Hindawi Publishing CorporationThe Scientific World JournalVolume 2013 Article ID 470646 10 pageshttpdxdoiorg1011552013470646

Research ArticlePositive Periodic Solutions of an EpidemicModel with Seasonality

Gui-Quan Sun123 Zhenguo Bai4 Zi-Ke Zhang567 Tao Zhou6 and Zhen Jin12

1 Complex Sciences Center Shanxi University Taiyuan 030006 China2 School of Mathematