) is a smooth function of 119909 1199101205791 and 1205792 at least of the third order and

1198921 = minus 2119890012058201198882+ 1198901

11989211 = 11989011 minus 2119890112058201198882+11989001205822

0

11988822

11989212 = 11989012 minus119890212058201198882

(75)

Now we assume that 11989211 = 0 and 11989212 = 0 when 120582119894 are smallSet 119909 = 119883 minus 119890211989212 and rewrite119883 as 119909 we can get

119889119909

119889119905= 119910

119889119910

119889119905= 1198910 + 1198911119909 + 11989211119909

2 + 11989212119909119910 + 1199084 (119909 119910 120579)

(76)

where 120579 = (1205791 1205792) 1199084(119909 119910 120579) is a smooth function of 119909 1199101205791 and 1205792 at least of the third order and

1198910 = 1198900 minus1198921119890211989212+119892111198902

2

119892212

1198911 = 1198921 minus211989211119890211989212

(77)

Mathematical Problems in Engineering 11

Making the final change of variables by 119883 = 11989221211990911989211 119884 =

119892312119910119892211 and 120591 = 1198921111990511989212 and then denoting them again by

119909 119910 and 119905 respectively we obtain

119889119909

119889119905= 119910

119889119910

119889119905= 1205911 (1205791 1205792) + 1205912 (1205791 1205792) 119909 + 119909

2 + 119909119910 + 1199085 (119909 119910 120579)

(78)

where 120579 = (1205791 1205792) and 1199085(119909 119910 120579) is a smooth function of 119909119910 1205791 and 1205792 at least of the third order

We substitute values of 1205720 = 1 1205820 = 14 1198890 = 1101205760 = 110 1205830 = 1100 1198600 = 537500 and 1199030 = 558 forthe above system and these values satisfy conditions (1198601) and(1198602) And we obtain the following equations

1205911 (1205791 1205792) =11989101198924

12

119892311

= minus237291605

851841205791 +

25485118377

8518401205792

+ 119874 (10038161003816100381610038161205791 1205792

10038161003816100381610038162)

1205912 (1205791 1205792) =11989111198922

12

119892211

= minus172225

19361205791 +

3983253

38721205792 + 119874 (

10038161003816100381610038161205791 120579210038161003816100381610038162)

(79)

where 11989211 = minus11500 minus (116)1205791 + (341800)1205792 + (14)1205792

2lt 0

and 11989212 = minus83200 + (14)1205792 lt 0 in a small neighborhood of(1205791 1205792) = (0 0) Let

119869 = (

12059712059111205971205791

12059712059111205971205792

12059712059121205971205791

12059712059121205971205792

) (80)

And after simple calculation we obtain that

det(119869)|120582=0 = minus16839398748825

82458112= 0 (81)

Thus 1205911 and 1205912 are regular maps in a small neighborhoodof (1205791 1205792) = (0 0) By the Bogdanov and Takens bifurcationtheorems [36] we obtain the following conclusion

Theorem 13 Suppose that1198600 1198890 1205820 1205760 1199030 1205720 and 1205830 satisfy(1198601) (1198602) 11989211 = 0 and 11989212 = 0 when 120579119894 are small Then (4)admits the following bifurcation behavior

(1) There is a saddle-node bifurcation curve 119878119873 =

(1205791 1205792) 4119891011989211 = 11989121+ 119900(|(1205791 1205792)|

2) 1198911 =0 = (1205791 1205792) (59076250)1205792 minus (11125)1205791 +(2313400)12057911205792 minus (516)120579

2

1minus (102863740000)1205792

2+

119900(|(1205791 1205792)|2) = 0 (4799116600)1205792 minus (14)1205791 + (25

83)12057911205792 minus (1364513778)1205792

2+ 119900(|(1205791 1205792)|

2) = 0

I

I

O

O 1205791

1205792

SN+

SN+

SNminus

SNminus

II H

III HL

IVIVHL

IIIII

H

Figure 6The bifurcation set and the corresponding phase portraitsfor system (68)

(2) There is a Hopf bifurcation curve 119867 = (1205791 1205792)

1198910 + 119900(|(1205791 1205792)|2) = 0 1198911 lt 0 = (1205791 1205792) minus(537

50)1205792+1205791 + (5083)12057911205792 minus (746676889)12057922 + 119900(|(12057911205792)|2) = 0 (4799116600)1205792 minus (14)1205791+(2583)12057911205792 minus

(1364513778)12057922+ 119900(|(1205791 1205792)|

2) lt 0

(3) There is a homoclinic bifurcation curve119867119871 = (1205791 1205792) 25119892111198910 + 6119891

2

1= 119900(|(1205791 1205792)|

2) 1198911 lt 0 = (1205791 1205792) (59071000)1205792 minus (1120)1205791 + (611979 33200)12057911205792 minus(1916)1205792

1minus (16075834839275560000)1205792

2+ 119900(|(1205791

1205792)|2) = 0 (4799116600)1205792minus(14)1205791+(2583)12057911205792minus

(1364513778)12057922+ 119900(|(1205791 1205792)|

2) lt 0

6 Numerical Simulations

When 120572 = 1 120582 = 14 119889 = 110 120576 = 110 120583 = 1100119860 = 537500 and 119903 = 558 120590 gt 0 By applying PPLANE8and Photoshop software the (1205791 1205792)-plane near the origin isdivided into 4 regions by these bifurcation curves as shown inFigure 6 Fix 1205791 lt 0 and decrease 1205792 from 0 our conclusionsare summarized as follows

(a) When (1205791 1205792) lies in region I there is no positiveequilibrium which implies that the positive orbitsof (4) meet the positive 119878-axis in finite time andtherefore the disease disappears

(b) There is a saddle-node point when (1205791 1205792) lies oncurve SN

(c) When (1205791 1205792) crosses SN into region II two positiveequilibria which are an unstable focus and a saddleappear

(d) When the parameters lie on the curve H there is alsoa saddle and an unstable focus An unstable limit cycleappears with the parameters crossing H into III

(e) A homoclinic cycle appears as the parameters passingIII into HL through the homoclinic bifurcation

12 Mathematical Problems in Engineering

0 5 10 15 20 25 30 35 40

0

1

2

3

4

5

6

7

8

9

10

S

I

Figure 7 When (1205791 1205792) = (0 0) the unique positive equilibrium isa cusp of codimension 2

3 35 45 55 65 754 5 6 7 8

1

2

3

4

5

6

7

8

S

I

Figure 8 When (1205791 1205792) = (minus01 minus00093) lies in region I there isno positive equilibrium

(f) The homoclinic loop breakswith the parameter cross-ing HL into IV and a saddle and a stable focus appear

Furthermore using PPLANE8 some numerical simu-lations of system (68) are depicted in Figures 7ndash10 When(1205791 1205792) = (0 0) that is (1198600 1198890) = (537500 110) there isa unique positive equilibrium (119878lowast 119868lowast) = (31750 4) whichis a cusp of codimension 2 (Figure 7) When (1205791 1205792) =(minus01 minus00093) lies in region I there is no positive equi-librium (Figure 8) When (1205791 1205792) = (minus002 minus0001863798)there is a homoclinic loop (Figure 9) When (1205791 1205792) =(minus002 minus00020) lies in region IV the homoclinic loopis broken and there is a stable focus and a saddle(Figure 10)

6 605 61 615 62 625 63 635 64 645 653536373839

44142434445

S

I

Figure 9 When (1205791 1205792) = (minus002 minus0001863798) there is ahomoclinic loop

0 1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

7

8

9

10

S

I

Figure 10 When (1205791 1205792) = (minus002 minus00020) lies in region IV thereis a stable focus and a saddle

7 Discussion

In this paper we focus on the bifurcation analysis of an SISepidemic model with bilinear incidence rate and saturatedtreatment Generally speaking in many epidemic modelsthe basic reproduction number which is the key conceptin epidemiology can be decreased below unity to eradicatethe disease However in our model the basic reproductionnumber below 1 is not enough to eradicate the diseaseAccording to our analysis in this paper we demonstrate notonly the global stability of the disease-free equilibrium when1198770 lt 119877

lowast

0and the local asymptotic stability of the endemic

equilibrium 1198642 when 119877lowast

0lt 1198770 lt 1 but also the global

stability of the unique endemic equilibrium 119864lowast when 1198770 gt 1Moreover it has been shown in Corollary 4 that backwardbifurcations occur if the effect of the infected being delayedfor treatment is strong That is to say we should get prompttreatment for patients Through Figure 4 we can see thatthere is a region such that the disease will persist if the initialposition lies in the region and disappear if the initial positionlies outside this region So in order to restrain the spread

Mathematical Problems in Engineering 13

of the disease governments should take timely measures tocontrol the initial infected individuals in a lower level Inaddition fromTheorem 8 we can see that the disease will dieout if 1198770 is small enough Therefore from the definition of1198770 one knows that we can decrease the incidence rate 120582 andincrease the cure rate 119903 so as to eradicate the diseases or tomake them controlled in a lower endemic steady state

The stability analysis of the model equilibria enables usto completely analyze their local bifurcation behavior suchas Hopf saddle-node and Bogdanov-Takens bifurcation Bycomputing the first Lyapunov coefficient we can determinethat the Hopf bifurcation is a supercritical Hopf bifurca-tion or a subcritical Hopf bifurcation We also show thatunder assumptions (1198601) and (1198602) the model undertakesBogdanov-Takens bifurcation that is there are saddle-nodebifurcation Hopf bifurcation and homoclinic bifurcationin the system The normal norm of the Bogdanov-Takensbifurcation is derived in Section 5 which is very helpful toobtain the three kinds of bifurcation curves By analyticaltechniques 119860 and 119889 are chosen as bifurcation parametersand other parameters are fixed and we easily get a clearpicture about the rich dynamics behaviors of our modelThrough studying the bifurcations of the SIS epidemicmodelwe are suggesting that in order to eradicate the diseasemore medical facilities as well as medical professionals areneeded and the medical standard needs to be improved aswell

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research was supported by the National Natural ScienceFoundation of China (no 11001041 no 10926105 no 11101283no 11271065 and no 11201360) SRFDP (no 200802001008)and Scientific Research Innovation Project of ShanghaiMunicipal Education Commission (12YZ173)

References

[1] M E Alexander and S M Moghadas ldquoPeriodicity in an epi-demic model with a generalized non-linear incidencerdquo Mathe-matical Biosciences vol 189 no 1 pp 75ndash96 2004

[2] M E Alexander and S M Moghadas ldquoBifurcation analysisof SIRS epidemic model with generalized incidencerdquo SIAMJournal on Applied Mathematics vol 65 no 5 pp 1794ndash18162005

[3] J Arino C C McCluskey and P van den Driessche ldquoGlobalresults for an epidemic model with vaccination that exhibitsbackward bifurcationrdquo SIAM Journal on Applied Mathematicsvol 64 no 1 pp 260ndash276 2003

[4] W R Derrick and P van den Driessche ldquoHomoclinic orbitsin a disease transmission model with nonlinear incidence andnonconstant populationrdquo Discrete and Continuous DynamicalSystems B vol 3 no 2 pp 299ndash309 2003

[5] F Brauer ldquoBackward bifurcations in simple vaccination mod-elsrdquo Journal of Mathematical Analysis and Applications vol 298no 2 pp 418ndash431 2004

[6] K P Hadeler and P van den Driessche ldquoBackward bifurcationin epidemic controlrdquo Mathematical Biosciences vol 146 no 1pp 15ndash35 1997

[7] Z X Hu S Liu and H Wang ldquoBackward bifurcation of anepidemic model with standard incidence rate and treatmentraterdquo Nonlinear Analysis Real World Applications vol 9 no 5pp 2302ndash2312 2008

[8] J Hui and D M Zhu ldquoGlobal stability and periodicity onSIS epidemic models with backward bifurcationrdquo Computers ampMathematics with Applications vol 50 no 8-9 pp 1271ndash12902005

[9] Y Jin W D Wang and S W Xiao ldquoAn SIRS model with anonlinear incidence raterdquoChaos Solitons amp Fractals vol 34 no5 pp 1482ndash1497 2007

[10] G H Li and W D Wang ldquoBifurcation analysis of an epidemicmodel with nonlinear incidencerdquo Applied Mathematics andComputation vol 214 no 2 pp 411ndash423 2009

[11] X Z Li W S Li and M Ghosh ldquoStability and bifurcation ofan SIS epidemic model with treatmentrdquo Chaos Solitons ampampFractals vol 42 no 5 pp 2822ndash2832 2009

[12] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1986

[13] X B Liu and L J Yang ldquoStability analysis of an SEIQV epidemicmodel with saturated incidence raterdquo Nonlinear Analysis RealWorld Applications vol 13 no 6 pp 2671ndash2679 2012

[14] M Martcheva and H R Thieme ldquoProgression age enhancedbackward bifurcation in an epidemic model with super-infectionrdquo Journal of Mathematical Biology vol 46 no 5 pp385ndash424 2003

[15] S G Ruan and W D Wang ldquoDynamical behavior of anepidemic model with a nonlinear incidence raterdquo Journal ofDifferential Equations vol 188 no 1 pp 135ndash163 2003

[16] W D Wang and S G Ruan ldquoBifurcation in an epidemicmodel with constant removal rate of the infectivesrdquo Journal ofMathematical Analysis and Applications vol 291 no 2 pp 775ndash793 2004

[17] Y L Tang D Q Huang S G Ruan and W N ZhangldquoCoexistence of limit cycles and homoclinic loops in a SIRSmodel with a nonlinear incidence raterdquo SIAM Journal onApplied Mathematics vol 69 no 2 pp 621ndash639 2008

[18] P van den Driessche and J Watmough ldquoA simple SIS epidemicmodel with a backward bifurcationrdquo Journal of MathematicalBiology vol 40 no 6 pp 525ndash540 2000

[19] J J Wei and J A Cui ldquoDynamics of SIS epidemic model withthe standard incidence rate and saturated treatment functionrdquoInternational Journal of Biomathematics vol 5 no 3 Article ID1260003 18 pages 2012

[20] W D Wang ldquoBackward bifurcation of an epidemic model withtreatmentrdquoMathematical Biosciences vol 201 no 1-2 pp 58ndash712006

[21] Z W Wang ldquoBackward bifurcation in simple SIS modelrdquo ActaMathematicae Applicatae Sinica vol 25 no 1 pp 127ndash136 2009

[22] L Xue and C Scoglio ldquoThe network level reproduction numberfor infectious diseases with both vertical and horizontal trans-missionrdquo Mathematical Biosciences vol 243 no 1 pp 67ndash802013

14 Mathematical Problems in Engineering

[23] X Zhang and X N Liu ldquoBackward bifurcation of an epidemicmodel with saturated treatment functionrdquo Journal ofMathemat-ical Analysis and Applications vol 348 no 1 pp 433ndash443 2008

[24] J Guckenheimer and P J Holmes Nonlinear OscillationsDynamical Systems and Bifurcation of Vector Fields SpringerNew York NY USA 1996

[25] X Zhang and X N Liu ldquoBackward bifurcation and globaldynamics of an SIS epidemic model with general incidence rateand treatmentrdquoNonlinearAnalysis RealWorldApplications vol10 no 2 pp 565ndash575 2009

[26] M A Safi A B Gumel and E H Elbasha ldquoQualitative analysisof an age-structured SEIR epidemic model with treatmentrdquoApplied Mathematics and Computation vol 219 no 22 pp10627ndash10642 2013

[27] L H Zhou and M Fan ldquoDynamics of an SIR epidemic modelwith limited medical resources revisitedrdquo Nonlinear AnalysisReal World Applications vol 13 no 1 pp 312ndash324 2012

[28] X Y Zhou and J A Cui ldquoAnalysis of stability and bifurcation foran SEIR epidemicmodel with saturated recovery raterdquoCommu-nications inNonlinear Science andNumerical Simulation vol 16no 11 pp 4438ndash4450 2011

[29] D Lacitignola ldquoSaturated treatments and measles resurgenceepisodes in South Africa a possible linkagerdquo MathematicalBiosciences and Engineering MBE vol 10 no 4 pp 1135ndash11572013

[30] J C Eckalbar and W L Eckalbar ldquoDynamics of an epidemicmodel with quadratic treatmentrdquo Nonlinear Analysis RealWorld Applications vol 12 no 1 pp 320ndash332 2011

[31] F Brauer ldquoBackward bifurcations in simple vaccinationtreat-ment modelsrdquo Journal of Biological Dynamics vol 5 no 5 pp410ndash418 2011

[32] M W Hirsch S Smale and R L Devaney Differential Equa-tions Dynamical Systems and an Introduction to Chaos Aca-demic Press New York NY USA 2003

[33] R Bogdanov ldquoBifurcations of a limit cycle for a family of vectorfields on the planrdquo SelectaMathematica vol 1 pp 373ndash388 1981

[34] R Bogdanov ldquoVersal deformations of a singular point on theplan in the case of zero eigen-valuesrdquo Selecta MathematicaSovietica vol 1 pp 389ndash421 1981

[35] F Takens ldquoForced oscillations and bifurcationrdquo in Applicationsof Global Analysis I vol 3 pp 1ndash59 Communications of theMathematical Institute Rijksuniversiteit Utrecht 1974

[36] Y A Kuznetsov Elements of Applied Bifurcation TheorySpringer New York NY USA 2nd edition 1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

and displayed all limit cycles and monoclinic loops of orderup to 2 Wei and Cui explored an SIS epidemic model withstandard incidence rate in [19] They took the incidence rateform 119891(119868)119878 = 120573119868119878(119868 + 119878) and showed the dynamics andbackward bifurcation of the SIS epidemic model

Recently in order to prevent and control the spread ofthe infectious diseases such as measles tuberculosis andflu many mathematicians (see [7 11 16 20 23 25ndash29])have begun to investigate the role of treatment functions inepidemiological models In some classical epidemic modelsthe treatment function is an important method to decreasethe spread of the epidemiological diseases Generally speak-ing the treatment function of the infective individuals isassumed to be proportional to the number of the infectiveindividuals But every community should have a maximalcapacity for the treatment of a disease and the resources fortreatment should be very largeTherefore it is very importantto adopt a suitable treatment function In [16] Wang andRuan introduced a constant treatment function of diseases inan SIR model that is

119879 (119868) = 119903 119868 gt 0

0 119868 = 0(2)

This means that they use the maximal treatment capacityto cure infective individuals so that the disease can beeradicatedThey also found that themodel undergoes saddle-node bifurcation Hopf bifurcation and Bogdanov-Takensbifurcation Further a piecewise linear treatment functionwas considered in [20] that is

119879 (119868) = 119896119868 0 le 119868 le 1198680

119898 119868 gt 1198680(3)

where 119898 = 1198961198680 and 119896 and 1198680 are positive constants Thismeans that the treatment rate is proportional to the numberof the infective individuals when the capacity of treatment hasnot been reached otherwise it takes the maximal capacity oftreatment 1198961198680 By considering the above treatment functionWang [20] found that a backward bifurcation takes place in anSIR epidemicmodel In [30] J C Eckalbar andW L Eckalbarconstructed an SIR epidemic model with a quadratic treat-ment function that is119879(119868) = max119903119868minus1198921198682 0 119903 119892 gt 0Theyfound that the system has as many as four equilibria withpossible bistability backward bifurcations and limit cycles

Recently saturated treatment function has been widelyapplied inmany epidemicmodels In particular in paper [23]Zhang and Liu took a continuous and differentiable saturatedtreatment function 119879(119868) = 119903119868(1 + 120572119868) where 119903 gt 0 120572 ge 0119903 stands for the cure rate and 120572 measures the extent of theeffect of the infected being delayed for treatment We cansee that the treatment function 119879(119868) sim 119903119868 when 119868 is smallenoughwhereas119879(119868) sim 119903120572when 119868 is large enough It ismorerealistic and it has the convenience of being continuous anddifferential compared to the previous ones Furthermore theauthors in [23] found that 1198770 = 1 is a critical threshold fordisease eradication when this delayed effect for treatment isweak and a backward bifurcation will take place when thiseffect is strong So it is really important to adequately stress

the interesting connection recently established between thechoice of saturated treatment functions in epidemic modelsand the occurrence of backward bifurcation in the relatedsystem dynamics In fact recently saturated-type treatmentfunctions have been indicated as responsible for the occur-rence of backward bifurcations for SIR [20 23] for SIS [1921] and for SEIR [28 29] models supporting the generalcircumstance that saturated-type treatments can be one ofthe causes of backward bifurcations in epidemic models Inparticular in [29] such connection has also been shown andvalidated in a specific concrete disease-control setting

In the real world some infectious diseases do not conferimmunity Such infections do not have a recovered state andindividuals become susceptible again after infection Thistype of disease can be modeled by the SIS type So the SISepidemic model has been adopted by many mathematicians(see [8 11 18 19 21 25]) And SIS models are appropriatefor some bacterial agent diseases such as meningitis plagueand venereal diseases and for protozoan agent diseases suchas malaria and sleeping sickness

Motivated by the above points we will consider the fol-lowing SIS model with bilinear incidence rate and saturatedtreatment function

119889119878

119889119905= 119860 minus 119889119878 minus 120582119878119868 + 120576119868 +

119903119868

1 + 120572119868

119889119868

119889119905= 120582119878119868 minus (119889 + 120576 + 120583) 119868 minus

119903119868

1 + 120572119868

(4)

where 119878 and 119868 denote the numbers of susceptible and infectiveindividuals respectively Positive constant 119860 is the recruit-ment rate of the population Positive constant 119889 is the naturedeath rate of population The bilinear incidence rate is 120582119878119868where 120582 is positive Positive constant 120576 is the natural recoveryrate of infective individuals Positive constant 120583 is the disease-related death rate The saturated treatment function ℎ(119868) ≜119903119868(1 + 120572119868) where 119903 is positive and 120572 is nonnegative

This paper focuses on the detailed dynamics analysisof the model (4) The local stability of these equilibria isinvestigated which enables us to classify the types of modelequilibria (eg attractor saddle or repeller) We show thatthe system has backward bifurcation and Bogdanov-Takensbifurcation (ie Hopf bifurcation saddle-node bifurcationand homoclinic bifurcation) under some certain conditionsFinally the three bifurcation curves and the complicatedglobal bifurcation phase portraits are derived by applyingthe Bogdanov-Takens normal form and the correspond-ing parameters which satisfy the conditions that ensureBogdanov-Takens bifurcation exists

The organization of this paper is as follows In Section 2we study the existence and local stability of equilibria andbackward bifurcation In Section 3 we investigate the globalstability of the model In Section 4 we give the supercriticaland subcritical bifurcation under two different conditionsin system (4) In Section 5 we show that the system (4)undergoes Bogdanov-Takens bifurcation under some certainconditions In Section 6 some numerical simulations aredisplayed in detail We close with a discussion in Section 7 onour mathematical results and epidemiological implications

Mathematical Problems in Engineering 3

0 5 10 15 20 25 30 35 4002468

101214161820

S

I

E0

Figure 1 The disease-free equilibrium 1198640 is locally asymptoticallystable when 1198770 lt 1 with the parameter values 119860 = 2 119889 = 01119896 = 01 120583 = 001 120582 = 001 120572 = 1 119903 = 2 and 120576 = 01

2 Equilibria and Backward Bifurcation

21 Disease-Free Equilibrium Obviously system (4) has adisease-free equilibrium 1198640 = (119860119889 0) The Jacobian matrixof (4) at 1198640 is

119872(1198640) = (minus119889

minus120582119860

119889+ 120576 + 119903

0120582119860

119889minus (119889 + 120576 + 120583) minus 119903

) (5)

By using the next generation matrix of (4) we get the basicreproduction number 1198770 = 120582119860119889(119889 + 120576 + 119903 + 120583)119872(1198640) hasnegative eigenvalues if 120582119860119889 minus (119889 + 120576 + 120583) minus 119903 lt 0 Then wehave the following result

Theorem 1 The disease-free equilibrium 1198640 is locally asymp-totically stable when 1198770 lt 1 (see Figure 1) and is unstable when1198770 gt 1 (see Figure 2)

22 Endemic Equilibria An endemic equilibrium alwayssatisfies

119860 minus 119889119878 minus 120582119878119868 + 120576119868 +119903119868

1 + 120572119868= 0

120582119878119868 minus (119889 + 120576 + 120583) 119868 minus119903119868

1 + 120572119868= 0

(6)

In view of 119889119878 + (119889 + 120583)119868 = 119860 we get 119878 = (119860 minus (119889 + 120583)119868)119889 andsubstitute it into the second equation of (6) When 119868 = 0 weobtain

120582[119860 minus (119889 + 120583) 119868

119889] minus (119889 + 120576 + 120583) minus

119903

1 + 120572119868= 0 (7)

Then we have an equation of the form

1198861198682 + 119887119868 + 119888 = 0 (8)

0 50 100 150 200 25002468

101214161820

S

I

E0

Figure 2 The disease-free equilibrium is unstable when 1198770 gt 1with the parameter values 119860 = 15 119889 = 01 119896 = 01 120583 = 001120582 = 001 120572 = 1 119903 = 08 and 120576 = 01

with

119886 = 120572120582 (119889 + 120583)

119887 = 120582 (119889 + 120583) + 120572119889 (119889 + 120576 + 120583) minus 120572120582119860

119888 = 119889 (119889 + 120576 + 119903 + 120583) minus 120582119860

(9)

This equation may admit positive solution

1198681 =minus119887 minus radic1198872 minus 4119886119888

2119886 1198682 =

minus119887 + radic1198872 minus 4119886119888

2119886 (10)

Obviously if 1198770 = 1 then 119888 = 0 if 1198770 gt 1 then 119888 lt 0 andif 1198770 lt 1 then 119888 gt 0 From (8) it is obvious that we have thefollowing results

Theorem 2 The following results hold

(1198671) Let 120572 = 0 Equation (8) is a linear equation with aunique solution 119868 = minus119888119887 Then the system (4) has aunique endemic equilibrium when 1198770 gt 1 and has noendemic equilibrium when 1198770 le 1

(1198672) Let 120572 gt 0 If 119887 gt 0 system (4) has a unique endemicequilibrium when 1198770 gt 1 and no endemic equilibriumwhen 1198770 le 1

(1198673) Let 120572 gt 0 If 119887 lt 0 system (4) has a unique endemicequilibrium when 1198770 ge 1 no endemic equilibriumwhen 1198770 lt 119877lowast0 and two endemic equilibria 1198641 and 1198642when 119877lowast

0le 1198770 lt 1 When 1198770 = 119877lowast0 and 1198641 = 1198642 one

has 1198872 minus 4119886119888 = 0 which is equivalent to

1198770 =4119886120582119860

1198872 + 4119886120582119860

=41205822120572 (119889 + 120583)119860

[120582 (119889 + 120583) + 120572119889 (119889 + 120576 + 120583) minus 120572120582119860]2+ 41205822120572 (119889 + 120583)119860

(11)

4 Mathematical Problems in Engineering

181614121080604020

10

20

30

40

50

60

I

Rlowast

0

R0

Figure 3 The figure of infective sizes at equilibria versus 1198770 when120572 = 1 119889 = 01 120583 = 001 120582 = 005 119903 = 2 and 120576 = 01

We define the right-hand side of (11) as

119877lowast0≜

41205822120572 (119889 + 120583)119860

[120582 (119889 + 120583) + 120572119889 (119889 + 120576 + 120583) minus 120572120582119860]2+ 41205822120572 (119889 + 120583)119860

(12)

Therefore according to the qualitative approach recentlyproposed by [31] which is based on the analysis of theequilibria curve in the neighboring of the critical threshold1198770 = 1 we have the following theorem

Theorem 3 If 120572 gt 0 119887 lt 0 then system (4) has a backwardbifurcation at 1198770 = 1 (see Figure 3)

In order to verify the bifurcation curve (the graph of 119868as a function of 1198770) in Figure 3 we think of 119903 as a variablewith the other parameters as constant Through implicitdifferentiation of (8) with respect to 119903 we get

(2119886119868 + 119887)119889119868

119889119903= minus119889 lt 0 (13)

From (13) we know that the sign of 119889119868119889119903 is opposite to thatof 2119886119868 + 119887 And from the definition of 1198770 we know that 1198770decreases when 119903 increases It implies that the bifurcationcurve has positive slope at equilibrium values with 2119886119868+119887 gt 0and negative slope at equilibrium values with 2119886119868 + 119887 lt 0 Ifthere is no backward bifurcation at 1198770 = 1 then the uniqueendemic equilibrium for 1198770 gt 1 satisfies

2119886119868 + 119887 = radic1198872 minus 4119886119888 gt 0 (14)

and the bifurcation curve has positive slope at all pointswhere119868 gt 0 If there is a backward bifurcation at 1198770 = 1 then thereis an interval onwhich there are two endemic equilibria givenby

2119886119868 + 119887 = plusmnradic1198872 minus 4119886119888 (15)

The bifurcation curve has negative slope at the smaller oneandpositive slope at the larger oneThus the bifurcation curveis as shown in Figure 3

Under the conditions of Theorem 3 if a backward bifur-cation takes place we can see from Figure 3 that there is acritical value 119877lowast

0at the turning point In this case the disease

will not die out when1198770 lt 1 However the diseasewill die outwhen 1198770 lt 119877

lowast

0 Therefore the critical value 119877lowast

0can be taken

as a new threshold for the control of the diseaseIn the following we give an explicit criterion of a

backward bifurcation at 1198770 = 1For convenience we define

1205720 =120582 (119889 + 120583)

119889119903 (16)

Corollary 4 When 120572 gt 1205720 system (4) has a backward bifur-cation at 1198770 = 1

Proof When 1198770 le 1 hArr 119888 ge 0

120582119860 le 119889 (119889 + 120576 + 120583 + 119903) (17)

The condition 119887 lt 0 is equivalent to

120582 (119889 + 120583) + 120572119889 (119889 + 120576 + 120583) lt 120572120582119860 (18)

From (17) and (18) we get 120582(119889 + 120583) + 120572119889(119889 + 120576 + 120583) lt 120572119889(119889 +120576 + 119903 + 120583) which reduces to

120572 gt120582 (119889 + 120583)

119889119903≜ 1205720 (19)

It means that 120572 is big enough to lead a backward bifurcationwith two endemic equilibria when 1198770 lt 1 Therefore theproof is complete

Next we consider the local stability of the uniqueendemic equilibrium when 1198770 gt 1

Theorem 5 When 1198770 gt 1 and 0 le 120572 lt 120582119903 the uniqueendemic equilibrium 119864lowast is locally asymptotically stable

Proof Firstly from Theorem 2 we can know that system(4) has a unique endemic equilibrium 119864lowast when 1198770 gt 1Moreover the Jacobian matrix of system (4) is

119872 =(

minus119889 minus 120582119868 minus120582119878 + 120576 +119903

1 + 120572119868minus

119903120572119868

(1 + 120572119868)2

120582119868 120582119878 minus (119889 + 120576 + 120583) minus119903

1 + 120572119868+

119903120572119868

(1 + 120572119868)2

)

(20)

From the second equation of (6) we have

minus120582119878 + 120576 +119903

1 + 120572119868= minus119889 minus 120583 (21)

From (21) the Jacobian matrix119872 reduces to

119872 =(

minus119889 minus 120582119868 minus119889 minus 120583 minus119903120572119868

(1 + 120572119868)2

120582119868119903120572119868

(1 + 120572119868)2

) (22)

Mathematical Problems in Engineering 5

We obtain

det (119872) = 119868

(1 + 120572119868)2[120582 (119889 + 120583) (1 + 120572119868)

2 minus 119903120572119889] (23)

In fact there is 120582(119889 + 120583)(1 + 120572119868)2 gt 120582119889 Since 1198770 gt 1 and0 le 120572 lt 120582119903 we have

120582 (119889 + 120583) (1 + 120572119868)2 minus 119903120572119889 gt 120582119889 minus 119903120572119889 gt 0 (24)

So we get det(119872) gt 0 The trace of119872 is given by

tr (119872) = 1

(1 + 120572119868)2[minus (119889 + 120582119868) (1 + 120572119868)

2 + 119903120572119868] (25)

In the same way as the above calculation of (24) we have

minus (119889 + 120582119868) (1 + 120572119868)2 + 119903120572119868 lt minus120582119868 + 119903120572119868 lt 0 (26)

So we get tr(119872) lt 0 The proof is complete

Now we consider the case that there are two endemicequilibria 1198641 and 1198642 let 119872119894 be the Jacobian matrix at 119864119894119894 = 1 2

Theorem 6 The endemic equilibrium 1198641 is a saddle wheneverit exists

Proof Since 1198681 = (minus119887 minus radic1198872 minus 4119886119888)2119886 and Δ = 1198872 minus 4119886119888 wehave 1198681 = (minus119887 minus radicΔ)2119886 Thus

det (1198721)

=1198681

(1 + 1205721198681)2[120582 (119889 + 120583) (1 + 1205721198681)

2minus 119903120572119889]

≜1198681

(1 + 1205721198681)2120595 (1198681)

(27)

From (1198673) of Theorem 2 we can know that if 120572 gt 0 119887 lt 0and 119877lowast

0lt 1198770 lt 1 then 1198641 exists Hence we can get 120595(0) =

120582(119889 + 120583) minus 119903120572119889 lt 0 and 1205951015840(1198681) = 2120582(119889 + 120583)(1 + 1205721198681)120572 gt 0 Itfollows that there exists a unique 119868lowast gt 0 such that

120595 (1198681) = 0 when 1198681 = 119868lowast

120595 (1198681) lt 0 when 0 lt 1198681 lt 119868lowast

120595 (1198681) gt 0 when 1198681 gt 119868lowast

(28)

where

119868lowast = radic119903119889

120572120582 (119889 + 120583)minus1

120572 (29)

On the other hand

1198681 = minus119887

2119886minusradicΔ

2119886

=120572120582119860 minus 120582 (119889 + 120583) minus 120572119889 (119889 + 120576 + 120583)

2120572120582 (119889 + 120583)minusradicΔ

2119886

= radic119903119889

120572120582 (119889 + 120583)minus1

120572+1

120572minus radic

119903119889

120572120582 (119889 + 120583)

+120572120582119860 minus 120582 (119889 + 120583) minus 120572119889 (119889 + 120576 + 120583)

2120572120582 (119889 + 120583)minusradicΔ

2119886

= 119868lowast + ([120582 (119889 + 120583) + 120572120582119860 minus 2radic119903119889120572120582 (119889 + 120583)

minus 120572119889 (119889 + 120576 + 120583)] minus radicΔ)times (2120572120582 (119889 + 120583))minus1

(30)

Δ = 120582 (119889 + 120583) minus [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]2

minus 4120572120582 (119889 + 120583) [119889 (119889 + 120576 + 119903 + 120583) minus 120582119860]

= 120582 (119889 + 120583) + [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

minus 2radic119903119889120572120582(119889 + 120583)2

minus 4119903119889120572120582 (119889 + 120583)

minus 4120582 (119889 + 120583) [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

+ 4radic119903119889120572120582 (119889 + 120583) 120582 (119889 + 120583) + [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

minus 4120572120582 (119889 + 120583) [minus120582119860 + 119889 (119889 + 120576 + 119903 + 120583)]

≜ 120582 (119889 + 120583) + [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

minus 2radic119903119889120572120582 (119889 + 120583)2

+ 119875

(31)

where

119875 = minus8119903119889120572120582 (119889 + 120583) + 4radic119903119889120572120582 (119889 + 120583)

times 120582 (119889 + 120583) + [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

(32)

In the following we will show that 119875 gt 0 Since 119877lowast0lt 1198770 we

have

4120572120582 (119889 + 120583)

[120582 (119889 + 120583) + 120572119889 (119889 + 120576 + 120583) minus 120572120582119860]2+ 41205721205822 (119889 + 120583)119860

lt1

119889 (119889 + 120576 + 119903 + 120583)

(33)

6 Mathematical Problems in Engineering

that is

[120582 (119889 + 120583) + 120572119889 (119889 + 120576 + 120583) minus 120572120582119860]2

gt 4120572120582 (119889 + 120583) 119889 (119889 + 120576 + 119903 + 120583) minus 41205721205822 (119889 + 120583)119860(34)

Therefore

120582 (119889 + 120583) + [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]2

= 120582 (119889 + 120583) minus [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]2

+ 4120582 (119889 + 120583) [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

gt 4120572120582 (119889 + 120583) 119889 (119889 + 120576 + 119903 + 120583) minus 41205721205822 (119889 + 120583)119860

+ 4120582 (119889 + 120583) [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

= 4119903119889120572120582 (119889 + 120583)

(35)

Obviously we have 119875 gt 0 From (30) one has 1198681 lt 119868lowast So

we get det(1198721) lt 0 Hence the endemic equilibrium 1198641 is asaddle The proof is complete

In order to explore the stability of the endemic equilib-rium 1198642 define

1198981 = (21198891205721198862 + 1205821198862 minus 1205761205721198862 minus 1198861198881205821205722)

minus 119887 (2120582120572119886 + 1198891205722119886 minus 1198871205821205722)

1198982 = 1198862119889 minus 119888 (2120582120572119886 + 1198891205722119886 minus 1198871205821205722)

(36)

Theorem 7 If 120578 gt 0 then endemic equilibrium 1198642 is locallyasymptotically stable if 120578 lt 0 then endemic equilibrium 1198642 isunstable where 120578 = 21198861198982 + 1198981(radic1198872 minus 4119886119888 minus 119887)

Proof Consider

det (1198722) =1198682

(1 + 1205721198682)2[120582 (119889 + 120583) (1 + 1205721198682)

2minus 120576120572119889]

=1198682

(1 + 1205721198682)2times 120595 (1198682)

(37)

By carrying out arguments similar to that of Theorem 6 wehave 1198682 gt 119868

lowast Therefore det(1198722) gt 0 In addition we have

tr (1198722) = minus(119889 + 1205821198682) (1 + 1205721198682)

2minus 1205761205721198682

(1 + 1205721198682)2

= minus120582120572211986832+ (2120582120572 + 1198891205722) 1198682

2+ (2119889120572 + 120582 minus 120576120572) 1198682 + 119889

(1 + 1205721198682)2

(38)

and then sgn(tr(1198722)) = minus sgn(119866(1198682)) where

119866 (119909) = 12058212057221199093 + (2120582120572 + 1198891205722) 1199092 + (2119889120572 + 120582 minus 120576120572) 119909 + 119889

(39)

I

0 50 100 1500

5

10

15

20

25

30

35

40

S

E0

E1

E2

Figure 4 One region of disease persistence and one region ofdisease extinction when 119860 = 10 120572 = 1 119889 = 01 120583 = 001 120582 = 001119903 = 01 and 120576 = 01

Using the expression of1198981 = (21198891205721198862 +1205821198862 minus 1205761205721198862 minus1198861198881205821205722) minus

119887(2120582120572119886+1198891205722119886minus1198871205821205722) and1198982 = 1198862119889minus119888(2120582120572119886+1198891205722119886minus1198871205821205722)

one has

119866 (1198682) = (1198861198682

2+ 1198871198682 + 119888) 1205930 +

11989811198682 + 11989821198862

(40)

where1205930 is a first degree polynomial of 1198682 Since 1198861198682

2+1198871198682+119888 =

0 sgn(tr(1198722)) = minus sgn(119866(1198682)) = minus sgn(11989811198682 + 1198982) From theexpression of 1198682 we have

sgn (11989811198682 + 1198982)

= sgn (21198861198982 + 1198981 (radic1198872 minus 4119886119888 minus 119887)) ≜ sgn (120578) (41)

Thus 1198642 is locally asymptotically stable if 120578 gt 0 and 1198642 isunstable if 120578 lt 0 The proof is complete

From the above discussion we have the following con-clusion If two endemic equilibria 1198641 and 1198642 exist the stablemanifolds of the saddle 1198641 split 119877

2

+into two regions The

disease is persistent in the upper region and dies out in thelower region (see Figure 4)

3 Global Analysis

Firstly we consider the global stability of the disease-freeequilibrium 1198640 Let 119873 = 119878 + 119868 be the total population sizeNow we note that the equation for total population is givenby 119889119873119889119905 = 119860 minus 119889119878 minus (119889 + 120583)119868 le 119860 minus 119889119873 It follows thatlim119905rarr+infin119873(119905) le 119860119889 Let

R = (119878 119868) isin 1198772

+ 119878 + 119868 le

119860

119889 (42)

which is positive invariant with respect to system (4)

Theorem 8 If 1198770 lt 119877lowast0 the disease-free equilibrium 1198640(119860119889 0) is globally asymptotically stable that is the disease diesout

Mathematical Problems in Engineering 7

Proof Suppose 1198770 lt 119877lowast0 From the (1198673) of Theorem 2 weknow that the model has no endemic equilibrium Fromthe corollary of Poincare-Bendixson theorem [32] we knowthat there is no periodic orbit in R as there is a disease-free equilibrium in R Since R is a bounded positivelyinvariant region and 1198640 is the only equilibrium in R thelocal stability of 1198640 implies that every solution initiating inR approaches 1198640 Thus the disease-free equilibrium 1198640 isglobally asymptotically stable The proof is complete

Now we analyze the global dynamics of the endemicequilibrium when 1198770 gt 1

Theorem 9 If 1198770 gt 1 and 0 le 120572 lt 120582119903 the system (4) has nolimit cycles

Proof We use Dulac theorem to exclude the limit cycle Let

119875 (119878 119868) = 119860 minus 119889119878 minus 120582119878119868 + 120576119868 +119903119868

1 + 120572119868

119876 (119878 119868) = 120582119878119868 minus (119889 + 120576 + 120583) 119868 minus119903119868

1 + 120572119868

(43)

and take the Dulac function

119863 =1

119868 (44)

According to 0 le 120572 lt 120582119903 we can get

120597 (119875119863)

120597119878+120597 (119876119863)

120597119868

= minus119889

119868minus 120582 +

120572119903

(1 + 120572119868)2

lt minus119889

119868minus 120582 +

120582

(1 + 120572119868)2

=1

119868 (1 + 120572119868)2minus119889 (1 + 120572119868)

2 minus 120582119868 [(1 + 120572119868)2 minus 1]

lt 0

(45)

Hence the system (4) has no limit cycles The proof iscomplete

Therefore we obtain the global result of the uniqueendemic equilibrium

Theorem 10 If 1198770 gt 1 and 0 le 120572 lt 120582119903 the unique endemicequilibrium 119864lowast is globally asymptotically stable (see Figure 5)

4 Hopf Bifurcation

In this section we study the Hopf bifurcation of system (4)From the above discussion we know that there is no closedorbit surrounding1198640 or1198641 because the 119878-axis is invariantwith

0 50 100 1500

50

100

150

S

I Elowast

Figure 5The unique endemic equilibrium 119864lowast is globally asymptot-ically stable when 119860 = 15 120572 = 001 119889 = 01 120583 = 001 120582 = 001119903 = 08 and 120576 = 01

respect to system (4) and 1198641 is always a saddle ThereforeHopf bifurcation can only occur at 1198642 Set

120590 = [minus120582120572 minus1198881 (1198882 + 21198884)

11988612]1198634

+ [minus 120582120572119886211minus 21205821205721198861111988612 + (1198882 minus

211988611119888111988612)

times (119886111198882 minus119886211119888111988612)

minus (21198862111198881

11988612minus 119886111198882 + 2119888411988611 minus 119888311988612)

times (1198884 minus11988611119888111988612)]1198632

+ 11988611 (119886211119888111988612minus 119886111198882 + 119888411988611 minus 119888311988612)

times (119886121198883 minus 2119886111198884 minus 119886111198882)

(46)

where 119886119894119895 (119894 119895 = 1 2) 119888119896 (119896 = 1 2 3 4) and 119863 are defined by(50) and (52)

Theorem 11 System (4) undergoes a Hopf bifurcation if 120578 = 0Moreover if 120590 lt 0 there is a family of stable period orbits ofsystem (4) as 120578 decreases from 0 that is a supercritical Hopfbifurcation occurs if 120590 gt 0 there is a family of unstable periodorbits of system (4) as 120578 increases from 0 that is a subcriticalHopf bifurcation occurs

Proof The proof of Theorem 7 shows that tr(1198722) = 0 if andonly if 120578 = 0 and det(1198722) gt 0 when 1198642 exists Therefore theeigenvalues of 1198722 are a pair of pure imaginary roots if andonly if 120578 = 0 The direct calculations show that

119889 (tr (1198722))119889120578

100381610038161003816100381610038161003816100381610038161003816120578=0= minus

1

21198863 (1 + 1205721198682)2lt 0 (47)

8 Mathematical Problems in Engineering

ByTheorem 342 in [24] 120578 = 0 is the Hopf bifurcation pointfor system (4)

To be concise in notations rescale (4) by 120591 = 119905(1 + 120572119868)For simplicity we still use 119905 instead of 120591 Then we obtain

119889119878

119889119905= (119860 minus 119889119878 minus 120582119878119868 + 120576119868) (1 + 120572119868) + 119903119868

119889119868

119889119905= 120582119878119868 (1 + 120572119868) minus (119889 + 120576 + 120583) 119868 (1 + 120572119868) minus 119903119868

(48)

Let 119909 = 119878 minus 1198782 and 119910 = 119868 minus 1198682 then (48) becomes

119889119909

119889119905= 11988611119909 + 11988612119910 + 1198881119910

2 + 1198882119909119910 minus 1205821205721199091199102

119889119910

119889119905= 11988621119909 + 11988622119910 + 1198883119909119910 + 1198884119910

2 + 1205821205721199091199102

(49)

where

11988611 = minus (119889 + 1205821198682) (1 + 1205721198682)

11988612 = (minus1205821198782 + 119903) (1 + 1205721198682)

+ 120572 (119860 minus 1198891198782 minus 12058211987821198682 + 1199031198682) + 120576

11988621 = 1205821198682 (1 + 1205721198682)

11988622 = [1205821198782 minus (119889 + 119903 + 120583)] (1 + 1205721198682)

+ 120572 [12058211987821198682 minus (119889 + 119903 + 120583) 1198682] minus 120576

1198881 = 120572 (minus1205821198782 + 119903)

1198882 = 120572 (minus119889 minus 1205821198682) minus 120582 (1 + 1205721198682)

1198883 = 120582 (1 + 1205721198682) + 1205821198682120572

1198884 = [1205821198782 minus (119889 + 119903 + 120583)] 120572

(50)

Let 119864lowast denote the origin of 119909-119910 plane Since 1198642 = (1198782 1198682)satisfies (6) we obtain

det (119872 (119864lowast))

= minus (119889 + 1205821198682) 1205721205761198682 + (119889 + 120583) (1 + 1205721198682

2)2

1205821198682 + 1205721205821205761198682

2

= 1198682 times 120595 (1198682)

(51)

From the proof of Theorem 7 it follows that 120595(1198682) is alwayspositive It is easy to verify that 11988611 + 11988622 = 0 if and only if120578 = 0 Set

119863 = radicdet (119872 (119864lowast)) (52)

and let 119906 = minus119909 and V = (11988611119863)119909+(11988612119863)119910 then the normalform of (48) for Hopf bifurcation reads

119889119906

119889119905= minus119863V + 119891 (119906 V)

119889V119889119905= 119863119906 + 119892 (119906 V)

(53)

where

119891 (119906 V) = (11988611119886121198882 minus119886211

119886212

1198881)1199062 minus11986321198881119886212

V2

+ (119863119888211988612minus 2119863119886111198881119886212

)119906V minus120582120572119886211

119886212

1199063

minus 211986311988611120582120572

119886212

1199062V minus1198632120582120572

119886212

119906V2

119892 (119906 V) =11988611119863(119886211

119886212

1198881 minus11988611119886121198882 +11988611119886121198884 minus 1198883)119906

2

+ (119863119886111198881119886212

+119863119888411988612) V2

+ (2119886211

119886212

1198881 minus11988611119886121198882 + 2

11988611119886121198884 minus 1198883)119906V

+120582120572119886211

11988612119863(1

11988612minus 1)1199063 +

21205821205721198861111988612

(1198861111988612minus 1)1199062V

+119863120582120572

11988612(1198861111988612minus 1)119906V2

(54)

Now we evaluate the first Lyapunov coefficient Γ of system(4) as follows

Γ =1

16[119891119906119906119906 + 119891119906VV + 119892119906119906V + 119892VVV] +

1

16119863

times [119891119906V (119891119906119906 + 119891VV) minus 119892119906V (119892119906119906 + 119892VV) minus 119891119906119906119892119906119906 + 119891VV119892VV]

(55)

where 119891119906V denotes (1205972119891120597119906120597V)(0 0) and so forth Then

Γ =1

81198862121198632[minus120582120572 minus

1198881 (1198882 + 21198884)

11988612]1198634

+ [minus 120582120572119886211minus 21205821205721198861111988612 + (1198882 minus

211988611119888111988612)

times (119886111198882 minus119886211119888111988612)

minus (21198862111198881

11988612minus 119886111198882 + 2119888411988611 minus 119888311988612)

times (1198884 minus11988611119888111988612)]1198632

+ 11988611 (119886211119888111988612minus 119886111198882 + 119888411988611 minus 119888311988612)

times (119886121198883 minus 2119886111198884 minus 119886111198882)

=120590

81198862121198632

(56)

Mathematical Problems in Engineering 9

Obviously the sign of Γ is determined by 120590 ByTheorem 342and (3411) in [24] the rest of the claims in Theorem 11 arevalid The proof is complete

5 Bogdanov-Takens Bifurcations

The purpose of this section is to study the Bogdanov-Takensbifurcation of (4) Now we assume the following

(1198601) 1198872 minus 4119886119888 = 0

(1198602) 21198861198892 + 119887120582120583 = 0

Then (6) admits a unique positive equilibrium 119864lowast = (119878lowast 119868lowast)where

119868lowast = minus119887

2119886 119878lowast =

119860 minus (119889 + 120583) 119868lowast

119889 (57)

The Jacobian matrix of system (4) at 119864lowast is

119872lowast = (

minus119889 minus 120582119868lowast minus120582119878lowast + 120576 +119903

(1 + 120572119868lowast)2

120582119868lowast 120582119878lowast minus (119889 + 120576 + 120583) minus119903

(1 + 120572119868lowast)2)

(58)

By (58) we have

det (119872lowast) = 119868lowast

(1 + 120572119868lowast)2[(119889 + 120583) 120582 (1 + 120572119868lowast)

2minus 119903119889120572]

= 0

(59)

because of

(1 + 120572119868lowast)2=41198862 minus 4119886120572119887 + 12057221198872

41198862

=41198862 minus 4119886120572119887 + 12057224119886119888

41198862

=119886 minus 120572119887 + 1205722119888

119886

=1205722119903119889

119886

(60)

Furthermore (1198602) implies that

tr (119872lowast) = 0 (61)

Thus (1198601) and (1198602) imply that the Jacobian matrix has azero eigenvalue withmultiplicity 2This suggests that (4)mayadmit a Bogdanov-Takens singularity

Theorem 12 Suppose that (1198601) (1198602) and 21198871 + 1198874 = 0 holdThen the endemic equilibrium 119864lowast = (119878lowast 119868lowast) of (4) is a cuspof codimension 2 that is it is a Bogdanov-Takens singularityHere 1198871 and 1198874 are defined by (65)

Proof Using the transformation of 119909 = 119868minus 119868lowast and 119910 = 119878minus 119878lowastsystem (4) becomes

119889119909

119889119905= 1198861119909 + 1198862119910 + 120582119909119910 + 11988611119909

2 minus 1198751 (119909)

119889119910

119889119905= minus11988621

1198862119909 minus 1198861119910 minus 120582119909119910 minus 11988611119909

2 + 1198751 (119909)

(62)

where 1198751(119909) is a smooth function of 119909 at least of the thirdorder and

1198861 = 120582119878lowast minus (119889 + 120576 + 120583) minus

119903

(1 + 120572119868lowast)2gt 0

1198862 = 120582119868lowast gt 0

11988611 =119903120572

(1 + 120572119868lowast)3gt 0

(63)

Set119883 = 119909 119884 = 1198861119909 + 1198862119910 Then (62) is transformed into

119889119883

119889119905= 119884 + 1198871119883

2 + 1198872119883119884 + 1198761 (119883)

119889119884

119889119905= 1198873119883

2 + 1198874119883119884 + 1198762 (119883)

(64)

where 119876119894(119883) are smooth functions of 119883 at least of the thirdorder and

1198871 = 11988611 minus12058211988611198862

1198872 =120582

1198862

1198873 = 119886111988611 minus12058211988621

1198862+ 1198861120582 minus 119886211988611

1198874 =12058211988611198862minus 120582

(65)

In order to obtain the canonical normal form we perform thetransformation of variables by

119906 = 119883 minus119887221198832 V = 119884 + 1198871119883

2 (66)

Then we obtain

119889119906

119889119905= V + 1198771 (119906)

119889V119889119905= 11988731199062 + (21198871 + 1198874) 119906V + 1198772 (119906)

(67)

where 119877119894(119906) are smooth functions of 119906 at least of the thirdorder Note that 1198873 lt 0 and 21198871 + 1198874 = 0 It follows from [33ndash35] that (4) admits a Bogdanov-Takens bifurcation

In the following wewill find the versal unfolding in termsof the original parameters in (4) In this way we will know theapproximate saddle-node Hopf and homoclinic bifurcation

10 Mathematical Problems in Engineering

curvesWe choose119860 and 119889 as bifurcation parameters Fix 120582 =1205820 120576 = 1205760 119903 = 1199030 120572 = 1205720 and 120583 = 1205830 Let 119860 = 1198600 + 1205791 and119889 = 1198890 + 1205792 where 1205791 and 1205792 are parameters which vary in asmall neighborhood of the origin

Suppose that 119860 = 1198600 119889 = 1198890 120582 = 1205820 120576 = 1205760 119903 = 1199030 120572 =1205720 and 120583 = 1205830 satisfy (1198601) and (1198602) Consider the followingsystem

119889119868

119889119905= 1205820119878119868 minus (1198890 + 1205792 + 1205760 + 1205830) 119868 minus

1199030119868

1 + 1205720119868

119889119878

119889119905= 1198600 + 1205791 minus (1198890 + 1205792) 119878 minus 1205820119878119868 + 1205760119868 +

1199030119868

1 + 1205720119868

(68)

By the transformations of 119909 = 119868 minus 119868lowast and 119910 = 119878 minus 119878lowast system(68) becomes

119889119909

119889119905= minus 1205792119868

lowast + 1198881119909 + 1198882119910 + 119888111199092 + 1205820119909119910 minus 1199081 (119909)

119889119910

119889119905= (1205791 minus 1205792119878

lowast) + 1198883119909 + 1198884119910 minus 119888111199092 minus 1205820119909119910 + 1199081 (119909)

(69)

where 1199081(119909) is a smooth function of 119909 at least of the thirdorder and

1198881 = 1205820119878lowast minus (1198890 + 1205792 + 1205760 + 1205830) minus

1199030

(1 + 1205720119868lowast)2

1198882 = 1205820119868lowast

1198883 = minus 1205820119878lowast + 1205760 +

1199030

(1 + 1205720119868lowast)2

1198884 = minus (1198890 + 1205792) minus 1205820119868lowast

11988811 =11990301205720

(1 + 1205720119868lowast)3

(70)

Making the change of variables 119883 = 119909 119884 = minus1205792119868lowast + 1198881119909 +

1198882119910 + 119888111199092 + 1205820119909119910 minus 1199081(119909) and rewriting 119883 119884 as 119909 and 119910

respectively we have

119889119909

119889119905= 119910

119889119910

119889119905= 1198900+ 1198901119909 + 1198902119910 + 11989011119909

2+ 11989012119909119910 + 119890221199102+ 1199082 (119909 119910 120579)

(71)

where 120579 = (1205791 1205792) 1199082(119909 119910 120579) is a smooth function of 119909 1199101205791 and 1205792 at least of the third order and

1198900 = 1198882 (1205791 minus 1205792119878lowast) + 11988841205792119868

lowast

1198901 = 1205820 (1205791 minus 1205792119878lowast) + 11988821198883 minus 11988811198884 minus 12058201205792119868

lowast

1198902 = 1198881 + 1198884 + 12058201205792119868lowast

1198882

11989011 = 12058201198883 minus 119888411988811 minus 119888111198882 + 11988811205820

11989012 = minus 1205820 + 211988811 + 1205820minus11988811198882 minus 12058201205792119868

lowast

11988822

11989022 =12058201198882

(72)

Next introduce a new time variable 120591 by 119889119905 = (1 minus(12058201198882)119909)119889120591 Rewriting 120591 as 119905 we obtain

119889119909

119889119905= 119910(1 minus

12058201198882119909)

119889119910

119889119905= (1 minus

12058201198882119909)

times (1198900 + 1198901119909 + 1198902119910 + 119890111199092

+ 11989012119909119910 + 119890221199102 + 1199082 (119909 119910 120579))

(73)

Let119883 = 119909 119884 = 119910(1minus(12058201198882)119909) and rename119883 and119884 as 119909 and119910 we have

119889119909

119889119905= 119910

119889119910

119889119905= 1198900 + 1198921119909 + 1198902119910 + 11989211119909

2 + 11989212119909119910 + 1199083 (119909 119910 120579)

(74)

where 120579 = (1205791 1205792) 1199083(119909 119910 120579) is a smooth function of 119909 1199101205791 and 1205792 at least of the third order and

1198921 = minus 2119890012058201198882+ 1198901

11989211 = 11989011 minus 2119890112058201198882+11989001205822

0

11988822

11989212 = 11989012 minus119890212058201198882

(75)

Now we assume that 11989211 = 0 and 11989212 = 0 when 120582119894 are smallSet 119909 = 119883 minus 119890211989212 and rewrite119883 as 119909 we can get

119889119909

119889119905= 119910

119889119910

119889119905= 1198910 + 1198911119909 + 11989211119909

2 + 11989212119909119910 + 1199084 (119909 119910 120579)

(76)

where 120579 = (1205791 1205792) 1199084(119909 119910 120579) is a smooth function of 119909 1199101205791 and 1205792 at least of the third order and

1198910 = 1198900 minus1198921119890211989212+119892111198902

2

119892212

1198911 = 1198921 minus211989211119890211989212

(77)

Mathematical Problems in Engineering 11

Making the final change of variables by 119883 = 11989221211990911989211 119884 =

119892312119910119892211 and 120591 = 1198921111990511989212 and then denoting them again by

119909 119910 and 119905 respectively we obtain

119889119909

119889119905= 119910

119889119910

119889119905= 1205911 (1205791 1205792) + 1205912 (1205791 1205792) 119909 + 119909

2 + 119909119910 + 1199085 (119909 119910 120579)

(78)

where 120579 = (1205791 1205792) and 1199085(119909 119910 120579) is a smooth function of 119909119910 1205791 and 1205792 at least of the third order

We substitute values of 1205720 = 1 1205820 = 14 1198890 = 1101205760 = 110 1205830 = 1100 1198600 = 537500 and 1199030 = 558 forthe above system and these values satisfy conditions (1198601) and(1198602) And we obtain the following equations

1205911 (1205791 1205792) =11989101198924

12

119892311

= minus237291605

851841205791 +

25485118377

8518401205792

+ 119874 (10038161003816100381610038161205791 1205792

10038161003816100381610038162)

1205912 (1205791 1205792) =11989111198922

12

119892211

= minus172225

19361205791 +

3983253

38721205792 + 119874 (

10038161003816100381610038161205791 120579210038161003816100381610038162)

(79)

where 11989211 = minus11500 minus (116)1205791 + (341800)1205792 + (14)1205792

2lt 0

and 11989212 = minus83200 + (14)1205792 lt 0 in a small neighborhood of(1205791 1205792) = (0 0) Let

119869 = (

12059712059111205971205791

12059712059111205971205792

12059712059121205971205791

12059712059121205971205792

) (80)

And after simple calculation we obtain that

det(119869)|120582=0 = minus16839398748825

82458112= 0 (81)

Thus 1205911 and 1205912 are regular maps in a small neighborhoodof (1205791 1205792) = (0 0) By the Bogdanov and Takens bifurcationtheorems [36] we obtain the following conclusion

Theorem 13 Suppose that1198600 1198890 1205820 1205760 1199030 1205720 and 1205830 satisfy(1198601) (1198602) 11989211 = 0 and 11989212 = 0 when 120579119894 are small Then (4)admits the following bifurcation behavior

(1) There is a saddle-node bifurcation curve 119878119873 =

(1205791 1205792) 4119891011989211 = 11989121+ 119900(|(1205791 1205792)|

2) 1198911 =0 = (1205791 1205792) (59076250)1205792 minus (11125)1205791 +(2313400)12057911205792 minus (516)120579

2

1minus (102863740000)1205792

2+

119900(|(1205791 1205792)|2) = 0 (4799116600)1205792 minus (14)1205791 + (25

83)12057911205792 minus (1364513778)1205792

2+ 119900(|(1205791 1205792)|

2) = 0

I

I

O

O 1205791

1205792

SN+

SN+

SNminus

SNminus

II H

III HL

IVIVHL

IIIII

H

Figure 6The bifurcation set and the corresponding phase portraitsfor system (68)

(2) There is a Hopf bifurcation curve 119867 = (1205791 1205792)

1198910 + 119900(|(1205791 1205792)|2) = 0 1198911 lt 0 = (1205791 1205792) minus(537

50)1205792+1205791 + (5083)12057911205792 minus (746676889)12057922 + 119900(|(12057911205792)|2) = 0 (4799116600)1205792 minus (14)1205791+(2583)12057911205792 minus

(1364513778)12057922+ 119900(|(1205791 1205792)|

2) lt 0

(3) There is a homoclinic bifurcation curve119867119871 = (1205791 1205792) 25119892111198910 + 6119891

2

1= 119900(|(1205791 1205792)|

2) 1198911 lt 0 = (1205791 1205792) (59071000)1205792 minus (1120)1205791 + (611979 33200)12057911205792 minus(1916)1205792

1minus (16075834839275560000)1205792

2+ 119900(|(1205791

1205792)|2) = 0 (4799116600)1205792minus(14)1205791+(2583)12057911205792minus

(1364513778)12057922+ 119900(|(1205791 1205792)|

2) lt 0

6 Numerical Simulations

When 120572 = 1 120582 = 14 119889 = 110 120576 = 110 120583 = 1100119860 = 537500 and 119903 = 558 120590 gt 0 By applying PPLANE8and Photoshop software the (1205791 1205792)-plane near the origin isdivided into 4 regions by these bifurcation curves as shown inFigure 6 Fix 1205791 lt 0 and decrease 1205792 from 0 our conclusionsare summarized as follows

(a) When (1205791 1205792) lies in region I there is no positiveequilibrium which implies that the positive orbitsof (4) meet the positive 119878-axis in finite time andtherefore the disease disappears

(b) There is a saddle-node point when (1205791 1205792) lies oncurve SN

(c) When (1205791 1205792) crosses SN into region II two positiveequilibria which are an unstable focus and a saddleappear

(d) When the parameters lie on the curve H there is alsoa saddle and an unstable focus An unstable limit cycleappears with the parameters crossing H into III

(e) A homoclinic cycle appears as the parameters passingIII into HL through the homoclinic bifurcation

12 Mathematical Problems in Engineering

0 5 10 15 20 25 30 35 40

0

1

2

3

4

5

6

7

8

9

10

S

I

Figure 7 When (1205791 1205792) = (0 0) the unique positive equilibrium isa cusp of codimension 2

3 35 45 55 65 754 5 6 7 8

1

2

3

4

5

6

7

8

S

I

Figure 8 When (1205791 1205792) = (minus01 minus00093) lies in region I there isno positive equilibrium

(f) The homoclinic loop breakswith the parameter cross-ing HL into IV and a saddle and a stable focus appear

Furthermore using PPLANE8 some numerical simu-lations of system (68) are depicted in Figures 7ndash10 When(1205791 1205792) = (0 0) that is (1198600 1198890) = (537500 110) there isa unique positive equilibrium (119878lowast 119868lowast) = (31750 4) whichis a cusp of codimension 2 (Figure 7) When (1205791 1205792) =(minus01 minus00093) lies in region I there is no positive equi-librium (Figure 8) When (1205791 1205792) = (minus002 minus0001863798)there is a homoclinic loop (Figure 9) When (1205791 1205792) =(minus002 minus00020) lies in region IV the homoclinic loopis broken and there is a stable focus and a saddle(Figure 10)

6 605 61 615 62 625 63 635 64 645 653536373839

44142434445

S

I

Figure 9 When (1205791 1205792) = (minus002 minus0001863798) there is ahomoclinic loop

0 1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

7

8

9

10

S

I

Figure 10 When (1205791 1205792) = (minus002 minus00020) lies in region IV thereis a stable focus and a saddle

7 Discussion

In this paper we focus on the bifurcation analysis of an SISepidemic model with bilinear incidence rate and saturatedtreatment Generally speaking in many epidemic modelsthe basic reproduction number which is the key conceptin epidemiology can be decreased below unity to eradicatethe disease However in our model the basic reproductionnumber below 1 is not enough to eradicate the diseaseAccording to our analysis in this paper we demonstrate notonly the global stability of the disease-free equilibrium when1198770 lt 119877

lowast

0and the local asymptotic stability of the endemic

equilibrium 1198642 when 119877lowast

0lt 1198770 lt 1 but also the global

stability of the unique endemic equilibrium 119864lowast when 1198770 gt 1Moreover it has been shown in Corollary 4 that backwardbifurcations occur if the effect of the infected being delayedfor treatment is strong That is to say we should get prompttreatment for patients Through Figure 4 we can see thatthere is a region such that the disease will persist if the initialposition lies in the region and disappear if the initial positionlies outside this region So in order to restrain the spread

Mathematical Problems in Engineering 13

of the disease governments should take timely measures tocontrol the initial infected individuals in a lower level Inaddition fromTheorem 8 we can see that the disease will dieout if 1198770 is small enough Therefore from the definition of1198770 one knows that we can decrease the incidence rate 120582 andincrease the cure rate 119903 so as to eradicate the diseases or tomake them controlled in a lower endemic steady state

The stability analysis of the model equilibria enables usto completely analyze their local bifurcation behavior suchas Hopf saddle-node and Bogdanov-Takens bifurcation Bycomputing the first Lyapunov coefficient we can determinethat the Hopf bifurcation is a supercritical Hopf bifurca-tion or a subcritical Hopf bifurcation We also show thatunder assumptions (1198601) and (1198602) the model undertakesBogdanov-Takens bifurcation that is there are saddle-nodebifurcation Hopf bifurcation and homoclinic bifurcationin the system The normal norm of the Bogdanov-Takensbifurcation is derived in Section 5 which is very helpful toobtain the three kinds of bifurcation curves By analyticaltechniques 119860 and 119889 are chosen as bifurcation parametersand other parameters are fixed and we easily get a clearpicture about the rich dynamics behaviors of our modelThrough studying the bifurcations of the SIS epidemicmodelwe are suggesting that in order to eradicate the diseasemore medical facilities as well as medical professionals areneeded and the medical standard needs to be improved aswell

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research was supported by the National Natural ScienceFoundation of China (no 11001041 no 10926105 no 11101283no 11271065 and no 11201360) SRFDP (no 200802001008)and Scientific Research Innovation Project of ShanghaiMunicipal Education Commission (12YZ173)

References

[1] M E Alexander and S M Moghadas ldquoPeriodicity in an epi-demic model with a generalized non-linear incidencerdquo Mathe-matical Biosciences vol 189 no 1 pp 75ndash96 2004

[2] M E Alexander and S M Moghadas ldquoBifurcation analysisof SIRS epidemic model with generalized incidencerdquo SIAMJournal on Applied Mathematics vol 65 no 5 pp 1794ndash18162005

[3] J Arino C C McCluskey and P van den Driessche ldquoGlobalresults for an epidemic model with vaccination that exhibitsbackward bifurcationrdquo SIAM Journal on Applied Mathematicsvol 64 no 1 pp 260ndash276 2003

[4] W R Derrick and P van den Driessche ldquoHomoclinic orbitsin a disease transmission model with nonlinear incidence andnonconstant populationrdquo Discrete and Continuous DynamicalSystems B vol 3 no 2 pp 299ndash309 2003

[5] F Brauer ldquoBackward bifurcations in simple vaccination mod-elsrdquo Journal of Mathematical Analysis and Applications vol 298no 2 pp 418ndash431 2004

[6] K P Hadeler and P van den Driessche ldquoBackward bifurcationin epidemic controlrdquo Mathematical Biosciences vol 146 no 1pp 15ndash35 1997

[7] Z X Hu S Liu and H Wang ldquoBackward bifurcation of anepidemic model with standard incidence rate and treatmentraterdquo Nonlinear Analysis Real World Applications vol 9 no 5pp 2302ndash2312 2008

[8] J Hui and D M Zhu ldquoGlobal stability and periodicity onSIS epidemic models with backward bifurcationrdquo Computers ampMathematics with Applications vol 50 no 8-9 pp 1271ndash12902005

[9] Y Jin W D Wang and S W Xiao ldquoAn SIRS model with anonlinear incidence raterdquoChaos Solitons amp Fractals vol 34 no5 pp 1482ndash1497 2007

[10] G H Li and W D Wang ldquoBifurcation analysis of an epidemicmodel with nonlinear incidencerdquo Applied Mathematics andComputation vol 214 no 2 pp 411ndash423 2009

[11] X Z Li W S Li and M Ghosh ldquoStability and bifurcation ofan SIS epidemic model with treatmentrdquo Chaos Solitons ampampFractals vol 42 no 5 pp 2822ndash2832 2009

[12] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1986

[13] X B Liu and L J Yang ldquoStability analysis of an SEIQV epidemicmodel with saturated incidence raterdquo Nonlinear Analysis RealWorld Applications vol 13 no 6 pp 2671ndash2679 2012

[14] M Martcheva and H R Thieme ldquoProgression age enhancedbackward bifurcation in an epidemic model with super-infectionrdquo Journal of Mathematical Biology vol 46 no 5 pp385ndash424 2003

[15] S G Ruan and W D Wang ldquoDynamical behavior of anepidemic model with a nonlinear incidence raterdquo Journal ofDifferential Equations vol 188 no 1 pp 135ndash163 2003

[16] W D Wang and S G Ruan ldquoBifurcation in an epidemicmodel with constant removal rate of the infectivesrdquo Journal ofMathematical Analysis and Applications vol 291 no 2 pp 775ndash793 2004

[17] Y L Tang D Q Huang S G Ruan and W N ZhangldquoCoexistence of limit cycles and homoclinic loops in a SIRSmodel with a nonlinear incidence raterdquo SIAM Journal onApplied Mathematics vol 69 no 2 pp 621ndash639 2008

[18] P van den Driessche and J Watmough ldquoA simple SIS epidemicmodel with a backward bifurcationrdquo Journal of MathematicalBiology vol 40 no 6 pp 525ndash540 2000

[19] J J Wei and J A Cui ldquoDynamics of SIS epidemic model withthe standard incidence rate and saturated treatment functionrdquoInternational Journal of Biomathematics vol 5 no 3 Article ID1260003 18 pages 2012

[20] W D Wang ldquoBackward bifurcation of an epidemic model withtreatmentrdquoMathematical Biosciences vol 201 no 1-2 pp 58ndash712006

[21] Z W Wang ldquoBackward bifurcation in simple SIS modelrdquo ActaMathematicae Applicatae Sinica vol 25 no 1 pp 127ndash136 2009

[22] L Xue and C Scoglio ldquoThe network level reproduction numberfor infectious diseases with both vertical and horizontal trans-missionrdquo Mathematical Biosciences vol 243 no 1 pp 67ndash802013

14 Mathematical Problems in Engineering

[23] X Zhang and X N Liu ldquoBackward bifurcation of an epidemicmodel with saturated treatment functionrdquo Journal ofMathemat-ical Analysis and Applications vol 348 no 1 pp 433ndash443 2008

[24] J Guckenheimer and P J Holmes Nonlinear OscillationsDynamical Systems and Bifurcation of Vector Fields SpringerNew York NY USA 1996

[25] X Zhang and X N Liu ldquoBackward bifurcation and globaldynamics of an SIS epidemic model with general incidence rateand treatmentrdquoNonlinearAnalysis RealWorldApplications vol10 no 2 pp 565ndash575 2009

[26] M A Safi A B Gumel and E H Elbasha ldquoQualitative analysisof an age-structured SEIR epidemic model with treatmentrdquoApplied Mathematics and Computation vol 219 no 22 pp10627ndash10642 2013

[27] L H Zhou and M Fan ldquoDynamics of an SIR epidemic modelwith limited medical resources revisitedrdquo Nonlinear AnalysisReal World Applications vol 13 no 1 pp 312ndash324 2012

[28] X Y Zhou and J A Cui ldquoAnalysis of stability and bifurcation foran SEIR epidemicmodel with saturated recovery raterdquoCommu-nications inNonlinear Science andNumerical Simulation vol 16no 11 pp 4438ndash4450 2011

[29] D Lacitignola ldquoSaturated treatments and measles resurgenceepisodes in South Africa a possible linkagerdquo MathematicalBiosciences and Engineering MBE vol 10 no 4 pp 1135ndash11572013

[30] J C Eckalbar and W L Eckalbar ldquoDynamics of an epidemicmodel with quadratic treatmentrdquo Nonlinear Analysis RealWorld Applications vol 12 no 1 pp 320ndash332 2011

[31] F Brauer ldquoBackward bifurcations in simple vaccinationtreat-ment modelsrdquo Journal of Biological Dynamics vol 5 no 5 pp410ndash418 2011

[32] M W Hirsch S Smale and R L Devaney Differential Equa-tions Dynamical Systems and an Introduction to Chaos Aca-demic Press New York NY USA 2003

[33] R Bogdanov ldquoBifurcations of a limit cycle for a family of vectorfields on the planrdquo SelectaMathematica vol 1 pp 373ndash388 1981

[34] R Bogdanov ldquoVersal deformations of a singular point on theplan in the case of zero eigen-valuesrdquo Selecta MathematicaSovietica vol 1 pp 389ndash421 1981

[35] F Takens ldquoForced oscillations and bifurcationrdquo in Applicationsof Global Analysis I vol 3 pp 1ndash59 Communications of theMathematical Institute Rijksuniversiteit Utrecht 1974

[36] Y A Kuznetsov Elements of Applied Bifurcation TheorySpringer New York NY USA 2nd edition 1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

0 5 10 15 20 25 30 35 4002468

101214161820

S

I

E0

Figure 1 The disease-free equilibrium 1198640 is locally asymptoticallystable when 1198770 lt 1 with the parameter values 119860 = 2 119889 = 01119896 = 01 120583 = 001 120582 = 001 120572 = 1 119903 = 2 and 120576 = 01

2 Equilibria and Backward Bifurcation

21 Disease-Free Equilibrium Obviously system (4) has adisease-free equilibrium 1198640 = (119860119889 0) The Jacobian matrixof (4) at 1198640 is

119872(1198640) = (minus119889

minus120582119860

119889+ 120576 + 119903

0120582119860

119889minus (119889 + 120576 + 120583) minus 119903

) (5)

By using the next generation matrix of (4) we get the basicreproduction number 1198770 = 120582119860119889(119889 + 120576 + 119903 + 120583)119872(1198640) hasnegative eigenvalues if 120582119860119889 minus (119889 + 120576 + 120583) minus 119903 lt 0 Then wehave the following result

Theorem 1 The disease-free equilibrium 1198640 is locally asymp-totically stable when 1198770 lt 1 (see Figure 1) and is unstable when1198770 gt 1 (see Figure 2)

22 Endemic Equilibria An endemic equilibrium alwayssatisfies

119860 minus 119889119878 minus 120582119878119868 + 120576119868 +119903119868

1 + 120572119868= 0

120582119878119868 minus (119889 + 120576 + 120583) 119868 minus119903119868

1 + 120572119868= 0

(6)

In view of 119889119878 + (119889 + 120583)119868 = 119860 we get 119878 = (119860 minus (119889 + 120583)119868)119889 andsubstitute it into the second equation of (6) When 119868 = 0 weobtain

120582[119860 minus (119889 + 120583) 119868

119889] minus (119889 + 120576 + 120583) minus

119903

1 + 120572119868= 0 (7)

Then we have an equation of the form

1198861198682 + 119887119868 + 119888 = 0 (8)

0 50 100 150 200 25002468

101214161820

S

I

E0

Figure 2 The disease-free equilibrium is unstable when 1198770 gt 1with the parameter values 119860 = 15 119889 = 01 119896 = 01 120583 = 001120582 = 001 120572 = 1 119903 = 08 and 120576 = 01

with

119886 = 120572120582 (119889 + 120583)

119887 = 120582 (119889 + 120583) + 120572119889 (119889 + 120576 + 120583) minus 120572120582119860

119888 = 119889 (119889 + 120576 + 119903 + 120583) minus 120582119860

(9)

This equation may admit positive solution

1198681 =minus119887 minus radic1198872 minus 4119886119888

2119886 1198682 =

minus119887 + radic1198872 minus 4119886119888

2119886 (10)

Obviously if 1198770 = 1 then 119888 = 0 if 1198770 gt 1 then 119888 lt 0 andif 1198770 lt 1 then 119888 gt 0 From (8) it is obvious that we have thefollowing results

Theorem 2 The following results hold

(1198671) Let 120572 = 0 Equation (8) is a linear equation with aunique solution 119868 = minus119888119887 Then the system (4) has aunique endemic equilibrium when 1198770 gt 1 and has noendemic equilibrium when 1198770 le 1

(1198672) Let 120572 gt 0 If 119887 gt 0 system (4) has a unique endemicequilibrium when 1198770 gt 1 and no endemic equilibriumwhen 1198770 le 1

(1198673) Let 120572 gt 0 If 119887 lt 0 system (4) has a unique endemicequilibrium when 1198770 ge 1 no endemic equilibriumwhen 1198770 lt 119877lowast0 and two endemic equilibria 1198641 and 1198642when 119877lowast

0le 1198770 lt 1 When 1198770 = 119877lowast0 and 1198641 = 1198642 one

has 1198872 minus 4119886119888 = 0 which is equivalent to

1198770 =4119886120582119860

1198872 + 4119886120582119860

=41205822120572 (119889 + 120583)119860

[120582 (119889 + 120583) + 120572119889 (119889 + 120576 + 120583) minus 120572120582119860]2+ 41205822120572 (119889 + 120583)119860

(11)

4 Mathematical Problems in Engineering

181614121080604020

10

20

30

40

50

60

I

Rlowast

0

R0

Figure 3 The figure of infective sizes at equilibria versus 1198770 when120572 = 1 119889 = 01 120583 = 001 120582 = 005 119903 = 2 and 120576 = 01

We define the right-hand side of (11) as

119877lowast0≜

41205822120572 (119889 + 120583)119860

[120582 (119889 + 120583) + 120572119889 (119889 + 120576 + 120583) minus 120572120582119860]2+ 41205822120572 (119889 + 120583)119860

(12)

Therefore according to the qualitative approach recentlyproposed by [31] which is based on the analysis of theequilibria curve in the neighboring of the critical threshold1198770 = 1 we have the following theorem

Theorem 3 If 120572 gt 0 119887 lt 0 then system (4) has a backwardbifurcation at 1198770 = 1 (see Figure 3)

In order to verify the bifurcation curve (the graph of 119868as a function of 1198770) in Figure 3 we think of 119903 as a variablewith the other parameters as constant Through implicitdifferentiation of (8) with respect to 119903 we get

(2119886119868 + 119887)119889119868

119889119903= minus119889 lt 0 (13)

From (13) we know that the sign of 119889119868119889119903 is opposite to thatof 2119886119868 + 119887 And from the definition of 1198770 we know that 1198770decreases when 119903 increases It implies that the bifurcationcurve has positive slope at equilibrium values with 2119886119868+119887 gt 0and negative slope at equilibrium values with 2119886119868 + 119887 lt 0 Ifthere is no backward bifurcation at 1198770 = 1 then the uniqueendemic equilibrium for 1198770 gt 1 satisfies

2119886119868 + 119887 = radic1198872 minus 4119886119888 gt 0 (14)

and the bifurcation curve has positive slope at all pointswhere119868 gt 0 If there is a backward bifurcation at 1198770 = 1 then thereis an interval onwhich there are two endemic equilibria givenby

2119886119868 + 119887 = plusmnradic1198872 minus 4119886119888 (15)

The bifurcation curve has negative slope at the smaller oneandpositive slope at the larger oneThus the bifurcation curveis as shown in Figure 3

Under the conditions of Theorem 3 if a backward bifur-cation takes place we can see from Figure 3 that there is acritical value 119877lowast

0at the turning point In this case the disease

will not die out when1198770 lt 1 However the diseasewill die outwhen 1198770 lt 119877

lowast

0 Therefore the critical value 119877lowast

0can be taken

as a new threshold for the control of the diseaseIn the following we give an explicit criterion of a

backward bifurcation at 1198770 = 1For convenience we define

1205720 =120582 (119889 + 120583)

119889119903 (16)

Corollary 4 When 120572 gt 1205720 system (4) has a backward bifur-cation at 1198770 = 1

Proof When 1198770 le 1 hArr 119888 ge 0

120582119860 le 119889 (119889 + 120576 + 120583 + 119903) (17)

The condition 119887 lt 0 is equivalent to

120582 (119889 + 120583) + 120572119889 (119889 + 120576 + 120583) lt 120572120582119860 (18)

From (17) and (18) we get 120582(119889 + 120583) + 120572119889(119889 + 120576 + 120583) lt 120572119889(119889 +120576 + 119903 + 120583) which reduces to

120572 gt120582 (119889 + 120583)

119889119903≜ 1205720 (19)

It means that 120572 is big enough to lead a backward bifurcationwith two endemic equilibria when 1198770 lt 1 Therefore theproof is complete

Next we consider the local stability of the uniqueendemic equilibrium when 1198770 gt 1

Theorem 5 When 1198770 gt 1 and 0 le 120572 lt 120582119903 the uniqueendemic equilibrium 119864lowast is locally asymptotically stable

Proof Firstly from Theorem 2 we can know that system(4) has a unique endemic equilibrium 119864lowast when 1198770 gt 1Moreover the Jacobian matrix of system (4) is

119872 =(

minus119889 minus 120582119868 minus120582119878 + 120576 +119903

1 + 120572119868minus

119903120572119868

(1 + 120572119868)2

120582119868 120582119878 minus (119889 + 120576 + 120583) minus119903

1 + 120572119868+

119903120572119868

(1 + 120572119868)2

)

(20)

From the second equation of (6) we have

minus120582119878 + 120576 +119903

1 + 120572119868= minus119889 minus 120583 (21)

From (21) the Jacobian matrix119872 reduces to

119872 =(

minus119889 minus 120582119868 minus119889 minus 120583 minus119903120572119868

(1 + 120572119868)2

120582119868119903120572119868

(1 + 120572119868)2

) (22)

Mathematical Problems in Engineering 5

We obtain

det (119872) = 119868

(1 + 120572119868)2[120582 (119889 + 120583) (1 + 120572119868)

2 minus 119903120572119889] (23)

In fact there is 120582(119889 + 120583)(1 + 120572119868)2 gt 120582119889 Since 1198770 gt 1 and0 le 120572 lt 120582119903 we have

120582 (119889 + 120583) (1 + 120572119868)2 minus 119903120572119889 gt 120582119889 minus 119903120572119889 gt 0 (24)

So we get det(119872) gt 0 The trace of119872 is given by

tr (119872) = 1

(1 + 120572119868)2[minus (119889 + 120582119868) (1 + 120572119868)

2 + 119903120572119868] (25)

In the same way as the above calculation of (24) we have

minus (119889 + 120582119868) (1 + 120572119868)2 + 119903120572119868 lt minus120582119868 + 119903120572119868 lt 0 (26)

So we get tr(119872) lt 0 The proof is complete

Now we consider the case that there are two endemicequilibria 1198641 and 1198642 let 119872119894 be the Jacobian matrix at 119864119894119894 = 1 2

Theorem 6 The endemic equilibrium 1198641 is a saddle wheneverit exists

Proof Since 1198681 = (minus119887 minus radic1198872 minus 4119886119888)2119886 and Δ = 1198872 minus 4119886119888 wehave 1198681 = (minus119887 minus radicΔ)2119886 Thus

det (1198721)

=1198681

(1 + 1205721198681)2[120582 (119889 + 120583) (1 + 1205721198681)

2minus 119903120572119889]

≜1198681

(1 + 1205721198681)2120595 (1198681)

(27)

From (1198673) of Theorem 2 we can know that if 120572 gt 0 119887 lt 0and 119877lowast

0lt 1198770 lt 1 then 1198641 exists Hence we can get 120595(0) =

120582(119889 + 120583) minus 119903120572119889 lt 0 and 1205951015840(1198681) = 2120582(119889 + 120583)(1 + 1205721198681)120572 gt 0 Itfollows that there exists a unique 119868lowast gt 0 such that

120595 (1198681) = 0 when 1198681 = 119868lowast

120595 (1198681) lt 0 when 0 lt 1198681 lt 119868lowast

120595 (1198681) gt 0 when 1198681 gt 119868lowast

(28)

where

119868lowast = radic119903119889

120572120582 (119889 + 120583)minus1

120572 (29)

On the other hand

1198681 = minus119887

2119886minusradicΔ

2119886

=120572120582119860 minus 120582 (119889 + 120583) minus 120572119889 (119889 + 120576 + 120583)

2120572120582 (119889 + 120583)minusradicΔ

2119886

= radic119903119889

120572120582 (119889 + 120583)minus1

120572+1

120572minus radic

119903119889

120572120582 (119889 + 120583)

+120572120582119860 minus 120582 (119889 + 120583) minus 120572119889 (119889 + 120576 + 120583)

2120572120582 (119889 + 120583)minusradicΔ

2119886

= 119868lowast + ([120582 (119889 + 120583) + 120572120582119860 minus 2radic119903119889120572120582 (119889 + 120583)

minus 120572119889 (119889 + 120576 + 120583)] minus radicΔ)times (2120572120582 (119889 + 120583))minus1

(30)

Δ = 120582 (119889 + 120583) minus [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]2

minus 4120572120582 (119889 + 120583) [119889 (119889 + 120576 + 119903 + 120583) minus 120582119860]

= 120582 (119889 + 120583) + [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

minus 2radic119903119889120572120582(119889 + 120583)2

minus 4119903119889120572120582 (119889 + 120583)

minus 4120582 (119889 + 120583) [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

+ 4radic119903119889120572120582 (119889 + 120583) 120582 (119889 + 120583) + [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

minus 4120572120582 (119889 + 120583) [minus120582119860 + 119889 (119889 + 120576 + 119903 + 120583)]

≜ 120582 (119889 + 120583) + [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

minus 2radic119903119889120572120582 (119889 + 120583)2

+ 119875

(31)

where

119875 = minus8119903119889120572120582 (119889 + 120583) + 4radic119903119889120572120582 (119889 + 120583)

times 120582 (119889 + 120583) + [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

(32)

In the following we will show that 119875 gt 0 Since 119877lowast0lt 1198770 we

have

4120572120582 (119889 + 120583)

[120582 (119889 + 120583) + 120572119889 (119889 + 120576 + 120583) minus 120572120582119860]2+ 41205721205822 (119889 + 120583)119860

lt1

119889 (119889 + 120576 + 119903 + 120583)

(33)

6 Mathematical Problems in Engineering

that is

[120582 (119889 + 120583) + 120572119889 (119889 + 120576 + 120583) minus 120572120582119860]2

gt 4120572120582 (119889 + 120583) 119889 (119889 + 120576 + 119903 + 120583) minus 41205721205822 (119889 + 120583)119860(34)

Therefore

120582 (119889 + 120583) + [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]2

= 120582 (119889 + 120583) minus [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]2

+ 4120582 (119889 + 120583) [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

gt 4120572120582 (119889 + 120583) 119889 (119889 + 120576 + 119903 + 120583) minus 41205721205822 (119889 + 120583)119860

+ 4120582 (119889 + 120583) [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

= 4119903119889120572120582 (119889 + 120583)

(35)

Obviously we have 119875 gt 0 From (30) one has 1198681 lt 119868lowast So

we get det(1198721) lt 0 Hence the endemic equilibrium 1198641 is asaddle The proof is complete

In order to explore the stability of the endemic equilib-rium 1198642 define

1198981 = (21198891205721198862 + 1205821198862 minus 1205761205721198862 minus 1198861198881205821205722)

minus 119887 (2120582120572119886 + 1198891205722119886 minus 1198871205821205722)

1198982 = 1198862119889 minus 119888 (2120582120572119886 + 1198891205722119886 minus 1198871205821205722)

(36)

Theorem 7 If 120578 gt 0 then endemic equilibrium 1198642 is locallyasymptotically stable if 120578 lt 0 then endemic equilibrium 1198642 isunstable where 120578 = 21198861198982 + 1198981(radic1198872 minus 4119886119888 minus 119887)

Proof Consider

det (1198722) =1198682

(1 + 1205721198682)2[120582 (119889 + 120583) (1 + 1205721198682)

2minus 120576120572119889]

=1198682

(1 + 1205721198682)2times 120595 (1198682)

(37)

By carrying out arguments similar to that of Theorem 6 wehave 1198682 gt 119868

lowast Therefore det(1198722) gt 0 In addition we have

tr (1198722) = minus(119889 + 1205821198682) (1 + 1205721198682)

2minus 1205761205721198682

(1 + 1205721198682)2

= minus120582120572211986832+ (2120582120572 + 1198891205722) 1198682

2+ (2119889120572 + 120582 minus 120576120572) 1198682 + 119889

(1 + 1205721198682)2

(38)

and then sgn(tr(1198722)) = minus sgn(119866(1198682)) where

119866 (119909) = 12058212057221199093 + (2120582120572 + 1198891205722) 1199092 + (2119889120572 + 120582 minus 120576120572) 119909 + 119889

(39)

I

0 50 100 1500

5

10

15

20

25

30

35

40

S

E0

E1

E2

Figure 4 One region of disease persistence and one region ofdisease extinction when 119860 = 10 120572 = 1 119889 = 01 120583 = 001 120582 = 001119903 = 01 and 120576 = 01

Using the expression of1198981 = (21198891205721198862 +1205821198862 minus 1205761205721198862 minus1198861198881205821205722) minus

119887(2120582120572119886+1198891205722119886minus1198871205821205722) and1198982 = 1198862119889minus119888(2120582120572119886+1198891205722119886minus1198871205821205722)

one has

119866 (1198682) = (1198861198682

2+ 1198871198682 + 119888) 1205930 +

11989811198682 + 11989821198862

(40)

where1205930 is a first degree polynomial of 1198682 Since 1198861198682

2+1198871198682+119888 =

0 sgn(tr(1198722)) = minus sgn(119866(1198682)) = minus sgn(11989811198682 + 1198982) From theexpression of 1198682 we have

sgn (11989811198682 + 1198982)

= sgn (21198861198982 + 1198981 (radic1198872 minus 4119886119888 minus 119887)) ≜ sgn (120578) (41)

Thus 1198642 is locally asymptotically stable if 120578 gt 0 and 1198642 isunstable if 120578 lt 0 The proof is complete

From the above discussion we have the following con-clusion If two endemic equilibria 1198641 and 1198642 exist the stablemanifolds of the saddle 1198641 split 119877

2

+into two regions The

disease is persistent in the upper region and dies out in thelower region (see Figure 4)

3 Global Analysis

Firstly we consider the global stability of the disease-freeequilibrium 1198640 Let 119873 = 119878 + 119868 be the total population sizeNow we note that the equation for total population is givenby 119889119873119889119905 = 119860 minus 119889119878 minus (119889 + 120583)119868 le 119860 minus 119889119873 It follows thatlim119905rarr+infin119873(119905) le 119860119889 Let

R = (119878 119868) isin 1198772

+ 119878 + 119868 le

119860

119889 (42)

which is positive invariant with respect to system (4)

Theorem 8 If 1198770 lt 119877lowast0 the disease-free equilibrium 1198640(119860119889 0) is globally asymptotically stable that is the disease diesout

Mathematical Problems in Engineering 7

Proof Suppose 1198770 lt 119877lowast0 From the (1198673) of Theorem 2 weknow that the model has no endemic equilibrium Fromthe corollary of Poincare-Bendixson theorem [32] we knowthat there is no periodic orbit in R as there is a disease-free equilibrium in R Since R is a bounded positivelyinvariant region and 1198640 is the only equilibrium in R thelocal stability of 1198640 implies that every solution initiating inR approaches 1198640 Thus the disease-free equilibrium 1198640 isglobally asymptotically stable The proof is complete

Now we analyze the global dynamics of the endemicequilibrium when 1198770 gt 1

Theorem 9 If 1198770 gt 1 and 0 le 120572 lt 120582119903 the system (4) has nolimit cycles

Proof We use Dulac theorem to exclude the limit cycle Let

119875 (119878 119868) = 119860 minus 119889119878 minus 120582119878119868 + 120576119868 +119903119868

1 + 120572119868

119876 (119878 119868) = 120582119878119868 minus (119889 + 120576 + 120583) 119868 minus119903119868

1 + 120572119868

(43)

and take the Dulac function

119863 =1

119868 (44)

According to 0 le 120572 lt 120582119903 we can get

120597 (119875119863)

120597119878+120597 (119876119863)

120597119868

= minus119889

119868minus 120582 +

120572119903

(1 + 120572119868)2

lt minus119889

119868minus 120582 +

120582

(1 + 120572119868)2

=1

119868 (1 + 120572119868)2minus119889 (1 + 120572119868)

2 minus 120582119868 [(1 + 120572119868)2 minus 1]

lt 0

(45)

Hence the system (4) has no limit cycles The proof iscomplete

Therefore we obtain the global result of the uniqueendemic equilibrium

Theorem 10 If 1198770 gt 1 and 0 le 120572 lt 120582119903 the unique endemicequilibrium 119864lowast is globally asymptotically stable (see Figure 5)

4 Hopf Bifurcation

In this section we study the Hopf bifurcation of system (4)From the above discussion we know that there is no closedorbit surrounding1198640 or1198641 because the 119878-axis is invariantwith

0 50 100 1500

50

100

150

S

I Elowast

Figure 5The unique endemic equilibrium 119864lowast is globally asymptot-ically stable when 119860 = 15 120572 = 001 119889 = 01 120583 = 001 120582 = 001119903 = 08 and 120576 = 01

respect to system (4) and 1198641 is always a saddle ThereforeHopf bifurcation can only occur at 1198642 Set

120590 = [minus120582120572 minus1198881 (1198882 + 21198884)

11988612]1198634

+ [minus 120582120572119886211minus 21205821205721198861111988612 + (1198882 minus

211988611119888111988612)

times (119886111198882 minus119886211119888111988612)

minus (21198862111198881

11988612minus 119886111198882 + 2119888411988611 minus 119888311988612)

times (1198884 minus11988611119888111988612)]1198632

+ 11988611 (119886211119888111988612minus 119886111198882 + 119888411988611 minus 119888311988612)

times (119886121198883 minus 2119886111198884 minus 119886111198882)

(46)

where 119886119894119895 (119894 119895 = 1 2) 119888119896 (119896 = 1 2 3 4) and 119863 are defined by(50) and (52)

Theorem 11 System (4) undergoes a Hopf bifurcation if 120578 = 0Moreover if 120590 lt 0 there is a family of stable period orbits ofsystem (4) as 120578 decreases from 0 that is a supercritical Hopfbifurcation occurs if 120590 gt 0 there is a family of unstable periodorbits of system (4) as 120578 increases from 0 that is a subcriticalHopf bifurcation occurs

Proof The proof of Theorem 7 shows that tr(1198722) = 0 if andonly if 120578 = 0 and det(1198722) gt 0 when 1198642 exists Therefore theeigenvalues of 1198722 are a pair of pure imaginary roots if andonly if 120578 = 0 The direct calculations show that

119889 (tr (1198722))119889120578

100381610038161003816100381610038161003816100381610038161003816120578=0= minus

1

21198863 (1 + 1205721198682)2lt 0 (47)

8 Mathematical Problems in Engineering

ByTheorem 342 in [24] 120578 = 0 is the Hopf bifurcation pointfor system (4)

To be concise in notations rescale (4) by 120591 = 119905(1 + 120572119868)For simplicity we still use 119905 instead of 120591 Then we obtain

119889119878

119889119905= (119860 minus 119889119878 minus 120582119878119868 + 120576119868) (1 + 120572119868) + 119903119868

119889119868

119889119905= 120582119878119868 (1 + 120572119868) minus (119889 + 120576 + 120583) 119868 (1 + 120572119868) minus 119903119868

(48)

Let 119909 = 119878 minus 1198782 and 119910 = 119868 minus 1198682 then (48) becomes

119889119909

119889119905= 11988611119909 + 11988612119910 + 1198881119910

2 + 1198882119909119910 minus 1205821205721199091199102

119889119910

119889119905= 11988621119909 + 11988622119910 + 1198883119909119910 + 1198884119910

2 + 1205821205721199091199102

(49)

where

11988611 = minus (119889 + 1205821198682) (1 + 1205721198682)

11988612 = (minus1205821198782 + 119903) (1 + 1205721198682)

+ 120572 (119860 minus 1198891198782 minus 12058211987821198682 + 1199031198682) + 120576

11988621 = 1205821198682 (1 + 1205721198682)

11988622 = [1205821198782 minus (119889 + 119903 + 120583)] (1 + 1205721198682)

+ 120572 [12058211987821198682 minus (119889 + 119903 + 120583) 1198682] minus 120576

1198881 = 120572 (minus1205821198782 + 119903)

1198882 = 120572 (minus119889 minus 1205821198682) minus 120582 (1 + 1205721198682)

1198883 = 120582 (1 + 1205721198682) + 1205821198682120572

1198884 = [1205821198782 minus (119889 + 119903 + 120583)] 120572

(50)

Let 119864lowast denote the origin of 119909-119910 plane Since 1198642 = (1198782 1198682)satisfies (6) we obtain

det (119872 (119864lowast))

= minus (119889 + 1205821198682) 1205721205761198682 + (119889 + 120583) (1 + 1205721198682

2)2

1205821198682 + 1205721205821205761198682

2

= 1198682 times 120595 (1198682)

(51)

From the proof of Theorem 7 it follows that 120595(1198682) is alwayspositive It is easy to verify that 11988611 + 11988622 = 0 if and only if120578 = 0 Set

119863 = radicdet (119872 (119864lowast)) (52)

and let 119906 = minus119909 and V = (11988611119863)119909+(11988612119863)119910 then the normalform of (48) for Hopf bifurcation reads

119889119906

119889119905= minus119863V + 119891 (119906 V)

119889V119889119905= 119863119906 + 119892 (119906 V)

(53)

where

119891 (119906 V) = (11988611119886121198882 minus119886211

119886212

1198881)1199062 minus11986321198881119886212

V2

+ (119863119888211988612minus 2119863119886111198881119886212

)119906V minus120582120572119886211

119886212

1199063

minus 211986311988611120582120572

119886212

1199062V minus1198632120582120572

119886212

119906V2

119892 (119906 V) =11988611119863(119886211

119886212

1198881 minus11988611119886121198882 +11988611119886121198884 minus 1198883)119906

2

+ (119863119886111198881119886212

+119863119888411988612) V2

+ (2119886211

119886212

1198881 minus11988611119886121198882 + 2

11988611119886121198884 minus 1198883)119906V

+120582120572119886211

11988612119863(1

11988612minus 1)1199063 +

21205821205721198861111988612

(1198861111988612minus 1)1199062V

+119863120582120572

11988612(1198861111988612minus 1)119906V2

(54)

Now we evaluate the first Lyapunov coefficient Γ of system(4) as follows

Γ =1

16[119891119906119906119906 + 119891119906VV + 119892119906119906V + 119892VVV] +

1

16119863

times [119891119906V (119891119906119906 + 119891VV) minus 119892119906V (119892119906119906 + 119892VV) minus 119891119906119906119892119906119906 + 119891VV119892VV]

(55)

where 119891119906V denotes (1205972119891120597119906120597V)(0 0) and so forth Then

Γ =1

81198862121198632[minus120582120572 minus

1198881 (1198882 + 21198884)

11988612]1198634

+ [minus 120582120572119886211minus 21205821205721198861111988612 + (1198882 minus

211988611119888111988612)

times (119886111198882 minus119886211119888111988612)

minus (21198862111198881

11988612minus 119886111198882 + 2119888411988611 minus 119888311988612)

times (1198884 minus11988611119888111988612)]1198632

+ 11988611 (119886211119888111988612minus 119886111198882 + 119888411988611 minus 119888311988612)

times (119886121198883 minus 2119886111198884 minus 119886111198882)

=120590

81198862121198632

(56)

Mathematical Problems in Engineering 9

Obviously the sign of Γ is determined by 120590 ByTheorem 342and (3411) in [24] the rest of the claims in Theorem 11 arevalid The proof is complete

5 Bogdanov-Takens Bifurcations

The purpose of this section is to study the Bogdanov-Takensbifurcation of (4) Now we assume the following

(1198601) 1198872 minus 4119886119888 = 0

(1198602) 21198861198892 + 119887120582120583 = 0

Then (6) admits a unique positive equilibrium 119864lowast = (119878lowast 119868lowast)where

119868lowast = minus119887

2119886 119878lowast =

119860 minus (119889 + 120583) 119868lowast

119889 (57)

The Jacobian matrix of system (4) at 119864lowast is

119872lowast = (

minus119889 minus 120582119868lowast minus120582119878lowast + 120576 +119903

(1 + 120572119868lowast)2

120582119868lowast 120582119878lowast minus (119889 + 120576 + 120583) minus119903

(1 + 120572119868lowast)2)

(58)

By (58) we have

det (119872lowast) = 119868lowast

(1 + 120572119868lowast)2[(119889 + 120583) 120582 (1 + 120572119868lowast)

2minus 119903119889120572]

= 0

(59)

because of

(1 + 120572119868lowast)2=41198862 minus 4119886120572119887 + 12057221198872

41198862

=41198862 minus 4119886120572119887 + 12057224119886119888

41198862

=119886 minus 120572119887 + 1205722119888

119886

=1205722119903119889

119886

(60)

Furthermore (1198602) implies that

tr (119872lowast) = 0 (61)

Thus (1198601) and (1198602) imply that the Jacobian matrix has azero eigenvalue withmultiplicity 2This suggests that (4)mayadmit a Bogdanov-Takens singularity

Theorem 12 Suppose that (1198601) (1198602) and 21198871 + 1198874 = 0 holdThen the endemic equilibrium 119864lowast = (119878lowast 119868lowast) of (4) is a cuspof codimension 2 that is it is a Bogdanov-Takens singularityHere 1198871 and 1198874 are defined by (65)

Proof Using the transformation of 119909 = 119868minus 119868lowast and 119910 = 119878minus 119878lowastsystem (4) becomes

119889119909

119889119905= 1198861119909 + 1198862119910 + 120582119909119910 + 11988611119909

2 minus 1198751 (119909)

119889119910

119889119905= minus11988621

1198862119909 minus 1198861119910 minus 120582119909119910 minus 11988611119909

2 + 1198751 (119909)

(62)

where 1198751(119909) is a smooth function of 119909 at least of the thirdorder and

1198861 = 120582119878lowast minus (119889 + 120576 + 120583) minus

119903

(1 + 120572119868lowast)2gt 0

1198862 = 120582119868lowast gt 0

11988611 =119903120572

(1 + 120572119868lowast)3gt 0

(63)

Set119883 = 119909 119884 = 1198861119909 + 1198862119910 Then (62) is transformed into

119889119883

119889119905= 119884 + 1198871119883

2 + 1198872119883119884 + 1198761 (119883)

119889119884

119889119905= 1198873119883

2 + 1198874119883119884 + 1198762 (119883)

(64)

where 119876119894(119883) are smooth functions of 119883 at least of the thirdorder and

1198871 = 11988611 minus12058211988611198862

1198872 =120582

1198862

1198873 = 119886111988611 minus12058211988621

1198862+ 1198861120582 minus 119886211988611

1198874 =12058211988611198862minus 120582

(65)

In order to obtain the canonical normal form we perform thetransformation of variables by

119906 = 119883 minus119887221198832 V = 119884 + 1198871119883

2 (66)

Then we obtain

119889119906

119889119905= V + 1198771 (119906)

119889V119889119905= 11988731199062 + (21198871 + 1198874) 119906V + 1198772 (119906)

(67)

where 119877119894(119906) are smooth functions of 119906 at least of the thirdorder Note that 1198873 lt 0 and 21198871 + 1198874 = 0 It follows from [33ndash35] that (4) admits a Bogdanov-Takens bifurcation

In the following wewill find the versal unfolding in termsof the original parameters in (4) In this way we will know theapproximate saddle-node Hopf and homoclinic bifurcation

10 Mathematical Problems in Engineering

curvesWe choose119860 and 119889 as bifurcation parameters Fix 120582 =1205820 120576 = 1205760 119903 = 1199030 120572 = 1205720 and 120583 = 1205830 Let 119860 = 1198600 + 1205791 and119889 = 1198890 + 1205792 where 1205791 and 1205792 are parameters which vary in asmall neighborhood of the origin

Suppose that 119860 = 1198600 119889 = 1198890 120582 = 1205820 120576 = 1205760 119903 = 1199030 120572 =1205720 and 120583 = 1205830 satisfy (1198601) and (1198602) Consider the followingsystem

119889119868

119889119905= 1205820119878119868 minus (1198890 + 1205792 + 1205760 + 1205830) 119868 minus

1199030119868

1 + 1205720119868

119889119878

119889119905= 1198600 + 1205791 minus (1198890 + 1205792) 119878 minus 1205820119878119868 + 1205760119868 +

1199030119868

1 + 1205720119868

(68)

By the transformations of 119909 = 119868 minus 119868lowast and 119910 = 119878 minus 119878lowast system(68) becomes

119889119909

119889119905= minus 1205792119868

lowast + 1198881119909 + 1198882119910 + 119888111199092 + 1205820119909119910 minus 1199081 (119909)

119889119910

119889119905= (1205791 minus 1205792119878

lowast) + 1198883119909 + 1198884119910 minus 119888111199092 minus 1205820119909119910 + 1199081 (119909)

(69)

where 1199081(119909) is a smooth function of 119909 at least of the thirdorder and

1198881 = 1205820119878lowast minus (1198890 + 1205792 + 1205760 + 1205830) minus

1199030

(1 + 1205720119868lowast)2

1198882 = 1205820119868lowast

1198883 = minus 1205820119878lowast + 1205760 +

1199030

(1 + 1205720119868lowast)2

1198884 = minus (1198890 + 1205792) minus 1205820119868lowast

11988811 =11990301205720

(1 + 1205720119868lowast)3

(70)

Making the change of variables 119883 = 119909 119884 = minus1205792119868lowast + 1198881119909 +

1198882119910 + 119888111199092 + 1205820119909119910 minus 1199081(119909) and rewriting 119883 119884 as 119909 and 119910

respectively we have

119889119909

119889119905= 119910

119889119910

119889119905= 1198900+ 1198901119909 + 1198902119910 + 11989011119909

2+ 11989012119909119910 + 119890221199102+ 1199082 (119909 119910 120579)

(71)

where 120579 = (1205791 1205792) 1199082(119909 119910 120579) is a smooth function of 119909 1199101205791 and 1205792 at least of the third order and

1198900 = 1198882 (1205791 minus 1205792119878lowast) + 11988841205792119868

lowast

1198901 = 1205820 (1205791 minus 1205792119878lowast) + 11988821198883 minus 11988811198884 minus 12058201205792119868

lowast

1198902 = 1198881 + 1198884 + 12058201205792119868lowast

1198882

11989011 = 12058201198883 minus 119888411988811 minus 119888111198882 + 11988811205820

11989012 = minus 1205820 + 211988811 + 1205820minus11988811198882 minus 12058201205792119868

lowast

11988822

11989022 =12058201198882

(72)

Next introduce a new time variable 120591 by 119889119905 = (1 minus(12058201198882)119909)119889120591 Rewriting 120591 as 119905 we obtain

119889119909

119889119905= 119910(1 minus

12058201198882119909)

119889119910

119889119905= (1 minus

12058201198882119909)

times (1198900 + 1198901119909 + 1198902119910 + 119890111199092

+ 11989012119909119910 + 119890221199102 + 1199082 (119909 119910 120579))

(73)

Let119883 = 119909 119884 = 119910(1minus(12058201198882)119909) and rename119883 and119884 as 119909 and119910 we have

119889119909

119889119905= 119910

119889119910

119889119905= 1198900 + 1198921119909 + 1198902119910 + 11989211119909

2 + 11989212119909119910 + 1199083 (119909 119910 120579)

(74)

where 120579 = (1205791 1205792) 1199083(119909 119910 120579) is a smooth function of 119909 1199101205791 and 1205792 at least of the third order and

1198921 = minus 2119890012058201198882+ 1198901

11989211 = 11989011 minus 2119890112058201198882+11989001205822

0

11988822

11989212 = 11989012 minus119890212058201198882

(75)

Now we assume that 11989211 = 0 and 11989212 = 0 when 120582119894 are smallSet 119909 = 119883 minus 119890211989212 and rewrite119883 as 119909 we can get

119889119909

119889119905= 119910

119889119910

119889119905= 1198910 + 1198911119909 + 11989211119909

2 + 11989212119909119910 + 1199084 (119909 119910 120579)

(76)

where 120579 = (1205791 1205792) 1199084(119909 119910 120579) is a smooth function of 119909 1199101205791 and 1205792 at least of the third order and

1198910 = 1198900 minus1198921119890211989212+119892111198902

2

119892212

1198911 = 1198921 minus211989211119890211989212

(77)

Mathematical Problems in Engineering 11

Making the final change of variables by 119883 = 11989221211990911989211 119884 =

119892312119910119892211 and 120591 = 1198921111990511989212 and then denoting them again by

119909 119910 and 119905 respectively we obtain

119889119909

119889119905= 119910

119889119910

119889119905= 1205911 (1205791 1205792) + 1205912 (1205791 1205792) 119909 + 119909

2 + 119909119910 + 1199085 (119909 119910 120579)

(78)

where 120579 = (1205791 1205792) and 1199085(119909 119910 120579) is a smooth function of 119909119910 1205791 and 1205792 at least of the third order

We substitute values of 1205720 = 1 1205820 = 14 1198890 = 1101205760 = 110 1205830 = 1100 1198600 = 537500 and 1199030 = 558 forthe above system and these values satisfy conditions (1198601) and(1198602) And we obtain the following equations

1205911 (1205791 1205792) =11989101198924

12

119892311

= minus237291605

851841205791 +

25485118377

8518401205792

+ 119874 (10038161003816100381610038161205791 1205792

10038161003816100381610038162)

1205912 (1205791 1205792) =11989111198922

12

119892211

= minus172225

19361205791 +

3983253

38721205792 + 119874 (

10038161003816100381610038161205791 120579210038161003816100381610038162)

(79)

where 11989211 = minus11500 minus (116)1205791 + (341800)1205792 + (14)1205792

2lt 0

and 11989212 = minus83200 + (14)1205792 lt 0 in a small neighborhood of(1205791 1205792) = (0 0) Let

119869 = (

12059712059111205971205791

12059712059111205971205792

12059712059121205971205791

12059712059121205971205792

) (80)

And after simple calculation we obtain that

det(119869)|120582=0 = minus16839398748825

82458112= 0 (81)

Thus 1205911 and 1205912 are regular maps in a small neighborhoodof (1205791 1205792) = (0 0) By the Bogdanov and Takens bifurcationtheorems [36] we obtain the following conclusion

Theorem 13 Suppose that1198600 1198890 1205820 1205760 1199030 1205720 and 1205830 satisfy(1198601) (1198602) 11989211 = 0 and 11989212 = 0 when 120579119894 are small Then (4)admits the following bifurcation behavior

(1) There is a saddle-node bifurcation curve 119878119873 =

(1205791 1205792) 4119891011989211 = 11989121+ 119900(|(1205791 1205792)|

2) 1198911 =0 = (1205791 1205792) (59076250)1205792 minus (11125)1205791 +(2313400)12057911205792 minus (516)120579

2

1minus (102863740000)1205792

2+

119900(|(1205791 1205792)|2) = 0 (4799116600)1205792 minus (14)1205791 + (25

83)12057911205792 minus (1364513778)1205792

2+ 119900(|(1205791 1205792)|

2) = 0

I

I

O

O 1205791

1205792

SN+

SN+

SNminus

SNminus

II H

III HL

IVIVHL

IIIII

H

Figure 6The bifurcation set and the corresponding phase portraitsfor system (68)

(2) There is a Hopf bifurcation curve 119867 = (1205791 1205792)

1198910 + 119900(|(1205791 1205792)|2) = 0 1198911 lt 0 = (1205791 1205792) minus(537

50)1205792+1205791 + (5083)12057911205792 minus (746676889)12057922 + 119900(|(12057911205792)|2) = 0 (4799116600)1205792 minus (14)1205791+(2583)12057911205792 minus

(1364513778)12057922+ 119900(|(1205791 1205792)|

2) lt 0

(3) There is a homoclinic bifurcation curve119867119871 = (1205791 1205792) 25119892111198910 + 6119891

2

1= 119900(|(1205791 1205792)|

2) 1198911 lt 0 = (1205791 1205792) (59071000)1205792 minus (1120)1205791 + (611979 33200)12057911205792 minus(1916)1205792

1minus (16075834839275560000)1205792

2+ 119900(|(1205791

1205792)|2) = 0 (4799116600)1205792minus(14)1205791+(2583)12057911205792minus

(1364513778)12057922+ 119900(|(1205791 1205792)|

2) lt 0

6 Numerical Simulations

When 120572 = 1 120582 = 14 119889 = 110 120576 = 110 120583 = 1100119860 = 537500 and 119903 = 558 120590 gt 0 By applying PPLANE8and Photoshop software the (1205791 1205792)-plane near the origin isdivided into 4 regions by these bifurcation curves as shown inFigure 6 Fix 1205791 lt 0 and decrease 1205792 from 0 our conclusionsare summarized as follows

(a) When (1205791 1205792) lies in region I there is no positiveequilibrium which implies that the positive orbitsof (4) meet the positive 119878-axis in finite time andtherefore the disease disappears

(b) There is a saddle-node point when (1205791 1205792) lies oncurve SN

(c) When (1205791 1205792) crosses SN into region II two positiveequilibria which are an unstable focus and a saddleappear

(d) When the parameters lie on the curve H there is alsoa saddle and an unstable focus An unstable limit cycleappears with the parameters crossing H into III

(e) A homoclinic cycle appears as the parameters passingIII into HL through the homoclinic bifurcation

12 Mathematical Problems in Engineering

0 5 10 15 20 25 30 35 40

0

1

2

3

4

5

6

7

8

9

10

S

I

Figure 7 When (1205791 1205792) = (0 0) the unique positive equilibrium isa cusp of codimension 2

3 35 45 55 65 754 5 6 7 8

1

2

3

4

5

6

7

8

S

I

Figure 8 When (1205791 1205792) = (minus01 minus00093) lies in region I there isno positive equilibrium

(f) The homoclinic loop breakswith the parameter cross-ing HL into IV and a saddle and a stable focus appear

Furthermore using PPLANE8 some numerical simu-lations of system (68) are depicted in Figures 7ndash10 When(1205791 1205792) = (0 0) that is (1198600 1198890) = (537500 110) there isa unique positive equilibrium (119878lowast 119868lowast) = (31750 4) whichis a cusp of codimension 2 (Figure 7) When (1205791 1205792) =(minus01 minus00093) lies in region I there is no positive equi-librium (Figure 8) When (1205791 1205792) = (minus002 minus0001863798)there is a homoclinic loop (Figure 9) When (1205791 1205792) =(minus002 minus00020) lies in region IV the homoclinic loopis broken and there is a stable focus and a saddle(Figure 10)

6 605 61 615 62 625 63 635 64 645 653536373839

44142434445

S

I

Figure 9 When (1205791 1205792) = (minus002 minus0001863798) there is ahomoclinic loop

0 1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

7

8

9

10

S

I

Figure 10 When (1205791 1205792) = (minus002 minus00020) lies in region IV thereis a stable focus and a saddle

7 Discussion

In this paper we focus on the bifurcation analysis of an SISepidemic model with bilinear incidence rate and saturatedtreatment Generally speaking in many epidemic modelsthe basic reproduction number which is the key conceptin epidemiology can be decreased below unity to eradicatethe disease However in our model the basic reproductionnumber below 1 is not enough to eradicate the diseaseAccording to our analysis in this paper we demonstrate notonly the global stability of the disease-free equilibrium when1198770 lt 119877

lowast

0and the local asymptotic stability of the endemic

equilibrium 1198642 when 119877lowast

0lt 1198770 lt 1 but also the global

stability of the unique endemic equilibrium 119864lowast when 1198770 gt 1Moreover it has been shown in Corollary 4 that backwardbifurcations occur if the effect of the infected being delayedfor treatment is strong That is to say we should get prompttreatment for patients Through Figure 4 we can see thatthere is a region such that the disease will persist if the initialposition lies in the region and disappear if the initial positionlies outside this region So in order to restrain the spread

Mathematical Problems in Engineering 13

of the disease governments should take timely measures tocontrol the initial infected individuals in a lower level Inaddition fromTheorem 8 we can see that the disease will dieout if 1198770 is small enough Therefore from the definition of1198770 one knows that we can decrease the incidence rate 120582 andincrease the cure rate 119903 so as to eradicate the diseases or tomake them controlled in a lower endemic steady state

The stability analysis of the model equilibria enables usto completely analyze their local bifurcation behavior suchas Hopf saddle-node and Bogdanov-Takens bifurcation Bycomputing the first Lyapunov coefficient we can determinethat the Hopf bifurcation is a supercritical Hopf bifurca-tion or a subcritical Hopf bifurcation We also show thatunder assumptions (1198601) and (1198602) the model undertakesBogdanov-Takens bifurcation that is there are saddle-nodebifurcation Hopf bifurcation and homoclinic bifurcationin the system The normal norm of the Bogdanov-Takensbifurcation is derived in Section 5 which is very helpful toobtain the three kinds of bifurcation curves By analyticaltechniques 119860 and 119889 are chosen as bifurcation parametersand other parameters are fixed and we easily get a clearpicture about the rich dynamics behaviors of our modelThrough studying the bifurcations of the SIS epidemicmodelwe are suggesting that in order to eradicate the diseasemore medical facilities as well as medical professionals areneeded and the medical standard needs to be improved aswell

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research was supported by the National Natural ScienceFoundation of China (no 11001041 no 10926105 no 11101283no 11271065 and no 11201360) SRFDP (no 200802001008)and Scientific Research Innovation Project of ShanghaiMunicipal Education Commission (12YZ173)

References

[1] M E Alexander and S M Moghadas ldquoPeriodicity in an epi-demic model with a generalized non-linear incidencerdquo Mathe-matical Biosciences vol 189 no 1 pp 75ndash96 2004

[2] M E Alexander and S M Moghadas ldquoBifurcation analysisof SIRS epidemic model with generalized incidencerdquo SIAMJournal on Applied Mathematics vol 65 no 5 pp 1794ndash18162005

[3] J Arino C C McCluskey and P van den Driessche ldquoGlobalresults for an epidemic model with vaccination that exhibitsbackward bifurcationrdquo SIAM Journal on Applied Mathematicsvol 64 no 1 pp 260ndash276 2003

[4] W R Derrick and P van den Driessche ldquoHomoclinic orbitsin a disease transmission model with nonlinear incidence andnonconstant populationrdquo Discrete and Continuous DynamicalSystems B vol 3 no 2 pp 299ndash309 2003

[5] F Brauer ldquoBackward bifurcations in simple vaccination mod-elsrdquo Journal of Mathematical Analysis and Applications vol 298no 2 pp 418ndash431 2004

[6] K P Hadeler and P van den Driessche ldquoBackward bifurcationin epidemic controlrdquo Mathematical Biosciences vol 146 no 1pp 15ndash35 1997

[7] Z X Hu S Liu and H Wang ldquoBackward bifurcation of anepidemic model with standard incidence rate and treatmentraterdquo Nonlinear Analysis Real World Applications vol 9 no 5pp 2302ndash2312 2008

[8] J Hui and D M Zhu ldquoGlobal stability and periodicity onSIS epidemic models with backward bifurcationrdquo Computers ampMathematics with Applications vol 50 no 8-9 pp 1271ndash12902005

[9] Y Jin W D Wang and S W Xiao ldquoAn SIRS model with anonlinear incidence raterdquoChaos Solitons amp Fractals vol 34 no5 pp 1482ndash1497 2007

[10] G H Li and W D Wang ldquoBifurcation analysis of an epidemicmodel with nonlinear incidencerdquo Applied Mathematics andComputation vol 214 no 2 pp 411ndash423 2009

[11] X Z Li W S Li and M Ghosh ldquoStability and bifurcation ofan SIS epidemic model with treatmentrdquo Chaos Solitons ampampFractals vol 42 no 5 pp 2822ndash2832 2009

[12] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1986

[13] X B Liu and L J Yang ldquoStability analysis of an SEIQV epidemicmodel with saturated incidence raterdquo Nonlinear Analysis RealWorld Applications vol 13 no 6 pp 2671ndash2679 2012

[14] M Martcheva and H R Thieme ldquoProgression age enhancedbackward bifurcation in an epidemic model with super-infectionrdquo Journal of Mathematical Biology vol 46 no 5 pp385ndash424 2003

[15] S G Ruan and W D Wang ldquoDynamical behavior of anepidemic model with a nonlinear incidence raterdquo Journal ofDifferential Equations vol 188 no 1 pp 135ndash163 2003

[16] W D Wang and S G Ruan ldquoBifurcation in an epidemicmodel with constant removal rate of the infectivesrdquo Journal ofMathematical Analysis and Applications vol 291 no 2 pp 775ndash793 2004

[17] Y L Tang D Q Huang S G Ruan and W N ZhangldquoCoexistence of limit cycles and homoclinic loops in a SIRSmodel with a nonlinear incidence raterdquo SIAM Journal onApplied Mathematics vol 69 no 2 pp 621ndash639 2008

[18] P van den Driessche and J Watmough ldquoA simple SIS epidemicmodel with a backward bifurcationrdquo Journal of MathematicalBiology vol 40 no 6 pp 525ndash540 2000

[19] J J Wei and J A Cui ldquoDynamics of SIS epidemic model withthe standard incidence rate and saturated treatment functionrdquoInternational Journal of Biomathematics vol 5 no 3 Article ID1260003 18 pages 2012

[20] W D Wang ldquoBackward bifurcation of an epidemic model withtreatmentrdquoMathematical Biosciences vol 201 no 1-2 pp 58ndash712006

[21] Z W Wang ldquoBackward bifurcation in simple SIS modelrdquo ActaMathematicae Applicatae Sinica vol 25 no 1 pp 127ndash136 2009

[22] L Xue and C Scoglio ldquoThe network level reproduction numberfor infectious diseases with both vertical and horizontal trans-missionrdquo Mathematical Biosciences vol 243 no 1 pp 67ndash802013

14 Mathematical Problems in Engineering

[23] X Zhang and X N Liu ldquoBackward bifurcation of an epidemicmodel with saturated treatment functionrdquo Journal ofMathemat-ical Analysis and Applications vol 348 no 1 pp 433ndash443 2008

[24] J Guckenheimer and P J Holmes Nonlinear OscillationsDynamical Systems and Bifurcation of Vector Fields SpringerNew York NY USA 1996

[25] X Zhang and X N Liu ldquoBackward bifurcation and globaldynamics of an SIS epidemic model with general incidence rateand treatmentrdquoNonlinearAnalysis RealWorldApplications vol10 no 2 pp 565ndash575 2009

[26] M A Safi A B Gumel and E H Elbasha ldquoQualitative analysisof an age-structured SEIR epidemic model with treatmentrdquoApplied Mathematics and Computation vol 219 no 22 pp10627ndash10642 2013

[27] L H Zhou and M Fan ldquoDynamics of an SIR epidemic modelwith limited medical resources revisitedrdquo Nonlinear AnalysisReal World Applications vol 13 no 1 pp 312ndash324 2012

[28] X Y Zhou and J A Cui ldquoAnalysis of stability and bifurcation foran SEIR epidemicmodel with saturated recovery raterdquoCommu-nications inNonlinear Science andNumerical Simulation vol 16no 11 pp 4438ndash4450 2011

[29] D Lacitignola ldquoSaturated treatments and measles resurgenceepisodes in South Africa a possible linkagerdquo MathematicalBiosciences and Engineering MBE vol 10 no 4 pp 1135ndash11572013

[30] J C Eckalbar and W L Eckalbar ldquoDynamics of an epidemicmodel with quadratic treatmentrdquo Nonlinear Analysis RealWorld Applications vol 12 no 1 pp 320ndash332 2011

[31] F Brauer ldquoBackward bifurcations in simple vaccinationtreat-ment modelsrdquo Journal of Biological Dynamics vol 5 no 5 pp410ndash418 2011

[32] M W Hirsch S Smale and R L Devaney Differential Equa-tions Dynamical Systems and an Introduction to Chaos Aca-demic Press New York NY USA 2003

[33] R Bogdanov ldquoBifurcations of a limit cycle for a family of vectorfields on the planrdquo SelectaMathematica vol 1 pp 373ndash388 1981

[34] R Bogdanov ldquoVersal deformations of a singular point on theplan in the case of zero eigen-valuesrdquo Selecta MathematicaSovietica vol 1 pp 389ndash421 1981

[35] F Takens ldquoForced oscillations and bifurcationrdquo in Applicationsof Global Analysis I vol 3 pp 1ndash59 Communications of theMathematical Institute Rijksuniversiteit Utrecht 1974

[36] Y A Kuznetsov Elements of Applied Bifurcation TheorySpringer New York NY USA 2nd edition 1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

181614121080604020

10

20

30

40

50

60

I

Rlowast

0

R0

Figure 3 The figure of infective sizes at equilibria versus 1198770 when120572 = 1 119889 = 01 120583 = 001 120582 = 005 119903 = 2 and 120576 = 01

We define the right-hand side of (11) as

119877lowast0≜

41205822120572 (119889 + 120583)119860

[120582 (119889 + 120583) + 120572119889 (119889 + 120576 + 120583) minus 120572120582119860]2+ 41205822120572 (119889 + 120583)119860

(12)

Therefore according to the qualitative approach recentlyproposed by [31] which is based on the analysis of theequilibria curve in the neighboring of the critical threshold1198770 = 1 we have the following theorem

Theorem 3 If 120572 gt 0 119887 lt 0 then system (4) has a backwardbifurcation at 1198770 = 1 (see Figure 3)

In order to verify the bifurcation curve (the graph of 119868as a function of 1198770) in Figure 3 we think of 119903 as a variablewith the other parameters as constant Through implicitdifferentiation of (8) with respect to 119903 we get

(2119886119868 + 119887)119889119868

119889119903= minus119889 lt 0 (13)

From (13) we know that the sign of 119889119868119889119903 is opposite to thatof 2119886119868 + 119887 And from the definition of 1198770 we know that 1198770decreases when 119903 increases It implies that the bifurcationcurve has positive slope at equilibrium values with 2119886119868+119887 gt 0and negative slope at equilibrium values with 2119886119868 + 119887 lt 0 Ifthere is no backward bifurcation at 1198770 = 1 then the uniqueendemic equilibrium for 1198770 gt 1 satisfies

2119886119868 + 119887 = radic1198872 minus 4119886119888 gt 0 (14)

and the bifurcation curve has positive slope at all pointswhere119868 gt 0 If there is a backward bifurcation at 1198770 = 1 then thereis an interval onwhich there are two endemic equilibria givenby

2119886119868 + 119887 = plusmnradic1198872 minus 4119886119888 (15)

The bifurcation curve has negative slope at the smaller oneandpositive slope at the larger oneThus the bifurcation curveis as shown in Figure 3

Under the conditions of Theorem 3 if a backward bifur-cation takes place we can see from Figure 3 that there is acritical value 119877lowast

0at the turning point In this case the disease

will not die out when1198770 lt 1 However the diseasewill die outwhen 1198770 lt 119877

lowast

0 Therefore the critical value 119877lowast

0can be taken

as a new threshold for the control of the diseaseIn the following we give an explicit criterion of a

backward bifurcation at 1198770 = 1For convenience we define

1205720 =120582 (119889 + 120583)

119889119903 (16)

Corollary 4 When 120572 gt 1205720 system (4) has a backward bifur-cation at 1198770 = 1

Proof When 1198770 le 1 hArr 119888 ge 0

120582119860 le 119889 (119889 + 120576 + 120583 + 119903) (17)

The condition 119887 lt 0 is equivalent to

120582 (119889 + 120583) + 120572119889 (119889 + 120576 + 120583) lt 120572120582119860 (18)

From (17) and (18) we get 120582(119889 + 120583) + 120572119889(119889 + 120576 + 120583) lt 120572119889(119889 +120576 + 119903 + 120583) which reduces to

120572 gt120582 (119889 + 120583)

119889119903≜ 1205720 (19)

It means that 120572 is big enough to lead a backward bifurcationwith two endemic equilibria when 1198770 lt 1 Therefore theproof is complete

Next we consider the local stability of the uniqueendemic equilibrium when 1198770 gt 1

Theorem 5 When 1198770 gt 1 and 0 le 120572 lt 120582119903 the uniqueendemic equilibrium 119864lowast is locally asymptotically stable

Proof Firstly from Theorem 2 we can know that system(4) has a unique endemic equilibrium 119864lowast when 1198770 gt 1Moreover the Jacobian matrix of system (4) is

119872 =(

minus119889 minus 120582119868 minus120582119878 + 120576 +119903

1 + 120572119868minus

119903120572119868

(1 + 120572119868)2

120582119868 120582119878 minus (119889 + 120576 + 120583) minus119903

1 + 120572119868+

119903120572119868

(1 + 120572119868)2

)

(20)

From the second equation of (6) we have

minus120582119878 + 120576 +119903

1 + 120572119868= minus119889 minus 120583 (21)

From (21) the Jacobian matrix119872 reduces to

119872 =(

minus119889 minus 120582119868 minus119889 minus 120583 minus119903120572119868

(1 + 120572119868)2

120582119868119903120572119868

(1 + 120572119868)2

) (22)

Mathematical Problems in Engineering 5

We obtain

det (119872) = 119868

(1 + 120572119868)2[120582 (119889 + 120583) (1 + 120572119868)

2 minus 119903120572119889] (23)

In fact there is 120582(119889 + 120583)(1 + 120572119868)2 gt 120582119889 Since 1198770 gt 1 and0 le 120572 lt 120582119903 we have

120582 (119889 + 120583) (1 + 120572119868)2 minus 119903120572119889 gt 120582119889 minus 119903120572119889 gt 0 (24)

So we get det(119872) gt 0 The trace of119872 is given by

tr (119872) = 1

(1 + 120572119868)2[minus (119889 + 120582119868) (1 + 120572119868)

2 + 119903120572119868] (25)

In the same way as the above calculation of (24) we have

minus (119889 + 120582119868) (1 + 120572119868)2 + 119903120572119868 lt minus120582119868 + 119903120572119868 lt 0 (26)

So we get tr(119872) lt 0 The proof is complete

Now we consider the case that there are two endemicequilibria 1198641 and 1198642 let 119872119894 be the Jacobian matrix at 119864119894119894 = 1 2

Theorem 6 The endemic equilibrium 1198641 is a saddle wheneverit exists

Proof Since 1198681 = (minus119887 minus radic1198872 minus 4119886119888)2119886 and Δ = 1198872 minus 4119886119888 wehave 1198681 = (minus119887 minus radicΔ)2119886 Thus

det (1198721)

=1198681

(1 + 1205721198681)2[120582 (119889 + 120583) (1 + 1205721198681)

2minus 119903120572119889]

≜1198681

(1 + 1205721198681)2120595 (1198681)

(27)

From (1198673) of Theorem 2 we can know that if 120572 gt 0 119887 lt 0and 119877lowast

0lt 1198770 lt 1 then 1198641 exists Hence we can get 120595(0) =

120582(119889 + 120583) minus 119903120572119889 lt 0 and 1205951015840(1198681) = 2120582(119889 + 120583)(1 + 1205721198681)120572 gt 0 Itfollows that there exists a unique 119868lowast gt 0 such that

120595 (1198681) = 0 when 1198681 = 119868lowast

120595 (1198681) lt 0 when 0 lt 1198681 lt 119868lowast

120595 (1198681) gt 0 when 1198681 gt 119868lowast

(28)

where

119868lowast = radic119903119889

120572120582 (119889 + 120583)minus1

120572 (29)

On the other hand

1198681 = minus119887

2119886minusradicΔ

2119886

=120572120582119860 minus 120582 (119889 + 120583) minus 120572119889 (119889 + 120576 + 120583)

2120572120582 (119889 + 120583)minusradicΔ

2119886

= radic119903119889

120572120582 (119889 + 120583)minus1

120572+1

120572minus radic

119903119889

120572120582 (119889 + 120583)

+120572120582119860 minus 120582 (119889 + 120583) minus 120572119889 (119889 + 120576 + 120583)

2120572120582 (119889 + 120583)minusradicΔ

2119886

= 119868lowast + ([120582 (119889 + 120583) + 120572120582119860 minus 2radic119903119889120572120582 (119889 + 120583)

minus 120572119889 (119889 + 120576 + 120583)] minus radicΔ)times (2120572120582 (119889 + 120583))minus1

(30)

Δ = 120582 (119889 + 120583) minus [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]2

minus 4120572120582 (119889 + 120583) [119889 (119889 + 120576 + 119903 + 120583) minus 120582119860]

= 120582 (119889 + 120583) + [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

minus 2radic119903119889120572120582(119889 + 120583)2

minus 4119903119889120572120582 (119889 + 120583)

minus 4120582 (119889 + 120583) [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

+ 4radic119903119889120572120582 (119889 + 120583) 120582 (119889 + 120583) + [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

minus 4120572120582 (119889 + 120583) [minus120582119860 + 119889 (119889 + 120576 + 119903 + 120583)]

≜ 120582 (119889 + 120583) + [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

minus 2radic119903119889120572120582 (119889 + 120583)2

+ 119875

(31)

where

119875 = minus8119903119889120572120582 (119889 + 120583) + 4radic119903119889120572120582 (119889 + 120583)

times 120582 (119889 + 120583) + [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

(32)

In the following we will show that 119875 gt 0 Since 119877lowast0lt 1198770 we

have

4120572120582 (119889 + 120583)

[120582 (119889 + 120583) + 120572119889 (119889 + 120576 + 120583) minus 120572120582119860]2+ 41205721205822 (119889 + 120583)119860

lt1

119889 (119889 + 120576 + 119903 + 120583)

(33)

6 Mathematical Problems in Engineering

that is

[120582 (119889 + 120583) + 120572119889 (119889 + 120576 + 120583) minus 120572120582119860]2

gt 4120572120582 (119889 + 120583) 119889 (119889 + 120576 + 119903 + 120583) minus 41205721205822 (119889 + 120583)119860(34)

Therefore

120582 (119889 + 120583) + [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]2

= 120582 (119889 + 120583) minus [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]2

+ 4120582 (119889 + 120583) [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

gt 4120572120582 (119889 + 120583) 119889 (119889 + 120576 + 119903 + 120583) minus 41205721205822 (119889 + 120583)119860

+ 4120582 (119889 + 120583) [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

= 4119903119889120572120582 (119889 + 120583)

(35)

Obviously we have 119875 gt 0 From (30) one has 1198681 lt 119868lowast So

we get det(1198721) lt 0 Hence the endemic equilibrium 1198641 is asaddle The proof is complete

In order to explore the stability of the endemic equilib-rium 1198642 define

1198981 = (21198891205721198862 + 1205821198862 minus 1205761205721198862 minus 1198861198881205821205722)

minus 119887 (2120582120572119886 + 1198891205722119886 minus 1198871205821205722)

1198982 = 1198862119889 minus 119888 (2120582120572119886 + 1198891205722119886 minus 1198871205821205722)

(36)

Theorem 7 If 120578 gt 0 then endemic equilibrium 1198642 is locallyasymptotically stable if 120578 lt 0 then endemic equilibrium 1198642 isunstable where 120578 = 21198861198982 + 1198981(radic1198872 minus 4119886119888 minus 119887)

Proof Consider

det (1198722) =1198682

(1 + 1205721198682)2[120582 (119889 + 120583) (1 + 1205721198682)

2minus 120576120572119889]

=1198682

(1 + 1205721198682)2times 120595 (1198682)

(37)

By carrying out arguments similar to that of Theorem 6 wehave 1198682 gt 119868

lowast Therefore det(1198722) gt 0 In addition we have

tr (1198722) = minus(119889 + 1205821198682) (1 + 1205721198682)

2minus 1205761205721198682

(1 + 1205721198682)2

= minus120582120572211986832+ (2120582120572 + 1198891205722) 1198682

2+ (2119889120572 + 120582 minus 120576120572) 1198682 + 119889

(1 + 1205721198682)2

(38)

and then sgn(tr(1198722)) = minus sgn(119866(1198682)) where

119866 (119909) = 12058212057221199093 + (2120582120572 + 1198891205722) 1199092 + (2119889120572 + 120582 minus 120576120572) 119909 + 119889

(39)

I

0 50 100 1500

5

10

15

20

25

30

35

40

S

E0

E1

E2

Figure 4 One region of disease persistence and one region ofdisease extinction when 119860 = 10 120572 = 1 119889 = 01 120583 = 001 120582 = 001119903 = 01 and 120576 = 01

Using the expression of1198981 = (21198891205721198862 +1205821198862 minus 1205761205721198862 minus1198861198881205821205722) minus

119887(2120582120572119886+1198891205722119886minus1198871205821205722) and1198982 = 1198862119889minus119888(2120582120572119886+1198891205722119886minus1198871205821205722)

one has

119866 (1198682) = (1198861198682

2+ 1198871198682 + 119888) 1205930 +

11989811198682 + 11989821198862

(40)

where1205930 is a first degree polynomial of 1198682 Since 1198861198682

2+1198871198682+119888 =

0 sgn(tr(1198722)) = minus sgn(119866(1198682)) = minus sgn(11989811198682 + 1198982) From theexpression of 1198682 we have

sgn (11989811198682 + 1198982)

= sgn (21198861198982 + 1198981 (radic1198872 minus 4119886119888 minus 119887)) ≜ sgn (120578) (41)

Thus 1198642 is locally asymptotically stable if 120578 gt 0 and 1198642 isunstable if 120578 lt 0 The proof is complete

From the above discussion we have the following con-clusion If two endemic equilibria 1198641 and 1198642 exist the stablemanifolds of the saddle 1198641 split 119877

2

+into two regions The

disease is persistent in the upper region and dies out in thelower region (see Figure 4)

3 Global Analysis

Firstly we consider the global stability of the disease-freeequilibrium 1198640 Let 119873 = 119878 + 119868 be the total population sizeNow we note that the equation for total population is givenby 119889119873119889119905 = 119860 minus 119889119878 minus (119889 + 120583)119868 le 119860 minus 119889119873 It follows thatlim119905rarr+infin119873(119905) le 119860119889 Let

R = (119878 119868) isin 1198772

+ 119878 + 119868 le

119860

119889 (42)

which is positive invariant with respect to system (4)

Theorem 8 If 1198770 lt 119877lowast0 the disease-free equilibrium 1198640(119860119889 0) is globally asymptotically stable that is the disease diesout

Mathematical Problems in Engineering 7

Proof Suppose 1198770 lt 119877lowast0 From the (1198673) of Theorem 2 weknow that the model has no endemic equilibrium Fromthe corollary of Poincare-Bendixson theorem [32] we knowthat there is no periodic orbit in R as there is a disease-free equilibrium in R Since R is a bounded positivelyinvariant region and 1198640 is the only equilibrium in R thelocal stability of 1198640 implies that every solution initiating inR approaches 1198640 Thus the disease-free equilibrium 1198640 isglobally asymptotically stable The proof is complete

Now we analyze the global dynamics of the endemicequilibrium when 1198770 gt 1

Theorem 9 If 1198770 gt 1 and 0 le 120572 lt 120582119903 the system (4) has nolimit cycles

Proof We use Dulac theorem to exclude the limit cycle Let

119875 (119878 119868) = 119860 minus 119889119878 minus 120582119878119868 + 120576119868 +119903119868

1 + 120572119868

119876 (119878 119868) = 120582119878119868 minus (119889 + 120576 + 120583) 119868 minus119903119868

1 + 120572119868

(43)

and take the Dulac function

119863 =1

119868 (44)

According to 0 le 120572 lt 120582119903 we can get

120597 (119875119863)

120597119878+120597 (119876119863)

120597119868

= minus119889

119868minus 120582 +

120572119903

(1 + 120572119868)2

lt minus119889

119868minus 120582 +

120582

(1 + 120572119868)2

=1

119868 (1 + 120572119868)2minus119889 (1 + 120572119868)

2 minus 120582119868 [(1 + 120572119868)2 minus 1]

lt 0

(45)

Hence the system (4) has no limit cycles The proof iscomplete

Therefore we obtain the global result of the uniqueendemic equilibrium

Theorem 10 If 1198770 gt 1 and 0 le 120572 lt 120582119903 the unique endemicequilibrium 119864lowast is globally asymptotically stable (see Figure 5)

4 Hopf Bifurcation

In this section we study the Hopf bifurcation of system (4)From the above discussion we know that there is no closedorbit surrounding1198640 or1198641 because the 119878-axis is invariantwith

0 50 100 1500

50

100

150

S

I Elowast

Figure 5The unique endemic equilibrium 119864lowast is globally asymptot-ically stable when 119860 = 15 120572 = 001 119889 = 01 120583 = 001 120582 = 001119903 = 08 and 120576 = 01

respect to system (4) and 1198641 is always a saddle ThereforeHopf bifurcation can only occur at 1198642 Set

120590 = [minus120582120572 minus1198881 (1198882 + 21198884)

11988612]1198634

+ [minus 120582120572119886211minus 21205821205721198861111988612 + (1198882 minus

211988611119888111988612)

times (119886111198882 minus119886211119888111988612)

minus (21198862111198881

11988612minus 119886111198882 + 2119888411988611 minus 119888311988612)

times (1198884 minus11988611119888111988612)]1198632

+ 11988611 (119886211119888111988612minus 119886111198882 + 119888411988611 minus 119888311988612)

times (119886121198883 minus 2119886111198884 minus 119886111198882)

(46)

where 119886119894119895 (119894 119895 = 1 2) 119888119896 (119896 = 1 2 3 4) and 119863 are defined by(50) and (52)

Theorem 11 System (4) undergoes a Hopf bifurcation if 120578 = 0Moreover if 120590 lt 0 there is a family of stable period orbits ofsystem (4) as 120578 decreases from 0 that is a supercritical Hopfbifurcation occurs if 120590 gt 0 there is a family of unstable periodorbits of system (4) as 120578 increases from 0 that is a subcriticalHopf bifurcation occurs

Proof The proof of Theorem 7 shows that tr(1198722) = 0 if andonly if 120578 = 0 and det(1198722) gt 0 when 1198642 exists Therefore theeigenvalues of 1198722 are a pair of pure imaginary roots if andonly if 120578 = 0 The direct calculations show that

119889 (tr (1198722))119889120578

100381610038161003816100381610038161003816100381610038161003816120578=0= minus

1

21198863 (1 + 1205721198682)2lt 0 (47)

8 Mathematical Problems in Engineering

ByTheorem 342 in [24] 120578 = 0 is the Hopf bifurcation pointfor system (4)

To be concise in notations rescale (4) by 120591 = 119905(1 + 120572119868)For simplicity we still use 119905 instead of 120591 Then we obtain

119889119878

119889119905= (119860 minus 119889119878 minus 120582119878119868 + 120576119868) (1 + 120572119868) + 119903119868

119889119868

119889119905= 120582119878119868 (1 + 120572119868) minus (119889 + 120576 + 120583) 119868 (1 + 120572119868) minus 119903119868

(48)

Let 119909 = 119878 minus 1198782 and 119910 = 119868 minus 1198682 then (48) becomes

119889119909

119889119905= 11988611119909 + 11988612119910 + 1198881119910

2 + 1198882119909119910 minus 1205821205721199091199102

119889119910

119889119905= 11988621119909 + 11988622119910 + 1198883119909119910 + 1198884119910

2 + 1205821205721199091199102

(49)

where

11988611 = minus (119889 + 1205821198682) (1 + 1205721198682)

11988612 = (minus1205821198782 + 119903) (1 + 1205721198682)

+ 120572 (119860 minus 1198891198782 minus 12058211987821198682 + 1199031198682) + 120576

11988621 = 1205821198682 (1 + 1205721198682)

11988622 = [1205821198782 minus (119889 + 119903 + 120583)] (1 + 1205721198682)

+ 120572 [12058211987821198682 minus (119889 + 119903 + 120583) 1198682] minus 120576

1198881 = 120572 (minus1205821198782 + 119903)

1198882 = 120572 (minus119889 minus 1205821198682) minus 120582 (1 + 1205721198682)

1198883 = 120582 (1 + 1205721198682) + 1205821198682120572

1198884 = [1205821198782 minus (119889 + 119903 + 120583)] 120572

(50)

Let 119864lowast denote the origin of 119909-119910 plane Since 1198642 = (1198782 1198682)satisfies (6) we obtain

det (119872 (119864lowast))

= minus (119889 + 1205821198682) 1205721205761198682 + (119889 + 120583) (1 + 1205721198682

2)2

1205821198682 + 1205721205821205761198682

2

= 1198682 times 120595 (1198682)

(51)

From the proof of Theorem 7 it follows that 120595(1198682) is alwayspositive It is easy to verify that 11988611 + 11988622 = 0 if and only if120578 = 0 Set

119863 = radicdet (119872 (119864lowast)) (52)

and let 119906 = minus119909 and V = (11988611119863)119909+(11988612119863)119910 then the normalform of (48) for Hopf bifurcation reads

119889119906

119889119905= minus119863V + 119891 (119906 V)

119889V119889119905= 119863119906 + 119892 (119906 V)

(53)

where

119891 (119906 V) = (11988611119886121198882 minus119886211

119886212

1198881)1199062 minus11986321198881119886212

V2

+ (119863119888211988612minus 2119863119886111198881119886212

)119906V minus120582120572119886211

119886212

1199063

minus 211986311988611120582120572

119886212

1199062V minus1198632120582120572

119886212

119906V2

119892 (119906 V) =11988611119863(119886211

119886212

1198881 minus11988611119886121198882 +11988611119886121198884 minus 1198883)119906

2

+ (119863119886111198881119886212

+119863119888411988612) V2

+ (2119886211

119886212

1198881 minus11988611119886121198882 + 2

11988611119886121198884 minus 1198883)119906V

+120582120572119886211

11988612119863(1

11988612minus 1)1199063 +

21205821205721198861111988612

(1198861111988612minus 1)1199062V

+119863120582120572

11988612(1198861111988612minus 1)119906V2

(54)

Now we evaluate the first Lyapunov coefficient Γ of system(4) as follows

Γ =1

16[119891119906119906119906 + 119891119906VV + 119892119906119906V + 119892VVV] +

1

16119863

times [119891119906V (119891119906119906 + 119891VV) minus 119892119906V (119892119906119906 + 119892VV) minus 119891119906119906119892119906119906 + 119891VV119892VV]

(55)

where 119891119906V denotes (1205972119891120597119906120597V)(0 0) and so forth Then

Γ =1

81198862121198632[minus120582120572 minus

1198881 (1198882 + 21198884)

11988612]1198634

+ [minus 120582120572119886211minus 21205821205721198861111988612 + (1198882 minus

211988611119888111988612)

times (119886111198882 minus119886211119888111988612)

minus (21198862111198881

11988612minus 119886111198882 + 2119888411988611 minus 119888311988612)

times (1198884 minus11988611119888111988612)]1198632

+ 11988611 (119886211119888111988612minus 119886111198882 + 119888411988611 minus 119888311988612)

times (119886121198883 minus 2119886111198884 minus 119886111198882)

=120590

81198862121198632

(56)

Mathematical Problems in Engineering 9

Obviously the sign of Γ is determined by 120590 ByTheorem 342and (3411) in [24] the rest of the claims in Theorem 11 arevalid The proof is complete

5 Bogdanov-Takens Bifurcations

The purpose of this section is to study the Bogdanov-Takensbifurcation of (4) Now we assume the following

(1198601) 1198872 minus 4119886119888 = 0

(1198602) 21198861198892 + 119887120582120583 = 0

Then (6) admits a unique positive equilibrium 119864lowast = (119878lowast 119868lowast)where

119868lowast = minus119887

2119886 119878lowast =

119860 minus (119889 + 120583) 119868lowast

119889 (57)

The Jacobian matrix of system (4) at 119864lowast is

119872lowast = (

minus119889 minus 120582119868lowast minus120582119878lowast + 120576 +119903

(1 + 120572119868lowast)2

120582119868lowast 120582119878lowast minus (119889 + 120576 + 120583) minus119903

(1 + 120572119868lowast)2)

(58)

By (58) we have

det (119872lowast) = 119868lowast

(1 + 120572119868lowast)2[(119889 + 120583) 120582 (1 + 120572119868lowast)

2minus 119903119889120572]

= 0

(59)

because of

(1 + 120572119868lowast)2=41198862 minus 4119886120572119887 + 12057221198872

41198862

=41198862 minus 4119886120572119887 + 12057224119886119888

41198862

=119886 minus 120572119887 + 1205722119888

119886

=1205722119903119889

119886

(60)

Furthermore (1198602) implies that

tr (119872lowast) = 0 (61)

Thus (1198601) and (1198602) imply that the Jacobian matrix has azero eigenvalue withmultiplicity 2This suggests that (4)mayadmit a Bogdanov-Takens singularity

Theorem 12 Suppose that (1198601) (1198602) and 21198871 + 1198874 = 0 holdThen the endemic equilibrium 119864lowast = (119878lowast 119868lowast) of (4) is a cuspof codimension 2 that is it is a Bogdanov-Takens singularityHere 1198871 and 1198874 are defined by (65)

Proof Using the transformation of 119909 = 119868minus 119868lowast and 119910 = 119878minus 119878lowastsystem (4) becomes

119889119909

119889119905= 1198861119909 + 1198862119910 + 120582119909119910 + 11988611119909

2 minus 1198751 (119909)

119889119910

119889119905= minus11988621

1198862119909 minus 1198861119910 minus 120582119909119910 minus 11988611119909

2 + 1198751 (119909)

(62)

where 1198751(119909) is a smooth function of 119909 at least of the thirdorder and

1198861 = 120582119878lowast minus (119889 + 120576 + 120583) minus

119903

(1 + 120572119868lowast)2gt 0

1198862 = 120582119868lowast gt 0

11988611 =119903120572

(1 + 120572119868lowast)3gt 0

(63)

Set119883 = 119909 119884 = 1198861119909 + 1198862119910 Then (62) is transformed into

119889119883

119889119905= 119884 + 1198871119883

2 + 1198872119883119884 + 1198761 (119883)

119889119884

119889119905= 1198873119883

2 + 1198874119883119884 + 1198762 (119883)

(64)

where 119876119894(119883) are smooth functions of 119883 at least of the thirdorder and

1198871 = 11988611 minus12058211988611198862

1198872 =120582

1198862

1198873 = 119886111988611 minus12058211988621

1198862+ 1198861120582 minus 119886211988611

1198874 =12058211988611198862minus 120582

(65)

In order to obtain the canonical normal form we perform thetransformation of variables by

119906 = 119883 minus119887221198832 V = 119884 + 1198871119883

2 (66)

Then we obtain

119889119906

119889119905= V + 1198771 (119906)

119889V119889119905= 11988731199062 + (21198871 + 1198874) 119906V + 1198772 (119906)

(67)

where 119877119894(119906) are smooth functions of 119906 at least of the thirdorder Note that 1198873 lt 0 and 21198871 + 1198874 = 0 It follows from [33ndash35] that (4) admits a Bogdanov-Takens bifurcation

In the following wewill find the versal unfolding in termsof the original parameters in (4) In this way we will know theapproximate saddle-node Hopf and homoclinic bifurcation

10 Mathematical Problems in Engineering

curvesWe choose119860 and 119889 as bifurcation parameters Fix 120582 =1205820 120576 = 1205760 119903 = 1199030 120572 = 1205720 and 120583 = 1205830 Let 119860 = 1198600 + 1205791 and119889 = 1198890 + 1205792 where 1205791 and 1205792 are parameters which vary in asmall neighborhood of the origin

Suppose that 119860 = 1198600 119889 = 1198890 120582 = 1205820 120576 = 1205760 119903 = 1199030 120572 =1205720 and 120583 = 1205830 satisfy (1198601) and (1198602) Consider the followingsystem

119889119868

119889119905= 1205820119878119868 minus (1198890 + 1205792 + 1205760 + 1205830) 119868 minus

1199030119868

1 + 1205720119868

119889119878

119889119905= 1198600 + 1205791 minus (1198890 + 1205792) 119878 minus 1205820119878119868 + 1205760119868 +

1199030119868

1 + 1205720119868

(68)

By the transformations of 119909 = 119868 minus 119868lowast and 119910 = 119878 minus 119878lowast system(68) becomes

119889119909

119889119905= minus 1205792119868

lowast + 1198881119909 + 1198882119910 + 119888111199092 + 1205820119909119910 minus 1199081 (119909)

119889119910

119889119905= (1205791 minus 1205792119878

lowast) + 1198883119909 + 1198884119910 minus 119888111199092 minus 1205820119909119910 + 1199081 (119909)

(69)

where 1199081(119909) is a smooth function of 119909 at least of the thirdorder and

1198881 = 1205820119878lowast minus (1198890 + 1205792 + 1205760 + 1205830) minus

1199030

(1 + 1205720119868lowast)2

1198882 = 1205820119868lowast

1198883 = minus 1205820119878lowast + 1205760 +

1199030

(1 + 1205720119868lowast)2

1198884 = minus (1198890 + 1205792) minus 1205820119868lowast

11988811 =11990301205720

(1 + 1205720119868lowast)3

(70)

Making the change of variables 119883 = 119909 119884 = minus1205792119868lowast + 1198881119909 +

1198882119910 + 119888111199092 + 1205820119909119910 minus 1199081(119909) and rewriting 119883 119884 as 119909 and 119910

respectively we have

119889119909

119889119905= 119910

119889119910

119889119905= 1198900+ 1198901119909 + 1198902119910 + 11989011119909

2+ 11989012119909119910 + 119890221199102+ 1199082 (119909 119910 120579)

(71)

where 120579 = (1205791 1205792) 1199082(119909 119910 120579) is a smooth function of 119909 1199101205791 and 1205792 at least of the third order and

1198900 = 1198882 (1205791 minus 1205792119878lowast) + 11988841205792119868

lowast

1198901 = 1205820 (1205791 minus 1205792119878lowast) + 11988821198883 minus 11988811198884 minus 12058201205792119868

lowast

1198902 = 1198881 + 1198884 + 12058201205792119868lowast

1198882

11989011 = 12058201198883 minus 119888411988811 minus 119888111198882 + 11988811205820

11989012 = minus 1205820 + 211988811 + 1205820minus11988811198882 minus 12058201205792119868

lowast

11988822

11989022 =12058201198882

(72)

Next introduce a new time variable 120591 by 119889119905 = (1 minus(12058201198882)119909)119889120591 Rewriting 120591 as 119905 we obtain

119889119909

119889119905= 119910(1 minus

12058201198882119909)

119889119910

119889119905= (1 minus

12058201198882119909)

times (1198900 + 1198901119909 + 1198902119910 + 119890111199092

+ 11989012119909119910 + 119890221199102 + 1199082 (119909 119910 120579))

(73)

Let119883 = 119909 119884 = 119910(1minus(12058201198882)119909) and rename119883 and119884 as 119909 and119910 we have

119889119909

119889119905= 119910

119889119910

119889119905= 1198900 + 1198921119909 + 1198902119910 + 11989211119909

2 + 11989212119909119910 + 1199083 (119909 119910 120579)

(74)

where 120579 = (1205791 1205792) 1199083(119909 119910 120579) is a smooth function of 119909 1199101205791 and 1205792 at least of the third order and

1198921 = minus 2119890012058201198882+ 1198901

11989211 = 11989011 minus 2119890112058201198882+11989001205822

0

11988822

11989212 = 11989012 minus119890212058201198882

(75)

Now we assume that 11989211 = 0 and 11989212 = 0 when 120582119894 are smallSet 119909 = 119883 minus 119890211989212 and rewrite119883 as 119909 we can get

119889119909

119889119905= 119910

119889119910

119889119905= 1198910 + 1198911119909 + 11989211119909

2 + 11989212119909119910 + 1199084 (119909 119910 120579)

(76)

where 120579 = (1205791 1205792) 1199084(119909 119910 120579) is a smooth function of 119909 1199101205791 and 1205792 at least of the third order and

1198910 = 1198900 minus1198921119890211989212+119892111198902

2

119892212

1198911 = 1198921 minus211989211119890211989212

(77)

Mathematical Problems in Engineering 11

Making the final change of variables by 119883 = 11989221211990911989211 119884 =

119892312119910119892211 and 120591 = 1198921111990511989212 and then denoting them again by

119909 119910 and 119905 respectively we obtain

119889119909

119889119905= 119910

119889119910

119889119905= 1205911 (1205791 1205792) + 1205912 (1205791 1205792) 119909 + 119909

2 + 119909119910 + 1199085 (119909 119910 120579)

(78)

where 120579 = (1205791 1205792) and 1199085(119909 119910 120579) is a smooth function of 119909119910 1205791 and 1205792 at least of the third order

We substitute values of 1205720 = 1 1205820 = 14 1198890 = 1101205760 = 110 1205830 = 1100 1198600 = 537500 and 1199030 = 558 forthe above system and these values satisfy conditions (1198601) and(1198602) And we obtain the following equations

1205911 (1205791 1205792) =11989101198924

12

119892311

= minus237291605

851841205791 +

25485118377

8518401205792

+ 119874 (10038161003816100381610038161205791 1205792

10038161003816100381610038162)

1205912 (1205791 1205792) =11989111198922

12

119892211

= minus172225

19361205791 +

3983253

38721205792 + 119874 (

10038161003816100381610038161205791 120579210038161003816100381610038162)

(79)

where 11989211 = minus11500 minus (116)1205791 + (341800)1205792 + (14)1205792

2lt 0

and 11989212 = minus83200 + (14)1205792 lt 0 in a small neighborhood of(1205791 1205792) = (0 0) Let

119869 = (

12059712059111205971205791

12059712059111205971205792

12059712059121205971205791

12059712059121205971205792

) (80)

And after simple calculation we obtain that

det(119869)|120582=0 = minus16839398748825

82458112= 0 (81)

Thus 1205911 and 1205912 are regular maps in a small neighborhoodof (1205791 1205792) = (0 0) By the Bogdanov and Takens bifurcationtheorems [36] we obtain the following conclusion

Theorem 13 Suppose that1198600 1198890 1205820 1205760 1199030 1205720 and 1205830 satisfy(1198601) (1198602) 11989211 = 0 and 11989212 = 0 when 120579119894 are small Then (4)admits the following bifurcation behavior

(1) There is a saddle-node bifurcation curve 119878119873 =

(1205791 1205792) 4119891011989211 = 11989121+ 119900(|(1205791 1205792)|

2) 1198911 =0 = (1205791 1205792) (59076250)1205792 minus (11125)1205791 +(2313400)12057911205792 minus (516)120579

2

1minus (102863740000)1205792

2+

119900(|(1205791 1205792)|2) = 0 (4799116600)1205792 minus (14)1205791 + (25

83)12057911205792 minus (1364513778)1205792

2+ 119900(|(1205791 1205792)|

2) = 0

I

I

O

O 1205791

1205792

SN+

SN+

SNminus

SNminus

II H

III HL

IVIVHL

IIIII

H

Figure 6The bifurcation set and the corresponding phase portraitsfor system (68)

(2) There is a Hopf bifurcation curve 119867 = (1205791 1205792)

1198910 + 119900(|(1205791 1205792)|2) = 0 1198911 lt 0 = (1205791 1205792) minus(537

50)1205792+1205791 + (5083)12057911205792 minus (746676889)12057922 + 119900(|(12057911205792)|2) = 0 (4799116600)1205792 minus (14)1205791+(2583)12057911205792 minus

(1364513778)12057922+ 119900(|(1205791 1205792)|

2) lt 0

(3) There is a homoclinic bifurcation curve119867119871 = (1205791 1205792) 25119892111198910 + 6119891

2

1= 119900(|(1205791 1205792)|

2) 1198911 lt 0 = (1205791 1205792) (59071000)1205792 minus (1120)1205791 + (611979 33200)12057911205792 minus(1916)1205792

1minus (16075834839275560000)1205792

2+ 119900(|(1205791

1205792)|2) = 0 (4799116600)1205792minus(14)1205791+(2583)12057911205792minus

(1364513778)12057922+ 119900(|(1205791 1205792)|

2) lt 0

6 Numerical Simulations

When 120572 = 1 120582 = 14 119889 = 110 120576 = 110 120583 = 1100119860 = 537500 and 119903 = 558 120590 gt 0 By applying PPLANE8and Photoshop software the (1205791 1205792)-plane near the origin isdivided into 4 regions by these bifurcation curves as shown inFigure 6 Fix 1205791 lt 0 and decrease 1205792 from 0 our conclusionsare summarized as follows

(a) When (1205791 1205792) lies in region I there is no positiveequilibrium which implies that the positive orbitsof (4) meet the positive 119878-axis in finite time andtherefore the disease disappears

(b) There is a saddle-node point when (1205791 1205792) lies oncurve SN

(c) When (1205791 1205792) crosses SN into region II two positiveequilibria which are an unstable focus and a saddleappear

(d) When the parameters lie on the curve H there is alsoa saddle and an unstable focus An unstable limit cycleappears with the parameters crossing H into III

(e) A homoclinic cycle appears as the parameters passingIII into HL through the homoclinic bifurcation

12 Mathematical Problems in Engineering

0 5 10 15 20 25 30 35 40

0

1

2

3

4

5

6

7

8

9

10

S

I

Figure 7 When (1205791 1205792) = (0 0) the unique positive equilibrium isa cusp of codimension 2

3 35 45 55 65 754 5 6 7 8

1

2

3

4

5

6

7

8

S

I

Figure 8 When (1205791 1205792) = (minus01 minus00093) lies in region I there isno positive equilibrium

(f) The homoclinic loop breakswith the parameter cross-ing HL into IV and a saddle and a stable focus appear

Furthermore using PPLANE8 some numerical simu-lations of system (68) are depicted in Figures 7ndash10 When(1205791 1205792) = (0 0) that is (1198600 1198890) = (537500 110) there isa unique positive equilibrium (119878lowast 119868lowast) = (31750 4) whichis a cusp of codimension 2 (Figure 7) When (1205791 1205792) =(minus01 minus00093) lies in region I there is no positive equi-librium (Figure 8) When (1205791 1205792) = (minus002 minus0001863798)there is a homoclinic loop (Figure 9) When (1205791 1205792) =(minus002 minus00020) lies in region IV the homoclinic loopis broken and there is a stable focus and a saddle(Figure 10)

6 605 61 615 62 625 63 635 64 645 653536373839

44142434445

S

I

Figure 9 When (1205791 1205792) = (minus002 minus0001863798) there is ahomoclinic loop

0 1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

7

8

9

10

S

I

Figure 10 When (1205791 1205792) = (minus002 minus00020) lies in region IV thereis a stable focus and a saddle

7 Discussion

In this paper we focus on the bifurcation analysis of an SISepidemic model with bilinear incidence rate and saturatedtreatment Generally speaking in many epidemic modelsthe basic reproduction number which is the key conceptin epidemiology can be decreased below unity to eradicatethe disease However in our model the basic reproductionnumber below 1 is not enough to eradicate the diseaseAccording to our analysis in this paper we demonstrate notonly the global stability of the disease-free equilibrium when1198770 lt 119877

lowast

0and the local asymptotic stability of the endemic

equilibrium 1198642 when 119877lowast

0lt 1198770 lt 1 but also the global

stability of the unique endemic equilibrium 119864lowast when 1198770 gt 1Moreover it has been shown in Corollary 4 that backwardbifurcations occur if the effect of the infected being delayedfor treatment is strong That is to say we should get prompttreatment for patients Through Figure 4 we can see thatthere is a region such that the disease will persist if the initialposition lies in the region and disappear if the initial positionlies outside this region So in order to restrain the spread

Mathematical Problems in Engineering 13

of the disease governments should take timely measures tocontrol the initial infected individuals in a lower level Inaddition fromTheorem 8 we can see that the disease will dieout if 1198770 is small enough Therefore from the definition of1198770 one knows that we can decrease the incidence rate 120582 andincrease the cure rate 119903 so as to eradicate the diseases or tomake them controlled in a lower endemic steady state

The stability analysis of the model equilibria enables usto completely analyze their local bifurcation behavior suchas Hopf saddle-node and Bogdanov-Takens bifurcation Bycomputing the first Lyapunov coefficient we can determinethat the Hopf bifurcation is a supercritical Hopf bifurca-tion or a subcritical Hopf bifurcation We also show thatunder assumptions (1198601) and (1198602) the model undertakesBogdanov-Takens bifurcation that is there are saddle-nodebifurcation Hopf bifurcation and homoclinic bifurcationin the system The normal norm of the Bogdanov-Takensbifurcation is derived in Section 5 which is very helpful toobtain the three kinds of bifurcation curves By analyticaltechniques 119860 and 119889 are chosen as bifurcation parametersand other parameters are fixed and we easily get a clearpicture about the rich dynamics behaviors of our modelThrough studying the bifurcations of the SIS epidemicmodelwe are suggesting that in order to eradicate the diseasemore medical facilities as well as medical professionals areneeded and the medical standard needs to be improved aswell

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research was supported by the National Natural ScienceFoundation of China (no 11001041 no 10926105 no 11101283no 11271065 and no 11201360) SRFDP (no 200802001008)and Scientific Research Innovation Project of ShanghaiMunicipal Education Commission (12YZ173)

References

[1] M E Alexander and S M Moghadas ldquoPeriodicity in an epi-demic model with a generalized non-linear incidencerdquo Mathe-matical Biosciences vol 189 no 1 pp 75ndash96 2004

[2] M E Alexander and S M Moghadas ldquoBifurcation analysisof SIRS epidemic model with generalized incidencerdquo SIAMJournal on Applied Mathematics vol 65 no 5 pp 1794ndash18162005

[3] J Arino C C McCluskey and P van den Driessche ldquoGlobalresults for an epidemic model with vaccination that exhibitsbackward bifurcationrdquo SIAM Journal on Applied Mathematicsvol 64 no 1 pp 260ndash276 2003

[4] W R Derrick and P van den Driessche ldquoHomoclinic orbitsin a disease transmission model with nonlinear incidence andnonconstant populationrdquo Discrete and Continuous DynamicalSystems B vol 3 no 2 pp 299ndash309 2003

[5] F Brauer ldquoBackward bifurcations in simple vaccination mod-elsrdquo Journal of Mathematical Analysis and Applications vol 298no 2 pp 418ndash431 2004

[6] K P Hadeler and P van den Driessche ldquoBackward bifurcationin epidemic controlrdquo Mathematical Biosciences vol 146 no 1pp 15ndash35 1997

[7] Z X Hu S Liu and H Wang ldquoBackward bifurcation of anepidemic model with standard incidence rate and treatmentraterdquo Nonlinear Analysis Real World Applications vol 9 no 5pp 2302ndash2312 2008

[8] J Hui and D M Zhu ldquoGlobal stability and periodicity onSIS epidemic models with backward bifurcationrdquo Computers ampMathematics with Applications vol 50 no 8-9 pp 1271ndash12902005

[9] Y Jin W D Wang and S W Xiao ldquoAn SIRS model with anonlinear incidence raterdquoChaos Solitons amp Fractals vol 34 no5 pp 1482ndash1497 2007

[10] G H Li and W D Wang ldquoBifurcation analysis of an epidemicmodel with nonlinear incidencerdquo Applied Mathematics andComputation vol 214 no 2 pp 411ndash423 2009

[11] X Z Li W S Li and M Ghosh ldquoStability and bifurcation ofan SIS epidemic model with treatmentrdquo Chaos Solitons ampampFractals vol 42 no 5 pp 2822ndash2832 2009

[12] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1986

[13] X B Liu and L J Yang ldquoStability analysis of an SEIQV epidemicmodel with saturated incidence raterdquo Nonlinear Analysis RealWorld Applications vol 13 no 6 pp 2671ndash2679 2012

[14] M Martcheva and H R Thieme ldquoProgression age enhancedbackward bifurcation in an epidemic model with super-infectionrdquo Journal of Mathematical Biology vol 46 no 5 pp385ndash424 2003

[15] S G Ruan and W D Wang ldquoDynamical behavior of anepidemic model with a nonlinear incidence raterdquo Journal ofDifferential Equations vol 188 no 1 pp 135ndash163 2003

[16] W D Wang and S G Ruan ldquoBifurcation in an epidemicmodel with constant removal rate of the infectivesrdquo Journal ofMathematical Analysis and Applications vol 291 no 2 pp 775ndash793 2004

[17] Y L Tang D Q Huang S G Ruan and W N ZhangldquoCoexistence of limit cycles and homoclinic loops in a SIRSmodel with a nonlinear incidence raterdquo SIAM Journal onApplied Mathematics vol 69 no 2 pp 621ndash639 2008

[18] P van den Driessche and J Watmough ldquoA simple SIS epidemicmodel with a backward bifurcationrdquo Journal of MathematicalBiology vol 40 no 6 pp 525ndash540 2000

[19] J J Wei and J A Cui ldquoDynamics of SIS epidemic model withthe standard incidence rate and saturated treatment functionrdquoInternational Journal of Biomathematics vol 5 no 3 Article ID1260003 18 pages 2012

[20] W D Wang ldquoBackward bifurcation of an epidemic model withtreatmentrdquoMathematical Biosciences vol 201 no 1-2 pp 58ndash712006

[21] Z W Wang ldquoBackward bifurcation in simple SIS modelrdquo ActaMathematicae Applicatae Sinica vol 25 no 1 pp 127ndash136 2009

[22] L Xue and C Scoglio ldquoThe network level reproduction numberfor infectious diseases with both vertical and horizontal trans-missionrdquo Mathematical Biosciences vol 243 no 1 pp 67ndash802013

14 Mathematical Problems in Engineering

[23] X Zhang and X N Liu ldquoBackward bifurcation of an epidemicmodel with saturated treatment functionrdquo Journal ofMathemat-ical Analysis and Applications vol 348 no 1 pp 433ndash443 2008

[24] J Guckenheimer and P J Holmes Nonlinear OscillationsDynamical Systems and Bifurcation of Vector Fields SpringerNew York NY USA 1996

[25] X Zhang and X N Liu ldquoBackward bifurcation and globaldynamics of an SIS epidemic model with general incidence rateand treatmentrdquoNonlinearAnalysis RealWorldApplications vol10 no 2 pp 565ndash575 2009

[26] M A Safi A B Gumel and E H Elbasha ldquoQualitative analysisof an age-structured SEIR epidemic model with treatmentrdquoApplied Mathematics and Computation vol 219 no 22 pp10627ndash10642 2013

[27] L H Zhou and M Fan ldquoDynamics of an SIR epidemic modelwith limited medical resources revisitedrdquo Nonlinear AnalysisReal World Applications vol 13 no 1 pp 312ndash324 2012

[28] X Y Zhou and J A Cui ldquoAnalysis of stability and bifurcation foran SEIR epidemicmodel with saturated recovery raterdquoCommu-nications inNonlinear Science andNumerical Simulation vol 16no 11 pp 4438ndash4450 2011

[29] D Lacitignola ldquoSaturated treatments and measles resurgenceepisodes in South Africa a possible linkagerdquo MathematicalBiosciences and Engineering MBE vol 10 no 4 pp 1135ndash11572013

[30] J C Eckalbar and W L Eckalbar ldquoDynamics of an epidemicmodel with quadratic treatmentrdquo Nonlinear Analysis RealWorld Applications vol 12 no 1 pp 320ndash332 2011

[31] F Brauer ldquoBackward bifurcations in simple vaccinationtreat-ment modelsrdquo Journal of Biological Dynamics vol 5 no 5 pp410ndash418 2011

[32] M W Hirsch S Smale and R L Devaney Differential Equa-tions Dynamical Systems and an Introduction to Chaos Aca-demic Press New York NY USA 2003

[33] R Bogdanov ldquoBifurcations of a limit cycle for a family of vectorfields on the planrdquo SelectaMathematica vol 1 pp 373ndash388 1981

[34] R Bogdanov ldquoVersal deformations of a singular point on theplan in the case of zero eigen-valuesrdquo Selecta MathematicaSovietica vol 1 pp 389ndash421 1981

[35] F Takens ldquoForced oscillations and bifurcationrdquo in Applicationsof Global Analysis I vol 3 pp 1ndash59 Communications of theMathematical Institute Rijksuniversiteit Utrecht 1974

[36] Y A Kuznetsov Elements of Applied Bifurcation TheorySpringer New York NY USA 2nd edition 1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

We obtain

det (119872) = 119868

(1 + 120572119868)2[120582 (119889 + 120583) (1 + 120572119868)

2 minus 119903120572119889] (23)

In fact there is 120582(119889 + 120583)(1 + 120572119868)2 gt 120582119889 Since 1198770 gt 1 and0 le 120572 lt 120582119903 we have

120582 (119889 + 120583) (1 + 120572119868)2 minus 119903120572119889 gt 120582119889 minus 119903120572119889 gt 0 (24)

So we get det(119872) gt 0 The trace of119872 is given by

tr (119872) = 1

(1 + 120572119868)2[minus (119889 + 120582119868) (1 + 120572119868)

2 + 119903120572119868] (25)

In the same way as the above calculation of (24) we have

minus (119889 + 120582119868) (1 + 120572119868)2 + 119903120572119868 lt minus120582119868 + 119903120572119868 lt 0 (26)

So we get tr(119872) lt 0 The proof is complete

Now we consider the case that there are two endemicequilibria 1198641 and 1198642 let 119872119894 be the Jacobian matrix at 119864119894119894 = 1 2

Theorem 6 The endemic equilibrium 1198641 is a saddle wheneverit exists

Proof Since 1198681 = (minus119887 minus radic1198872 minus 4119886119888)2119886 and Δ = 1198872 minus 4119886119888 wehave 1198681 = (minus119887 minus radicΔ)2119886 Thus

det (1198721)

=1198681

(1 + 1205721198681)2[120582 (119889 + 120583) (1 + 1205721198681)

2minus 119903120572119889]

≜1198681

(1 + 1205721198681)2120595 (1198681)

(27)

From (1198673) of Theorem 2 we can know that if 120572 gt 0 119887 lt 0and 119877lowast

0lt 1198770 lt 1 then 1198641 exists Hence we can get 120595(0) =

120582(119889 + 120583) minus 119903120572119889 lt 0 and 1205951015840(1198681) = 2120582(119889 + 120583)(1 + 1205721198681)120572 gt 0 Itfollows that there exists a unique 119868lowast gt 0 such that

120595 (1198681) = 0 when 1198681 = 119868lowast

120595 (1198681) lt 0 when 0 lt 1198681 lt 119868lowast

120595 (1198681) gt 0 when 1198681 gt 119868lowast

(28)

where

119868lowast = radic119903119889

120572120582 (119889 + 120583)minus1

120572 (29)

On the other hand

1198681 = minus119887

2119886minusradicΔ

2119886

=120572120582119860 minus 120582 (119889 + 120583) minus 120572119889 (119889 + 120576 + 120583)

2120572120582 (119889 + 120583)minusradicΔ

2119886

= radic119903119889

120572120582 (119889 + 120583)minus1

120572+1

120572minus radic

119903119889

120572120582 (119889 + 120583)

+120572120582119860 minus 120582 (119889 + 120583) minus 120572119889 (119889 + 120576 + 120583)

2120572120582 (119889 + 120583)minusradicΔ

2119886

= 119868lowast + ([120582 (119889 + 120583) + 120572120582119860 minus 2radic119903119889120572120582 (119889 + 120583)

minus 120572119889 (119889 + 120576 + 120583)] minus radicΔ)times (2120572120582 (119889 + 120583))minus1

(30)

Δ = 120582 (119889 + 120583) minus [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]2

minus 4120572120582 (119889 + 120583) [119889 (119889 + 120576 + 119903 + 120583) minus 120582119860]

= 120582 (119889 + 120583) + [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

minus 2radic119903119889120572120582(119889 + 120583)2

minus 4119903119889120572120582 (119889 + 120583)

minus 4120582 (119889 + 120583) [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

+ 4radic119903119889120572120582 (119889 + 120583) 120582 (119889 + 120583) + [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

minus 4120572120582 (119889 + 120583) [minus120582119860 + 119889 (119889 + 120576 + 119903 + 120583)]

≜ 120582 (119889 + 120583) + [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

minus 2radic119903119889120572120582 (119889 + 120583)2

+ 119875

(31)

where

119875 = minus8119903119889120572120582 (119889 + 120583) + 4radic119903119889120572120582 (119889 + 120583)

times 120582 (119889 + 120583) + [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

(32)

In the following we will show that 119875 gt 0 Since 119877lowast0lt 1198770 we

have

4120572120582 (119889 + 120583)

[120582 (119889 + 120583) + 120572119889 (119889 + 120576 + 120583) minus 120572120582119860]2+ 41205721205822 (119889 + 120583)119860

lt1

119889 (119889 + 120576 + 119903 + 120583)

(33)

6 Mathematical Problems in Engineering

that is

[120582 (119889 + 120583) + 120572119889 (119889 + 120576 + 120583) minus 120572120582119860]2

gt 4120572120582 (119889 + 120583) 119889 (119889 + 120576 + 119903 + 120583) minus 41205721205822 (119889 + 120583)119860(34)

Therefore

120582 (119889 + 120583) + [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]2

= 120582 (119889 + 120583) minus [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]2

+ 4120582 (119889 + 120583) [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

gt 4120572120582 (119889 + 120583) 119889 (119889 + 120576 + 119903 + 120583) minus 41205721205822 (119889 + 120583)119860

+ 4120582 (119889 + 120583) [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

= 4119903119889120572120582 (119889 + 120583)

(35)

Obviously we have 119875 gt 0 From (30) one has 1198681 lt 119868lowast So

we get det(1198721) lt 0 Hence the endemic equilibrium 1198641 is asaddle The proof is complete

In order to explore the stability of the endemic equilib-rium 1198642 define

1198981 = (21198891205721198862 + 1205821198862 minus 1205761205721198862 minus 1198861198881205821205722)

minus 119887 (2120582120572119886 + 1198891205722119886 minus 1198871205821205722)

1198982 = 1198862119889 minus 119888 (2120582120572119886 + 1198891205722119886 minus 1198871205821205722)

(36)

Theorem 7 If 120578 gt 0 then endemic equilibrium 1198642 is locallyasymptotically stable if 120578 lt 0 then endemic equilibrium 1198642 isunstable where 120578 = 21198861198982 + 1198981(radic1198872 minus 4119886119888 minus 119887)

Proof Consider

det (1198722) =1198682

(1 + 1205721198682)2[120582 (119889 + 120583) (1 + 1205721198682)

2minus 120576120572119889]

=1198682

(1 + 1205721198682)2times 120595 (1198682)

(37)

By carrying out arguments similar to that of Theorem 6 wehave 1198682 gt 119868

lowast Therefore det(1198722) gt 0 In addition we have

tr (1198722) = minus(119889 + 1205821198682) (1 + 1205721198682)

2minus 1205761205721198682

(1 + 1205721198682)2

= minus120582120572211986832+ (2120582120572 + 1198891205722) 1198682

2+ (2119889120572 + 120582 minus 120576120572) 1198682 + 119889

(1 + 1205721198682)2

(38)

and then sgn(tr(1198722)) = minus sgn(119866(1198682)) where

119866 (119909) = 12058212057221199093 + (2120582120572 + 1198891205722) 1199092 + (2119889120572 + 120582 minus 120576120572) 119909 + 119889

(39)

I

0 50 100 1500

5

10

15

20

25

30

35

40

S

E0

E1

E2

Figure 4 One region of disease persistence and one region ofdisease extinction when 119860 = 10 120572 = 1 119889 = 01 120583 = 001 120582 = 001119903 = 01 and 120576 = 01

Using the expression of1198981 = (21198891205721198862 +1205821198862 minus 1205761205721198862 minus1198861198881205821205722) minus

119887(2120582120572119886+1198891205722119886minus1198871205821205722) and1198982 = 1198862119889minus119888(2120582120572119886+1198891205722119886minus1198871205821205722)

one has

119866 (1198682) = (1198861198682

2+ 1198871198682 + 119888) 1205930 +

11989811198682 + 11989821198862

(40)

where1205930 is a first degree polynomial of 1198682 Since 1198861198682

2+1198871198682+119888 =

0 sgn(tr(1198722)) = minus sgn(119866(1198682)) = minus sgn(11989811198682 + 1198982) From theexpression of 1198682 we have

sgn (11989811198682 + 1198982)

= sgn (21198861198982 + 1198981 (radic1198872 minus 4119886119888 minus 119887)) ≜ sgn (120578) (41)

Thus 1198642 is locally asymptotically stable if 120578 gt 0 and 1198642 isunstable if 120578 lt 0 The proof is complete

From the above discussion we have the following con-clusion If two endemic equilibria 1198641 and 1198642 exist the stablemanifolds of the saddle 1198641 split 119877

2

+into two regions The

disease is persistent in the upper region and dies out in thelower region (see Figure 4)

3 Global Analysis

Firstly we consider the global stability of the disease-freeequilibrium 1198640 Let 119873 = 119878 + 119868 be the total population sizeNow we note that the equation for total population is givenby 119889119873119889119905 = 119860 minus 119889119878 minus (119889 + 120583)119868 le 119860 minus 119889119873 It follows thatlim119905rarr+infin119873(119905) le 119860119889 Let

R = (119878 119868) isin 1198772

+ 119878 + 119868 le

119860

119889 (42)

which is positive invariant with respect to system (4)

Theorem 8 If 1198770 lt 119877lowast0 the disease-free equilibrium 1198640(119860119889 0) is globally asymptotically stable that is the disease diesout

Mathematical Problems in Engineering 7

Proof Suppose 1198770 lt 119877lowast0 From the (1198673) of Theorem 2 weknow that the model has no endemic equilibrium Fromthe corollary of Poincare-Bendixson theorem [32] we knowthat there is no periodic orbit in R as there is a disease-free equilibrium in R Since R is a bounded positivelyinvariant region and 1198640 is the only equilibrium in R thelocal stability of 1198640 implies that every solution initiating inR approaches 1198640 Thus the disease-free equilibrium 1198640 isglobally asymptotically stable The proof is complete

Now we analyze the global dynamics of the endemicequilibrium when 1198770 gt 1

Theorem 9 If 1198770 gt 1 and 0 le 120572 lt 120582119903 the system (4) has nolimit cycles

Proof We use Dulac theorem to exclude the limit cycle Let

119875 (119878 119868) = 119860 minus 119889119878 minus 120582119878119868 + 120576119868 +119903119868

1 + 120572119868

119876 (119878 119868) = 120582119878119868 minus (119889 + 120576 + 120583) 119868 minus119903119868

1 + 120572119868

(43)

and take the Dulac function

119863 =1

119868 (44)

According to 0 le 120572 lt 120582119903 we can get

120597 (119875119863)

120597119878+120597 (119876119863)

120597119868

= minus119889

119868minus 120582 +

120572119903

(1 + 120572119868)2

lt minus119889

119868minus 120582 +

120582

(1 + 120572119868)2

=1

119868 (1 + 120572119868)2minus119889 (1 + 120572119868)

2 minus 120582119868 [(1 + 120572119868)2 minus 1]

lt 0

(45)

Hence the system (4) has no limit cycles The proof iscomplete

Therefore we obtain the global result of the uniqueendemic equilibrium

Theorem 10 If 1198770 gt 1 and 0 le 120572 lt 120582119903 the unique endemicequilibrium 119864lowast is globally asymptotically stable (see Figure 5)

4 Hopf Bifurcation

In this section we study the Hopf bifurcation of system (4)From the above discussion we know that there is no closedorbit surrounding1198640 or1198641 because the 119878-axis is invariantwith

0 50 100 1500

50

100

150

S

I Elowast

Figure 5The unique endemic equilibrium 119864lowast is globally asymptot-ically stable when 119860 = 15 120572 = 001 119889 = 01 120583 = 001 120582 = 001119903 = 08 and 120576 = 01

respect to system (4) and 1198641 is always a saddle ThereforeHopf bifurcation can only occur at 1198642 Set

120590 = [minus120582120572 minus1198881 (1198882 + 21198884)

11988612]1198634

+ [minus 120582120572119886211minus 21205821205721198861111988612 + (1198882 minus

211988611119888111988612)

times (119886111198882 minus119886211119888111988612)

minus (21198862111198881

11988612minus 119886111198882 + 2119888411988611 minus 119888311988612)

times (1198884 minus11988611119888111988612)]1198632

+ 11988611 (119886211119888111988612minus 119886111198882 + 119888411988611 minus 119888311988612)

times (119886121198883 minus 2119886111198884 minus 119886111198882)

(46)

where 119886119894119895 (119894 119895 = 1 2) 119888119896 (119896 = 1 2 3 4) and 119863 are defined by(50) and (52)

Theorem 11 System (4) undergoes a Hopf bifurcation if 120578 = 0Moreover if 120590 lt 0 there is a family of stable period orbits ofsystem (4) as 120578 decreases from 0 that is a supercritical Hopfbifurcation occurs if 120590 gt 0 there is a family of unstable periodorbits of system (4) as 120578 increases from 0 that is a subcriticalHopf bifurcation occurs

Proof The proof of Theorem 7 shows that tr(1198722) = 0 if andonly if 120578 = 0 and det(1198722) gt 0 when 1198642 exists Therefore theeigenvalues of 1198722 are a pair of pure imaginary roots if andonly if 120578 = 0 The direct calculations show that

119889 (tr (1198722))119889120578

100381610038161003816100381610038161003816100381610038161003816120578=0= minus

1

21198863 (1 + 1205721198682)2lt 0 (47)

8 Mathematical Problems in Engineering

ByTheorem 342 in [24] 120578 = 0 is the Hopf bifurcation pointfor system (4)

To be concise in notations rescale (4) by 120591 = 119905(1 + 120572119868)For simplicity we still use 119905 instead of 120591 Then we obtain

119889119878

119889119905= (119860 minus 119889119878 minus 120582119878119868 + 120576119868) (1 + 120572119868) + 119903119868

119889119868

119889119905= 120582119878119868 (1 + 120572119868) minus (119889 + 120576 + 120583) 119868 (1 + 120572119868) minus 119903119868

(48)

Let 119909 = 119878 minus 1198782 and 119910 = 119868 minus 1198682 then (48) becomes

119889119909

119889119905= 11988611119909 + 11988612119910 + 1198881119910

2 + 1198882119909119910 minus 1205821205721199091199102

119889119910

119889119905= 11988621119909 + 11988622119910 + 1198883119909119910 + 1198884119910

2 + 1205821205721199091199102

(49)

where

11988611 = minus (119889 + 1205821198682) (1 + 1205721198682)

11988612 = (minus1205821198782 + 119903) (1 + 1205721198682)

+ 120572 (119860 minus 1198891198782 minus 12058211987821198682 + 1199031198682) + 120576

11988621 = 1205821198682 (1 + 1205721198682)

11988622 = [1205821198782 minus (119889 + 119903 + 120583)] (1 + 1205721198682)

+ 120572 [12058211987821198682 minus (119889 + 119903 + 120583) 1198682] minus 120576

1198881 = 120572 (minus1205821198782 + 119903)

1198882 = 120572 (minus119889 minus 1205821198682) minus 120582 (1 + 1205721198682)

1198883 = 120582 (1 + 1205721198682) + 1205821198682120572

1198884 = [1205821198782 minus (119889 + 119903 + 120583)] 120572

(50)

Let 119864lowast denote the origin of 119909-119910 plane Since 1198642 = (1198782 1198682)satisfies (6) we obtain

det (119872 (119864lowast))

= minus (119889 + 1205821198682) 1205721205761198682 + (119889 + 120583) (1 + 1205721198682

2)2

1205821198682 + 1205721205821205761198682

2

= 1198682 times 120595 (1198682)

(51)

From the proof of Theorem 7 it follows that 120595(1198682) is alwayspositive It is easy to verify that 11988611 + 11988622 = 0 if and only if120578 = 0 Set

119863 = radicdet (119872 (119864lowast)) (52)

and let 119906 = minus119909 and V = (11988611119863)119909+(11988612119863)119910 then the normalform of (48) for Hopf bifurcation reads

119889119906

119889119905= minus119863V + 119891 (119906 V)

119889V119889119905= 119863119906 + 119892 (119906 V)

(53)

where

119891 (119906 V) = (11988611119886121198882 minus119886211

119886212

1198881)1199062 minus11986321198881119886212

V2

+ (119863119888211988612minus 2119863119886111198881119886212

)119906V minus120582120572119886211

119886212

1199063

minus 211986311988611120582120572

119886212

1199062V minus1198632120582120572

119886212

119906V2

119892 (119906 V) =11988611119863(119886211

119886212

1198881 minus11988611119886121198882 +11988611119886121198884 minus 1198883)119906

2

+ (119863119886111198881119886212

+119863119888411988612) V2

+ (2119886211

119886212

1198881 minus11988611119886121198882 + 2

11988611119886121198884 minus 1198883)119906V

+120582120572119886211

11988612119863(1

11988612minus 1)1199063 +

21205821205721198861111988612

(1198861111988612minus 1)1199062V

+119863120582120572

11988612(1198861111988612minus 1)119906V2

(54)

Now we evaluate the first Lyapunov coefficient Γ of system(4) as follows

Γ =1

16[119891119906119906119906 + 119891119906VV + 119892119906119906V + 119892VVV] +

1

16119863

times [119891119906V (119891119906119906 + 119891VV) minus 119892119906V (119892119906119906 + 119892VV) minus 119891119906119906119892119906119906 + 119891VV119892VV]

(55)

where 119891119906V denotes (1205972119891120597119906120597V)(0 0) and so forth Then

Γ =1

81198862121198632[minus120582120572 minus

1198881 (1198882 + 21198884)

11988612]1198634

+ [minus 120582120572119886211minus 21205821205721198861111988612 + (1198882 minus

211988611119888111988612)

times (119886111198882 minus119886211119888111988612)

minus (21198862111198881

11988612minus 119886111198882 + 2119888411988611 minus 119888311988612)

times (1198884 minus11988611119888111988612)]1198632

+ 11988611 (119886211119888111988612minus 119886111198882 + 119888411988611 minus 119888311988612)

times (119886121198883 minus 2119886111198884 minus 119886111198882)

=120590

81198862121198632

(56)

Mathematical Problems in Engineering 9

Obviously the sign of Γ is determined by 120590 ByTheorem 342and (3411) in [24] the rest of the claims in Theorem 11 arevalid The proof is complete

5 Bogdanov-Takens Bifurcations

The purpose of this section is to study the Bogdanov-Takensbifurcation of (4) Now we assume the following

(1198601) 1198872 minus 4119886119888 = 0

(1198602) 21198861198892 + 119887120582120583 = 0

Then (6) admits a unique positive equilibrium 119864lowast = (119878lowast 119868lowast)where

119868lowast = minus119887

2119886 119878lowast =

119860 minus (119889 + 120583) 119868lowast

119889 (57)

The Jacobian matrix of system (4) at 119864lowast is

119872lowast = (

minus119889 minus 120582119868lowast minus120582119878lowast + 120576 +119903

(1 + 120572119868lowast)2

120582119868lowast 120582119878lowast minus (119889 + 120576 + 120583) minus119903

(1 + 120572119868lowast)2)

(58)

By (58) we have

det (119872lowast) = 119868lowast

(1 + 120572119868lowast)2[(119889 + 120583) 120582 (1 + 120572119868lowast)

2minus 119903119889120572]

= 0

(59)

because of

(1 + 120572119868lowast)2=41198862 minus 4119886120572119887 + 12057221198872

41198862

=41198862 minus 4119886120572119887 + 12057224119886119888

41198862

=119886 minus 120572119887 + 1205722119888

119886

=1205722119903119889

119886

(60)

Furthermore (1198602) implies that

tr (119872lowast) = 0 (61)

Thus (1198601) and (1198602) imply that the Jacobian matrix has azero eigenvalue withmultiplicity 2This suggests that (4)mayadmit a Bogdanov-Takens singularity

Theorem 12 Suppose that (1198601) (1198602) and 21198871 + 1198874 = 0 holdThen the endemic equilibrium 119864lowast = (119878lowast 119868lowast) of (4) is a cuspof codimension 2 that is it is a Bogdanov-Takens singularityHere 1198871 and 1198874 are defined by (65)

Proof Using the transformation of 119909 = 119868minus 119868lowast and 119910 = 119878minus 119878lowastsystem (4) becomes

119889119909

119889119905= 1198861119909 + 1198862119910 + 120582119909119910 + 11988611119909

2 minus 1198751 (119909)

119889119910

119889119905= minus11988621

1198862119909 minus 1198861119910 minus 120582119909119910 minus 11988611119909

2 + 1198751 (119909)

(62)

where 1198751(119909) is a smooth function of 119909 at least of the thirdorder and

1198861 = 120582119878lowast minus (119889 + 120576 + 120583) minus

119903

(1 + 120572119868lowast)2gt 0

1198862 = 120582119868lowast gt 0

11988611 =119903120572

(1 + 120572119868lowast)3gt 0

(63)

Set119883 = 119909 119884 = 1198861119909 + 1198862119910 Then (62) is transformed into

119889119883

119889119905= 119884 + 1198871119883

2 + 1198872119883119884 + 1198761 (119883)

119889119884

119889119905= 1198873119883

2 + 1198874119883119884 + 1198762 (119883)

(64)

where 119876119894(119883) are smooth functions of 119883 at least of the thirdorder and

1198871 = 11988611 minus12058211988611198862

1198872 =120582

1198862

1198873 = 119886111988611 minus12058211988621

1198862+ 1198861120582 minus 119886211988611

1198874 =12058211988611198862minus 120582

(65)

In order to obtain the canonical normal form we perform thetransformation of variables by

119906 = 119883 minus119887221198832 V = 119884 + 1198871119883

2 (66)

Then we obtain

119889119906

119889119905= V + 1198771 (119906)

119889V119889119905= 11988731199062 + (21198871 + 1198874) 119906V + 1198772 (119906)

(67)

where 119877119894(119906) are smooth functions of 119906 at least of the thirdorder Note that 1198873 lt 0 and 21198871 + 1198874 = 0 It follows from [33ndash35] that (4) admits a Bogdanov-Takens bifurcation

In the following wewill find the versal unfolding in termsof the original parameters in (4) In this way we will know theapproximate saddle-node Hopf and homoclinic bifurcation

10 Mathematical Problems in Engineering

curvesWe choose119860 and 119889 as bifurcation parameters Fix 120582 =1205820 120576 = 1205760 119903 = 1199030 120572 = 1205720 and 120583 = 1205830 Let 119860 = 1198600 + 1205791 and119889 = 1198890 + 1205792 where 1205791 and 1205792 are parameters which vary in asmall neighborhood of the origin

Suppose that 119860 = 1198600 119889 = 1198890 120582 = 1205820 120576 = 1205760 119903 = 1199030 120572 =1205720 and 120583 = 1205830 satisfy (1198601) and (1198602) Consider the followingsystem

119889119868

119889119905= 1205820119878119868 minus (1198890 + 1205792 + 1205760 + 1205830) 119868 minus

1199030119868

1 + 1205720119868

119889119878

119889119905= 1198600 + 1205791 minus (1198890 + 1205792) 119878 minus 1205820119878119868 + 1205760119868 +

1199030119868

1 + 1205720119868

(68)

By the transformations of 119909 = 119868 minus 119868lowast and 119910 = 119878 minus 119878lowast system(68) becomes

119889119909

119889119905= minus 1205792119868

lowast + 1198881119909 + 1198882119910 + 119888111199092 + 1205820119909119910 minus 1199081 (119909)

119889119910

119889119905= (1205791 minus 1205792119878

lowast) + 1198883119909 + 1198884119910 minus 119888111199092 minus 1205820119909119910 + 1199081 (119909)

(69)

where 1199081(119909) is a smooth function of 119909 at least of the thirdorder and

1198881 = 1205820119878lowast minus (1198890 + 1205792 + 1205760 + 1205830) minus

1199030

(1 + 1205720119868lowast)2

1198882 = 1205820119868lowast

1198883 = minus 1205820119878lowast + 1205760 +

1199030

(1 + 1205720119868lowast)2

1198884 = minus (1198890 + 1205792) minus 1205820119868lowast

11988811 =11990301205720

(1 + 1205720119868lowast)3

(70)

Making the change of variables 119883 = 119909 119884 = minus1205792119868lowast + 1198881119909 +

1198882119910 + 119888111199092 + 1205820119909119910 minus 1199081(119909) and rewriting 119883 119884 as 119909 and 119910

respectively we have

119889119909

119889119905= 119910

119889119910

119889119905= 1198900+ 1198901119909 + 1198902119910 + 11989011119909

2+ 11989012119909119910 + 119890221199102+ 1199082 (119909 119910 120579)

(71)

where 120579 = (1205791 1205792) 1199082(119909 119910 120579) is a smooth function of 119909 1199101205791 and 1205792 at least of the third order and

1198900 = 1198882 (1205791 minus 1205792119878lowast) + 11988841205792119868

lowast

1198901 = 1205820 (1205791 minus 1205792119878lowast) + 11988821198883 minus 11988811198884 minus 12058201205792119868

lowast

1198902 = 1198881 + 1198884 + 12058201205792119868lowast

1198882

11989011 = 12058201198883 minus 119888411988811 minus 119888111198882 + 11988811205820

11989012 = minus 1205820 + 211988811 + 1205820minus11988811198882 minus 12058201205792119868

lowast

11988822

11989022 =12058201198882

(72)

Next introduce a new time variable 120591 by 119889119905 = (1 minus(12058201198882)119909)119889120591 Rewriting 120591 as 119905 we obtain

119889119909

119889119905= 119910(1 minus

12058201198882119909)

119889119910

119889119905= (1 minus

12058201198882119909)

times (1198900 + 1198901119909 + 1198902119910 + 119890111199092

+ 11989012119909119910 + 119890221199102 + 1199082 (119909 119910 120579))

(73)

Let119883 = 119909 119884 = 119910(1minus(12058201198882)119909) and rename119883 and119884 as 119909 and119910 we have

119889119909

119889119905= 119910

119889119910

119889119905= 1198900 + 1198921119909 + 1198902119910 + 11989211119909

2 + 11989212119909119910 + 1199083 (119909 119910 120579)

(74)

where 120579 = (1205791 1205792) 1199083(119909 119910 120579) is a smooth function of 119909 1199101205791 and 1205792 at least of the third order and

1198921 = minus 2119890012058201198882+ 1198901

11989211 = 11989011 minus 2119890112058201198882+11989001205822

0

11988822

11989212 = 11989012 minus119890212058201198882

(75)

Now we assume that 11989211 = 0 and 11989212 = 0 when 120582119894 are smallSet 119909 = 119883 minus 119890211989212 and rewrite119883 as 119909 we can get

119889119909

119889119905= 119910

119889119910

119889119905= 1198910 + 1198911119909 + 11989211119909

2 + 11989212119909119910 + 1199084 (119909 119910 120579)

(76)

where 120579 = (1205791 1205792) 1199084(119909 119910 120579) is a smooth function of 119909 1199101205791 and 1205792 at least of the third order and

1198910 = 1198900 minus1198921119890211989212+119892111198902

2

119892212

1198911 = 1198921 minus211989211119890211989212

(77)

Mathematical Problems in Engineering 11

Making the final change of variables by 119883 = 11989221211990911989211 119884 =

119892312119910119892211 and 120591 = 1198921111990511989212 and then denoting them again by

119909 119910 and 119905 respectively we obtain

119889119909

119889119905= 119910

119889119910

119889119905= 1205911 (1205791 1205792) + 1205912 (1205791 1205792) 119909 + 119909

2 + 119909119910 + 1199085 (119909 119910 120579)

(78)

where 120579 = (1205791 1205792) and 1199085(119909 119910 120579) is a smooth function of 119909119910 1205791 and 1205792 at least of the third order

We substitute values of 1205720 = 1 1205820 = 14 1198890 = 1101205760 = 110 1205830 = 1100 1198600 = 537500 and 1199030 = 558 forthe above system and these values satisfy conditions (1198601) and(1198602) And we obtain the following equations

1205911 (1205791 1205792) =11989101198924

12

119892311

= minus237291605

851841205791 +

25485118377

8518401205792

+ 119874 (10038161003816100381610038161205791 1205792

10038161003816100381610038162)

1205912 (1205791 1205792) =11989111198922

12

119892211

= minus172225

19361205791 +

3983253

38721205792 + 119874 (

10038161003816100381610038161205791 120579210038161003816100381610038162)

(79)

where 11989211 = minus11500 minus (116)1205791 + (341800)1205792 + (14)1205792

2lt 0

and 11989212 = minus83200 + (14)1205792 lt 0 in a small neighborhood of(1205791 1205792) = (0 0) Let

119869 = (

12059712059111205971205791

12059712059111205971205792

12059712059121205971205791

12059712059121205971205792

) (80)

And after simple calculation we obtain that

det(119869)|120582=0 = minus16839398748825

82458112= 0 (81)

Thus 1205911 and 1205912 are regular maps in a small neighborhoodof (1205791 1205792) = (0 0) By the Bogdanov and Takens bifurcationtheorems [36] we obtain the following conclusion

Theorem 13 Suppose that1198600 1198890 1205820 1205760 1199030 1205720 and 1205830 satisfy(1198601) (1198602) 11989211 = 0 and 11989212 = 0 when 120579119894 are small Then (4)admits the following bifurcation behavior

(1) There is a saddle-node bifurcation curve 119878119873 =

(1205791 1205792) 4119891011989211 = 11989121+ 119900(|(1205791 1205792)|

2) 1198911 =0 = (1205791 1205792) (59076250)1205792 minus (11125)1205791 +(2313400)12057911205792 minus (516)120579

2

1minus (102863740000)1205792

2+

119900(|(1205791 1205792)|2) = 0 (4799116600)1205792 minus (14)1205791 + (25

83)12057911205792 minus (1364513778)1205792

2+ 119900(|(1205791 1205792)|

2) = 0

I

I

O

O 1205791

1205792

SN+

SN+

SNminus

SNminus

II H

III HL

IVIVHL

IIIII

H

Figure 6The bifurcation set and the corresponding phase portraitsfor system (68)

(2) There is a Hopf bifurcation curve 119867 = (1205791 1205792)

1198910 + 119900(|(1205791 1205792)|2) = 0 1198911 lt 0 = (1205791 1205792) minus(537

50)1205792+1205791 + (5083)12057911205792 minus (746676889)12057922 + 119900(|(12057911205792)|2) = 0 (4799116600)1205792 minus (14)1205791+(2583)12057911205792 minus

(1364513778)12057922+ 119900(|(1205791 1205792)|

2) lt 0

(3) There is a homoclinic bifurcation curve119867119871 = (1205791 1205792) 25119892111198910 + 6119891

2

1= 119900(|(1205791 1205792)|

2) 1198911 lt 0 = (1205791 1205792) (59071000)1205792 minus (1120)1205791 + (611979 33200)12057911205792 minus(1916)1205792

1minus (16075834839275560000)1205792

2+ 119900(|(1205791

1205792)|2) = 0 (4799116600)1205792minus(14)1205791+(2583)12057911205792minus

(1364513778)12057922+ 119900(|(1205791 1205792)|

2) lt 0

6 Numerical Simulations

When 120572 = 1 120582 = 14 119889 = 110 120576 = 110 120583 = 1100119860 = 537500 and 119903 = 558 120590 gt 0 By applying PPLANE8and Photoshop software the (1205791 1205792)-plane near the origin isdivided into 4 regions by these bifurcation curves as shown inFigure 6 Fix 1205791 lt 0 and decrease 1205792 from 0 our conclusionsare summarized as follows

(a) When (1205791 1205792) lies in region I there is no positiveequilibrium which implies that the positive orbitsof (4) meet the positive 119878-axis in finite time andtherefore the disease disappears

(b) There is a saddle-node point when (1205791 1205792) lies oncurve SN

(c) When (1205791 1205792) crosses SN into region II two positiveequilibria which are an unstable focus and a saddleappear

(d) When the parameters lie on the curve H there is alsoa saddle and an unstable focus An unstable limit cycleappears with the parameters crossing H into III

(e) A homoclinic cycle appears as the parameters passingIII into HL through the homoclinic bifurcation

12 Mathematical Problems in Engineering

0 5 10 15 20 25 30 35 40

0

1

2

3

4

5

6

7

8

9

10

S

I

Figure 7 When (1205791 1205792) = (0 0) the unique positive equilibrium isa cusp of codimension 2

3 35 45 55 65 754 5 6 7 8

1

2

3

4

5

6

7

8

S

I

Figure 8 When (1205791 1205792) = (minus01 minus00093) lies in region I there isno positive equilibrium

(f) The homoclinic loop breakswith the parameter cross-ing HL into IV and a saddle and a stable focus appear

Furthermore using PPLANE8 some numerical simu-lations of system (68) are depicted in Figures 7ndash10 When(1205791 1205792) = (0 0) that is (1198600 1198890) = (537500 110) there isa unique positive equilibrium (119878lowast 119868lowast) = (31750 4) whichis a cusp of codimension 2 (Figure 7) When (1205791 1205792) =(minus01 minus00093) lies in region I there is no positive equi-librium (Figure 8) When (1205791 1205792) = (minus002 minus0001863798)there is a homoclinic loop (Figure 9) When (1205791 1205792) =(minus002 minus00020) lies in region IV the homoclinic loopis broken and there is a stable focus and a saddle(Figure 10)

6 605 61 615 62 625 63 635 64 645 653536373839

44142434445

S

I

Figure 9 When (1205791 1205792) = (minus002 minus0001863798) there is ahomoclinic loop

0 1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

7

8

9

10

S

I

Figure 10 When (1205791 1205792) = (minus002 minus00020) lies in region IV thereis a stable focus and a saddle

7 Discussion

In this paper we focus on the bifurcation analysis of an SISepidemic model with bilinear incidence rate and saturatedtreatment Generally speaking in many epidemic modelsthe basic reproduction number which is the key conceptin epidemiology can be decreased below unity to eradicatethe disease However in our model the basic reproductionnumber below 1 is not enough to eradicate the diseaseAccording to our analysis in this paper we demonstrate notonly the global stability of the disease-free equilibrium when1198770 lt 119877

lowast

0and the local asymptotic stability of the endemic

equilibrium 1198642 when 119877lowast

0lt 1198770 lt 1 but also the global

stability of the unique endemic equilibrium 119864lowast when 1198770 gt 1Moreover it has been shown in Corollary 4 that backwardbifurcations occur if the effect of the infected being delayedfor treatment is strong That is to say we should get prompttreatment for patients Through Figure 4 we can see thatthere is a region such that the disease will persist if the initialposition lies in the region and disappear if the initial positionlies outside this region So in order to restrain the spread

Mathematical Problems in Engineering 13

of the disease governments should take timely measures tocontrol the initial infected individuals in a lower level Inaddition fromTheorem 8 we can see that the disease will dieout if 1198770 is small enough Therefore from the definition of1198770 one knows that we can decrease the incidence rate 120582 andincrease the cure rate 119903 so as to eradicate the diseases or tomake them controlled in a lower endemic steady state

The stability analysis of the model equilibria enables usto completely analyze their local bifurcation behavior suchas Hopf saddle-node and Bogdanov-Takens bifurcation Bycomputing the first Lyapunov coefficient we can determinethat the Hopf bifurcation is a supercritical Hopf bifurca-tion or a subcritical Hopf bifurcation We also show thatunder assumptions (1198601) and (1198602) the model undertakesBogdanov-Takens bifurcation that is there are saddle-nodebifurcation Hopf bifurcation and homoclinic bifurcationin the system The normal norm of the Bogdanov-Takensbifurcation is derived in Section 5 which is very helpful toobtain the three kinds of bifurcation curves By analyticaltechniques 119860 and 119889 are chosen as bifurcation parametersand other parameters are fixed and we easily get a clearpicture about the rich dynamics behaviors of our modelThrough studying the bifurcations of the SIS epidemicmodelwe are suggesting that in order to eradicate the diseasemore medical facilities as well as medical professionals areneeded and the medical standard needs to be improved aswell

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research was supported by the National Natural ScienceFoundation of China (no 11001041 no 10926105 no 11101283no 11271065 and no 11201360) SRFDP (no 200802001008)and Scientific Research Innovation Project of ShanghaiMunicipal Education Commission (12YZ173)

References

[1] M E Alexander and S M Moghadas ldquoPeriodicity in an epi-demic model with a generalized non-linear incidencerdquo Mathe-matical Biosciences vol 189 no 1 pp 75ndash96 2004

[2] M E Alexander and S M Moghadas ldquoBifurcation analysisof SIRS epidemic model with generalized incidencerdquo SIAMJournal on Applied Mathematics vol 65 no 5 pp 1794ndash18162005

[3] J Arino C C McCluskey and P van den Driessche ldquoGlobalresults for an epidemic model with vaccination that exhibitsbackward bifurcationrdquo SIAM Journal on Applied Mathematicsvol 64 no 1 pp 260ndash276 2003

[4] W R Derrick and P van den Driessche ldquoHomoclinic orbitsin a disease transmission model with nonlinear incidence andnonconstant populationrdquo Discrete and Continuous DynamicalSystems B vol 3 no 2 pp 299ndash309 2003

[5] F Brauer ldquoBackward bifurcations in simple vaccination mod-elsrdquo Journal of Mathematical Analysis and Applications vol 298no 2 pp 418ndash431 2004

[6] K P Hadeler and P van den Driessche ldquoBackward bifurcationin epidemic controlrdquo Mathematical Biosciences vol 146 no 1pp 15ndash35 1997

[7] Z X Hu S Liu and H Wang ldquoBackward bifurcation of anepidemic model with standard incidence rate and treatmentraterdquo Nonlinear Analysis Real World Applications vol 9 no 5pp 2302ndash2312 2008

[8] J Hui and D M Zhu ldquoGlobal stability and periodicity onSIS epidemic models with backward bifurcationrdquo Computers ampMathematics with Applications vol 50 no 8-9 pp 1271ndash12902005

[9] Y Jin W D Wang and S W Xiao ldquoAn SIRS model with anonlinear incidence raterdquoChaos Solitons amp Fractals vol 34 no5 pp 1482ndash1497 2007

[10] G H Li and W D Wang ldquoBifurcation analysis of an epidemicmodel with nonlinear incidencerdquo Applied Mathematics andComputation vol 214 no 2 pp 411ndash423 2009

[11] X Z Li W S Li and M Ghosh ldquoStability and bifurcation ofan SIS epidemic model with treatmentrdquo Chaos Solitons ampampFractals vol 42 no 5 pp 2822ndash2832 2009

[12] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1986

[13] X B Liu and L J Yang ldquoStability analysis of an SEIQV epidemicmodel with saturated incidence raterdquo Nonlinear Analysis RealWorld Applications vol 13 no 6 pp 2671ndash2679 2012

[14] M Martcheva and H R Thieme ldquoProgression age enhancedbackward bifurcation in an epidemic model with super-infectionrdquo Journal of Mathematical Biology vol 46 no 5 pp385ndash424 2003

[15] S G Ruan and W D Wang ldquoDynamical behavior of anepidemic model with a nonlinear incidence raterdquo Journal ofDifferential Equations vol 188 no 1 pp 135ndash163 2003

[16] W D Wang and S G Ruan ldquoBifurcation in an epidemicmodel with constant removal rate of the infectivesrdquo Journal ofMathematical Analysis and Applications vol 291 no 2 pp 775ndash793 2004

[17] Y L Tang D Q Huang S G Ruan and W N ZhangldquoCoexistence of limit cycles and homoclinic loops in a SIRSmodel with a nonlinear incidence raterdquo SIAM Journal onApplied Mathematics vol 69 no 2 pp 621ndash639 2008

[18] P van den Driessche and J Watmough ldquoA simple SIS epidemicmodel with a backward bifurcationrdquo Journal of MathematicalBiology vol 40 no 6 pp 525ndash540 2000

[19] J J Wei and J A Cui ldquoDynamics of SIS epidemic model withthe standard incidence rate and saturated treatment functionrdquoInternational Journal of Biomathematics vol 5 no 3 Article ID1260003 18 pages 2012

[20] W D Wang ldquoBackward bifurcation of an epidemic model withtreatmentrdquoMathematical Biosciences vol 201 no 1-2 pp 58ndash712006

[21] Z W Wang ldquoBackward bifurcation in simple SIS modelrdquo ActaMathematicae Applicatae Sinica vol 25 no 1 pp 127ndash136 2009

[22] L Xue and C Scoglio ldquoThe network level reproduction numberfor infectious diseases with both vertical and horizontal trans-missionrdquo Mathematical Biosciences vol 243 no 1 pp 67ndash802013

14 Mathematical Problems in Engineering

[23] X Zhang and X N Liu ldquoBackward bifurcation of an epidemicmodel with saturated treatment functionrdquo Journal ofMathemat-ical Analysis and Applications vol 348 no 1 pp 433ndash443 2008

[24] J Guckenheimer and P J Holmes Nonlinear OscillationsDynamical Systems and Bifurcation of Vector Fields SpringerNew York NY USA 1996

[25] X Zhang and X N Liu ldquoBackward bifurcation and globaldynamics of an SIS epidemic model with general incidence rateand treatmentrdquoNonlinearAnalysis RealWorldApplications vol10 no 2 pp 565ndash575 2009

[26] M A Safi A B Gumel and E H Elbasha ldquoQualitative analysisof an age-structured SEIR epidemic model with treatmentrdquoApplied Mathematics and Computation vol 219 no 22 pp10627ndash10642 2013

[27] L H Zhou and M Fan ldquoDynamics of an SIR epidemic modelwith limited medical resources revisitedrdquo Nonlinear AnalysisReal World Applications vol 13 no 1 pp 312ndash324 2012

[28] X Y Zhou and J A Cui ldquoAnalysis of stability and bifurcation foran SEIR epidemicmodel with saturated recovery raterdquoCommu-nications inNonlinear Science andNumerical Simulation vol 16no 11 pp 4438ndash4450 2011

[29] D Lacitignola ldquoSaturated treatments and measles resurgenceepisodes in South Africa a possible linkagerdquo MathematicalBiosciences and Engineering MBE vol 10 no 4 pp 1135ndash11572013

[30] J C Eckalbar and W L Eckalbar ldquoDynamics of an epidemicmodel with quadratic treatmentrdquo Nonlinear Analysis RealWorld Applications vol 12 no 1 pp 320ndash332 2011

[31] F Brauer ldquoBackward bifurcations in simple vaccinationtreat-ment modelsrdquo Journal of Biological Dynamics vol 5 no 5 pp410ndash418 2011

[32] M W Hirsch S Smale and R L Devaney Differential Equa-tions Dynamical Systems and an Introduction to Chaos Aca-demic Press New York NY USA 2003

[33] R Bogdanov ldquoBifurcations of a limit cycle for a family of vectorfields on the planrdquo SelectaMathematica vol 1 pp 373ndash388 1981

[34] R Bogdanov ldquoVersal deformations of a singular point on theplan in the case of zero eigen-valuesrdquo Selecta MathematicaSovietica vol 1 pp 389ndash421 1981

[35] F Takens ldquoForced oscillations and bifurcationrdquo in Applicationsof Global Analysis I vol 3 pp 1ndash59 Communications of theMathematical Institute Rijksuniversiteit Utrecht 1974

[36] Y A Kuznetsov Elements of Applied Bifurcation TheorySpringer New York NY USA 2nd edition 1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

that is

[120582 (119889 + 120583) + 120572119889 (119889 + 120576 + 120583) minus 120572120582119860]2

gt 4120572120582 (119889 + 120583) 119889 (119889 + 120576 + 119903 + 120583) minus 41205721205822 (119889 + 120583)119860(34)

Therefore

120582 (119889 + 120583) + [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]2

= 120582 (119889 + 120583) minus [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]2

+ 4120582 (119889 + 120583) [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

gt 4120572120582 (119889 + 120583) 119889 (119889 + 120576 + 119903 + 120583) minus 41205721205822 (119889 + 120583)119860

+ 4120582 (119889 + 120583) [120572120582119860 minus 120572119889 (119889 + 120576 + 120583)]

= 4119903119889120572120582 (119889 + 120583)

(35)

Obviously we have 119875 gt 0 From (30) one has 1198681 lt 119868lowast So

we get det(1198721) lt 0 Hence the endemic equilibrium 1198641 is asaddle The proof is complete

In order to explore the stability of the endemic equilib-rium 1198642 define

1198981 = (21198891205721198862 + 1205821198862 minus 1205761205721198862 minus 1198861198881205821205722)

minus 119887 (2120582120572119886 + 1198891205722119886 minus 1198871205821205722)

1198982 = 1198862119889 minus 119888 (2120582120572119886 + 1198891205722119886 minus 1198871205821205722)

(36)

Theorem 7 If 120578 gt 0 then endemic equilibrium 1198642 is locallyasymptotically stable if 120578 lt 0 then endemic equilibrium 1198642 isunstable where 120578 = 21198861198982 + 1198981(radic1198872 minus 4119886119888 minus 119887)

Proof Consider

det (1198722) =1198682

(1 + 1205721198682)2[120582 (119889 + 120583) (1 + 1205721198682)

2minus 120576120572119889]

=1198682

(1 + 1205721198682)2times 120595 (1198682)

(37)

By carrying out arguments similar to that of Theorem 6 wehave 1198682 gt 119868

lowast Therefore det(1198722) gt 0 In addition we have

tr (1198722) = minus(119889 + 1205821198682) (1 + 1205721198682)

2minus 1205761205721198682

(1 + 1205721198682)2

= minus120582120572211986832+ (2120582120572 + 1198891205722) 1198682

2+ (2119889120572 + 120582 minus 120576120572) 1198682 + 119889

(1 + 1205721198682)2

(38)

and then sgn(tr(1198722)) = minus sgn(119866(1198682)) where

119866 (119909) = 12058212057221199093 + (2120582120572 + 1198891205722) 1199092 + (2119889120572 + 120582 minus 120576120572) 119909 + 119889

(39)

I

0 50 100 1500

5

10

15

20

25

30

35

40

S

E0

E1

E2

Figure 4 One region of disease persistence and one region ofdisease extinction when 119860 = 10 120572 = 1 119889 = 01 120583 = 001 120582 = 001119903 = 01 and 120576 = 01

Using the expression of1198981 = (21198891205721198862 +1205821198862 minus 1205761205721198862 minus1198861198881205821205722) minus

119887(2120582120572119886+1198891205722119886minus1198871205821205722) and1198982 = 1198862119889minus119888(2120582120572119886+1198891205722119886minus1198871205821205722)

one has

119866 (1198682) = (1198861198682

2+ 1198871198682 + 119888) 1205930 +

11989811198682 + 11989821198862

(40)

where1205930 is a first degree polynomial of 1198682 Since 1198861198682

2+1198871198682+119888 =

0 sgn(tr(1198722)) = minus sgn(119866(1198682)) = minus sgn(11989811198682 + 1198982) From theexpression of 1198682 we have

sgn (11989811198682 + 1198982)

= sgn (21198861198982 + 1198981 (radic1198872 minus 4119886119888 minus 119887)) ≜ sgn (120578) (41)

Thus 1198642 is locally asymptotically stable if 120578 gt 0 and 1198642 isunstable if 120578 lt 0 The proof is complete

From the above discussion we have the following con-clusion If two endemic equilibria 1198641 and 1198642 exist the stablemanifolds of the saddle 1198641 split 119877

2

+into two regions The

disease is persistent in the upper region and dies out in thelower region (see Figure 4)

3 Global Analysis

Firstly we consider the global stability of the disease-freeequilibrium 1198640 Let 119873 = 119878 + 119868 be the total population sizeNow we note that the equation for total population is givenby 119889119873119889119905 = 119860 minus 119889119878 minus (119889 + 120583)119868 le 119860 minus 119889119873 It follows thatlim119905rarr+infin119873(119905) le 119860119889 Let

R = (119878 119868) isin 1198772

+ 119878 + 119868 le

119860

119889 (42)

which is positive invariant with respect to system (4)

Theorem 8 If 1198770 lt 119877lowast0 the disease-free equilibrium 1198640(119860119889 0) is globally asymptotically stable that is the disease diesout

Mathematical Problems in Engineering 7

Proof Suppose 1198770 lt 119877lowast0 From the (1198673) of Theorem 2 weknow that the model has no endemic equilibrium Fromthe corollary of Poincare-Bendixson theorem [32] we knowthat there is no periodic orbit in R as there is a disease-free equilibrium in R Since R is a bounded positivelyinvariant region and 1198640 is the only equilibrium in R thelocal stability of 1198640 implies that every solution initiating inR approaches 1198640 Thus the disease-free equilibrium 1198640 isglobally asymptotically stable The proof is complete

Now we analyze the global dynamics of the endemicequilibrium when 1198770 gt 1

Theorem 9 If 1198770 gt 1 and 0 le 120572 lt 120582119903 the system (4) has nolimit cycles

Proof We use Dulac theorem to exclude the limit cycle Let

119875 (119878 119868) = 119860 minus 119889119878 minus 120582119878119868 + 120576119868 +119903119868

1 + 120572119868

119876 (119878 119868) = 120582119878119868 minus (119889 + 120576 + 120583) 119868 minus119903119868

1 + 120572119868

(43)

and take the Dulac function

119863 =1

119868 (44)

According to 0 le 120572 lt 120582119903 we can get

120597 (119875119863)

120597119878+120597 (119876119863)

120597119868

= minus119889

119868minus 120582 +

120572119903

(1 + 120572119868)2

lt minus119889

119868minus 120582 +

120582

(1 + 120572119868)2

=1

119868 (1 + 120572119868)2minus119889 (1 + 120572119868)

2 minus 120582119868 [(1 + 120572119868)2 minus 1]

lt 0

(45)

Hence the system (4) has no limit cycles The proof iscomplete

Therefore we obtain the global result of the uniqueendemic equilibrium

Theorem 10 If 1198770 gt 1 and 0 le 120572 lt 120582119903 the unique endemicequilibrium 119864lowast is globally asymptotically stable (see Figure 5)

4 Hopf Bifurcation

In this section we study the Hopf bifurcation of system (4)From the above discussion we know that there is no closedorbit surrounding1198640 or1198641 because the 119878-axis is invariantwith

0 50 100 1500

50

100

150

S

I Elowast

Figure 5The unique endemic equilibrium 119864lowast is globally asymptot-ically stable when 119860 = 15 120572 = 001 119889 = 01 120583 = 001 120582 = 001119903 = 08 and 120576 = 01

respect to system (4) and 1198641 is always a saddle ThereforeHopf bifurcation can only occur at 1198642 Set

120590 = [minus120582120572 minus1198881 (1198882 + 21198884)

11988612]1198634

+ [minus 120582120572119886211minus 21205821205721198861111988612 + (1198882 minus

211988611119888111988612)

times (119886111198882 minus119886211119888111988612)

minus (21198862111198881

11988612minus 119886111198882 + 2119888411988611 minus 119888311988612)

times (1198884 minus11988611119888111988612)]1198632

+ 11988611 (119886211119888111988612minus 119886111198882 + 119888411988611 minus 119888311988612)

times (119886121198883 minus 2119886111198884 minus 119886111198882)

(46)

where 119886119894119895 (119894 119895 = 1 2) 119888119896 (119896 = 1 2 3 4) and 119863 are defined by(50) and (52)

Theorem 11 System (4) undergoes a Hopf bifurcation if 120578 = 0Moreover if 120590 lt 0 there is a family of stable period orbits ofsystem (4) as 120578 decreases from 0 that is a supercritical Hopfbifurcation occurs if 120590 gt 0 there is a family of unstable periodorbits of system (4) as 120578 increases from 0 that is a subcriticalHopf bifurcation occurs

Proof The proof of Theorem 7 shows that tr(1198722) = 0 if andonly if 120578 = 0 and det(1198722) gt 0 when 1198642 exists Therefore theeigenvalues of 1198722 are a pair of pure imaginary roots if andonly if 120578 = 0 The direct calculations show that

119889 (tr (1198722))119889120578

100381610038161003816100381610038161003816100381610038161003816120578=0= minus

1

21198863 (1 + 1205721198682)2lt 0 (47)

8 Mathematical Problems in Engineering

ByTheorem 342 in [24] 120578 = 0 is the Hopf bifurcation pointfor system (4)

To be concise in notations rescale (4) by 120591 = 119905(1 + 120572119868)For simplicity we still use 119905 instead of 120591 Then we obtain

119889119878

119889119905= (119860 minus 119889119878 minus 120582119878119868 + 120576119868) (1 + 120572119868) + 119903119868

119889119868

119889119905= 120582119878119868 (1 + 120572119868) minus (119889 + 120576 + 120583) 119868 (1 + 120572119868) minus 119903119868

(48)

Let 119909 = 119878 minus 1198782 and 119910 = 119868 minus 1198682 then (48) becomes

119889119909

119889119905= 11988611119909 + 11988612119910 + 1198881119910

2 + 1198882119909119910 minus 1205821205721199091199102

119889119910

119889119905= 11988621119909 + 11988622119910 + 1198883119909119910 + 1198884119910

2 + 1205821205721199091199102

(49)

where

11988611 = minus (119889 + 1205821198682) (1 + 1205721198682)

11988612 = (minus1205821198782 + 119903) (1 + 1205721198682)

+ 120572 (119860 minus 1198891198782 minus 12058211987821198682 + 1199031198682) + 120576

11988621 = 1205821198682 (1 + 1205721198682)

11988622 = [1205821198782 minus (119889 + 119903 + 120583)] (1 + 1205721198682)

+ 120572 [12058211987821198682 minus (119889 + 119903 + 120583) 1198682] minus 120576

1198881 = 120572 (minus1205821198782 + 119903)

1198882 = 120572 (minus119889 minus 1205821198682) minus 120582 (1 + 1205721198682)

1198883 = 120582 (1 + 1205721198682) + 1205821198682120572

1198884 = [1205821198782 minus (119889 + 119903 + 120583)] 120572

(50)

Let 119864lowast denote the origin of 119909-119910 plane Since 1198642 = (1198782 1198682)satisfies (6) we obtain

det (119872 (119864lowast))

= minus (119889 + 1205821198682) 1205721205761198682 + (119889 + 120583) (1 + 1205721198682

2)2

1205821198682 + 1205721205821205761198682

2

= 1198682 times 120595 (1198682)

(51)

From the proof of Theorem 7 it follows that 120595(1198682) is alwayspositive It is easy to verify that 11988611 + 11988622 = 0 if and only if120578 = 0 Set

119863 = radicdet (119872 (119864lowast)) (52)

and let 119906 = minus119909 and V = (11988611119863)119909+(11988612119863)119910 then the normalform of (48) for Hopf bifurcation reads

119889119906

119889119905= minus119863V + 119891 (119906 V)

119889V119889119905= 119863119906 + 119892 (119906 V)

(53)

where

119891 (119906 V) = (11988611119886121198882 minus119886211

119886212

1198881)1199062 minus11986321198881119886212

V2

+ (119863119888211988612minus 2119863119886111198881119886212

)119906V minus120582120572119886211

119886212

1199063

minus 211986311988611120582120572

119886212

1199062V minus1198632120582120572

119886212

119906V2

119892 (119906 V) =11988611119863(119886211

119886212

1198881 minus11988611119886121198882 +11988611119886121198884 minus 1198883)119906

2

+ (119863119886111198881119886212

+119863119888411988612) V2

+ (2119886211

119886212

1198881 minus11988611119886121198882 + 2

11988611119886121198884 minus 1198883)119906V

+120582120572119886211

11988612119863(1

11988612minus 1)1199063 +

21205821205721198861111988612

(1198861111988612minus 1)1199062V

+119863120582120572

11988612(1198861111988612minus 1)119906V2

(54)

Now we evaluate the first Lyapunov coefficient Γ of system(4) as follows

Γ =1

16[119891119906119906119906 + 119891119906VV + 119892119906119906V + 119892VVV] +

1

16119863

times [119891119906V (119891119906119906 + 119891VV) minus 119892119906V (119892119906119906 + 119892VV) minus 119891119906119906119892119906119906 + 119891VV119892VV]

(55)

where 119891119906V denotes (1205972119891120597119906120597V)(0 0) and so forth Then

Γ =1

81198862121198632[minus120582120572 minus

1198881 (1198882 + 21198884)

11988612]1198634

+ [minus 120582120572119886211minus 21205821205721198861111988612 + (1198882 minus

211988611119888111988612)

times (119886111198882 minus119886211119888111988612)

minus (21198862111198881

11988612minus 119886111198882 + 2119888411988611 minus 119888311988612)

times (1198884 minus11988611119888111988612)]1198632

+ 11988611 (119886211119888111988612minus 119886111198882 + 119888411988611 minus 119888311988612)

times (119886121198883 minus 2119886111198884 minus 119886111198882)

=120590

81198862121198632

(56)

Mathematical Problems in Engineering 9

Obviously the sign of Γ is determined by 120590 ByTheorem 342and (3411) in [24] the rest of the claims in Theorem 11 arevalid The proof is complete

5 Bogdanov-Takens Bifurcations

The purpose of this section is to study the Bogdanov-Takensbifurcation of (4) Now we assume the following

(1198601) 1198872 minus 4119886119888 = 0

(1198602) 21198861198892 + 119887120582120583 = 0

Then (6) admits a unique positive equilibrium 119864lowast = (119878lowast 119868lowast)where

119868lowast = minus119887

2119886 119878lowast =

119860 minus (119889 + 120583) 119868lowast

119889 (57)

The Jacobian matrix of system (4) at 119864lowast is

119872lowast = (

minus119889 minus 120582119868lowast minus120582119878lowast + 120576 +119903

(1 + 120572119868lowast)2

120582119868lowast 120582119878lowast minus (119889 + 120576 + 120583) minus119903

(1 + 120572119868lowast)2)

(58)

By (58) we have

det (119872lowast) = 119868lowast

(1 + 120572119868lowast)2[(119889 + 120583) 120582 (1 + 120572119868lowast)

2minus 119903119889120572]

= 0

(59)

because of

(1 + 120572119868lowast)2=41198862 minus 4119886120572119887 + 12057221198872

41198862

=41198862 minus 4119886120572119887 + 12057224119886119888

41198862

=119886 minus 120572119887 + 1205722119888

119886

=1205722119903119889

119886

(60)

Furthermore (1198602) implies that

tr (119872lowast) = 0 (61)

Thus (1198601) and (1198602) imply that the Jacobian matrix has azero eigenvalue withmultiplicity 2This suggests that (4)mayadmit a Bogdanov-Takens singularity

Theorem 12 Suppose that (1198601) (1198602) and 21198871 + 1198874 = 0 holdThen the endemic equilibrium 119864lowast = (119878lowast 119868lowast) of (4) is a cuspof codimension 2 that is it is a Bogdanov-Takens singularityHere 1198871 and 1198874 are defined by (65)

Proof Using the transformation of 119909 = 119868minus 119868lowast and 119910 = 119878minus 119878lowastsystem (4) becomes

119889119909

119889119905= 1198861119909 + 1198862119910 + 120582119909119910 + 11988611119909

2 minus 1198751 (119909)

119889119910

119889119905= minus11988621

1198862119909 minus 1198861119910 minus 120582119909119910 minus 11988611119909

2 + 1198751 (119909)

(62)

where 1198751(119909) is a smooth function of 119909 at least of the thirdorder and

1198861 = 120582119878lowast minus (119889 + 120576 + 120583) minus

119903

(1 + 120572119868lowast)2gt 0

1198862 = 120582119868lowast gt 0

11988611 =119903120572

(1 + 120572119868lowast)3gt 0

(63)

Set119883 = 119909 119884 = 1198861119909 + 1198862119910 Then (62) is transformed into

119889119883

119889119905= 119884 + 1198871119883

2 + 1198872119883119884 + 1198761 (119883)

119889119884

119889119905= 1198873119883

2 + 1198874119883119884 + 1198762 (119883)

(64)

where 119876119894(119883) are smooth functions of 119883 at least of the thirdorder and

1198871 = 11988611 minus12058211988611198862

1198872 =120582

1198862

1198873 = 119886111988611 minus12058211988621

1198862+ 1198861120582 minus 119886211988611

1198874 =12058211988611198862minus 120582

(65)

In order to obtain the canonical normal form we perform thetransformation of variables by

119906 = 119883 minus119887221198832 V = 119884 + 1198871119883

2 (66)

Then we obtain

119889119906

119889119905= V + 1198771 (119906)

119889V119889119905= 11988731199062 + (21198871 + 1198874) 119906V + 1198772 (119906)

(67)

where 119877119894(119906) are smooth functions of 119906 at least of the thirdorder Note that 1198873 lt 0 and 21198871 + 1198874 = 0 It follows from [33ndash35] that (4) admits a Bogdanov-Takens bifurcation

In the following wewill find the versal unfolding in termsof the original parameters in (4) In this way we will know theapproximate saddle-node Hopf and homoclinic bifurcation

10 Mathematical Problems in Engineering

curvesWe choose119860 and 119889 as bifurcation parameters Fix 120582 =1205820 120576 = 1205760 119903 = 1199030 120572 = 1205720 and 120583 = 1205830 Let 119860 = 1198600 + 1205791 and119889 = 1198890 + 1205792 where 1205791 and 1205792 are parameters which vary in asmall neighborhood of the origin

Suppose that 119860 = 1198600 119889 = 1198890 120582 = 1205820 120576 = 1205760 119903 = 1199030 120572 =1205720 and 120583 = 1205830 satisfy (1198601) and (1198602) Consider the followingsystem

119889119868

119889119905= 1205820119878119868 minus (1198890 + 1205792 + 1205760 + 1205830) 119868 minus

1199030119868

1 + 1205720119868

119889119878

119889119905= 1198600 + 1205791 minus (1198890 + 1205792) 119878 minus 1205820119878119868 + 1205760119868 +

1199030119868

1 + 1205720119868

(68)

By the transformations of 119909 = 119868 minus 119868lowast and 119910 = 119878 minus 119878lowast system(68) becomes

119889119909

119889119905= minus 1205792119868

lowast + 1198881119909 + 1198882119910 + 119888111199092 + 1205820119909119910 minus 1199081 (119909)

119889119910

119889119905= (1205791 minus 1205792119878

lowast) + 1198883119909 + 1198884119910 minus 119888111199092 minus 1205820119909119910 + 1199081 (119909)

(69)

where 1199081(119909) is a smooth function of 119909 at least of the thirdorder and

1198881 = 1205820119878lowast minus (1198890 + 1205792 + 1205760 + 1205830) minus

1199030

(1 + 1205720119868lowast)2

1198882 = 1205820119868lowast

1198883 = minus 1205820119878lowast + 1205760 +

1199030

(1 + 1205720119868lowast)2

1198884 = minus (1198890 + 1205792) minus 1205820119868lowast

11988811 =11990301205720

(1 + 1205720119868lowast)3

(70)

Making the change of variables 119883 = 119909 119884 = minus1205792119868lowast + 1198881119909 +

1198882119910 + 119888111199092 + 1205820119909119910 minus 1199081(119909) and rewriting 119883 119884 as 119909 and 119910

respectively we have

119889119909

119889119905= 119910

119889119910

119889119905= 1198900+ 1198901119909 + 1198902119910 + 11989011119909

2+ 11989012119909119910 + 119890221199102+ 1199082 (119909 119910 120579)

(71)

where 120579 = (1205791 1205792) 1199082(119909 119910 120579) is a smooth function of 119909 1199101205791 and 1205792 at least of the third order and

1198900 = 1198882 (1205791 minus 1205792119878lowast) + 11988841205792119868

lowast

1198901 = 1205820 (1205791 minus 1205792119878lowast) + 11988821198883 minus 11988811198884 minus 12058201205792119868

lowast

1198902 = 1198881 + 1198884 + 12058201205792119868lowast

1198882

11989011 = 12058201198883 minus 119888411988811 minus 119888111198882 + 11988811205820

11989012 = minus 1205820 + 211988811 + 1205820minus11988811198882 minus 12058201205792119868

lowast

11988822

11989022 =12058201198882

(72)

Next introduce a new time variable 120591 by 119889119905 = (1 minus(12058201198882)119909)119889120591 Rewriting 120591 as 119905 we obtain

119889119909

119889119905= 119910(1 minus

12058201198882119909)

119889119910

119889119905= (1 minus

12058201198882119909)

times (1198900 + 1198901119909 + 1198902119910 + 119890111199092

+ 11989012119909119910 + 119890221199102 + 1199082 (119909 119910 120579))

(73)

Let119883 = 119909 119884 = 119910(1minus(12058201198882)119909) and rename119883 and119884 as 119909 and119910 we have

119889119909

119889119905= 119910

119889119910

119889119905= 1198900 + 1198921119909 + 1198902119910 + 11989211119909

2 + 11989212119909119910 + 1199083 (119909 119910 120579)

(74)

where 120579 = (1205791 1205792) 1199083(119909 119910 120579) is a smooth function of 119909 1199101205791 and 1205792 at least of the third order and

1198921 = minus 2119890012058201198882+ 1198901

11989211 = 11989011 minus 2119890112058201198882+11989001205822

0

11988822

11989212 = 11989012 minus119890212058201198882

(75)

Now we assume that 11989211 = 0 and 11989212 = 0 when 120582119894 are smallSet 119909 = 119883 minus 119890211989212 and rewrite119883 as 119909 we can get

119889119909

119889119905= 119910

119889119910

119889119905= 1198910 + 1198911119909 + 11989211119909

2 + 11989212119909119910 + 1199084 (119909 119910 120579)

(76)

where 120579 = (1205791 1205792) 1199084(119909 119910 120579) is a smooth function of 119909 1199101205791 and 1205792 at least of the third order and

1198910 = 1198900 minus1198921119890211989212+119892111198902

2

119892212

1198911 = 1198921 minus211989211119890211989212

(77)

Mathematical Problems in Engineering 11

Making the final change of variables by 119883 = 11989221211990911989211 119884 =

119892312119910119892211 and 120591 = 1198921111990511989212 and then denoting them again by

119909 119910 and 119905 respectively we obtain

119889119909

119889119905= 119910

119889119910

119889119905= 1205911 (1205791 1205792) + 1205912 (1205791 1205792) 119909 + 119909

2 + 119909119910 + 1199085 (119909 119910 120579)

(78)

where 120579 = (1205791 1205792) and 1199085(119909 119910 120579) is a smooth function of 119909119910 1205791 and 1205792 at least of the third order

We substitute values of 1205720 = 1 1205820 = 14 1198890 = 1101205760 = 110 1205830 = 1100 1198600 = 537500 and 1199030 = 558 forthe above system and these values satisfy conditions (1198601) and(1198602) And we obtain the following equations

1205911 (1205791 1205792) =11989101198924

12

119892311

= minus237291605

851841205791 +

25485118377

8518401205792

+ 119874 (10038161003816100381610038161205791 1205792

10038161003816100381610038162)

1205912 (1205791 1205792) =11989111198922

12

119892211

= minus172225

19361205791 +

3983253

38721205792 + 119874 (

10038161003816100381610038161205791 120579210038161003816100381610038162)

(79)

where 11989211 = minus11500 minus (116)1205791 + (341800)1205792 + (14)1205792

2lt 0

and 11989212 = minus83200 + (14)1205792 lt 0 in a small neighborhood of(1205791 1205792) = (0 0) Let

119869 = (

12059712059111205971205791

12059712059111205971205792

12059712059121205971205791

12059712059121205971205792

) (80)

And after simple calculation we obtain that

det(119869)|120582=0 = minus16839398748825

82458112= 0 (81)

Thus 1205911 and 1205912 are regular maps in a small neighborhoodof (1205791 1205792) = (0 0) By the Bogdanov and Takens bifurcationtheorems [36] we obtain the following conclusion

Theorem 13 Suppose that1198600 1198890 1205820 1205760 1199030 1205720 and 1205830 satisfy(1198601) (1198602) 11989211 = 0 and 11989212 = 0 when 120579119894 are small Then (4)admits the following bifurcation behavior

(1) There is a saddle-node bifurcation curve 119878119873 =

(1205791 1205792) 4119891011989211 = 11989121+ 119900(|(1205791 1205792)|

2) 1198911 =0 = (1205791 1205792) (59076250)1205792 minus (11125)1205791 +(2313400)12057911205792 minus (516)120579

2

1minus (102863740000)1205792

2+

119900(|(1205791 1205792)|2) = 0 (4799116600)1205792 minus (14)1205791 + (25

83)12057911205792 minus (1364513778)1205792

2+ 119900(|(1205791 1205792)|

2) = 0

I

I

O

O 1205791

1205792

SN+

SN+

SNminus

SNminus

II H

III HL

IVIVHL

IIIII

H

Figure 6The bifurcation set and the corresponding phase portraitsfor system (68)

(2) There is a Hopf bifurcation curve 119867 = (1205791 1205792)

1198910 + 119900(|(1205791 1205792)|2) = 0 1198911 lt 0 = (1205791 1205792) minus(537

50)1205792+1205791 + (5083)12057911205792 minus (746676889)12057922 + 119900(|(12057911205792)|2) = 0 (4799116600)1205792 minus (14)1205791+(2583)12057911205792 minus

(1364513778)12057922+ 119900(|(1205791 1205792)|

2) lt 0

(3) There is a homoclinic bifurcation curve119867119871 = (1205791 1205792) 25119892111198910 + 6119891

2

1= 119900(|(1205791 1205792)|

2) 1198911 lt 0 = (1205791 1205792) (59071000)1205792 minus (1120)1205791 + (611979 33200)12057911205792 minus(1916)1205792

1minus (16075834839275560000)1205792

2+ 119900(|(1205791

1205792)|2) = 0 (4799116600)1205792minus(14)1205791+(2583)12057911205792minus

(1364513778)12057922+ 119900(|(1205791 1205792)|

2) lt 0

6 Numerical Simulations

When 120572 = 1 120582 = 14 119889 = 110 120576 = 110 120583 = 1100119860 = 537500 and 119903 = 558 120590 gt 0 By applying PPLANE8and Photoshop software the (1205791 1205792)-plane near the origin isdivided into 4 regions by these bifurcation curves as shown inFigure 6 Fix 1205791 lt 0 and decrease 1205792 from 0 our conclusionsare summarized as follows

(a) When (1205791 1205792) lies in region I there is no positiveequilibrium which implies that the positive orbitsof (4) meet the positive 119878-axis in finite time andtherefore the disease disappears

(b) There is a saddle-node point when (1205791 1205792) lies oncurve SN

(c) When (1205791 1205792) crosses SN into region II two positiveequilibria which are an unstable focus and a saddleappear

(d) When the parameters lie on the curve H there is alsoa saddle and an unstable focus An unstable limit cycleappears with the parameters crossing H into III

(e) A homoclinic cycle appears as the parameters passingIII into HL through the homoclinic bifurcation

12 Mathematical Problems in Engineering

0 5 10 15 20 25 30 35 40

0

1

2

3

4

5

6

7

8

9

10

S

I

Figure 7 When (1205791 1205792) = (0 0) the unique positive equilibrium isa cusp of codimension 2

3 35 45 55 65 754 5 6 7 8

1

2

3

4

5

6

7

8

S

I

Figure 8 When (1205791 1205792) = (minus01 minus00093) lies in region I there isno positive equilibrium

(f) The homoclinic loop breakswith the parameter cross-ing HL into IV and a saddle and a stable focus appear

Furthermore using PPLANE8 some numerical simu-lations of system (68) are depicted in Figures 7ndash10 When(1205791 1205792) = (0 0) that is (1198600 1198890) = (537500 110) there isa unique positive equilibrium (119878lowast 119868lowast) = (31750 4) whichis a cusp of codimension 2 (Figure 7) When (1205791 1205792) =(minus01 minus00093) lies in region I there is no positive equi-librium (Figure 8) When (1205791 1205792) = (minus002 minus0001863798)there is a homoclinic loop (Figure 9) When (1205791 1205792) =(minus002 minus00020) lies in region IV the homoclinic loopis broken and there is a stable focus and a saddle(Figure 10)

6 605 61 615 62 625 63 635 64 645 653536373839

44142434445

S

I

Figure 9 When (1205791 1205792) = (minus002 minus0001863798) there is ahomoclinic loop

0 1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

7

8

9

10

S

I

Figure 10 When (1205791 1205792) = (minus002 minus00020) lies in region IV thereis a stable focus and a saddle

7 Discussion

In this paper we focus on the bifurcation analysis of an SISepidemic model with bilinear incidence rate and saturatedtreatment Generally speaking in many epidemic modelsthe basic reproduction number which is the key conceptin epidemiology can be decreased below unity to eradicatethe disease However in our model the basic reproductionnumber below 1 is not enough to eradicate the diseaseAccording to our analysis in this paper we demonstrate notonly the global stability of the disease-free equilibrium when1198770 lt 119877

lowast

0and the local asymptotic stability of the endemic

equilibrium 1198642 when 119877lowast

0lt 1198770 lt 1 but also the global

stability of the unique endemic equilibrium 119864lowast when 1198770 gt 1Moreover it has been shown in Corollary 4 that backwardbifurcations occur if the effect of the infected being delayedfor treatment is strong That is to say we should get prompttreatment for patients Through Figure 4 we can see thatthere is a region such that the disease will persist if the initialposition lies in the region and disappear if the initial positionlies outside this region So in order to restrain the spread

Mathematical Problems in Engineering 13

of the disease governments should take timely measures tocontrol the initial infected individuals in a lower level Inaddition fromTheorem 8 we can see that the disease will dieout if 1198770 is small enough Therefore from the definition of1198770 one knows that we can decrease the incidence rate 120582 andincrease the cure rate 119903 so as to eradicate the diseases or tomake them controlled in a lower endemic steady state

The stability analysis of the model equilibria enables usto completely analyze their local bifurcation behavior suchas Hopf saddle-node and Bogdanov-Takens bifurcation Bycomputing the first Lyapunov coefficient we can determinethat the Hopf bifurcation is a supercritical Hopf bifurca-tion or a subcritical Hopf bifurcation We also show thatunder assumptions (1198601) and (1198602) the model undertakesBogdanov-Takens bifurcation that is there are saddle-nodebifurcation Hopf bifurcation and homoclinic bifurcationin the system The normal norm of the Bogdanov-Takensbifurcation is derived in Section 5 which is very helpful toobtain the three kinds of bifurcation curves By analyticaltechniques 119860 and 119889 are chosen as bifurcation parametersand other parameters are fixed and we easily get a clearpicture about the rich dynamics behaviors of our modelThrough studying the bifurcations of the SIS epidemicmodelwe are suggesting that in order to eradicate the diseasemore medical facilities as well as medical professionals areneeded and the medical standard needs to be improved aswell

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research was supported by the National Natural ScienceFoundation of China (no 11001041 no 10926105 no 11101283no 11271065 and no 11201360) SRFDP (no 200802001008)and Scientific Research Innovation Project of ShanghaiMunicipal Education Commission (12YZ173)

References

[1] M E Alexander and S M Moghadas ldquoPeriodicity in an epi-demic model with a generalized non-linear incidencerdquo Mathe-matical Biosciences vol 189 no 1 pp 75ndash96 2004

[2] M E Alexander and S M Moghadas ldquoBifurcation analysisof SIRS epidemic model with generalized incidencerdquo SIAMJournal on Applied Mathematics vol 65 no 5 pp 1794ndash18162005

[3] J Arino C C McCluskey and P van den Driessche ldquoGlobalresults for an epidemic model with vaccination that exhibitsbackward bifurcationrdquo SIAM Journal on Applied Mathematicsvol 64 no 1 pp 260ndash276 2003

[4] W R Derrick and P van den Driessche ldquoHomoclinic orbitsin a disease transmission model with nonlinear incidence andnonconstant populationrdquo Discrete and Continuous DynamicalSystems B vol 3 no 2 pp 299ndash309 2003

[5] F Brauer ldquoBackward bifurcations in simple vaccination mod-elsrdquo Journal of Mathematical Analysis and Applications vol 298no 2 pp 418ndash431 2004

[6] K P Hadeler and P van den Driessche ldquoBackward bifurcationin epidemic controlrdquo Mathematical Biosciences vol 146 no 1pp 15ndash35 1997

[7] Z X Hu S Liu and H Wang ldquoBackward bifurcation of anepidemic model with standard incidence rate and treatmentraterdquo Nonlinear Analysis Real World Applications vol 9 no 5pp 2302ndash2312 2008

[8] J Hui and D M Zhu ldquoGlobal stability and periodicity onSIS epidemic models with backward bifurcationrdquo Computers ampMathematics with Applications vol 50 no 8-9 pp 1271ndash12902005

[9] Y Jin W D Wang and S W Xiao ldquoAn SIRS model with anonlinear incidence raterdquoChaos Solitons amp Fractals vol 34 no5 pp 1482ndash1497 2007

[10] G H Li and W D Wang ldquoBifurcation analysis of an epidemicmodel with nonlinear incidencerdquo Applied Mathematics andComputation vol 214 no 2 pp 411ndash423 2009

[11] X Z Li W S Li and M Ghosh ldquoStability and bifurcation ofan SIS epidemic model with treatmentrdquo Chaos Solitons ampampFractals vol 42 no 5 pp 2822ndash2832 2009

[12] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1986

[13] X B Liu and L J Yang ldquoStability analysis of an SEIQV epidemicmodel with saturated incidence raterdquo Nonlinear Analysis RealWorld Applications vol 13 no 6 pp 2671ndash2679 2012

[14] M Martcheva and H R Thieme ldquoProgression age enhancedbackward bifurcation in an epidemic model with super-infectionrdquo Journal of Mathematical Biology vol 46 no 5 pp385ndash424 2003

[15] S G Ruan and W D Wang ldquoDynamical behavior of anepidemic model with a nonlinear incidence raterdquo Journal ofDifferential Equations vol 188 no 1 pp 135ndash163 2003

[16] W D Wang and S G Ruan ldquoBifurcation in an epidemicmodel with constant removal rate of the infectivesrdquo Journal ofMathematical Analysis and Applications vol 291 no 2 pp 775ndash793 2004

[17] Y L Tang D Q Huang S G Ruan and W N ZhangldquoCoexistence of limit cycles and homoclinic loops in a SIRSmodel with a nonlinear incidence raterdquo SIAM Journal onApplied Mathematics vol 69 no 2 pp 621ndash639 2008

[18] P van den Driessche and J Watmough ldquoA simple SIS epidemicmodel with a backward bifurcationrdquo Journal of MathematicalBiology vol 40 no 6 pp 525ndash540 2000

[19] J J Wei and J A Cui ldquoDynamics of SIS epidemic model withthe standard incidence rate and saturated treatment functionrdquoInternational Journal of Biomathematics vol 5 no 3 Article ID1260003 18 pages 2012

[20] W D Wang ldquoBackward bifurcation of an epidemic model withtreatmentrdquoMathematical Biosciences vol 201 no 1-2 pp 58ndash712006

[21] Z W Wang ldquoBackward bifurcation in simple SIS modelrdquo ActaMathematicae Applicatae Sinica vol 25 no 1 pp 127ndash136 2009

[22] L Xue and C Scoglio ldquoThe network level reproduction numberfor infectious diseases with both vertical and horizontal trans-missionrdquo Mathematical Biosciences vol 243 no 1 pp 67ndash802013

14 Mathematical Problems in Engineering

[23] X Zhang and X N Liu ldquoBackward bifurcation of an epidemicmodel with saturated treatment functionrdquo Journal ofMathemat-ical Analysis and Applications vol 348 no 1 pp 433ndash443 2008

[24] J Guckenheimer and P J Holmes Nonlinear OscillationsDynamical Systems and Bifurcation of Vector Fields SpringerNew York NY USA 1996

[25] X Zhang and X N Liu ldquoBackward bifurcation and globaldynamics of an SIS epidemic model with general incidence rateand treatmentrdquoNonlinearAnalysis RealWorldApplications vol10 no 2 pp 565ndash575 2009

[26] M A Safi A B Gumel and E H Elbasha ldquoQualitative analysisof an age-structured SEIR epidemic model with treatmentrdquoApplied Mathematics and Computation vol 219 no 22 pp10627ndash10642 2013

[27] L H Zhou and M Fan ldquoDynamics of an SIR epidemic modelwith limited medical resources revisitedrdquo Nonlinear AnalysisReal World Applications vol 13 no 1 pp 312ndash324 2012

[28] X Y Zhou and J A Cui ldquoAnalysis of stability and bifurcation foran SEIR epidemicmodel with saturated recovery raterdquoCommu-nications inNonlinear Science andNumerical Simulation vol 16no 11 pp 4438ndash4450 2011

[29] D Lacitignola ldquoSaturated treatments and measles resurgenceepisodes in South Africa a possible linkagerdquo MathematicalBiosciences and Engineering MBE vol 10 no 4 pp 1135ndash11572013

[30] J C Eckalbar and W L Eckalbar ldquoDynamics of an epidemicmodel with quadratic treatmentrdquo Nonlinear Analysis RealWorld Applications vol 12 no 1 pp 320ndash332 2011

[31] F Brauer ldquoBackward bifurcations in simple vaccinationtreat-ment modelsrdquo Journal of Biological Dynamics vol 5 no 5 pp410ndash418 2011

[32] M W Hirsch S Smale and R L Devaney Differential Equa-tions Dynamical Systems and an Introduction to Chaos Aca-demic Press New York NY USA 2003

[33] R Bogdanov ldquoBifurcations of a limit cycle for a family of vectorfields on the planrdquo SelectaMathematica vol 1 pp 373ndash388 1981

[34] R Bogdanov ldquoVersal deformations of a singular point on theplan in the case of zero eigen-valuesrdquo Selecta MathematicaSovietica vol 1 pp 389ndash421 1981

[35] F Takens ldquoForced oscillations and bifurcationrdquo in Applicationsof Global Analysis I vol 3 pp 1ndash59 Communications of theMathematical Institute Rijksuniversiteit Utrecht 1974

[36] Y A Kuznetsov Elements of Applied Bifurcation TheorySpringer New York NY USA 2nd edition 1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Proof Suppose 1198770 lt 119877lowast0 From the (1198673) of Theorem 2 weknow that the model has no endemic equilibrium Fromthe corollary of Poincare-Bendixson theorem [32] we knowthat there is no periodic orbit in R as there is a disease-free equilibrium in R Since R is a bounded positivelyinvariant region and 1198640 is the only equilibrium in R thelocal stability of 1198640 implies that every solution initiating inR approaches 1198640 Thus the disease-free equilibrium 1198640 isglobally asymptotically stable The proof is complete

Now we analyze the global dynamics of the endemicequilibrium when 1198770 gt 1

Theorem 9 If 1198770 gt 1 and 0 le 120572 lt 120582119903 the system (4) has nolimit cycles

Proof We use Dulac theorem to exclude the limit cycle Let

119875 (119878 119868) = 119860 minus 119889119878 minus 120582119878119868 + 120576119868 +119903119868

1 + 120572119868

119876 (119878 119868) = 120582119878119868 minus (119889 + 120576 + 120583) 119868 minus119903119868

1 + 120572119868

(43)

and take the Dulac function

119863 =1

119868 (44)

According to 0 le 120572 lt 120582119903 we can get

120597 (119875119863)

120597119878+120597 (119876119863)

120597119868

= minus119889

119868minus 120582 +

120572119903

(1 + 120572119868)2

lt minus119889

119868minus 120582 +

120582

(1 + 120572119868)2

=1

119868 (1 + 120572119868)2minus119889 (1 + 120572119868)

2 minus 120582119868 [(1 + 120572119868)2 minus 1]

lt 0

(45)

Hence the system (4) has no limit cycles The proof iscomplete

Therefore we obtain the global result of the uniqueendemic equilibrium

Theorem 10 If 1198770 gt 1 and 0 le 120572 lt 120582119903 the unique endemicequilibrium 119864lowast is globally asymptotically stable (see Figure 5)

4 Hopf Bifurcation

In this section we study the Hopf bifurcation of system (4)From the above discussion we know that there is no closedorbit surrounding1198640 or1198641 because the 119878-axis is invariantwith

0 50 100 1500

50

100

150

S

I Elowast

Figure 5The unique endemic equilibrium 119864lowast is globally asymptot-ically stable when 119860 = 15 120572 = 001 119889 = 01 120583 = 001 120582 = 001119903 = 08 and 120576 = 01

respect to system (4) and 1198641 is always a saddle ThereforeHopf bifurcation can only occur at 1198642 Set

120590 = [minus120582120572 minus1198881 (1198882 + 21198884)

11988612]1198634

+ [minus 120582120572119886211minus 21205821205721198861111988612 + (1198882 minus

211988611119888111988612)

times (119886111198882 minus119886211119888111988612)

minus (21198862111198881

11988612minus 119886111198882 + 2119888411988611 minus 119888311988612)

times (1198884 minus11988611119888111988612)]1198632

+ 11988611 (119886211119888111988612minus 119886111198882 + 119888411988611 minus 119888311988612)

times (119886121198883 minus 2119886111198884 minus 119886111198882)

(46)

where 119886119894119895 (119894 119895 = 1 2) 119888119896 (119896 = 1 2 3 4) and 119863 are defined by(50) and (52)

Theorem 11 System (4) undergoes a Hopf bifurcation if 120578 = 0Moreover if 120590 lt 0 there is a family of stable period orbits ofsystem (4) as 120578 decreases from 0 that is a supercritical Hopfbifurcation occurs if 120590 gt 0 there is a family of unstable periodorbits of system (4) as 120578 increases from 0 that is a subcriticalHopf bifurcation occurs

Proof The proof of Theorem 7 shows that tr(1198722) = 0 if andonly if 120578 = 0 and det(1198722) gt 0 when 1198642 exists Therefore theeigenvalues of 1198722 are a pair of pure imaginary roots if andonly if 120578 = 0 The direct calculations show that

119889 (tr (1198722))119889120578

100381610038161003816100381610038161003816100381610038161003816120578=0= minus

1

21198863 (1 + 1205721198682)2lt 0 (47)

8 Mathematical Problems in Engineering

ByTheorem 342 in [24] 120578 = 0 is the Hopf bifurcation pointfor system (4)

To be concise in notations rescale (4) by 120591 = 119905(1 + 120572119868)For simplicity we still use 119905 instead of 120591 Then we obtain

119889119878

119889119905= (119860 minus 119889119878 minus 120582119878119868 + 120576119868) (1 + 120572119868) + 119903119868

119889119868

119889119905= 120582119878119868 (1 + 120572119868) minus (119889 + 120576 + 120583) 119868 (1 + 120572119868) minus 119903119868

(48)

Let 119909 = 119878 minus 1198782 and 119910 = 119868 minus 1198682 then (48) becomes

119889119909

119889119905= 11988611119909 + 11988612119910 + 1198881119910

2 + 1198882119909119910 minus 1205821205721199091199102

119889119910

119889119905= 11988621119909 + 11988622119910 + 1198883119909119910 + 1198884119910

2 + 1205821205721199091199102

(49)

where

11988611 = minus (119889 + 1205821198682) (1 + 1205721198682)

11988612 = (minus1205821198782 + 119903) (1 + 1205721198682)

+ 120572 (119860 minus 1198891198782 minus 12058211987821198682 + 1199031198682) + 120576

11988621 = 1205821198682 (1 + 1205721198682)

11988622 = [1205821198782 minus (119889 + 119903 + 120583)] (1 + 1205721198682)

+ 120572 [12058211987821198682 minus (119889 + 119903 + 120583) 1198682] minus 120576

1198881 = 120572 (minus1205821198782 + 119903)

1198882 = 120572 (minus119889 minus 1205821198682) minus 120582 (1 + 1205721198682)

1198883 = 120582 (1 + 1205721198682) + 1205821198682120572

1198884 = [1205821198782 minus (119889 + 119903 + 120583)] 120572

(50)

Let 119864lowast denote the origin of 119909-119910 plane Since 1198642 = (1198782 1198682)satisfies (6) we obtain

det (119872 (119864lowast))

= minus (119889 + 1205821198682) 1205721205761198682 + (119889 + 120583) (1 + 1205721198682

2)2

1205821198682 + 1205721205821205761198682

2

= 1198682 times 120595 (1198682)

(51)

From the proof of Theorem 7 it follows that 120595(1198682) is alwayspositive It is easy to verify that 11988611 + 11988622 = 0 if and only if120578 = 0 Set

119863 = radicdet (119872 (119864lowast)) (52)

and let 119906 = minus119909 and V = (11988611119863)119909+(11988612119863)119910 then the normalform of (48) for Hopf bifurcation reads

119889119906

119889119905= minus119863V + 119891 (119906 V)

119889V119889119905= 119863119906 + 119892 (119906 V)

(53)

where

119891 (119906 V) = (11988611119886121198882 minus119886211

119886212

1198881)1199062 minus11986321198881119886212

V2

+ (119863119888211988612minus 2119863119886111198881119886212

)119906V minus120582120572119886211

119886212

1199063

minus 211986311988611120582120572

119886212

1199062V minus1198632120582120572

119886212

119906V2

119892 (119906 V) =11988611119863(119886211

119886212

1198881 minus11988611119886121198882 +11988611119886121198884 minus 1198883)119906

2

+ (119863119886111198881119886212

+119863119888411988612) V2

+ (2119886211

119886212

1198881 minus11988611119886121198882 + 2

11988611119886121198884 minus 1198883)119906V

+120582120572119886211

11988612119863(1

11988612minus 1)1199063 +

21205821205721198861111988612

(1198861111988612minus 1)1199062V

+119863120582120572

11988612(1198861111988612minus 1)119906V2

(54)

Now we evaluate the first Lyapunov coefficient Γ of system(4) as follows

Γ =1

16[119891119906119906119906 + 119891119906VV + 119892119906119906V + 119892VVV] +

1

16119863

times [119891119906V (119891119906119906 + 119891VV) minus 119892119906V (119892119906119906 + 119892VV) minus 119891119906119906119892119906119906 + 119891VV119892VV]

(55)

where 119891119906V denotes (1205972119891120597119906120597V)(0 0) and so forth Then

Γ =1

81198862121198632[minus120582120572 minus

1198881 (1198882 + 21198884)

11988612]1198634

+ [minus 120582120572119886211minus 21205821205721198861111988612 + (1198882 minus

211988611119888111988612)

times (119886111198882 minus119886211119888111988612)

minus (21198862111198881

11988612minus 119886111198882 + 2119888411988611 minus 119888311988612)

times (1198884 minus11988611119888111988612)]1198632

+ 11988611 (119886211119888111988612minus 119886111198882 + 119888411988611 minus 119888311988612)

times (119886121198883 minus 2119886111198884 minus 119886111198882)

=120590

81198862121198632

(56)

Mathematical Problems in Engineering 9

Obviously the sign of Γ is determined by 120590 ByTheorem 342and (3411) in [24] the rest of the claims in Theorem 11 arevalid The proof is complete

5 Bogdanov-Takens Bifurcations

The purpose of this section is to study the Bogdanov-Takensbifurcation of (4) Now we assume the following

(1198601) 1198872 minus 4119886119888 = 0

(1198602) 21198861198892 + 119887120582120583 = 0

Then (6) admits a unique positive equilibrium 119864lowast = (119878lowast 119868lowast)where

119868lowast = minus119887

2119886 119878lowast =

119860 minus (119889 + 120583) 119868lowast

119889 (57)

The Jacobian matrix of system (4) at 119864lowast is

119872lowast = (

minus119889 minus 120582119868lowast minus120582119878lowast + 120576 +119903

(1 + 120572119868lowast)2

120582119868lowast 120582119878lowast minus (119889 + 120576 + 120583) minus119903

(1 + 120572119868lowast)2)

(58)

By (58) we have

det (119872lowast) = 119868lowast

(1 + 120572119868lowast)2[(119889 + 120583) 120582 (1 + 120572119868lowast)

2minus 119903119889120572]

= 0

(59)

because of

(1 + 120572119868lowast)2=41198862 minus 4119886120572119887 + 12057221198872

41198862

=41198862 minus 4119886120572119887 + 12057224119886119888

41198862

=119886 minus 120572119887 + 1205722119888

119886

=1205722119903119889

119886

(60)

Furthermore (1198602) implies that

tr (119872lowast) = 0 (61)

Thus (1198601) and (1198602) imply that the Jacobian matrix has azero eigenvalue withmultiplicity 2This suggests that (4)mayadmit a Bogdanov-Takens singularity

Theorem 12 Suppose that (1198601) (1198602) and 21198871 + 1198874 = 0 holdThen the endemic equilibrium 119864lowast = (119878lowast 119868lowast) of (4) is a cuspof codimension 2 that is it is a Bogdanov-Takens singularityHere 1198871 and 1198874 are defined by (65)

Proof Using the transformation of 119909 = 119868minus 119868lowast and 119910 = 119878minus 119878lowastsystem (4) becomes

119889119909

119889119905= 1198861119909 + 1198862119910 + 120582119909119910 + 11988611119909

2 minus 1198751 (119909)

119889119910

119889119905= minus11988621

1198862119909 minus 1198861119910 minus 120582119909119910 minus 11988611119909

2 + 1198751 (119909)

(62)

where 1198751(119909) is a smooth function of 119909 at least of the thirdorder and

1198861 = 120582119878lowast minus (119889 + 120576 + 120583) minus

119903

(1 + 120572119868lowast)2gt 0

1198862 = 120582119868lowast gt 0

11988611 =119903120572

(1 + 120572119868lowast)3gt 0

(63)

Set119883 = 119909 119884 = 1198861119909 + 1198862119910 Then (62) is transformed into

119889119883

119889119905= 119884 + 1198871119883

2 + 1198872119883119884 + 1198761 (119883)

119889119884

119889119905= 1198873119883

2 + 1198874119883119884 + 1198762 (119883)

(64)

where 119876119894(119883) are smooth functions of 119883 at least of the thirdorder and

1198871 = 11988611 minus12058211988611198862

1198872 =120582

1198862

1198873 = 119886111988611 minus12058211988621

1198862+ 1198861120582 minus 119886211988611

1198874 =12058211988611198862minus 120582

(65)

In order to obtain the canonical normal form we perform thetransformation of variables by

119906 = 119883 minus119887221198832 V = 119884 + 1198871119883

2 (66)

Then we obtain

119889119906

119889119905= V + 1198771 (119906)

119889V119889119905= 11988731199062 + (21198871 + 1198874) 119906V + 1198772 (119906)

(67)

where 119877119894(119906) are smooth functions of 119906 at least of the thirdorder Note that 1198873 lt 0 and 21198871 + 1198874 = 0 It follows from [33ndash35] that (4) admits a Bogdanov-Takens bifurcation

In the following wewill find the versal unfolding in termsof the original parameters in (4) In this way we will know theapproximate saddle-node Hopf and homoclinic bifurcation

10 Mathematical Problems in Engineering

curvesWe choose119860 and 119889 as bifurcation parameters Fix 120582 =1205820 120576 = 1205760 119903 = 1199030 120572 = 1205720 and 120583 = 1205830 Let 119860 = 1198600 + 1205791 and119889 = 1198890 + 1205792 where 1205791 and 1205792 are parameters which vary in asmall neighborhood of the origin

Suppose that 119860 = 1198600 119889 = 1198890 120582 = 1205820 120576 = 1205760 119903 = 1199030 120572 =1205720 and 120583 = 1205830 satisfy (1198601) and (1198602) Consider the followingsystem

119889119868

119889119905= 1205820119878119868 minus (1198890 + 1205792 + 1205760 + 1205830) 119868 minus

1199030119868

1 + 1205720119868

119889119878

119889119905= 1198600 + 1205791 minus (1198890 + 1205792) 119878 minus 1205820119878119868 + 1205760119868 +

1199030119868

1 + 1205720119868

(68)

By the transformations of 119909 = 119868 minus 119868lowast and 119910 = 119878 minus 119878lowast system(68) becomes

119889119909

119889119905= minus 1205792119868

lowast + 1198881119909 + 1198882119910 + 119888111199092 + 1205820119909119910 minus 1199081 (119909)

119889119910

119889119905= (1205791 minus 1205792119878

lowast) + 1198883119909 + 1198884119910 minus 119888111199092 minus 1205820119909119910 + 1199081 (119909)

(69)

where 1199081(119909) is a smooth function of 119909 at least of the thirdorder and

1198881 = 1205820119878lowast minus (1198890 + 1205792 + 1205760 + 1205830) minus

1199030

(1 + 1205720119868lowast)2

1198882 = 1205820119868lowast

1198883 = minus 1205820119878lowast + 1205760 +

1199030

(1 + 1205720119868lowast)2

1198884 = minus (1198890 + 1205792) minus 1205820119868lowast

11988811 =11990301205720

(1 + 1205720119868lowast)3

(70)

Making the change of variables 119883 = 119909 119884 = minus1205792119868lowast + 1198881119909 +

1198882119910 + 119888111199092 + 1205820119909119910 minus 1199081(119909) and rewriting 119883 119884 as 119909 and 119910

respectively we have

119889119909

119889119905= 119910

119889119910

119889119905= 1198900+ 1198901119909 + 1198902119910 + 11989011119909

2+ 11989012119909119910 + 119890221199102+ 1199082 (119909 119910 120579)

(71)

where 120579 = (1205791 1205792) 1199082(119909 119910 120579) is a smooth function of 119909 1199101205791 and 1205792 at least of the third order and

1198900 = 1198882 (1205791 minus 1205792119878lowast) + 11988841205792119868

lowast

1198901 = 1205820 (1205791 minus 1205792119878lowast) + 11988821198883 minus 11988811198884 minus 12058201205792119868

lowast

1198902 = 1198881 + 1198884 + 12058201205792119868lowast

1198882

11989011 = 12058201198883 minus 119888411988811 minus 119888111198882 + 11988811205820

11989012 = minus 1205820 + 211988811 + 1205820minus11988811198882 minus 12058201205792119868

lowast

11988822

11989022 =12058201198882

(72)

Next introduce a new time variable 120591 by 119889119905 = (1 minus(12058201198882)119909)119889120591 Rewriting 120591 as 119905 we obtain

119889119909

119889119905= 119910(1 minus

12058201198882119909)

119889119910

119889119905= (1 minus

12058201198882119909)

times (1198900 + 1198901119909 + 1198902119910 + 119890111199092

+ 11989012119909119910 + 119890221199102 + 1199082 (119909 119910 120579))

(73)

Let119883 = 119909 119884 = 119910(1minus(12058201198882)119909) and rename119883 and119884 as 119909 and119910 we have

119889119909

119889119905= 119910

119889119910

119889119905= 1198900 + 1198921119909 + 1198902119910 + 11989211119909

2 + 11989212119909119910 + 1199083 (119909 119910 120579)

(74)

where 120579 = (1205791 1205792) 1199083(119909 119910 120579) is a smooth function of 119909 1199101205791 and 1205792 at least of the third order and

1198921 = minus 2119890012058201198882+ 1198901

11989211 = 11989011 minus 2119890112058201198882+11989001205822

0

11988822

11989212 = 11989012 minus119890212058201198882

(75)

Now we assume that 11989211 = 0 and 11989212 = 0 when 120582119894 are smallSet 119909 = 119883 minus 119890211989212 and rewrite119883 as 119909 we can get

119889119909

119889119905= 119910

119889119910

119889119905= 1198910 + 1198911119909 + 11989211119909

2 + 11989212119909119910 + 1199084 (119909 119910 120579)

(76)

where 120579 = (1205791 1205792) 1199084(119909 119910 120579) is a smooth function of 119909 1199101205791 and 1205792 at least of the third order and

1198910 = 1198900 minus1198921119890211989212+119892111198902

2

119892212

1198911 = 1198921 minus211989211119890211989212

(77)

Mathematical Problems in Engineering 11

Making the final change of variables by 119883 = 11989221211990911989211 119884 =

119892312119910119892211 and 120591 = 1198921111990511989212 and then denoting them again by

119909 119910 and 119905 respectively we obtain

119889119909

119889119905= 119910

119889119910

119889119905= 1205911 (1205791 1205792) + 1205912 (1205791 1205792) 119909 + 119909

2 + 119909119910 + 1199085 (119909 119910 120579)

(78)

where 120579 = (1205791 1205792) and 1199085(119909 119910 120579) is a smooth function of 119909119910 1205791 and 1205792 at least of the third order

We substitute values of 1205720 = 1 1205820 = 14 1198890 = 1101205760 = 110 1205830 = 1100 1198600 = 537500 and 1199030 = 558 forthe above system and these values satisfy conditions (1198601) and(1198602) And we obtain the following equations

1205911 (1205791 1205792) =11989101198924

12

119892311

= minus237291605

851841205791 +

25485118377

8518401205792

+ 119874 (10038161003816100381610038161205791 1205792

10038161003816100381610038162)

1205912 (1205791 1205792) =11989111198922

12

119892211

= minus172225

19361205791 +

3983253

38721205792 + 119874 (

10038161003816100381610038161205791 120579210038161003816100381610038162)

(79)

where 11989211 = minus11500 minus (116)1205791 + (341800)1205792 + (14)1205792

2lt 0

and 11989212 = minus83200 + (14)1205792 lt 0 in a small neighborhood of(1205791 1205792) = (0 0) Let

119869 = (

12059712059111205971205791

12059712059111205971205792

12059712059121205971205791

12059712059121205971205792

) (80)

And after simple calculation we obtain that

det(119869)|120582=0 = minus16839398748825

82458112= 0 (81)

Thus 1205911 and 1205912 are regular maps in a small neighborhoodof (1205791 1205792) = (0 0) By the Bogdanov and Takens bifurcationtheorems [36] we obtain the following conclusion

Theorem 13 Suppose that1198600 1198890 1205820 1205760 1199030 1205720 and 1205830 satisfy(1198601) (1198602) 11989211 = 0 and 11989212 = 0 when 120579119894 are small Then (4)admits the following bifurcation behavior

(1) There is a saddle-node bifurcation curve 119878119873 =

(1205791 1205792) 4119891011989211 = 11989121+ 119900(|(1205791 1205792)|

2) 1198911 =0 = (1205791 1205792) (59076250)1205792 minus (11125)1205791 +(2313400)12057911205792 minus (516)120579

2

1minus (102863740000)1205792

2+

119900(|(1205791 1205792)|2) = 0 (4799116600)1205792 minus (14)1205791 + (25

83)12057911205792 minus (1364513778)1205792

2+ 119900(|(1205791 1205792)|

2) = 0

I

I

O

O 1205791

1205792

SN+

SN+

SNminus

SNminus

II H

III HL

IVIVHL

IIIII

H

Figure 6The bifurcation set and the corresponding phase portraitsfor system (68)

(2) There is a Hopf bifurcation curve 119867 = (1205791 1205792)

1198910 + 119900(|(1205791 1205792)|2) = 0 1198911 lt 0 = (1205791 1205792) minus(537

50)1205792+1205791 + (5083)12057911205792 minus (746676889)12057922 + 119900(|(12057911205792)|2) = 0 (4799116600)1205792 minus (14)1205791+(2583)12057911205792 minus

(1364513778)12057922+ 119900(|(1205791 1205792)|

2) lt 0

(3) There is a homoclinic bifurcation curve119867119871 = (1205791 1205792) 25119892111198910 + 6119891

2

1= 119900(|(1205791 1205792)|

2) 1198911 lt 0 = (1205791 1205792) (59071000)1205792 minus (1120)1205791 + (611979 33200)12057911205792 minus(1916)1205792

1minus (16075834839275560000)1205792

2+ 119900(|(1205791

1205792)|2) = 0 (4799116600)1205792minus(14)1205791+(2583)12057911205792minus

(1364513778)12057922+ 119900(|(1205791 1205792)|

2) lt 0

6 Numerical Simulations

When 120572 = 1 120582 = 14 119889 = 110 120576 = 110 120583 = 1100119860 = 537500 and 119903 = 558 120590 gt 0 By applying PPLANE8and Photoshop software the (1205791 1205792)-plane near the origin isdivided into 4 regions by these bifurcation curves as shown inFigure 6 Fix 1205791 lt 0 and decrease 1205792 from 0 our conclusionsare summarized as follows

(a) When (1205791 1205792) lies in region I there is no positiveequilibrium which implies that the positive orbitsof (4) meet the positive 119878-axis in finite time andtherefore the disease disappears

(b) There is a saddle-node point when (1205791 1205792) lies oncurve SN

(c) When (1205791 1205792) crosses SN into region II two positiveequilibria which are an unstable focus and a saddleappear

(d) When the parameters lie on the curve H there is alsoa saddle and an unstable focus An unstable limit cycleappears with the parameters crossing H into III

(e) A homoclinic cycle appears as the parameters passingIII into HL through the homoclinic bifurcation

12 Mathematical Problems in Engineering

0 5 10 15 20 25 30 35 40

0

1

2

3

4

5

6

7

8

9

10

S

I

Figure 7 When (1205791 1205792) = (0 0) the unique positive equilibrium isa cusp of codimension 2

3 35 45 55 65 754 5 6 7 8

1

2

3

4

5

6

7

8

S

I

Figure 8 When (1205791 1205792) = (minus01 minus00093) lies in region I there isno positive equilibrium

(f) The homoclinic loop breakswith the parameter cross-ing HL into IV and a saddle and a stable focus appear

Furthermore using PPLANE8 some numerical simu-lations of system (68) are depicted in Figures 7ndash10 When(1205791 1205792) = (0 0) that is (1198600 1198890) = (537500 110) there isa unique positive equilibrium (119878lowast 119868lowast) = (31750 4) whichis a cusp of codimension 2 (Figure 7) When (1205791 1205792) =(minus01 minus00093) lies in region I there is no positive equi-librium (Figure 8) When (1205791 1205792) = (minus002 minus0001863798)there is a homoclinic loop (Figure 9) When (1205791 1205792) =(minus002 minus00020) lies in region IV the homoclinic loopis broken and there is a stable focus and a saddle(Figure 10)

6 605 61 615 62 625 63 635 64 645 653536373839

44142434445

S

I

Figure 9 When (1205791 1205792) = (minus002 minus0001863798) there is ahomoclinic loop

0 1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

7

8

9

10

S

I

Figure 10 When (1205791 1205792) = (minus002 minus00020) lies in region IV thereis a stable focus and a saddle

7 Discussion

In this paper we focus on the bifurcation analysis of an SISepidemic model with bilinear incidence rate and saturatedtreatment Generally speaking in many epidemic modelsthe basic reproduction number which is the key conceptin epidemiology can be decreased below unity to eradicatethe disease However in our model the basic reproductionnumber below 1 is not enough to eradicate the diseaseAccording to our analysis in this paper we demonstrate notonly the global stability of the disease-free equilibrium when1198770 lt 119877

lowast

0and the local asymptotic stability of the endemic

equilibrium 1198642 when 119877lowast

0lt 1198770 lt 1 but also the global

stability of the unique endemic equilibrium 119864lowast when 1198770 gt 1Moreover it has been shown in Corollary 4 that backwardbifurcations occur if the effect of the infected being delayedfor treatment is strong That is to say we should get prompttreatment for patients Through Figure 4 we can see thatthere is a region such that the disease will persist if the initialposition lies in the region and disappear if the initial positionlies outside this region So in order to restrain the spread

Mathematical Problems in Engineering 13

of the disease governments should take timely measures tocontrol the initial infected individuals in a lower level Inaddition fromTheorem 8 we can see that the disease will dieout if 1198770 is small enough Therefore from the definition of1198770 one knows that we can decrease the incidence rate 120582 andincrease the cure rate 119903 so as to eradicate the diseases or tomake them controlled in a lower endemic steady state

The stability analysis of the model equilibria enables usto completely analyze their local bifurcation behavior suchas Hopf saddle-node and Bogdanov-Takens bifurcation Bycomputing the first Lyapunov coefficient we can determinethat the Hopf bifurcation is a supercritical Hopf bifurca-tion or a subcritical Hopf bifurcation We also show thatunder assumptions (1198601) and (1198602) the model undertakesBogdanov-Takens bifurcation that is there are saddle-nodebifurcation Hopf bifurcation and homoclinic bifurcationin the system The normal norm of the Bogdanov-Takensbifurcation is derived in Section 5 which is very helpful toobtain the three kinds of bifurcation curves By analyticaltechniques 119860 and 119889 are chosen as bifurcation parametersand other parameters are fixed and we easily get a clearpicture about the rich dynamics behaviors of our modelThrough studying the bifurcations of the SIS epidemicmodelwe are suggesting that in order to eradicate the diseasemore medical facilities as well as medical professionals areneeded and the medical standard needs to be improved aswell

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research was supported by the National Natural ScienceFoundation of China (no 11001041 no 10926105 no 11101283no 11271065 and no 11201360) SRFDP (no 200802001008)and Scientific Research Innovation Project of ShanghaiMunicipal Education Commission (12YZ173)

References

[1] M E Alexander and S M Moghadas ldquoPeriodicity in an epi-demic model with a generalized non-linear incidencerdquo Mathe-matical Biosciences vol 189 no 1 pp 75ndash96 2004

[2] M E Alexander and S M Moghadas ldquoBifurcation analysisof SIRS epidemic model with generalized incidencerdquo SIAMJournal on Applied Mathematics vol 65 no 5 pp 1794ndash18162005

[3] J Arino C C McCluskey and P van den Driessche ldquoGlobalresults for an epidemic model with vaccination that exhibitsbackward bifurcationrdquo SIAM Journal on Applied Mathematicsvol 64 no 1 pp 260ndash276 2003

[4] W R Derrick and P van den Driessche ldquoHomoclinic orbitsin a disease transmission model with nonlinear incidence andnonconstant populationrdquo Discrete and Continuous DynamicalSystems B vol 3 no 2 pp 299ndash309 2003

[5] F Brauer ldquoBackward bifurcations in simple vaccination mod-elsrdquo Journal of Mathematical Analysis and Applications vol 298no 2 pp 418ndash431 2004

[6] K P Hadeler and P van den Driessche ldquoBackward bifurcationin epidemic controlrdquo Mathematical Biosciences vol 146 no 1pp 15ndash35 1997

[7] Z X Hu S Liu and H Wang ldquoBackward bifurcation of anepidemic model with standard incidence rate and treatmentraterdquo Nonlinear Analysis Real World Applications vol 9 no 5pp 2302ndash2312 2008

[8] J Hui and D M Zhu ldquoGlobal stability and periodicity onSIS epidemic models with backward bifurcationrdquo Computers ampMathematics with Applications vol 50 no 8-9 pp 1271ndash12902005

[9] Y Jin W D Wang and S W Xiao ldquoAn SIRS model with anonlinear incidence raterdquoChaos Solitons amp Fractals vol 34 no5 pp 1482ndash1497 2007

[10] G H Li and W D Wang ldquoBifurcation analysis of an epidemicmodel with nonlinear incidencerdquo Applied Mathematics andComputation vol 214 no 2 pp 411ndash423 2009

[11] X Z Li W S Li and M Ghosh ldquoStability and bifurcation ofan SIS epidemic model with treatmentrdquo Chaos Solitons ampampFractals vol 42 no 5 pp 2822ndash2832 2009

[12] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1986

[13] X B Liu and L J Yang ldquoStability analysis of an SEIQV epidemicmodel with saturated incidence raterdquo Nonlinear Analysis RealWorld Applications vol 13 no 6 pp 2671ndash2679 2012

[14] M Martcheva and H R Thieme ldquoProgression age enhancedbackward bifurcation in an epidemic model with super-infectionrdquo Journal of Mathematical Biology vol 46 no 5 pp385ndash424 2003

[15] S G Ruan and W D Wang ldquoDynamical behavior of anepidemic model with a nonlinear incidence raterdquo Journal ofDifferential Equations vol 188 no 1 pp 135ndash163 2003

[16] W D Wang and S G Ruan ldquoBifurcation in an epidemicmodel with constant removal rate of the infectivesrdquo Journal ofMathematical Analysis and Applications vol 291 no 2 pp 775ndash793 2004

[17] Y L Tang D Q Huang S G Ruan and W N ZhangldquoCoexistence of limit cycles and homoclinic loops in a SIRSmodel with a nonlinear incidence raterdquo SIAM Journal onApplied Mathematics vol 69 no 2 pp 621ndash639 2008

[18] P van den Driessche and J Watmough ldquoA simple SIS epidemicmodel with a backward bifurcationrdquo Journal of MathematicalBiology vol 40 no 6 pp 525ndash540 2000

[19] J J Wei and J A Cui ldquoDynamics of SIS epidemic model withthe standard incidence rate and saturated treatment functionrdquoInternational Journal of Biomathematics vol 5 no 3 Article ID1260003 18 pages 2012

[20] W D Wang ldquoBackward bifurcation of an epidemic model withtreatmentrdquoMathematical Biosciences vol 201 no 1-2 pp 58ndash712006

[21] Z W Wang ldquoBackward bifurcation in simple SIS modelrdquo ActaMathematicae Applicatae Sinica vol 25 no 1 pp 127ndash136 2009

[22] L Xue and C Scoglio ldquoThe network level reproduction numberfor infectious diseases with both vertical and horizontal trans-missionrdquo Mathematical Biosciences vol 243 no 1 pp 67ndash802013

14 Mathematical Problems in Engineering

[23] X Zhang and X N Liu ldquoBackward bifurcation of an epidemicmodel with saturated treatment functionrdquo Journal ofMathemat-ical Analysis and Applications vol 348 no 1 pp 433ndash443 2008

[24] J Guckenheimer and P J Holmes Nonlinear OscillationsDynamical Systems and Bifurcation of Vector Fields SpringerNew York NY USA 1996

[25] X Zhang and X N Liu ldquoBackward bifurcation and globaldynamics of an SIS epidemic model with general incidence rateand treatmentrdquoNonlinearAnalysis RealWorldApplications vol10 no 2 pp 565ndash575 2009

[26] M A Safi A B Gumel and E H Elbasha ldquoQualitative analysisof an age-structured SEIR epidemic model with treatmentrdquoApplied Mathematics and Computation vol 219 no 22 pp10627ndash10642 2013

[27] L H Zhou and M Fan ldquoDynamics of an SIR epidemic modelwith limited medical resources revisitedrdquo Nonlinear AnalysisReal World Applications vol 13 no 1 pp 312ndash324 2012

[28] X Y Zhou and J A Cui ldquoAnalysis of stability and bifurcation foran SEIR epidemicmodel with saturated recovery raterdquoCommu-nications inNonlinear Science andNumerical Simulation vol 16no 11 pp 4438ndash4450 2011

[29] D Lacitignola ldquoSaturated treatments and measles resurgenceepisodes in South Africa a possible linkagerdquo MathematicalBiosciences and Engineering MBE vol 10 no 4 pp 1135ndash11572013

[30] J C Eckalbar and W L Eckalbar ldquoDynamics of an epidemicmodel with quadratic treatmentrdquo Nonlinear Analysis RealWorld Applications vol 12 no 1 pp 320ndash332 2011

[31] F Brauer ldquoBackward bifurcations in simple vaccinationtreat-ment modelsrdquo Journal of Biological Dynamics vol 5 no 5 pp410ndash418 2011

[32] M W Hirsch S Smale and R L Devaney Differential Equa-tions Dynamical Systems and an Introduction to Chaos Aca-demic Press New York NY USA 2003

[33] R Bogdanov ldquoBifurcations of a limit cycle for a family of vectorfields on the planrdquo SelectaMathematica vol 1 pp 373ndash388 1981

[34] R Bogdanov ldquoVersal deformations of a singular point on theplan in the case of zero eigen-valuesrdquo Selecta MathematicaSovietica vol 1 pp 389ndash421 1981

[35] F Takens ldquoForced oscillations and bifurcationrdquo in Applicationsof Global Analysis I vol 3 pp 1ndash59 Communications of theMathematical Institute Rijksuniversiteit Utrecht 1974

[36] Y A Kuznetsov Elements of Applied Bifurcation TheorySpringer New York NY USA 2nd edition 1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

ByTheorem 342 in [24] 120578 = 0 is the Hopf bifurcation pointfor system (4)

To be concise in notations rescale (4) by 120591 = 119905(1 + 120572119868)For simplicity we still use 119905 instead of 120591 Then we obtain

119889119878

119889119905= (119860 minus 119889119878 minus 120582119878119868 + 120576119868) (1 + 120572119868) + 119903119868

119889119868

119889119905= 120582119878119868 (1 + 120572119868) minus (119889 + 120576 + 120583) 119868 (1 + 120572119868) minus 119903119868

(48)

Let 119909 = 119878 minus 1198782 and 119910 = 119868 minus 1198682 then (48) becomes

119889119909

119889119905= 11988611119909 + 11988612119910 + 1198881119910

2 + 1198882119909119910 minus 1205821205721199091199102

119889119910

119889119905= 11988621119909 + 11988622119910 + 1198883119909119910 + 1198884119910

2 + 1205821205721199091199102

(49)

where

11988611 = minus (119889 + 1205821198682) (1 + 1205721198682)

11988612 = (minus1205821198782 + 119903) (1 + 1205721198682)

+ 120572 (119860 minus 1198891198782 minus 12058211987821198682 + 1199031198682) + 120576

11988621 = 1205821198682 (1 + 1205721198682)

11988622 = [1205821198782 minus (119889 + 119903 + 120583)] (1 + 1205721198682)

+ 120572 [12058211987821198682 minus (119889 + 119903 + 120583) 1198682] minus 120576

1198881 = 120572 (minus1205821198782 + 119903)

1198882 = 120572 (minus119889 minus 1205821198682) minus 120582 (1 + 1205721198682)

1198883 = 120582 (1 + 1205721198682) + 1205821198682120572

1198884 = [1205821198782 minus (119889 + 119903 + 120583)] 120572

(50)

Let 119864lowast denote the origin of 119909-119910 plane Since 1198642 = (1198782 1198682)satisfies (6) we obtain

det (119872 (119864lowast))

= minus (119889 + 1205821198682) 1205721205761198682 + (119889 + 120583) (1 + 1205721198682

2)2

1205821198682 + 1205721205821205761198682

2

= 1198682 times 120595 (1198682)

(51)

From the proof of Theorem 7 it follows that 120595(1198682) is alwayspositive It is easy to verify that 11988611 + 11988622 = 0 if and only if120578 = 0 Set

119863 = radicdet (119872 (119864lowast)) (52)

and let 119906 = minus119909 and V = (11988611119863)119909+(11988612119863)119910 then the normalform of (48) for Hopf bifurcation reads

119889119906

119889119905= minus119863V + 119891 (119906 V)

119889V119889119905= 119863119906 + 119892 (119906 V)

(53)

where

119891 (119906 V) = (11988611119886121198882 minus119886211

119886212

1198881)1199062 minus11986321198881119886212

V2

+ (119863119888211988612minus 2119863119886111198881119886212

)119906V minus120582120572119886211

119886212

1199063

minus 211986311988611120582120572

119886212

1199062V minus1198632120582120572

119886212

119906V2

119892 (119906 V) =11988611119863(119886211

119886212

1198881 minus11988611119886121198882 +11988611119886121198884 minus 1198883)119906

2

+ (119863119886111198881119886212

+119863119888411988612) V2

+ (2119886211

119886212

1198881 minus11988611119886121198882 + 2

11988611119886121198884 minus 1198883)119906V

+120582120572119886211

11988612119863(1

11988612minus 1)1199063 +

21205821205721198861111988612

(1198861111988612minus 1)1199062V

+119863120582120572

11988612(1198861111988612minus 1)119906V2

(54)

Now we evaluate the first Lyapunov coefficient Γ of system(4) as follows

Γ =1

16[119891119906119906119906 + 119891119906VV + 119892119906119906V + 119892VVV] +

1

16119863

times [119891119906V (119891119906119906 + 119891VV) minus 119892119906V (119892119906119906 + 119892VV) minus 119891119906119906119892119906119906 + 119891VV119892VV]

(55)

where 119891119906V denotes (1205972119891120597119906120597V)(0 0) and so forth Then

Γ =1

81198862121198632[minus120582120572 minus

1198881 (1198882 + 21198884)

11988612]1198634

+ [minus 120582120572119886211minus 21205821205721198861111988612 + (1198882 minus

211988611119888111988612)

times (119886111198882 minus119886211119888111988612)

minus (21198862111198881

11988612minus 119886111198882 + 2119888411988611 minus 119888311988612)

times (1198884 minus11988611119888111988612)]1198632

+ 11988611 (119886211119888111988612minus 119886111198882 + 119888411988611 minus 119888311988612)

times (119886121198883 minus 2119886111198884 minus 119886111198882)

=120590

81198862121198632

(56)

Mathematical Problems in Engineering 9

Obviously the sign of Γ is determined by 120590 ByTheorem 342and (3411) in [24] the rest of the claims in Theorem 11 arevalid The proof is complete

5 Bogdanov-Takens Bifurcations

The purpose of this section is to study the Bogdanov-Takensbifurcation of (4) Now we assume the following

(1198601) 1198872 minus 4119886119888 = 0

(1198602) 21198861198892 + 119887120582120583 = 0

Then (6) admits a unique positive equilibrium 119864lowast = (119878lowast 119868lowast)where

119868lowast = minus119887

2119886 119878lowast =

119860 minus (119889 + 120583) 119868lowast

119889 (57)

The Jacobian matrix of system (4) at 119864lowast is

119872lowast = (

minus119889 minus 120582119868lowast minus120582119878lowast + 120576 +119903

(1 + 120572119868lowast)2

120582119868lowast 120582119878lowast minus (119889 + 120576 + 120583) minus119903

(1 + 120572119868lowast)2)

(58)

By (58) we have

det (119872lowast) = 119868lowast

(1 + 120572119868lowast)2[(119889 + 120583) 120582 (1 + 120572119868lowast)

2minus 119903119889120572]

= 0

(59)

because of

(1 + 120572119868lowast)2=41198862 minus 4119886120572119887 + 12057221198872

41198862

=41198862 minus 4119886120572119887 + 12057224119886119888

41198862

=119886 minus 120572119887 + 1205722119888

119886

=1205722119903119889

119886

(60)

Furthermore (1198602) implies that

tr (119872lowast) = 0 (61)

Thus (1198601) and (1198602) imply that the Jacobian matrix has azero eigenvalue withmultiplicity 2This suggests that (4)mayadmit a Bogdanov-Takens singularity

Theorem 12 Suppose that (1198601) (1198602) and 21198871 + 1198874 = 0 holdThen the endemic equilibrium 119864lowast = (119878lowast 119868lowast) of (4) is a cuspof codimension 2 that is it is a Bogdanov-Takens singularityHere 1198871 and 1198874 are defined by (65)

Proof Using the transformation of 119909 = 119868minus 119868lowast and 119910 = 119878minus 119878lowastsystem (4) becomes

119889119909

119889119905= 1198861119909 + 1198862119910 + 120582119909119910 + 11988611119909

2 minus 1198751 (119909)

119889119910

119889119905= minus11988621

1198862119909 minus 1198861119910 minus 120582119909119910 minus 11988611119909

2 + 1198751 (119909)

(62)

where 1198751(119909) is a smooth function of 119909 at least of the thirdorder and

1198861 = 120582119878lowast minus (119889 + 120576 + 120583) minus

119903

(1 + 120572119868lowast)2gt 0

1198862 = 120582119868lowast gt 0

11988611 =119903120572

(1 + 120572119868lowast)3gt 0

(63)

Set119883 = 119909 119884 = 1198861119909 + 1198862119910 Then (62) is transformed into

119889119883

119889119905= 119884 + 1198871119883

2 + 1198872119883119884 + 1198761 (119883)

119889119884

119889119905= 1198873119883

2 + 1198874119883119884 + 1198762 (119883)

(64)

where 119876119894(119883) are smooth functions of 119883 at least of the thirdorder and

1198871 = 11988611 minus12058211988611198862

1198872 =120582

1198862

1198873 = 119886111988611 minus12058211988621

1198862+ 1198861120582 minus 119886211988611

1198874 =12058211988611198862minus 120582

(65)

In order to obtain the canonical normal form we perform thetransformation of variables by

119906 = 119883 minus119887221198832 V = 119884 + 1198871119883

2 (66)

Then we obtain

119889119906

119889119905= V + 1198771 (119906)

119889V119889119905= 11988731199062 + (21198871 + 1198874) 119906V + 1198772 (119906)

(67)

where 119877119894(119906) are smooth functions of 119906 at least of the thirdorder Note that 1198873 lt 0 and 21198871 + 1198874 = 0 It follows from [33ndash35] that (4) admits a Bogdanov-Takens bifurcation

In the following wewill find the versal unfolding in termsof the original parameters in (4) In this way we will know theapproximate saddle-node Hopf and homoclinic bifurcation

10 Mathematical Problems in Engineering

curvesWe choose119860 and 119889 as bifurcation parameters Fix 120582 =1205820 120576 = 1205760 119903 = 1199030 120572 = 1205720 and 120583 = 1205830 Let 119860 = 1198600 + 1205791 and119889 = 1198890 + 1205792 where 1205791 and 1205792 are parameters which vary in asmall neighborhood of the origin

Suppose that 119860 = 1198600 119889 = 1198890 120582 = 1205820 120576 = 1205760 119903 = 1199030 120572 =1205720 and 120583 = 1205830 satisfy (1198601) and (1198602) Consider the followingsystem

119889119868

119889119905= 1205820119878119868 minus (1198890 + 1205792 + 1205760 + 1205830) 119868 minus

1199030119868

1 + 1205720119868

119889119878

119889119905= 1198600 + 1205791 minus (1198890 + 1205792) 119878 minus 1205820119878119868 + 1205760119868 +

1199030119868

1 + 1205720119868

(68)

By the transformations of 119909 = 119868 minus 119868lowast and 119910 = 119878 minus 119878lowast system(68) becomes

119889119909

119889119905= minus 1205792119868

lowast + 1198881119909 + 1198882119910 + 119888111199092 + 1205820119909119910 minus 1199081 (119909)

119889119910

119889119905= (1205791 minus 1205792119878

lowast) + 1198883119909 + 1198884119910 minus 119888111199092 minus 1205820119909119910 + 1199081 (119909)

(69)

where 1199081(119909) is a smooth function of 119909 at least of the thirdorder and

1198881 = 1205820119878lowast minus (1198890 + 1205792 + 1205760 + 1205830) minus

1199030

(1 + 1205720119868lowast)2

1198882 = 1205820119868lowast

1198883 = minus 1205820119878lowast + 1205760 +

1199030

(1 + 1205720119868lowast)2

1198884 = minus (1198890 + 1205792) minus 1205820119868lowast

11988811 =11990301205720

(1 + 1205720119868lowast)3

(70)

Making the change of variables 119883 = 119909 119884 = minus1205792119868lowast + 1198881119909 +

1198882119910 + 119888111199092 + 1205820119909119910 minus 1199081(119909) and rewriting 119883 119884 as 119909 and 119910

respectively we have

119889119909

119889119905= 119910

119889119910

119889119905= 1198900+ 1198901119909 + 1198902119910 + 11989011119909

2+ 11989012119909119910 + 119890221199102+ 1199082 (119909 119910 120579)

(71)

where 120579 = (1205791 1205792) 1199082(119909 119910 120579) is a smooth function of 119909 1199101205791 and 1205792 at least of the third order and

1198900 = 1198882 (1205791 minus 1205792119878lowast) + 11988841205792119868

lowast

1198901 = 1205820 (1205791 minus 1205792119878lowast) + 11988821198883 minus 11988811198884 minus 12058201205792119868

lowast

1198902 = 1198881 + 1198884 + 12058201205792119868lowast

1198882

11989011 = 12058201198883 minus 119888411988811 minus 119888111198882 + 11988811205820

11989012 = minus 1205820 + 211988811 + 1205820minus11988811198882 minus 12058201205792119868

lowast

11988822

11989022 =12058201198882

(72)

Next introduce a new time variable 120591 by 119889119905 = (1 minus(12058201198882)119909)119889120591 Rewriting 120591 as 119905 we obtain

119889119909

119889119905= 119910(1 minus

12058201198882119909)

119889119910

119889119905= (1 minus

12058201198882119909)

times (1198900 + 1198901119909 + 1198902119910 + 119890111199092

+ 11989012119909119910 + 119890221199102 + 1199082 (119909 119910 120579))

(73)

Let119883 = 119909 119884 = 119910(1minus(12058201198882)119909) and rename119883 and119884 as 119909 and119910 we have

119889119909

119889119905= 119910

119889119910

119889119905= 1198900 + 1198921119909 + 1198902119910 + 11989211119909

2 + 11989212119909119910 + 1199083 (119909 119910 120579)

(74)

where 120579 = (1205791 1205792) 1199083(119909 119910 120579) is a smooth function of 119909 1199101205791 and 1205792 at least of the third order and

1198921 = minus 2119890012058201198882+ 1198901

11989211 = 11989011 minus 2119890112058201198882+11989001205822

0

11988822

11989212 = 11989012 minus119890212058201198882

(75)

Now we assume that 11989211 = 0 and 11989212 = 0 when 120582119894 are smallSet 119909 = 119883 minus 119890211989212 and rewrite119883 as 119909 we can get

119889119909

119889119905= 119910

119889119910

119889119905= 1198910 + 1198911119909 + 11989211119909

2 + 11989212119909119910 + 1199084 (119909 119910 120579)

(76)

where 120579 = (1205791 1205792) 1199084(119909 119910 120579) is a smooth function of 119909 1199101205791 and 1205792 at least of the third order and

1198910 = 1198900 minus1198921119890211989212+119892111198902

2

119892212

1198911 = 1198921 minus211989211119890211989212

(77)

Mathematical Problems in Engineering 11

Making the final change of variables by 119883 = 11989221211990911989211 119884 =

119892312119910119892211 and 120591 = 1198921111990511989212 and then denoting them again by

119909 119910 and 119905 respectively we obtain

119889119909

119889119905= 119910

119889119910

119889119905= 1205911 (1205791 1205792) + 1205912 (1205791 1205792) 119909 + 119909

2 + 119909119910 + 1199085 (119909 119910 120579)

(78)

where 120579 = (1205791 1205792) and 1199085(119909 119910 120579) is a smooth function of 119909119910 1205791 and 1205792 at least of the third order

We substitute values of 1205720 = 1 1205820 = 14 1198890 = 1101205760 = 110 1205830 = 1100 1198600 = 537500 and 1199030 = 558 forthe above system and these values satisfy conditions (1198601) and(1198602) And we obtain the following equations

1205911 (1205791 1205792) =11989101198924

12

119892311

= minus237291605

851841205791 +

25485118377

8518401205792

+ 119874 (10038161003816100381610038161205791 1205792

10038161003816100381610038162)

1205912 (1205791 1205792) =11989111198922

12

119892211

= minus172225

19361205791 +

3983253

38721205792 + 119874 (

10038161003816100381610038161205791 120579210038161003816100381610038162)

(79)

where 11989211 = minus11500 minus (116)1205791 + (341800)1205792 + (14)1205792

2lt 0

and 11989212 = minus83200 + (14)1205792 lt 0 in a small neighborhood of(1205791 1205792) = (0 0) Let

119869 = (

12059712059111205971205791

12059712059111205971205792

12059712059121205971205791

12059712059121205971205792

) (80)

And after simple calculation we obtain that

det(119869)|120582=0 = minus16839398748825

82458112= 0 (81)

Thus 1205911 and 1205912 are regular maps in a small neighborhoodof (1205791 1205792) = (0 0) By the Bogdanov and Takens bifurcationtheorems [36] we obtain the following conclusion

Theorem 13 Suppose that1198600 1198890 1205820 1205760 1199030 1205720 and 1205830 satisfy(1198601) (1198602) 11989211 = 0 and 11989212 = 0 when 120579119894 are small Then (4)admits the following bifurcation behavior

(1) There is a saddle-node bifurcation curve 119878119873 =

(1205791 1205792) 4119891011989211 = 11989121+ 119900(|(1205791 1205792)|

2) 1198911 =0 = (1205791 1205792) (59076250)1205792 minus (11125)1205791 +(2313400)12057911205792 minus (516)120579

2

1minus (102863740000)1205792

2+

119900(|(1205791 1205792)|2) = 0 (4799116600)1205792 minus (14)1205791 + (25

83)12057911205792 minus (1364513778)1205792

2+ 119900(|(1205791 1205792)|

2) = 0

I

I

O

O 1205791

1205792

SN+

SN+

SNminus

SNminus

II H

III HL

IVIVHL

IIIII

H

Figure 6The bifurcation set and the corresponding phase portraitsfor system (68)

(2) There is a Hopf bifurcation curve 119867 = (1205791 1205792)

1198910 + 119900(|(1205791 1205792)|2) = 0 1198911 lt 0 = (1205791 1205792) minus(537

50)1205792+1205791 + (5083)12057911205792 minus (746676889)12057922 + 119900(|(12057911205792)|2) = 0 (4799116600)1205792 minus (14)1205791+(2583)12057911205792 minus

(1364513778)12057922+ 119900(|(1205791 1205792)|

2) lt 0

(3) There is a homoclinic bifurcation curve119867119871 = (1205791 1205792) 25119892111198910 + 6119891

2

1= 119900(|(1205791 1205792)|

2) 1198911 lt 0 = (1205791 1205792) (59071000)1205792 minus (1120)1205791 + (611979 33200)12057911205792 minus(1916)1205792

1minus (16075834839275560000)1205792

2+ 119900(|(1205791

1205792)|2) = 0 (4799116600)1205792minus(14)1205791+(2583)12057911205792minus

(1364513778)12057922+ 119900(|(1205791 1205792)|

2) lt 0

6 Numerical Simulations

When 120572 = 1 120582 = 14 119889 = 110 120576 = 110 120583 = 1100119860 = 537500 and 119903 = 558 120590 gt 0 By applying PPLANE8and Photoshop software the (1205791 1205792)-plane near the origin isdivided into 4 regions by these bifurcation curves as shown inFigure 6 Fix 1205791 lt 0 and decrease 1205792 from 0 our conclusionsare summarized as follows

(a) When (1205791 1205792) lies in region I there is no positiveequilibrium which implies that the positive orbitsof (4) meet the positive 119878-axis in finite time andtherefore the disease disappears

(b) There is a saddle-node point when (1205791 1205792) lies oncurve SN

(c) When (1205791 1205792) crosses SN into region II two positiveequilibria which are an unstable focus and a saddleappear

(d) When the parameters lie on the curve H there is alsoa saddle and an unstable focus An unstable limit cycleappears with the parameters crossing H into III

(e) A homoclinic cycle appears as the parameters passingIII into HL through the homoclinic bifurcation

12 Mathematical Problems in Engineering

0 5 10 15 20 25 30 35 40

0

1

2

3

4

5

6

7

8

9

10

S

I

Figure 7 When (1205791 1205792) = (0 0) the unique positive equilibrium isa cusp of codimension 2

3 35 45 55 65 754 5 6 7 8

1

2

3

4

5

6

7

8

S

I

Figure 8 When (1205791 1205792) = (minus01 minus00093) lies in region I there isno positive equilibrium

(f) The homoclinic loop breakswith the parameter cross-ing HL into IV and a saddle and a stable focus appear

Furthermore using PPLANE8 some numerical simu-lations of system (68) are depicted in Figures 7ndash10 When(1205791 1205792) = (0 0) that is (1198600 1198890) = (537500 110) there isa unique positive equilibrium (119878lowast 119868lowast) = (31750 4) whichis a cusp of codimension 2 (Figure 7) When (1205791 1205792) =(minus01 minus00093) lies in region I there is no positive equi-librium (Figure 8) When (1205791 1205792) = (minus002 minus0001863798)there is a homoclinic loop (Figure 9) When (1205791 1205792) =(minus002 minus00020) lies in region IV the homoclinic loopis broken and there is a stable focus and a saddle(Figure 10)

6 605 61 615 62 625 63 635 64 645 653536373839

44142434445

S

I

Figure 9 When (1205791 1205792) = (minus002 minus0001863798) there is ahomoclinic loop

0 1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

7

8

9

10

S

I

Figure 10 When (1205791 1205792) = (minus002 minus00020) lies in region IV thereis a stable focus and a saddle

7 Discussion

In this paper we focus on the bifurcation analysis of an SISepidemic model with bilinear incidence rate and saturatedtreatment Generally speaking in many epidemic modelsthe basic reproduction number which is the key conceptin epidemiology can be decreased below unity to eradicatethe disease However in our model the basic reproductionnumber below 1 is not enough to eradicate the diseaseAccording to our analysis in this paper we demonstrate notonly the global stability of the disease-free equilibrium when1198770 lt 119877

lowast

0and the local asymptotic stability of the endemic

equilibrium 1198642 when 119877lowast

0lt 1198770 lt 1 but also the global

stability of the unique endemic equilibrium 119864lowast when 1198770 gt 1Moreover it has been shown in Corollary 4 that backwardbifurcations occur if the effect of the infected being delayedfor treatment is strong That is to say we should get prompttreatment for patients Through Figure 4 we can see thatthere is a region such that the disease will persist if the initialposition lies in the region and disappear if the initial positionlies outside this region So in order to restrain the spread

Mathematical Problems in Engineering 13

of the disease governments should take timely measures tocontrol the initial infected individuals in a lower level Inaddition fromTheorem 8 we can see that the disease will dieout if 1198770 is small enough Therefore from the definition of1198770 one knows that we can decrease the incidence rate 120582 andincrease the cure rate 119903 so as to eradicate the diseases or tomake them controlled in a lower endemic steady state

The stability analysis of the model equilibria enables usto completely analyze their local bifurcation behavior suchas Hopf saddle-node and Bogdanov-Takens bifurcation Bycomputing the first Lyapunov coefficient we can determinethat the Hopf bifurcation is a supercritical Hopf bifurca-tion or a subcritical Hopf bifurcation We also show thatunder assumptions (1198601) and (1198602) the model undertakesBogdanov-Takens bifurcation that is there are saddle-nodebifurcation Hopf bifurcation and homoclinic bifurcationin the system The normal norm of the Bogdanov-Takensbifurcation is derived in Section 5 which is very helpful toobtain the three kinds of bifurcation curves By analyticaltechniques 119860 and 119889 are chosen as bifurcation parametersand other parameters are fixed and we easily get a clearpicture about the rich dynamics behaviors of our modelThrough studying the bifurcations of the SIS epidemicmodelwe are suggesting that in order to eradicate the diseasemore medical facilities as well as medical professionals areneeded and the medical standard needs to be improved aswell

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research was supported by the National Natural ScienceFoundation of China (no 11001041 no 10926105 no 11101283no 11271065 and no 11201360) SRFDP (no 200802001008)and Scientific Research Innovation Project of ShanghaiMunicipal Education Commission (12YZ173)

References

[1] M E Alexander and S M Moghadas ldquoPeriodicity in an epi-demic model with a generalized non-linear incidencerdquo Mathe-matical Biosciences vol 189 no 1 pp 75ndash96 2004

[2] M E Alexander and S M Moghadas ldquoBifurcation analysisof SIRS epidemic model with generalized incidencerdquo SIAMJournal on Applied Mathematics vol 65 no 5 pp 1794ndash18162005

[3] J Arino C C McCluskey and P van den Driessche ldquoGlobalresults for an epidemic model with vaccination that exhibitsbackward bifurcationrdquo SIAM Journal on Applied Mathematicsvol 64 no 1 pp 260ndash276 2003

[4] W R Derrick and P van den Driessche ldquoHomoclinic orbitsin a disease transmission model with nonlinear incidence andnonconstant populationrdquo Discrete and Continuous DynamicalSystems B vol 3 no 2 pp 299ndash309 2003

[5] F Brauer ldquoBackward bifurcations in simple vaccination mod-elsrdquo Journal of Mathematical Analysis and Applications vol 298no 2 pp 418ndash431 2004

[6] K P Hadeler and P van den Driessche ldquoBackward bifurcationin epidemic controlrdquo Mathematical Biosciences vol 146 no 1pp 15ndash35 1997

[7] Z X Hu S Liu and H Wang ldquoBackward bifurcation of anepidemic model with standard incidence rate and treatmentraterdquo Nonlinear Analysis Real World Applications vol 9 no 5pp 2302ndash2312 2008

[8] J Hui and D M Zhu ldquoGlobal stability and periodicity onSIS epidemic models with backward bifurcationrdquo Computers ampMathematics with Applications vol 50 no 8-9 pp 1271ndash12902005

[9] Y Jin W D Wang and S W Xiao ldquoAn SIRS model with anonlinear incidence raterdquoChaos Solitons amp Fractals vol 34 no5 pp 1482ndash1497 2007

[10] G H Li and W D Wang ldquoBifurcation analysis of an epidemicmodel with nonlinear incidencerdquo Applied Mathematics andComputation vol 214 no 2 pp 411ndash423 2009

[11] X Z Li W S Li and M Ghosh ldquoStability and bifurcation ofan SIS epidemic model with treatmentrdquo Chaos Solitons ampampFractals vol 42 no 5 pp 2822ndash2832 2009

[12] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1986

[13] X B Liu and L J Yang ldquoStability analysis of an SEIQV epidemicmodel with saturated incidence raterdquo Nonlinear Analysis RealWorld Applications vol 13 no 6 pp 2671ndash2679 2012

[14] M Martcheva and H R Thieme ldquoProgression age enhancedbackward bifurcation in an epidemic model with super-infectionrdquo Journal of Mathematical Biology vol 46 no 5 pp385ndash424 2003

[15] S G Ruan and W D Wang ldquoDynamical behavior of anepidemic model with a nonlinear incidence raterdquo Journal ofDifferential Equations vol 188 no 1 pp 135ndash163 2003

[16] W D Wang and S G Ruan ldquoBifurcation in an epidemicmodel with constant removal rate of the infectivesrdquo Journal ofMathematical Analysis and Applications vol 291 no 2 pp 775ndash793 2004

[17] Y L Tang D Q Huang S G Ruan and W N ZhangldquoCoexistence of limit cycles and homoclinic loops in a SIRSmodel with a nonlinear incidence raterdquo SIAM Journal onApplied Mathematics vol 69 no 2 pp 621ndash639 2008

[18] P van den Driessche and J Watmough ldquoA simple SIS epidemicmodel with a backward bifurcationrdquo Journal of MathematicalBiology vol 40 no 6 pp 525ndash540 2000

[19] J J Wei and J A Cui ldquoDynamics of SIS epidemic model withthe standard incidence rate and saturated treatment functionrdquoInternational Journal of Biomathematics vol 5 no 3 Article ID1260003 18 pages 2012

[20] W D Wang ldquoBackward bifurcation of an epidemic model withtreatmentrdquoMathematical Biosciences vol 201 no 1-2 pp 58ndash712006

[21] Z W Wang ldquoBackward bifurcation in simple SIS modelrdquo ActaMathematicae Applicatae Sinica vol 25 no 1 pp 127ndash136 2009

[22] L Xue and C Scoglio ldquoThe network level reproduction numberfor infectious diseases with both vertical and horizontal trans-missionrdquo Mathematical Biosciences vol 243 no 1 pp 67ndash802013

14 Mathematical Problems in Engineering

[23] X Zhang and X N Liu ldquoBackward bifurcation of an epidemicmodel with saturated treatment functionrdquo Journal ofMathemat-ical Analysis and Applications vol 348 no 1 pp 433ndash443 2008

[24] J Guckenheimer and P J Holmes Nonlinear OscillationsDynamical Systems and Bifurcation of Vector Fields SpringerNew York NY USA 1996

[25] X Zhang and X N Liu ldquoBackward bifurcation and globaldynamics of an SIS epidemic model with general incidence rateand treatmentrdquoNonlinearAnalysis RealWorldApplications vol10 no 2 pp 565ndash575 2009

[26] M A Safi A B Gumel and E H Elbasha ldquoQualitative analysisof an age-structured SEIR epidemic model with treatmentrdquoApplied Mathematics and Computation vol 219 no 22 pp10627ndash10642 2013

[27] L H Zhou and M Fan ldquoDynamics of an SIR epidemic modelwith limited medical resources revisitedrdquo Nonlinear AnalysisReal World Applications vol 13 no 1 pp 312ndash324 2012

[28] X Y Zhou and J A Cui ldquoAnalysis of stability and bifurcation foran SEIR epidemicmodel with saturated recovery raterdquoCommu-nications inNonlinear Science andNumerical Simulation vol 16no 11 pp 4438ndash4450 2011

[29] D Lacitignola ldquoSaturated treatments and measles resurgenceepisodes in South Africa a possible linkagerdquo MathematicalBiosciences and Engineering MBE vol 10 no 4 pp 1135ndash11572013

[30] J C Eckalbar and W L Eckalbar ldquoDynamics of an epidemicmodel with quadratic treatmentrdquo Nonlinear Analysis RealWorld Applications vol 12 no 1 pp 320ndash332 2011

[31] F Brauer ldquoBackward bifurcations in simple vaccinationtreat-ment modelsrdquo Journal of Biological Dynamics vol 5 no 5 pp410ndash418 2011

[32] M W Hirsch S Smale and R L Devaney Differential Equa-tions Dynamical Systems and an Introduction to Chaos Aca-demic Press New York NY USA 2003

[33] R Bogdanov ldquoBifurcations of a limit cycle for a family of vectorfields on the planrdquo SelectaMathematica vol 1 pp 373ndash388 1981

[34] R Bogdanov ldquoVersal deformations of a singular point on theplan in the case of zero eigen-valuesrdquo Selecta MathematicaSovietica vol 1 pp 389ndash421 1981

[35] F Takens ldquoForced oscillations and bifurcationrdquo in Applicationsof Global Analysis I vol 3 pp 1ndash59 Communications of theMathematical Institute Rijksuniversiteit Utrecht 1974

[36] Y A Kuznetsov Elements of Applied Bifurcation TheorySpringer New York NY USA 2nd edition 1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Obviously the sign of Γ is determined by 120590 ByTheorem 342and (3411) in [24] the rest of the claims in Theorem 11 arevalid The proof is complete

5 Bogdanov-Takens Bifurcations

The purpose of this section is to study the Bogdanov-Takensbifurcation of (4) Now we assume the following

(1198601) 1198872 minus 4119886119888 = 0

(1198602) 21198861198892 + 119887120582120583 = 0

Then (6) admits a unique positive equilibrium 119864lowast = (119878lowast 119868lowast)where

119868lowast = minus119887

2119886 119878lowast =

119860 minus (119889 + 120583) 119868lowast

119889 (57)

The Jacobian matrix of system (4) at 119864lowast is

119872lowast = (

minus119889 minus 120582119868lowast minus120582119878lowast + 120576 +119903

(1 + 120572119868lowast)2

120582119868lowast 120582119878lowast minus (119889 + 120576 + 120583) minus119903

(1 + 120572119868lowast)2)

(58)

By (58) we have

det (119872lowast) = 119868lowast

(1 + 120572119868lowast)2[(119889 + 120583) 120582 (1 + 120572119868lowast)

2minus 119903119889120572]

= 0

(59)

because of

(1 + 120572119868lowast)2=41198862 minus 4119886120572119887 + 12057221198872

41198862

=41198862 minus 4119886120572119887 + 12057224119886119888

41198862

=119886 minus 120572119887 + 1205722119888

119886

=1205722119903119889

119886

(60)

Furthermore (1198602) implies that

tr (119872lowast) = 0 (61)

Thus (1198601) and (1198602) imply that the Jacobian matrix has azero eigenvalue withmultiplicity 2This suggests that (4)mayadmit a Bogdanov-Takens singularity

Theorem 12 Suppose that (1198601) (1198602) and 21198871 + 1198874 = 0 holdThen the endemic equilibrium 119864lowast = (119878lowast 119868lowast) of (4) is a cuspof codimension 2 that is it is a Bogdanov-Takens singularityHere 1198871 and 1198874 are defined by (65)

Proof Using the transformation of 119909 = 119868minus 119868lowast and 119910 = 119878minus 119878lowastsystem (4) becomes

119889119909

119889119905= 1198861119909 + 1198862119910 + 120582119909119910 + 11988611119909

2 minus 1198751 (119909)

119889119910

119889119905= minus11988621

1198862119909 minus 1198861119910 minus 120582119909119910 minus 11988611119909

2 + 1198751 (119909)

(62)

where 1198751(119909) is a smooth function of 119909 at least of the thirdorder and

1198861 = 120582119878lowast minus (119889 + 120576 + 120583) minus

119903

(1 + 120572119868lowast)2gt 0

1198862 = 120582119868lowast gt 0

11988611 =119903120572

(1 + 120572119868lowast)3gt 0

(63)

Set119883 = 119909 119884 = 1198861119909 + 1198862119910 Then (62) is transformed into

119889119883

119889119905= 119884 + 1198871119883

2 + 1198872119883119884 + 1198761 (119883)

119889119884

119889119905= 1198873119883

2 + 1198874119883119884 + 1198762 (119883)

(64)

where 119876119894(119883) are smooth functions of 119883 at least of the thirdorder and

1198871 = 11988611 minus12058211988611198862

1198872 =120582

1198862

1198873 = 119886111988611 minus12058211988621

1198862+ 1198861120582 minus 119886211988611

1198874 =12058211988611198862minus 120582

(65)

In order to obtain the canonical normal form we perform thetransformation of variables by

119906 = 119883 minus119887221198832 V = 119884 + 1198871119883

2 (66)

Then we obtain

119889119906

119889119905= V + 1198771 (119906)

119889V119889119905= 11988731199062 + (21198871 + 1198874) 119906V + 1198772 (119906)

(67)

where 119877119894(119906) are smooth functions of 119906 at least of the thirdorder Note that 1198873 lt 0 and 21198871 + 1198874 = 0 It follows from [33ndash35] that (4) admits a Bogdanov-Takens bifurcation

In the following wewill find the versal unfolding in termsof the original parameters in (4) In this way we will know theapproximate saddle-node Hopf and homoclinic bifurcation

10 Mathematical Problems in Engineering

curvesWe choose119860 and 119889 as bifurcation parameters Fix 120582 =1205820 120576 = 1205760 119903 = 1199030 120572 = 1205720 and 120583 = 1205830 Let 119860 = 1198600 + 1205791 and119889 = 1198890 + 1205792 where 1205791 and 1205792 are parameters which vary in asmall neighborhood of the origin

Suppose that 119860 = 1198600 119889 = 1198890 120582 = 1205820 120576 = 1205760 119903 = 1199030 120572 =1205720 and 120583 = 1205830 satisfy (1198601) and (1198602) Consider the followingsystem

119889119868

119889119905= 1205820119878119868 minus (1198890 + 1205792 + 1205760 + 1205830) 119868 minus

1199030119868

1 + 1205720119868

119889119878

119889119905= 1198600 + 1205791 minus (1198890 + 1205792) 119878 minus 1205820119878119868 + 1205760119868 +

1199030119868

1 + 1205720119868

(68)

By the transformations of 119909 = 119868 minus 119868lowast and 119910 = 119878 minus 119878lowast system(68) becomes

119889119909

119889119905= minus 1205792119868

lowast + 1198881119909 + 1198882119910 + 119888111199092 + 1205820119909119910 minus 1199081 (119909)

119889119910

119889119905= (1205791 minus 1205792119878

lowast) + 1198883119909 + 1198884119910 minus 119888111199092 minus 1205820119909119910 + 1199081 (119909)

(69)

where 1199081(119909) is a smooth function of 119909 at least of the thirdorder and

1198881 = 1205820119878lowast minus (1198890 + 1205792 + 1205760 + 1205830) minus

1199030

(1 + 1205720119868lowast)2

1198882 = 1205820119868lowast

1198883 = minus 1205820119878lowast + 1205760 +

1199030

(1 + 1205720119868lowast)2

1198884 = minus (1198890 + 1205792) minus 1205820119868lowast

11988811 =11990301205720

(1 + 1205720119868lowast)3

(70)

Making the change of variables 119883 = 119909 119884 = minus1205792119868lowast + 1198881119909 +

1198882119910 + 119888111199092 + 1205820119909119910 minus 1199081(119909) and rewriting 119883 119884 as 119909 and 119910

respectively we have

119889119909

119889119905= 119910

119889119910

119889119905= 1198900+ 1198901119909 + 1198902119910 + 11989011119909

2+ 11989012119909119910 + 119890221199102+ 1199082 (119909 119910 120579)

(71)

where 120579 = (1205791 1205792) 1199082(119909 119910 120579) is a smooth function of 119909 1199101205791 and 1205792 at least of the third order and

1198900 = 1198882 (1205791 minus 1205792119878lowast) + 11988841205792119868

lowast

1198901 = 1205820 (1205791 minus 1205792119878lowast) + 11988821198883 minus 11988811198884 minus 12058201205792119868

lowast

1198902 = 1198881 + 1198884 + 12058201205792119868lowast

1198882

11989011 = 12058201198883 minus 119888411988811 minus 119888111198882 + 11988811205820

11989012 = minus 1205820 + 211988811 + 1205820minus11988811198882 minus 12058201205792119868

lowast

11988822

11989022 =12058201198882

(72)

Next introduce a new time variable 120591 by 119889119905 = (1 minus(12058201198882)119909)119889120591 Rewriting 120591 as 119905 we obtain

119889119909

119889119905= 119910(1 minus

12058201198882119909)

119889119910

119889119905= (1 minus

12058201198882119909)

times (1198900 + 1198901119909 + 1198902119910 + 119890111199092

+ 11989012119909119910 + 119890221199102 + 1199082 (119909 119910 120579))

(73)

Let119883 = 119909 119884 = 119910(1minus(12058201198882)119909) and rename119883 and119884 as 119909 and119910 we have

119889119909

119889119905= 119910

119889119910

119889119905= 1198900 + 1198921119909 + 1198902119910 + 11989211119909

2 + 11989212119909119910 + 1199083 (119909 119910 120579)

(74)

where 120579 = (1205791 1205792) 1199083(119909 119910 120579) is a smooth function of 119909 1199101205791 and 1205792 at least of the third order and

1198921 = minus 2119890012058201198882+ 1198901

11989211 = 11989011 minus 2119890112058201198882+11989001205822

0

11988822

11989212 = 11989012 minus119890212058201198882

(75)

Now we assume that 11989211 = 0 and 11989212 = 0 when 120582119894 are smallSet 119909 = 119883 minus 119890211989212 and rewrite119883 as 119909 we can get

119889119909

119889119905= 119910

119889119910

119889119905= 1198910 + 1198911119909 + 11989211119909

2 + 11989212119909119910 + 1199084 (119909 119910 120579)

(76)

where 120579 = (1205791 1205792) 1199084(119909 119910 120579) is a smooth function of 119909 1199101205791 and 1205792 at least of the third order and

1198910 = 1198900 minus1198921119890211989212+119892111198902

2

119892212

1198911 = 1198921 minus211989211119890211989212

(77)

Mathematical Problems in Engineering 11

Making the final change of variables by 119883 = 11989221211990911989211 119884 =

119892312119910119892211 and 120591 = 1198921111990511989212 and then denoting them again by

119909 119910 and 119905 respectively we obtain

119889119909

119889119905= 119910

119889119910

119889119905= 1205911 (1205791 1205792) + 1205912 (1205791 1205792) 119909 + 119909

2 + 119909119910 + 1199085 (119909 119910 120579)

(78)

where 120579 = (1205791 1205792) and 1199085(119909 119910 120579) is a smooth function of 119909119910 1205791 and 1205792 at least of the third order

We substitute values of 1205720 = 1 1205820 = 14 1198890 = 1101205760 = 110 1205830 = 1100 1198600 = 537500 and 1199030 = 558 forthe above system and these values satisfy conditions (1198601) and(1198602) And we obtain the following equations

1205911 (1205791 1205792) =11989101198924

12

119892311

= minus237291605

851841205791 +

25485118377

8518401205792

+ 119874 (10038161003816100381610038161205791 1205792

10038161003816100381610038162)

1205912 (1205791 1205792) =11989111198922

12

119892211

= minus172225

19361205791 +

3983253

38721205792 + 119874 (

10038161003816100381610038161205791 120579210038161003816100381610038162)

(79)

where 11989211 = minus11500 minus (116)1205791 + (341800)1205792 + (14)1205792

2lt 0

and 11989212 = minus83200 + (14)1205792 lt 0 in a small neighborhood of(1205791 1205792) = (0 0) Let

119869 = (

12059712059111205971205791

12059712059111205971205792

12059712059121205971205791

12059712059121205971205792

) (80)

And after simple calculation we obtain that

det(119869)|120582=0 = minus16839398748825

82458112= 0 (81)

Thus 1205911 and 1205912 are regular maps in a small neighborhoodof (1205791 1205792) = (0 0) By the Bogdanov and Takens bifurcationtheorems [36] we obtain the following conclusion

Theorem 13 Suppose that1198600 1198890 1205820 1205760 1199030 1205720 and 1205830 satisfy(1198601) (1198602) 11989211 = 0 and 11989212 = 0 when 120579119894 are small Then (4)admits the following bifurcation behavior

(1) There is a saddle-node bifurcation curve 119878119873 =

(1205791 1205792) 4119891011989211 = 11989121+ 119900(|(1205791 1205792)|

2) 1198911 =0 = (1205791 1205792) (59076250)1205792 minus (11125)1205791 +(2313400)12057911205792 minus (516)120579

2

1minus (102863740000)1205792

2+

119900(|(1205791 1205792)|2) = 0 (4799116600)1205792 minus (14)1205791 + (25

83)12057911205792 minus (1364513778)1205792

2+ 119900(|(1205791 1205792)|

2) = 0

I

I

O

O 1205791

1205792

SN+

SN+

SNminus

SNminus

II H

III HL

IVIVHL

IIIII

H

Figure 6The bifurcation set and the corresponding phase portraitsfor system (68)

(2) There is a Hopf bifurcation curve 119867 = (1205791 1205792)

1198910 + 119900(|(1205791 1205792)|2) = 0 1198911 lt 0 = (1205791 1205792) minus(537

50)1205792+1205791 + (5083)12057911205792 minus (746676889)12057922 + 119900(|(12057911205792)|2) = 0 (4799116600)1205792 minus (14)1205791+(2583)12057911205792 minus

(1364513778)12057922+ 119900(|(1205791 1205792)|

2) lt 0

(3) There is a homoclinic bifurcation curve119867119871 = (1205791 1205792) 25119892111198910 + 6119891

2

1= 119900(|(1205791 1205792)|

2) 1198911 lt 0 = (1205791 1205792) (59071000)1205792 minus (1120)1205791 + (611979 33200)12057911205792 minus(1916)1205792

1minus (16075834839275560000)1205792

2+ 119900(|(1205791

1205792)|2) = 0 (4799116600)1205792minus(14)1205791+(2583)12057911205792minus

(1364513778)12057922+ 119900(|(1205791 1205792)|

2) lt 0

6 Numerical Simulations

When 120572 = 1 120582 = 14 119889 = 110 120576 = 110 120583 = 1100119860 = 537500 and 119903 = 558 120590 gt 0 By applying PPLANE8and Photoshop software the (1205791 1205792)-plane near the origin isdivided into 4 regions by these bifurcation curves as shown inFigure 6 Fix 1205791 lt 0 and decrease 1205792 from 0 our conclusionsare summarized as follows

(a) When (1205791 1205792) lies in region I there is no positiveequilibrium which implies that the positive orbitsof (4) meet the positive 119878-axis in finite time andtherefore the disease disappears

(b) There is a saddle-node point when (1205791 1205792) lies oncurve SN

(c) When (1205791 1205792) crosses SN into region II two positiveequilibria which are an unstable focus and a saddleappear

(d) When the parameters lie on the curve H there is alsoa saddle and an unstable focus An unstable limit cycleappears with the parameters crossing H into III

(e) A homoclinic cycle appears as the parameters passingIII into HL through the homoclinic bifurcation

12 Mathematical Problems in Engineering

0 5 10 15 20 25 30 35 40

0

1

2

3

4

5

6

7

8

9

10

S

I

Figure 7 When (1205791 1205792) = (0 0) the unique positive equilibrium isa cusp of codimension 2

3 35 45 55 65 754 5 6 7 8

1

2

3

4

5

6

7

8

S

I

Figure 8 When (1205791 1205792) = (minus01 minus00093) lies in region I there isno positive equilibrium

(f) The homoclinic loop breakswith the parameter cross-ing HL into IV and a saddle and a stable focus appear

Furthermore using PPLANE8 some numerical simu-lations of system (68) are depicted in Figures 7ndash10 When(1205791 1205792) = (0 0) that is (1198600 1198890) = (537500 110) there isa unique positive equilibrium (119878lowast 119868lowast) = (31750 4) whichis a cusp of codimension 2 (Figure 7) When (1205791 1205792) =(minus01 minus00093) lies in region I there is no positive equi-librium (Figure 8) When (1205791 1205792) = (minus002 minus0001863798)there is a homoclinic loop (Figure 9) When (1205791 1205792) =(minus002 minus00020) lies in region IV the homoclinic loopis broken and there is a stable focus and a saddle(Figure 10)

6 605 61 615 62 625 63 635 64 645 653536373839

44142434445

S

I

Figure 9 When (1205791 1205792) = (minus002 minus0001863798) there is ahomoclinic loop

0 1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

7

8

9

10

S

I

Figure 10 When (1205791 1205792) = (minus002 minus00020) lies in region IV thereis a stable focus and a saddle

7 Discussion

In this paper we focus on the bifurcation analysis of an SISepidemic model with bilinear incidence rate and saturatedtreatment Generally speaking in many epidemic modelsthe basic reproduction number which is the key conceptin epidemiology can be decreased below unity to eradicatethe disease However in our model the basic reproductionnumber below 1 is not enough to eradicate the diseaseAccording to our analysis in this paper we demonstrate notonly the global stability of the disease-free equilibrium when1198770 lt 119877

lowast

0and the local asymptotic stability of the endemic

equilibrium 1198642 when 119877lowast

0lt 1198770 lt 1 but also the global

stability of the unique endemic equilibrium 119864lowast when 1198770 gt 1Moreover it has been shown in Corollary 4 that backwardbifurcations occur if the effect of the infected being delayedfor treatment is strong That is to say we should get prompttreatment for patients Through Figure 4 we can see thatthere is a region such that the disease will persist if the initialposition lies in the region and disappear if the initial positionlies outside this region So in order to restrain the spread

Mathematical Problems in Engineering 13

of the disease governments should take timely measures tocontrol the initial infected individuals in a lower level Inaddition fromTheorem 8 we can see that the disease will dieout if 1198770 is small enough Therefore from the definition of1198770 one knows that we can decrease the incidence rate 120582 andincrease the cure rate 119903 so as to eradicate the diseases or tomake them controlled in a lower endemic steady state

The stability analysis of the model equilibria enables usto completely analyze their local bifurcation behavior suchas Hopf saddle-node and Bogdanov-Takens bifurcation Bycomputing the first Lyapunov coefficient we can determinethat the Hopf bifurcation is a supercritical Hopf bifurca-tion or a subcritical Hopf bifurcation We also show thatunder assumptions (1198601) and (1198602) the model undertakesBogdanov-Takens bifurcation that is there are saddle-nodebifurcation Hopf bifurcation and homoclinic bifurcationin the system The normal norm of the Bogdanov-Takensbifurcation is derived in Section 5 which is very helpful toobtain the three kinds of bifurcation curves By analyticaltechniques 119860 and 119889 are chosen as bifurcation parametersand other parameters are fixed and we easily get a clearpicture about the rich dynamics behaviors of our modelThrough studying the bifurcations of the SIS epidemicmodelwe are suggesting that in order to eradicate the diseasemore medical facilities as well as medical professionals areneeded and the medical standard needs to be improved aswell

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research was supported by the National Natural ScienceFoundation of China (no 11001041 no 10926105 no 11101283no 11271065 and no 11201360) SRFDP (no 200802001008)and Scientific Research Innovation Project of ShanghaiMunicipal Education Commission (12YZ173)

References

[1] M E Alexander and S M Moghadas ldquoPeriodicity in an epi-demic model with a generalized non-linear incidencerdquo Mathe-matical Biosciences vol 189 no 1 pp 75ndash96 2004

[2] M E Alexander and S M Moghadas ldquoBifurcation analysisof SIRS epidemic model with generalized incidencerdquo SIAMJournal on Applied Mathematics vol 65 no 5 pp 1794ndash18162005

[3] J Arino C C McCluskey and P van den Driessche ldquoGlobalresults for an epidemic model with vaccination that exhibitsbackward bifurcationrdquo SIAM Journal on Applied Mathematicsvol 64 no 1 pp 260ndash276 2003

[4] W R Derrick and P van den Driessche ldquoHomoclinic orbitsin a disease transmission model with nonlinear incidence andnonconstant populationrdquo Discrete and Continuous DynamicalSystems B vol 3 no 2 pp 299ndash309 2003

[5] F Brauer ldquoBackward bifurcations in simple vaccination mod-elsrdquo Journal of Mathematical Analysis and Applications vol 298no 2 pp 418ndash431 2004

[6] K P Hadeler and P van den Driessche ldquoBackward bifurcationin epidemic controlrdquo Mathematical Biosciences vol 146 no 1pp 15ndash35 1997

[7] Z X Hu S Liu and H Wang ldquoBackward bifurcation of anepidemic model with standard incidence rate and treatmentraterdquo Nonlinear Analysis Real World Applications vol 9 no 5pp 2302ndash2312 2008

[8] J Hui and D M Zhu ldquoGlobal stability and periodicity onSIS epidemic models with backward bifurcationrdquo Computers ampMathematics with Applications vol 50 no 8-9 pp 1271ndash12902005

[9] Y Jin W D Wang and S W Xiao ldquoAn SIRS model with anonlinear incidence raterdquoChaos Solitons amp Fractals vol 34 no5 pp 1482ndash1497 2007

[10] G H Li and W D Wang ldquoBifurcation analysis of an epidemicmodel with nonlinear incidencerdquo Applied Mathematics andComputation vol 214 no 2 pp 411ndash423 2009

[11] X Z Li W S Li and M Ghosh ldquoStability and bifurcation ofan SIS epidemic model with treatmentrdquo Chaos Solitons ampampFractals vol 42 no 5 pp 2822ndash2832 2009

[12] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1986

[13] X B Liu and L J Yang ldquoStability analysis of an SEIQV epidemicmodel with saturated incidence raterdquo Nonlinear Analysis RealWorld Applications vol 13 no 6 pp 2671ndash2679 2012

[14] M Martcheva and H R Thieme ldquoProgression age enhancedbackward bifurcation in an epidemic model with super-infectionrdquo Journal of Mathematical Biology vol 46 no 5 pp385ndash424 2003

[15] S G Ruan and W D Wang ldquoDynamical behavior of anepidemic model with a nonlinear incidence raterdquo Journal ofDifferential Equations vol 188 no 1 pp 135ndash163 2003

[16] W D Wang and S G Ruan ldquoBifurcation in an epidemicmodel with constant removal rate of the infectivesrdquo Journal ofMathematical Analysis and Applications vol 291 no 2 pp 775ndash793 2004

[17] Y L Tang D Q Huang S G Ruan and W N ZhangldquoCoexistence of limit cycles and homoclinic loops in a SIRSmodel with a nonlinear incidence raterdquo SIAM Journal onApplied Mathematics vol 69 no 2 pp 621ndash639 2008

[18] P van den Driessche and J Watmough ldquoA simple SIS epidemicmodel with a backward bifurcationrdquo Journal of MathematicalBiology vol 40 no 6 pp 525ndash540 2000

[19] J J Wei and J A Cui ldquoDynamics of SIS epidemic model withthe standard incidence rate and saturated treatment functionrdquoInternational Journal of Biomathematics vol 5 no 3 Article ID1260003 18 pages 2012

[20] W D Wang ldquoBackward bifurcation of an epidemic model withtreatmentrdquoMathematical Biosciences vol 201 no 1-2 pp 58ndash712006

[21] Z W Wang ldquoBackward bifurcation in simple SIS modelrdquo ActaMathematicae Applicatae Sinica vol 25 no 1 pp 127ndash136 2009

[22] L Xue and C Scoglio ldquoThe network level reproduction numberfor infectious diseases with both vertical and horizontal trans-missionrdquo Mathematical Biosciences vol 243 no 1 pp 67ndash802013

14 Mathematical Problems in Engineering

[23] X Zhang and X N Liu ldquoBackward bifurcation of an epidemicmodel with saturated treatment functionrdquo Journal ofMathemat-ical Analysis and Applications vol 348 no 1 pp 433ndash443 2008

[24] J Guckenheimer and P J Holmes Nonlinear OscillationsDynamical Systems and Bifurcation of Vector Fields SpringerNew York NY USA 1996

[25] X Zhang and X N Liu ldquoBackward bifurcation and globaldynamics of an SIS epidemic model with general incidence rateand treatmentrdquoNonlinearAnalysis RealWorldApplications vol10 no 2 pp 565ndash575 2009

[26] M A Safi A B Gumel and E H Elbasha ldquoQualitative analysisof an age-structured SEIR epidemic model with treatmentrdquoApplied Mathematics and Computation vol 219 no 22 pp10627ndash10642 2013

[27] L H Zhou and M Fan ldquoDynamics of an SIR epidemic modelwith limited medical resources revisitedrdquo Nonlinear AnalysisReal World Applications vol 13 no 1 pp 312ndash324 2012

[28] X Y Zhou and J A Cui ldquoAnalysis of stability and bifurcation foran SEIR epidemicmodel with saturated recovery raterdquoCommu-nications inNonlinear Science andNumerical Simulation vol 16no 11 pp 4438ndash4450 2011

[29] D Lacitignola ldquoSaturated treatments and measles resurgenceepisodes in South Africa a possible linkagerdquo MathematicalBiosciences and Engineering MBE vol 10 no 4 pp 1135ndash11572013

[30] J C Eckalbar and W L Eckalbar ldquoDynamics of an epidemicmodel with quadratic treatmentrdquo Nonlinear Analysis RealWorld Applications vol 12 no 1 pp 320ndash332 2011

[31] F Brauer ldquoBackward bifurcations in simple vaccinationtreat-ment modelsrdquo Journal of Biological Dynamics vol 5 no 5 pp410ndash418 2011

[32] M W Hirsch S Smale and R L Devaney Differential Equa-tions Dynamical Systems and an Introduction to Chaos Aca-demic Press New York NY USA 2003

[33] R Bogdanov ldquoBifurcations of a limit cycle for a family of vectorfields on the planrdquo SelectaMathematica vol 1 pp 373ndash388 1981

[34] R Bogdanov ldquoVersal deformations of a singular point on theplan in the case of zero eigen-valuesrdquo Selecta MathematicaSovietica vol 1 pp 389ndash421 1981

[35] F Takens ldquoForced oscillations and bifurcationrdquo in Applicationsof Global Analysis I vol 3 pp 1ndash59 Communications of theMathematical Institute Rijksuniversiteit Utrecht 1974

[36] Y A Kuznetsov Elements of Applied Bifurcation TheorySpringer New York NY USA 2nd edition 1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

curvesWe choose119860 and 119889 as bifurcation parameters Fix 120582 =1205820 120576 = 1205760 119903 = 1199030 120572 = 1205720 and 120583 = 1205830 Let 119860 = 1198600 + 1205791 and119889 = 1198890 + 1205792 where 1205791 and 1205792 are parameters which vary in asmall neighborhood of the origin

Suppose that 119860 = 1198600 119889 = 1198890 120582 = 1205820 120576 = 1205760 119903 = 1199030 120572 =1205720 and 120583 = 1205830 satisfy (1198601) and (1198602) Consider the followingsystem

119889119868

119889119905= 1205820119878119868 minus (1198890 + 1205792 + 1205760 + 1205830) 119868 minus

1199030119868

1 + 1205720119868

119889119878

119889119905= 1198600 + 1205791 minus (1198890 + 1205792) 119878 minus 1205820119878119868 + 1205760119868 +

1199030119868

1 + 1205720119868

(68)

By the transformations of 119909 = 119868 minus 119868lowast and 119910 = 119878 minus 119878lowast system(68) becomes

119889119909

119889119905= minus 1205792119868

lowast + 1198881119909 + 1198882119910 + 119888111199092 + 1205820119909119910 minus 1199081 (119909)

119889119910

119889119905= (1205791 minus 1205792119878

lowast) + 1198883119909 + 1198884119910 minus 119888111199092 minus 1205820119909119910 + 1199081 (119909)

(69)

where 1199081(119909) is a smooth function of 119909 at least of the thirdorder and

1198881 = 1205820119878lowast minus (1198890 + 1205792 + 1205760 + 1205830) minus

1199030

(1 + 1205720119868lowast)2

1198882 = 1205820119868lowast

1198883 = minus 1205820119878lowast + 1205760 +

1199030

(1 + 1205720119868lowast)2

1198884 = minus (1198890 + 1205792) minus 1205820119868lowast

11988811 =11990301205720

(1 + 1205720119868lowast)3

(70)

Making the change of variables 119883 = 119909 119884 = minus1205792119868lowast + 1198881119909 +

1198882119910 + 119888111199092 + 1205820119909119910 minus 1199081(119909) and rewriting 119883 119884 as 119909 and 119910

respectively we have

119889119909

119889119905= 119910

119889119910

119889119905= 1198900+ 1198901119909 + 1198902119910 + 11989011119909

2+ 11989012119909119910 + 119890221199102+ 1199082 (119909 119910 120579)

(71)

where 120579 = (1205791 1205792) 1199082(119909 119910 120579) is a smooth function of 119909 1199101205791 and 1205792 at least of the third order and

1198900 = 1198882 (1205791 minus 1205792119878lowast) + 11988841205792119868

lowast

1198901 = 1205820 (1205791 minus 1205792119878lowast) + 11988821198883 minus 11988811198884 minus 12058201205792119868

lowast

1198902 = 1198881 + 1198884 + 12058201205792119868lowast

1198882

11989011 = 12058201198883 minus 119888411988811 minus 119888111198882 + 11988811205820

11989012 = minus 1205820 + 211988811 + 1205820minus11988811198882 minus 12058201205792119868

lowast

11988822

11989022 =12058201198882

(72)

Next introduce a new time variable 120591 by 119889119905 = (1 minus(12058201198882)119909)119889120591 Rewriting 120591 as 119905 we obtain

119889119909

119889119905= 119910(1 minus

12058201198882119909)

119889119910

119889119905= (1 minus

12058201198882119909)

times (1198900 + 1198901119909 + 1198902119910 + 119890111199092

+ 11989012119909119910 + 119890221199102 + 1199082 (119909 119910 120579))

(73)

Let119883 = 119909 119884 = 119910(1minus(12058201198882)119909) and rename119883 and119884 as 119909 and119910 we have

119889119909

119889119905= 119910

119889119910

119889119905= 1198900 + 1198921119909 + 1198902119910 + 11989211119909

2 + 11989212119909119910 + 1199083 (119909 119910 120579)

(74)

where 120579 = (1205791 1205792) 1199083(119909 119910 120579) is a smooth function of 119909 1199101205791 and 1205792 at least of the third order and

1198921 = minus 2119890012058201198882+ 1198901

11989211 = 11989011 minus 2119890112058201198882+11989001205822

0

11988822

11989212 = 11989012 minus119890212058201198882

(75)

Now we assume that 11989211 = 0 and 11989212 = 0 when 120582119894 are smallSet 119909 = 119883 minus 119890211989212 and rewrite119883 as 119909 we can get

119889119909

119889119905= 119910

119889119910

119889119905= 1198910 + 1198911119909 + 11989211119909

2 + 11989212119909119910 + 1199084 (119909 119910 120579)

(76)

where 120579 = (1205791 1205792) 1199084(119909 119910 120579) is a smooth function of 119909 1199101205791 and 1205792 at least of the third order and

1198910 = 1198900 minus1198921119890211989212+119892111198902

2

119892212

1198911 = 1198921 minus211989211119890211989212

(77)

Mathematical Problems in Engineering 11

Making the final change of variables by 119883 = 11989221211990911989211 119884 =

119892312119910119892211 and 120591 = 1198921111990511989212 and then denoting them again by

119909 119910 and 119905 respectively we obtain

119889119909

119889119905= 119910

119889119910

119889119905= 1205911 (1205791 1205792) + 1205912 (1205791 1205792) 119909 + 119909

2 + 119909119910 + 1199085 (119909 119910 120579)

(78)

where 120579 = (1205791 1205792) and 1199085(119909 119910 120579) is a smooth function of 119909119910 1205791 and 1205792 at least of the third order

We substitute values of 1205720 = 1 1205820 = 14 1198890 = 1101205760 = 110 1205830 = 1100 1198600 = 537500 and 1199030 = 558 forthe above system and these values satisfy conditions (1198601) and(1198602) And we obtain the following equations

1205911 (1205791 1205792) =11989101198924

12

119892311

= minus237291605

851841205791 +

25485118377

8518401205792

+ 119874 (10038161003816100381610038161205791 1205792

10038161003816100381610038162)

1205912 (1205791 1205792) =11989111198922

12

119892211

= minus172225

19361205791 +

3983253

38721205792 + 119874 (

10038161003816100381610038161205791 120579210038161003816100381610038162)

(79)

where 11989211 = minus11500 minus (116)1205791 + (341800)1205792 + (14)1205792

2lt 0

and 11989212 = minus83200 + (14)1205792 lt 0 in a small neighborhood of(1205791 1205792) = (0 0) Let

119869 = (

12059712059111205971205791

12059712059111205971205792

12059712059121205971205791

12059712059121205971205792

) (80)

And after simple calculation we obtain that

det(119869)|120582=0 = minus16839398748825

82458112= 0 (81)

Thus 1205911 and 1205912 are regular maps in a small neighborhoodof (1205791 1205792) = (0 0) By the Bogdanov and Takens bifurcationtheorems [36] we obtain the following conclusion

Theorem 13 Suppose that1198600 1198890 1205820 1205760 1199030 1205720 and 1205830 satisfy(1198601) (1198602) 11989211 = 0 and 11989212 = 0 when 120579119894 are small Then (4)admits the following bifurcation behavior

(1) There is a saddle-node bifurcation curve 119878119873 =

(1205791 1205792) 4119891011989211 = 11989121+ 119900(|(1205791 1205792)|

2) 1198911 =0 = (1205791 1205792) (59076250)1205792 minus (11125)1205791 +(2313400)12057911205792 minus (516)120579

2

1minus (102863740000)1205792

2+

119900(|(1205791 1205792)|2) = 0 (4799116600)1205792 minus (14)1205791 + (25

83)12057911205792 minus (1364513778)1205792

2+ 119900(|(1205791 1205792)|

2) = 0

I

I

O

O 1205791

1205792

SN+

SN+

SNminus

SNminus

II H

III HL

IVIVHL

IIIII

H

Figure 6The bifurcation set and the corresponding phase portraitsfor system (68)

(2) There is a Hopf bifurcation curve 119867 = (1205791 1205792)

1198910 + 119900(|(1205791 1205792)|2) = 0 1198911 lt 0 = (1205791 1205792) minus(537

50)1205792+1205791 + (5083)12057911205792 minus (746676889)12057922 + 119900(|(12057911205792)|2) = 0 (4799116600)1205792 minus (14)1205791+(2583)12057911205792 minus

(1364513778)12057922+ 119900(|(1205791 1205792)|

2) lt 0

(3) There is a homoclinic bifurcation curve119867119871 = (1205791 1205792) 25119892111198910 + 6119891

2

1= 119900(|(1205791 1205792)|

2) 1198911 lt 0 = (1205791 1205792) (59071000)1205792 minus (1120)1205791 + (611979 33200)12057911205792 minus(1916)1205792

1minus (16075834839275560000)1205792

2+ 119900(|(1205791

1205792)|2) = 0 (4799116600)1205792minus(14)1205791+(2583)12057911205792minus

(1364513778)12057922+ 119900(|(1205791 1205792)|

2) lt 0

6 Numerical Simulations

When 120572 = 1 120582 = 14 119889 = 110 120576 = 110 120583 = 1100119860 = 537500 and 119903 = 558 120590 gt 0 By applying PPLANE8and Photoshop software the (1205791 1205792)-plane near the origin isdivided into 4 regions by these bifurcation curves as shown inFigure 6 Fix 1205791 lt 0 and decrease 1205792 from 0 our conclusionsare summarized as follows

(a) When (1205791 1205792) lies in region I there is no positiveequilibrium which implies that the positive orbitsof (4) meet the positive 119878-axis in finite time andtherefore the disease disappears

(b) There is a saddle-node point when (1205791 1205792) lies oncurve SN

(c) When (1205791 1205792) crosses SN into region II two positiveequilibria which are an unstable focus and a saddleappear

(d) When the parameters lie on the curve H there is alsoa saddle and an unstable focus An unstable limit cycleappears with the parameters crossing H into III

(e) A homoclinic cycle appears as the parameters passingIII into HL through the homoclinic bifurcation

12 Mathematical Problems in Engineering

0 5 10 15 20 25 30 35 40

0

1

2

3

4

5

6

7

8

9

10

S

I

Figure 7 When (1205791 1205792) = (0 0) the unique positive equilibrium isa cusp of codimension 2

3 35 45 55 65 754 5 6 7 8

1

2

3

4

5

6

7

8

S

I

Figure 8 When (1205791 1205792) = (minus01 minus00093) lies in region I there isno positive equilibrium

(f) The homoclinic loop breakswith the parameter cross-ing HL into IV and a saddle and a stable focus appear

Furthermore using PPLANE8 some numerical simu-lations of system (68) are depicted in Figures 7ndash10 When(1205791 1205792) = (0 0) that is (1198600 1198890) = (537500 110) there isa unique positive equilibrium (119878lowast 119868lowast) = (31750 4) whichis a cusp of codimension 2 (Figure 7) When (1205791 1205792) =(minus01 minus00093) lies in region I there is no positive equi-librium (Figure 8) When (1205791 1205792) = (minus002 minus0001863798)there is a homoclinic loop (Figure 9) When (1205791 1205792) =(minus002 minus00020) lies in region IV the homoclinic loopis broken and there is a stable focus and a saddle(Figure 10)

6 605 61 615 62 625 63 635 64 645 653536373839

44142434445

S

I

Figure 9 When (1205791 1205792) = (minus002 minus0001863798) there is ahomoclinic loop

0 1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

7

8

9

10

S

I

Figure 10 When (1205791 1205792) = (minus002 minus00020) lies in region IV thereis a stable focus and a saddle

7 Discussion

In this paper we focus on the bifurcation analysis of an SISepidemic model with bilinear incidence rate and saturatedtreatment Generally speaking in many epidemic modelsthe basic reproduction number which is the key conceptin epidemiology can be decreased below unity to eradicatethe disease However in our model the basic reproductionnumber below 1 is not enough to eradicate the diseaseAccording to our analysis in this paper we demonstrate notonly the global stability of the disease-free equilibrium when1198770 lt 119877

lowast

0and the local asymptotic stability of the endemic

equilibrium 1198642 when 119877lowast

0lt 1198770 lt 1 but also the global

stability of the unique endemic equilibrium 119864lowast when 1198770 gt 1Moreover it has been shown in Corollary 4 that backwardbifurcations occur if the effect of the infected being delayedfor treatment is strong That is to say we should get prompttreatment for patients Through Figure 4 we can see thatthere is a region such that the disease will persist if the initialposition lies in the region and disappear if the initial positionlies outside this region So in order to restrain the spread

Mathematical Problems in Engineering 13

of the disease governments should take timely measures tocontrol the initial infected individuals in a lower level Inaddition fromTheorem 8 we can see that the disease will dieout if 1198770 is small enough Therefore from the definition of1198770 one knows that we can decrease the incidence rate 120582 andincrease the cure rate 119903 so as to eradicate the diseases or tomake them controlled in a lower endemic steady state

The stability analysis of the model equilibria enables usto completely analyze their local bifurcation behavior suchas Hopf saddle-node and Bogdanov-Takens bifurcation Bycomputing the first Lyapunov coefficient we can determinethat the Hopf bifurcation is a supercritical Hopf bifurca-tion or a subcritical Hopf bifurcation We also show thatunder assumptions (1198601) and (1198602) the model undertakesBogdanov-Takens bifurcation that is there are saddle-nodebifurcation Hopf bifurcation and homoclinic bifurcationin the system The normal norm of the Bogdanov-Takensbifurcation is derived in Section 5 which is very helpful toobtain the three kinds of bifurcation curves By analyticaltechniques 119860 and 119889 are chosen as bifurcation parametersand other parameters are fixed and we easily get a clearpicture about the rich dynamics behaviors of our modelThrough studying the bifurcations of the SIS epidemicmodelwe are suggesting that in order to eradicate the diseasemore medical facilities as well as medical professionals areneeded and the medical standard needs to be improved aswell

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research was supported by the National Natural ScienceFoundation of China (no 11001041 no 10926105 no 11101283no 11271065 and no 11201360) SRFDP (no 200802001008)and Scientific Research Innovation Project of ShanghaiMunicipal Education Commission (12YZ173)

References

[1] M E Alexander and S M Moghadas ldquoPeriodicity in an epi-demic model with a generalized non-linear incidencerdquo Mathe-matical Biosciences vol 189 no 1 pp 75ndash96 2004

[2] M E Alexander and S M Moghadas ldquoBifurcation analysisof SIRS epidemic model with generalized incidencerdquo SIAMJournal on Applied Mathematics vol 65 no 5 pp 1794ndash18162005

[3] J Arino C C McCluskey and P van den Driessche ldquoGlobalresults for an epidemic model with vaccination that exhibitsbackward bifurcationrdquo SIAM Journal on Applied Mathematicsvol 64 no 1 pp 260ndash276 2003

[4] W R Derrick and P van den Driessche ldquoHomoclinic orbitsin a disease transmission model with nonlinear incidence andnonconstant populationrdquo Discrete and Continuous DynamicalSystems B vol 3 no 2 pp 299ndash309 2003

[5] F Brauer ldquoBackward bifurcations in simple vaccination mod-elsrdquo Journal of Mathematical Analysis and Applications vol 298no 2 pp 418ndash431 2004

[6] K P Hadeler and P van den Driessche ldquoBackward bifurcationin epidemic controlrdquo Mathematical Biosciences vol 146 no 1pp 15ndash35 1997

[7] Z X Hu S Liu and H Wang ldquoBackward bifurcation of anepidemic model with standard incidence rate and treatmentraterdquo Nonlinear Analysis Real World Applications vol 9 no 5pp 2302ndash2312 2008

[8] J Hui and D M Zhu ldquoGlobal stability and periodicity onSIS epidemic models with backward bifurcationrdquo Computers ampMathematics with Applications vol 50 no 8-9 pp 1271ndash12902005

[9] Y Jin W D Wang and S W Xiao ldquoAn SIRS model with anonlinear incidence raterdquoChaos Solitons amp Fractals vol 34 no5 pp 1482ndash1497 2007

[10] G H Li and W D Wang ldquoBifurcation analysis of an epidemicmodel with nonlinear incidencerdquo Applied Mathematics andComputation vol 214 no 2 pp 411ndash423 2009

[11] X Z Li W S Li and M Ghosh ldquoStability and bifurcation ofan SIS epidemic model with treatmentrdquo Chaos Solitons ampampFractals vol 42 no 5 pp 2822ndash2832 2009

[12] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1986

[13] X B Liu and L J Yang ldquoStability analysis of an SEIQV epidemicmodel with saturated incidence raterdquo Nonlinear Analysis RealWorld Applications vol 13 no 6 pp 2671ndash2679 2012

[14] M Martcheva and H R Thieme ldquoProgression age enhancedbackward bifurcation in an epidemic model with super-infectionrdquo Journal of Mathematical Biology vol 46 no 5 pp385ndash424 2003

[15] S G Ruan and W D Wang ldquoDynamical behavior of anepidemic model with a nonlinear incidence raterdquo Journal ofDifferential Equations vol 188 no 1 pp 135ndash163 2003

[16] W D Wang and S G Ruan ldquoBifurcation in an epidemicmodel with constant removal rate of the infectivesrdquo Journal ofMathematical Analysis and Applications vol 291 no 2 pp 775ndash793 2004

[17] Y L Tang D Q Huang S G Ruan and W N ZhangldquoCoexistence of limit cycles and homoclinic loops in a SIRSmodel with a nonlinear incidence raterdquo SIAM Journal onApplied Mathematics vol 69 no 2 pp 621ndash639 2008

[18] P van den Driessche and J Watmough ldquoA simple SIS epidemicmodel with a backward bifurcationrdquo Journal of MathematicalBiology vol 40 no 6 pp 525ndash540 2000

[19] J J Wei and J A Cui ldquoDynamics of SIS epidemic model withthe standard incidence rate and saturated treatment functionrdquoInternational Journal of Biomathematics vol 5 no 3 Article ID1260003 18 pages 2012

[20] W D Wang ldquoBackward bifurcation of an epidemic model withtreatmentrdquoMathematical Biosciences vol 201 no 1-2 pp 58ndash712006

[21] Z W Wang ldquoBackward bifurcation in simple SIS modelrdquo ActaMathematicae Applicatae Sinica vol 25 no 1 pp 127ndash136 2009

[22] L Xue and C Scoglio ldquoThe network level reproduction numberfor infectious diseases with both vertical and horizontal trans-missionrdquo Mathematical Biosciences vol 243 no 1 pp 67ndash802013

14 Mathematical Problems in Engineering

[23] X Zhang and X N Liu ldquoBackward bifurcation of an epidemicmodel with saturated treatment functionrdquo Journal ofMathemat-ical Analysis and Applications vol 348 no 1 pp 433ndash443 2008

[24] J Guckenheimer and P J Holmes Nonlinear OscillationsDynamical Systems and Bifurcation of Vector Fields SpringerNew York NY USA 1996

[25] X Zhang and X N Liu ldquoBackward bifurcation and globaldynamics of an SIS epidemic model with general incidence rateand treatmentrdquoNonlinearAnalysis RealWorldApplications vol10 no 2 pp 565ndash575 2009

[26] M A Safi A B Gumel and E H Elbasha ldquoQualitative analysisof an age-structured SEIR epidemic model with treatmentrdquoApplied Mathematics and Computation vol 219 no 22 pp10627ndash10642 2013

[27] L H Zhou and M Fan ldquoDynamics of an SIR epidemic modelwith limited medical resources revisitedrdquo Nonlinear AnalysisReal World Applications vol 13 no 1 pp 312ndash324 2012

[28] X Y Zhou and J A Cui ldquoAnalysis of stability and bifurcation foran SEIR epidemicmodel with saturated recovery raterdquoCommu-nications inNonlinear Science andNumerical Simulation vol 16no 11 pp 4438ndash4450 2011

[29] D Lacitignola ldquoSaturated treatments and measles resurgenceepisodes in South Africa a possible linkagerdquo MathematicalBiosciences and Engineering MBE vol 10 no 4 pp 1135ndash11572013

[30] J C Eckalbar and W L Eckalbar ldquoDynamics of an epidemicmodel with quadratic treatmentrdquo Nonlinear Analysis RealWorld Applications vol 12 no 1 pp 320ndash332 2011

[31] F Brauer ldquoBackward bifurcations in simple vaccinationtreat-ment modelsrdquo Journal of Biological Dynamics vol 5 no 5 pp410ndash418 2011

[32] M W Hirsch S Smale and R L Devaney Differential Equa-tions Dynamical Systems and an Introduction to Chaos Aca-demic Press New York NY USA 2003

[33] R Bogdanov ldquoBifurcations of a limit cycle for a family of vectorfields on the planrdquo SelectaMathematica vol 1 pp 373ndash388 1981

[34] R Bogdanov ldquoVersal deformations of a singular point on theplan in the case of zero eigen-valuesrdquo Selecta MathematicaSovietica vol 1 pp 389ndash421 1981

[35] F Takens ldquoForced oscillations and bifurcationrdquo in Applicationsof Global Analysis I vol 3 pp 1ndash59 Communications of theMathematical Institute Rijksuniversiteit Utrecht 1974

[36] Y A Kuznetsov Elements of Applied Bifurcation TheorySpringer New York NY USA 2nd edition 1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

Making the final change of variables by 119883 = 11989221211990911989211 119884 =

119892312119910119892211 and 120591 = 1198921111990511989212 and then denoting them again by

119909 119910 and 119905 respectively we obtain

119889119909

119889119905= 119910

119889119910

119889119905= 1205911 (1205791 1205792) + 1205912 (1205791 1205792) 119909 + 119909

2 + 119909119910 + 1199085 (119909 119910 120579)

(78)

where 120579 = (1205791 1205792) and 1199085(119909 119910 120579) is a smooth function of 119909119910 1205791 and 1205792 at least of the third order

We substitute values of 1205720 = 1 1205820 = 14 1198890 = 1101205760 = 110 1205830 = 1100 1198600 = 537500 and 1199030 = 558 forthe above system and these values satisfy conditions (1198601) and(1198602) And we obtain the following equations

1205911 (1205791 1205792) =11989101198924

12

119892311

= minus237291605

851841205791 +

25485118377

8518401205792

+ 119874 (10038161003816100381610038161205791 1205792

10038161003816100381610038162)

1205912 (1205791 1205792) =11989111198922

12

119892211

= minus172225

19361205791 +

3983253

38721205792 + 119874 (

10038161003816100381610038161205791 120579210038161003816100381610038162)

(79)

where 11989211 = minus11500 minus (116)1205791 + (341800)1205792 + (14)1205792

2lt 0

and 11989212 = minus83200 + (14)1205792 lt 0 in a small neighborhood of(1205791 1205792) = (0 0) Let

119869 = (

12059712059111205971205791

12059712059111205971205792

12059712059121205971205791

12059712059121205971205792

) (80)

And after simple calculation we obtain that

det(119869)|120582=0 = minus16839398748825

82458112= 0 (81)

Thus 1205911 and 1205912 are regular maps in a small neighborhoodof (1205791 1205792) = (0 0) By the Bogdanov and Takens bifurcationtheorems [36] we obtain the following conclusion

Theorem 13 Suppose that1198600 1198890 1205820 1205760 1199030 1205720 and 1205830 satisfy(1198601) (1198602) 11989211 = 0 and 11989212 = 0 when 120579119894 are small Then (4)admits the following bifurcation behavior

(1) There is a saddle-node bifurcation curve 119878119873 =

(1205791 1205792) 4119891011989211 = 11989121+ 119900(|(1205791 1205792)|

2) 1198911 =0 = (1205791 1205792) (59076250)1205792 minus (11125)1205791 +(2313400)12057911205792 minus (516)120579

2

1minus (102863740000)1205792

2+

119900(|(1205791 1205792)|2) = 0 (4799116600)1205792 minus (14)1205791 + (25

83)12057911205792 minus (1364513778)1205792

2+ 119900(|(1205791 1205792)|

2) = 0

I

I

O

O 1205791

1205792

SN+

SN+

SNminus

SNminus

II H

III HL

IVIVHL

IIIII

H

Figure 6The bifurcation set and the corresponding phase portraitsfor system (68)

(2) There is a Hopf bifurcation curve 119867 = (1205791 1205792)

1198910 + 119900(|(1205791 1205792)|2) = 0 1198911 lt 0 = (1205791 1205792) minus(537

50)1205792+1205791 + (5083)12057911205792 minus (746676889)12057922 + 119900(|(12057911205792)|2) = 0 (4799116600)1205792 minus (14)1205791+(2583)12057911205792 minus

(1364513778)12057922+ 119900(|(1205791 1205792)|

2) lt 0

(3) There is a homoclinic bifurcation curve119867119871 = (1205791 1205792) 25119892111198910 + 6119891

2

1= 119900(|(1205791 1205792)|

2) 1198911 lt 0 = (1205791 1205792) (59071000)1205792 minus (1120)1205791 + (611979 33200)12057911205792 minus(1916)1205792

1minus (16075834839275560000)1205792

2+ 119900(|(1205791

1205792)|2) = 0 (4799116600)1205792minus(14)1205791+(2583)12057911205792minus

(1364513778)12057922+ 119900(|(1205791 1205792)|

2) lt 0

6 Numerical Simulations

When 120572 = 1 120582 = 14 119889 = 110 120576 = 110 120583 = 1100119860 = 537500 and 119903 = 558 120590 gt 0 By applying PPLANE8and Photoshop software the (1205791 1205792)-plane near the origin isdivided into 4 regions by these bifurcation curves as shown inFigure 6 Fix 1205791 lt 0 and decrease 1205792 from 0 our conclusionsare summarized as follows

(a) When (1205791 1205792) lies in region I there is no positiveequilibrium which implies that the positive orbitsof (4) meet the positive 119878-axis in finite time andtherefore the disease disappears

(b) There is a saddle-node point when (1205791 1205792) lies oncurve SN

(c) When (1205791 1205792) crosses SN into region II two positiveequilibria which are an unstable focus and a saddleappear

(d) When the parameters lie on the curve H there is alsoa saddle and an unstable focus An unstable limit cycleappears with the parameters crossing H into III

(e) A homoclinic cycle appears as the parameters passingIII into HL through the homoclinic bifurcation

12 Mathematical Problems in Engineering

0 5 10 15 20 25 30 35 40

0

1

2

3

4

5

6

7

8

9

10

S

I

Figure 7 When (1205791 1205792) = (0 0) the unique positive equilibrium isa cusp of codimension 2

3 35 45 55 65 754 5 6 7 8

1

2

3

4

5

6

7

8

S

I

Figure 8 When (1205791 1205792) = (minus01 minus00093) lies in region I there isno positive equilibrium

(f) The homoclinic loop breakswith the parameter cross-ing HL into IV and a saddle and a stable focus appear

Furthermore using PPLANE8 some numerical simu-lations of system (68) are depicted in Figures 7ndash10 When(1205791 1205792) = (0 0) that is (1198600 1198890) = (537500 110) there isa unique positive equilibrium (119878lowast 119868lowast) = (31750 4) whichis a cusp of codimension 2 (Figure 7) When (1205791 1205792) =(minus01 minus00093) lies in region I there is no positive equi-librium (Figure 8) When (1205791 1205792) = (minus002 minus0001863798)there is a homoclinic loop (Figure 9) When (1205791 1205792) =(minus002 minus00020) lies in region IV the homoclinic loopis broken and there is a stable focus and a saddle(Figure 10)

6 605 61 615 62 625 63 635 64 645 653536373839

44142434445

S

I

Figure 9 When (1205791 1205792) = (minus002 minus0001863798) there is ahomoclinic loop

0 1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

7

8

9

10

S

I

Figure 10 When (1205791 1205792) = (minus002 minus00020) lies in region IV thereis a stable focus and a saddle

7 Discussion

In this paper we focus on the bifurcation analysis of an SISepidemic model with bilinear incidence rate and saturatedtreatment Generally speaking in many epidemic modelsthe basic reproduction number which is the key conceptin epidemiology can be decreased below unity to eradicatethe disease However in our model the basic reproductionnumber below 1 is not enough to eradicate the diseaseAccording to our analysis in this paper we demonstrate notonly the global stability of the disease-free equilibrium when1198770 lt 119877

lowast

0and the local asymptotic stability of the endemic

equilibrium 1198642 when 119877lowast

0lt 1198770 lt 1 but also the global

stability of the unique endemic equilibrium 119864lowast when 1198770 gt 1Moreover it has been shown in Corollary 4 that backwardbifurcations occur if the effect of the infected being delayedfor treatment is strong That is to say we should get prompttreatment for patients Through Figure 4 we can see thatthere is a region such that the disease will persist if the initialposition lies in the region and disappear if the initial positionlies outside this region So in order to restrain the spread

Mathematical Problems in Engineering 13

of the disease governments should take timely measures tocontrol the initial infected individuals in a lower level Inaddition fromTheorem 8 we can see that the disease will dieout if 1198770 is small enough Therefore from the definition of1198770 one knows that we can decrease the incidence rate 120582 andincrease the cure rate 119903 so as to eradicate the diseases or tomake them controlled in a lower endemic steady state

The stability analysis of the model equilibria enables usto completely analyze their local bifurcation behavior suchas Hopf saddle-node and Bogdanov-Takens bifurcation Bycomputing the first Lyapunov coefficient we can determinethat the Hopf bifurcation is a supercritical Hopf bifurca-tion or a subcritical Hopf bifurcation We also show thatunder assumptions (1198601) and (1198602) the model undertakesBogdanov-Takens bifurcation that is there are saddle-nodebifurcation Hopf bifurcation and homoclinic bifurcationin the system The normal norm of the Bogdanov-Takensbifurcation is derived in Section 5 which is very helpful toobtain the three kinds of bifurcation curves By analyticaltechniques 119860 and 119889 are chosen as bifurcation parametersand other parameters are fixed and we easily get a clearpicture about the rich dynamics behaviors of our modelThrough studying the bifurcations of the SIS epidemicmodelwe are suggesting that in order to eradicate the diseasemore medical facilities as well as medical professionals areneeded and the medical standard needs to be improved aswell

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research was supported by the National Natural ScienceFoundation of China (no 11001041 no 10926105 no 11101283no 11271065 and no 11201360) SRFDP (no 200802001008)and Scientific Research Innovation Project of ShanghaiMunicipal Education Commission (12YZ173)

References

[1] M E Alexander and S M Moghadas ldquoPeriodicity in an epi-demic model with a generalized non-linear incidencerdquo Mathe-matical Biosciences vol 189 no 1 pp 75ndash96 2004

[2] M E Alexander and S M Moghadas ldquoBifurcation analysisof SIRS epidemic model with generalized incidencerdquo SIAMJournal on Applied Mathematics vol 65 no 5 pp 1794ndash18162005

[3] J Arino C C McCluskey and P van den Driessche ldquoGlobalresults for an epidemic model with vaccination that exhibitsbackward bifurcationrdquo SIAM Journal on Applied Mathematicsvol 64 no 1 pp 260ndash276 2003

[4] W R Derrick and P van den Driessche ldquoHomoclinic orbitsin a disease transmission model with nonlinear incidence andnonconstant populationrdquo Discrete and Continuous DynamicalSystems B vol 3 no 2 pp 299ndash309 2003

[5] F Brauer ldquoBackward bifurcations in simple vaccination mod-elsrdquo Journal of Mathematical Analysis and Applications vol 298no 2 pp 418ndash431 2004

[6] K P Hadeler and P van den Driessche ldquoBackward bifurcationin epidemic controlrdquo Mathematical Biosciences vol 146 no 1pp 15ndash35 1997

[7] Z X Hu S Liu and H Wang ldquoBackward bifurcation of anepidemic model with standard incidence rate and treatmentraterdquo Nonlinear Analysis Real World Applications vol 9 no 5pp 2302ndash2312 2008

[8] J Hui and D M Zhu ldquoGlobal stability and periodicity onSIS epidemic models with backward bifurcationrdquo Computers ampMathematics with Applications vol 50 no 8-9 pp 1271ndash12902005

[9] Y Jin W D Wang and S W Xiao ldquoAn SIRS model with anonlinear incidence raterdquoChaos Solitons amp Fractals vol 34 no5 pp 1482ndash1497 2007

[10] G H Li and W D Wang ldquoBifurcation analysis of an epidemicmodel with nonlinear incidencerdquo Applied Mathematics andComputation vol 214 no 2 pp 411ndash423 2009

[11] X Z Li W S Li and M Ghosh ldquoStability and bifurcation ofan SIS epidemic model with treatmentrdquo Chaos Solitons ampampFractals vol 42 no 5 pp 2822ndash2832 2009

[12] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1986

[13] X B Liu and L J Yang ldquoStability analysis of an SEIQV epidemicmodel with saturated incidence raterdquo Nonlinear Analysis RealWorld Applications vol 13 no 6 pp 2671ndash2679 2012

[14] M Martcheva and H R Thieme ldquoProgression age enhancedbackward bifurcation in an epidemic model with super-infectionrdquo Journal of Mathematical Biology vol 46 no 5 pp385ndash424 2003

[15] S G Ruan and W D Wang ldquoDynamical behavior of anepidemic model with a nonlinear incidence raterdquo Journal ofDifferential Equations vol 188 no 1 pp 135ndash163 2003

[16] W D Wang and S G Ruan ldquoBifurcation in an epidemicmodel with constant removal rate of the infectivesrdquo Journal ofMathematical Analysis and Applications vol 291 no 2 pp 775ndash793 2004

[17] Y L Tang D Q Huang S G Ruan and W N ZhangldquoCoexistence of limit cycles and homoclinic loops in a SIRSmodel with a nonlinear incidence raterdquo SIAM Journal onApplied Mathematics vol 69 no 2 pp 621ndash639 2008

[18] P van den Driessche and J Watmough ldquoA simple SIS epidemicmodel with a backward bifurcationrdquo Journal of MathematicalBiology vol 40 no 6 pp 525ndash540 2000

[19] J J Wei and J A Cui ldquoDynamics of SIS epidemic model withthe standard incidence rate and saturated treatment functionrdquoInternational Journal of Biomathematics vol 5 no 3 Article ID1260003 18 pages 2012

[20] W D Wang ldquoBackward bifurcation of an epidemic model withtreatmentrdquoMathematical Biosciences vol 201 no 1-2 pp 58ndash712006

[21] Z W Wang ldquoBackward bifurcation in simple SIS modelrdquo ActaMathematicae Applicatae Sinica vol 25 no 1 pp 127ndash136 2009

[22] L Xue and C Scoglio ldquoThe network level reproduction numberfor infectious diseases with both vertical and horizontal trans-missionrdquo Mathematical Biosciences vol 243 no 1 pp 67ndash802013

14 Mathematical Problems in Engineering

[23] X Zhang and X N Liu ldquoBackward bifurcation of an epidemicmodel with saturated treatment functionrdquo Journal ofMathemat-ical Analysis and Applications vol 348 no 1 pp 433ndash443 2008

[24] J Guckenheimer and P J Holmes Nonlinear OscillationsDynamical Systems and Bifurcation of Vector Fields SpringerNew York NY USA 1996

[25] X Zhang and X N Liu ldquoBackward bifurcation and globaldynamics of an SIS epidemic model with general incidence rateand treatmentrdquoNonlinearAnalysis RealWorldApplications vol10 no 2 pp 565ndash575 2009

[26] M A Safi A B Gumel and E H Elbasha ldquoQualitative analysisof an age-structured SEIR epidemic model with treatmentrdquoApplied Mathematics and Computation vol 219 no 22 pp10627ndash10642 2013

[27] L H Zhou and M Fan ldquoDynamics of an SIR epidemic modelwith limited medical resources revisitedrdquo Nonlinear AnalysisReal World Applications vol 13 no 1 pp 312ndash324 2012

[28] X Y Zhou and J A Cui ldquoAnalysis of stability and bifurcation foran SEIR epidemicmodel with saturated recovery raterdquoCommu-nications inNonlinear Science andNumerical Simulation vol 16no 11 pp 4438ndash4450 2011

[29] D Lacitignola ldquoSaturated treatments and measles resurgenceepisodes in South Africa a possible linkagerdquo MathematicalBiosciences and Engineering MBE vol 10 no 4 pp 1135ndash11572013

[30] J C Eckalbar and W L Eckalbar ldquoDynamics of an epidemicmodel with quadratic treatmentrdquo Nonlinear Analysis RealWorld Applications vol 12 no 1 pp 320ndash332 2011

[31] F Brauer ldquoBackward bifurcations in simple vaccinationtreat-ment modelsrdquo Journal of Biological Dynamics vol 5 no 5 pp410ndash418 2011

[32] M W Hirsch S Smale and R L Devaney Differential Equa-tions Dynamical Systems and an Introduction to Chaos Aca-demic Press New York NY USA 2003

[33] R Bogdanov ldquoBifurcations of a limit cycle for a family of vectorfields on the planrdquo SelectaMathematica vol 1 pp 373ndash388 1981

[34] R Bogdanov ldquoVersal deformations of a singular point on theplan in the case of zero eigen-valuesrdquo Selecta MathematicaSovietica vol 1 pp 389ndash421 1981

[35] F Takens ldquoForced oscillations and bifurcationrdquo in Applicationsof Global Analysis I vol 3 pp 1ndash59 Communications of theMathematical Institute Rijksuniversiteit Utrecht 1974

[36] Y A Kuznetsov Elements of Applied Bifurcation TheorySpringer New York NY USA 2nd edition 1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

0 5 10 15 20 25 30 35 40

0

1

2

3

4

5

6

7

8

9

10

S

I

Figure 7 When (1205791 1205792) = (0 0) the unique positive equilibrium isa cusp of codimension 2

3 35 45 55 65 754 5 6 7 8

1

2

3

4

5

6

7

8

S

I

Figure 8 When (1205791 1205792) = (minus01 minus00093) lies in region I there isno positive equilibrium

(f) The homoclinic loop breakswith the parameter cross-ing HL into IV and a saddle and a stable focus appear

Furthermore using PPLANE8 some numerical simu-lations of system (68) are depicted in Figures 7ndash10 When(1205791 1205792) = (0 0) that is (1198600 1198890) = (537500 110) there isa unique positive equilibrium (119878lowast 119868lowast) = (31750 4) whichis a cusp of codimension 2 (Figure 7) When (1205791 1205792) =(minus01 minus00093) lies in region I there is no positive equi-librium (Figure 8) When (1205791 1205792) = (minus002 minus0001863798)there is a homoclinic loop (Figure 9) When (1205791 1205792) =(minus002 minus00020) lies in region IV the homoclinic loopis broken and there is a stable focus and a saddle(Figure 10)

6 605 61 615 62 625 63 635 64 645 653536373839

44142434445

S

I

Figure 9 When (1205791 1205792) = (minus002 minus0001863798) there is ahomoclinic loop

0 1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

7

8

9

10

S

I

Figure 10 When (1205791 1205792) = (minus002 minus00020) lies in region IV thereis a stable focus and a saddle

7 Discussion

In this paper we focus on the bifurcation analysis of an SISepidemic model with bilinear incidence rate and saturatedtreatment Generally speaking in many epidemic modelsthe basic reproduction number which is the key conceptin epidemiology can be decreased below unity to eradicatethe disease However in our model the basic reproductionnumber below 1 is not enough to eradicate the diseaseAccording to our analysis in this paper we demonstrate notonly the global stability of the disease-free equilibrium when1198770 lt 119877

lowast

0and the local asymptotic stability of the endemic

equilibrium 1198642 when 119877lowast

0lt 1198770 lt 1 but also the global

stability of the unique endemic equilibrium 119864lowast when 1198770 gt 1Moreover it has been shown in Corollary 4 that backwardbifurcations occur if the effect of the infected being delayedfor treatment is strong That is to say we should get prompttreatment for patients Through Figure 4 we can see thatthere is a region such that the disease will persist if the initialposition lies in the region and disappear if the initial positionlies outside this region So in order to restrain the spread

Mathematical Problems in Engineering 13

of the disease governments should take timely measures tocontrol the initial infected individuals in a lower level Inaddition fromTheorem 8 we can see that the disease will dieout if 1198770 is small enough Therefore from the definition of1198770 one knows that we can decrease the incidence rate 120582 andincrease the cure rate 119903 so as to eradicate the diseases or tomake them controlled in a lower endemic steady state

The stability analysis of the model equilibria enables usto completely analyze their local bifurcation behavior suchas Hopf saddle-node and Bogdanov-Takens bifurcation Bycomputing the first Lyapunov coefficient we can determinethat the Hopf bifurcation is a supercritical Hopf bifurca-tion or a subcritical Hopf bifurcation We also show thatunder assumptions (1198601) and (1198602) the model undertakesBogdanov-Takens bifurcation that is there are saddle-nodebifurcation Hopf bifurcation and homoclinic bifurcationin the system The normal norm of the Bogdanov-Takensbifurcation is derived in Section 5 which is very helpful toobtain the three kinds of bifurcation curves By analyticaltechniques 119860 and 119889 are chosen as bifurcation parametersand other parameters are fixed and we easily get a clearpicture about the rich dynamics behaviors of our modelThrough studying the bifurcations of the SIS epidemicmodelwe are suggesting that in order to eradicate the diseasemore medical facilities as well as medical professionals areneeded and the medical standard needs to be improved aswell

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research was supported by the National Natural ScienceFoundation of China (no 11001041 no 10926105 no 11101283no 11271065 and no 11201360) SRFDP (no 200802001008)and Scientific Research Innovation Project of ShanghaiMunicipal Education Commission (12YZ173)

References

[1] M E Alexander and S M Moghadas ldquoPeriodicity in an epi-demic model with a generalized non-linear incidencerdquo Mathe-matical Biosciences vol 189 no 1 pp 75ndash96 2004

[2] M E Alexander and S M Moghadas ldquoBifurcation analysisof SIRS epidemic model with generalized incidencerdquo SIAMJournal on Applied Mathematics vol 65 no 5 pp 1794ndash18162005

[3] J Arino C C McCluskey and P van den Driessche ldquoGlobalresults for an epidemic model with vaccination that exhibitsbackward bifurcationrdquo SIAM Journal on Applied Mathematicsvol 64 no 1 pp 260ndash276 2003

[4] W R Derrick and P van den Driessche ldquoHomoclinic orbitsin a disease transmission model with nonlinear incidence andnonconstant populationrdquo Discrete and Continuous DynamicalSystems B vol 3 no 2 pp 299ndash309 2003

[5] F Brauer ldquoBackward bifurcations in simple vaccination mod-elsrdquo Journal of Mathematical Analysis and Applications vol 298no 2 pp 418ndash431 2004

[6] K P Hadeler and P van den Driessche ldquoBackward bifurcationin epidemic controlrdquo Mathematical Biosciences vol 146 no 1pp 15ndash35 1997

[7] Z X Hu S Liu and H Wang ldquoBackward bifurcation of anepidemic model with standard incidence rate and treatmentraterdquo Nonlinear Analysis Real World Applications vol 9 no 5pp 2302ndash2312 2008

[8] J Hui and D M Zhu ldquoGlobal stability and periodicity onSIS epidemic models with backward bifurcationrdquo Computers ampMathematics with Applications vol 50 no 8-9 pp 1271ndash12902005

[9] Y Jin W D Wang and S W Xiao ldquoAn SIRS model with anonlinear incidence raterdquoChaos Solitons amp Fractals vol 34 no5 pp 1482ndash1497 2007

[10] G H Li and W D Wang ldquoBifurcation analysis of an epidemicmodel with nonlinear incidencerdquo Applied Mathematics andComputation vol 214 no 2 pp 411ndash423 2009

[11] X Z Li W S Li and M Ghosh ldquoStability and bifurcation ofan SIS epidemic model with treatmentrdquo Chaos Solitons ampampFractals vol 42 no 5 pp 2822ndash2832 2009

[12] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1986

[13] X B Liu and L J Yang ldquoStability analysis of an SEIQV epidemicmodel with saturated incidence raterdquo Nonlinear Analysis RealWorld Applications vol 13 no 6 pp 2671ndash2679 2012

[14] M Martcheva and H R Thieme ldquoProgression age enhancedbackward bifurcation in an epidemic model with super-infectionrdquo Journal of Mathematical Biology vol 46 no 5 pp385ndash424 2003

[15] S G Ruan and W D Wang ldquoDynamical behavior of anepidemic model with a nonlinear incidence raterdquo Journal ofDifferential Equations vol 188 no 1 pp 135ndash163 2003

[16] W D Wang and S G Ruan ldquoBifurcation in an epidemicmodel with constant removal rate of the infectivesrdquo Journal ofMathematical Analysis and Applications vol 291 no 2 pp 775ndash793 2004

[17] Y L Tang D Q Huang S G Ruan and W N ZhangldquoCoexistence of limit cycles and homoclinic loops in a SIRSmodel with a nonlinear incidence raterdquo SIAM Journal onApplied Mathematics vol 69 no 2 pp 621ndash639 2008

[18] P van den Driessche and J Watmough ldquoA simple SIS epidemicmodel with a backward bifurcationrdquo Journal of MathematicalBiology vol 40 no 6 pp 525ndash540 2000

[19] J J Wei and J A Cui ldquoDynamics of SIS epidemic model withthe standard incidence rate and saturated treatment functionrdquoInternational Journal of Biomathematics vol 5 no 3 Article ID1260003 18 pages 2012

[20] W D Wang ldquoBackward bifurcation of an epidemic model withtreatmentrdquoMathematical Biosciences vol 201 no 1-2 pp 58ndash712006

[21] Z W Wang ldquoBackward bifurcation in simple SIS modelrdquo ActaMathematicae Applicatae Sinica vol 25 no 1 pp 127ndash136 2009

[22] L Xue and C Scoglio ldquoThe network level reproduction numberfor infectious diseases with both vertical and horizontal trans-missionrdquo Mathematical Biosciences vol 243 no 1 pp 67ndash802013

14 Mathematical Problems in Engineering

[23] X Zhang and X N Liu ldquoBackward bifurcation of an epidemicmodel with saturated treatment functionrdquo Journal ofMathemat-ical Analysis and Applications vol 348 no 1 pp 433ndash443 2008

[24] J Guckenheimer and P J Holmes Nonlinear OscillationsDynamical Systems and Bifurcation of Vector Fields SpringerNew York NY USA 1996

[25] X Zhang and X N Liu ldquoBackward bifurcation and globaldynamics of an SIS epidemic model with general incidence rateand treatmentrdquoNonlinearAnalysis RealWorldApplications vol10 no 2 pp 565ndash575 2009

[26] M A Safi A B Gumel and E H Elbasha ldquoQualitative analysisof an age-structured SEIR epidemic model with treatmentrdquoApplied Mathematics and Computation vol 219 no 22 pp10627ndash10642 2013

[27] L H Zhou and M Fan ldquoDynamics of an SIR epidemic modelwith limited medical resources revisitedrdquo Nonlinear AnalysisReal World Applications vol 13 no 1 pp 312ndash324 2012

[28] X Y Zhou and J A Cui ldquoAnalysis of stability and bifurcation foran SEIR epidemicmodel with saturated recovery raterdquoCommu-nications inNonlinear Science andNumerical Simulation vol 16no 11 pp 4438ndash4450 2011

[29] D Lacitignola ldquoSaturated treatments and measles resurgenceepisodes in South Africa a possible linkagerdquo MathematicalBiosciences and Engineering MBE vol 10 no 4 pp 1135ndash11572013

[30] J C Eckalbar and W L Eckalbar ldquoDynamics of an epidemicmodel with quadratic treatmentrdquo Nonlinear Analysis RealWorld Applications vol 12 no 1 pp 320ndash332 2011

[31] F Brauer ldquoBackward bifurcations in simple vaccinationtreat-ment modelsrdquo Journal of Biological Dynamics vol 5 no 5 pp410ndash418 2011

[32] M W Hirsch S Smale and R L Devaney Differential Equa-tions Dynamical Systems and an Introduction to Chaos Aca-demic Press New York NY USA 2003

[33] R Bogdanov ldquoBifurcations of a limit cycle for a family of vectorfields on the planrdquo SelectaMathematica vol 1 pp 373ndash388 1981

[34] R Bogdanov ldquoVersal deformations of a singular point on theplan in the case of zero eigen-valuesrdquo Selecta MathematicaSovietica vol 1 pp 389ndash421 1981

[35] F Takens ldquoForced oscillations and bifurcationrdquo in Applicationsof Global Analysis I vol 3 pp 1ndash59 Communications of theMathematical Institute Rijksuniversiteit Utrecht 1974

[36] Y A Kuznetsov Elements of Applied Bifurcation TheorySpringer New York NY USA 2nd edition 1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

of the disease governments should take timely measures tocontrol the initial infected individuals in a lower level Inaddition fromTheorem 8 we can see that the disease will dieout if 1198770 is small enough Therefore from the definition of1198770 one knows that we can decrease the incidence rate 120582 andincrease the cure rate 119903 so as to eradicate the diseases or tomake them controlled in a lower endemic steady state

The stability analysis of the model equilibria enables usto completely analyze their local bifurcation behavior suchas Hopf saddle-node and Bogdanov-Takens bifurcation Bycomputing the first Lyapunov coefficient we can determinethat the Hopf bifurcation is a supercritical Hopf bifurca-tion or a subcritical Hopf bifurcation We also show thatunder assumptions (1198601) and (1198602) the model undertakesBogdanov-Takens bifurcation that is there are saddle-nodebifurcation Hopf bifurcation and homoclinic bifurcationin the system The normal norm of the Bogdanov-Takensbifurcation is derived in Section 5 which is very helpful toobtain the three kinds of bifurcation curves By analyticaltechniques 119860 and 119889 are chosen as bifurcation parametersand other parameters are fixed and we easily get a clearpicture about the rich dynamics behaviors of our modelThrough studying the bifurcations of the SIS epidemicmodelwe are suggesting that in order to eradicate the diseasemore medical facilities as well as medical professionals areneeded and the medical standard needs to be improved aswell

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research was supported by the National Natural ScienceFoundation of China (no 11001041 no 10926105 no 11101283no 11271065 and no 11201360) SRFDP (no 200802001008)and Scientific Research Innovation Project of ShanghaiMunicipal Education Commission (12YZ173)

References

[1] M E Alexander and S M Moghadas ldquoPeriodicity in an epi-demic model with a generalized non-linear incidencerdquo Mathe-matical Biosciences vol 189 no 1 pp 75ndash96 2004

[2] M E Alexander and S M Moghadas ldquoBifurcation analysisof SIRS epidemic model with generalized incidencerdquo SIAMJournal on Applied Mathematics vol 65 no 5 pp 1794ndash18162005

[3] J Arino C C McCluskey and P van den Driessche ldquoGlobalresults for an epidemic model with vaccination that exhibitsbackward bifurcationrdquo SIAM Journal on Applied Mathematicsvol 64 no 1 pp 260ndash276 2003

[4] W R Derrick and P van den Driessche ldquoHomoclinic orbitsin a disease transmission model with nonlinear incidence andnonconstant populationrdquo Discrete and Continuous DynamicalSystems B vol 3 no 2 pp 299ndash309 2003

[5] F Brauer ldquoBackward bifurcations in simple vaccination mod-elsrdquo Journal of Mathematical Analysis and Applications vol 298no 2 pp 418ndash431 2004

[6] K P Hadeler and P van den Driessche ldquoBackward bifurcationin epidemic controlrdquo Mathematical Biosciences vol 146 no 1pp 15ndash35 1997

[7] Z X Hu S Liu and H Wang ldquoBackward bifurcation of anepidemic model with standard incidence rate and treatmentraterdquo Nonlinear Analysis Real World Applications vol 9 no 5pp 2302ndash2312 2008

[8] J Hui and D M Zhu ldquoGlobal stability and periodicity onSIS epidemic models with backward bifurcationrdquo Computers ampMathematics with Applications vol 50 no 8-9 pp 1271ndash12902005

[9] Y Jin W D Wang and S W Xiao ldquoAn SIRS model with anonlinear incidence raterdquoChaos Solitons amp Fractals vol 34 no5 pp 1482ndash1497 2007

[10] G H Li and W D Wang ldquoBifurcation analysis of an epidemicmodel with nonlinear incidencerdquo Applied Mathematics andComputation vol 214 no 2 pp 411ndash423 2009

[11] X Z Li W S Li and M Ghosh ldquoStability and bifurcation ofan SIS epidemic model with treatmentrdquo Chaos Solitons ampampFractals vol 42 no 5 pp 2822ndash2832 2009

[12] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1986

[13] X B Liu and L J Yang ldquoStability analysis of an SEIQV epidemicmodel with saturated incidence raterdquo Nonlinear Analysis RealWorld Applications vol 13 no 6 pp 2671ndash2679 2012

[14] M Martcheva and H R Thieme ldquoProgression age enhancedbackward bifurcation in an epidemic model with super-infectionrdquo Journal of Mathematical Biology vol 46 no 5 pp385ndash424 2003

[15] S G Ruan and W D Wang ldquoDynamical behavior of anepidemic model with a nonlinear incidence raterdquo Journal ofDifferential Equations vol 188 no 1 pp 135ndash163 2003

[16] W D Wang and S G Ruan ldquoBifurcation in an epidemicmodel with constant removal rate of the infectivesrdquo Journal ofMathematical Analysis and Applications vol 291 no 2 pp 775ndash793 2004

[17] Y L Tang D Q Huang S G Ruan and W N ZhangldquoCoexistence of limit cycles and homoclinic loops in a SIRSmodel with a nonlinear incidence raterdquo SIAM Journal onApplied Mathematics vol 69 no 2 pp 621ndash639 2008

[18] P van den Driessche and J Watmough ldquoA simple SIS epidemicmodel with a backward bifurcationrdquo Journal of MathematicalBiology vol 40 no 6 pp 525ndash540 2000

[19] J J Wei and J A Cui ldquoDynamics of SIS epidemic model withthe standard incidence rate and saturated treatment functionrdquoInternational Journal of Biomathematics vol 5 no 3 Article ID1260003 18 pages 2012

[20] W D Wang ldquoBackward bifurcation of an epidemic model withtreatmentrdquoMathematical Biosciences vol 201 no 1-2 pp 58ndash712006

[21] Z W Wang ldquoBackward bifurcation in simple SIS modelrdquo ActaMathematicae Applicatae Sinica vol 25 no 1 pp 127ndash136 2009

[22] L Xue and C Scoglio ldquoThe network level reproduction numberfor infectious diseases with both vertical and horizontal trans-missionrdquo Mathematical Biosciences vol 243 no 1 pp 67ndash802013

14 Mathematical Problems in Engineering

[23] X Zhang and X N Liu ldquoBackward bifurcation of an epidemicmodel with saturated treatment functionrdquo Journal ofMathemat-ical Analysis and Applications vol 348 no 1 pp 433ndash443 2008

[24] J Guckenheimer and P J Holmes Nonlinear OscillationsDynamical Systems and Bifurcation of Vector Fields SpringerNew York NY USA 1996

[25] X Zhang and X N Liu ldquoBackward bifurcation and globaldynamics of an SIS epidemic model with general incidence rateand treatmentrdquoNonlinearAnalysis RealWorldApplications vol10 no 2 pp 565ndash575 2009

[26] M A Safi A B Gumel and E H Elbasha ldquoQualitative analysisof an age-structured SEIR epidemic model with treatmentrdquoApplied Mathematics and Computation vol 219 no 22 pp10627ndash10642 2013

[27] L H Zhou and M Fan ldquoDynamics of an SIR epidemic modelwith limited medical resources revisitedrdquo Nonlinear AnalysisReal World Applications vol 13 no 1 pp 312ndash324 2012

[28] X Y Zhou and J A Cui ldquoAnalysis of stability and bifurcation foran SEIR epidemicmodel with saturated recovery raterdquoCommu-nications inNonlinear Science andNumerical Simulation vol 16no 11 pp 4438ndash4450 2011

[29] D Lacitignola ldquoSaturated treatments and measles resurgenceepisodes in South Africa a possible linkagerdquo MathematicalBiosciences and Engineering MBE vol 10 no 4 pp 1135ndash11572013

[30] J C Eckalbar and W L Eckalbar ldquoDynamics of an epidemicmodel with quadratic treatmentrdquo Nonlinear Analysis RealWorld Applications vol 12 no 1 pp 320ndash332 2011

[31] F Brauer ldquoBackward bifurcations in simple vaccinationtreat-ment modelsrdquo Journal of Biological Dynamics vol 5 no 5 pp410ndash418 2011

[32] M W Hirsch S Smale and R L Devaney Differential Equa-tions Dynamical Systems and an Introduction to Chaos Aca-demic Press New York NY USA 2003

[33] R Bogdanov ldquoBifurcations of a limit cycle for a family of vectorfields on the planrdquo SelectaMathematica vol 1 pp 373ndash388 1981

[34] R Bogdanov ldquoVersal deformations of a singular point on theplan in the case of zero eigen-valuesrdquo Selecta MathematicaSovietica vol 1 pp 389ndash421 1981

[35] F Takens ldquoForced oscillations and bifurcationrdquo in Applicationsof Global Analysis I vol 3 pp 1ndash59 Communications of theMathematical Institute Rijksuniversiteit Utrecht 1974

[36] Y A Kuznetsov Elements of Applied Bifurcation TheorySpringer New York NY USA 2nd edition 1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 Mathematical Problems in Engineering

[23] X Zhang and X N Liu ldquoBackward bifurcation of an epidemicmodel with saturated treatment functionrdquo Journal ofMathemat-ical Analysis and Applications vol 348 no 1 pp 433ndash443 2008

[24] J Guckenheimer and P J Holmes Nonlinear OscillationsDynamical Systems and Bifurcation of Vector Fields SpringerNew York NY USA 1996

[25] X Zhang and X N Liu ldquoBackward bifurcation and globaldynamics of an SIS epidemic model with general incidence rateand treatmentrdquoNonlinearAnalysis RealWorldApplications vol10 no 2 pp 565ndash575 2009

[26] M A Safi A B Gumel and E H Elbasha ldquoQualitative analysisof an age-structured SEIR epidemic model with treatmentrdquoApplied Mathematics and Computation vol 219 no 22 pp10627ndash10642 2013

[27] L H Zhou and M Fan ldquoDynamics of an SIR epidemic modelwith limited medical resources revisitedrdquo Nonlinear AnalysisReal World Applications vol 13 no 1 pp 312ndash324 2012

[28] X Y Zhou and J A Cui ldquoAnalysis of stability and bifurcation foran SEIR epidemicmodel with saturated recovery raterdquoCommu-nications inNonlinear Science andNumerical Simulation vol 16no 11 pp 4438ndash4450 2011

[29] D Lacitignola ldquoSaturated treatments and measles resurgenceepisodes in South Africa a possible linkagerdquo MathematicalBiosciences and Engineering MBE vol 10 no 4 pp 1135ndash11572013

[30] J C Eckalbar and W L Eckalbar ldquoDynamics of an epidemicmodel with quadratic treatmentrdquo Nonlinear Analysis RealWorld Applications vol 12 no 1 pp 320ndash332 2011

[31] F Brauer ldquoBackward bifurcations in simple vaccinationtreat-ment modelsrdquo Journal of Biological Dynamics vol 5 no 5 pp410ndash418 2011

[32] M W Hirsch S Smale and R L Devaney Differential Equa-tions Dynamical Systems and an Introduction to Chaos Aca-demic Press New York NY USA 2003

[33] R Bogdanov ldquoBifurcations of a limit cycle for a family of vectorfields on the planrdquo SelectaMathematica vol 1 pp 373ndash388 1981

[34] R Bogdanov ldquoVersal deformations of a singular point on theplan in the case of zero eigen-valuesrdquo Selecta MathematicaSovietica vol 1 pp 389ndash421 1981

[35] F Takens ldquoForced oscillations and bifurcationrdquo in Applicationsof Global Analysis I vol 3 pp 1ndash59 Communications of theMathematical Institute Rijksuniversiteit Utrecht 1974

[36] Y A Kuznetsov Elements of Applied Bifurcation TheorySpringer New York NY USA 2nd edition 1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of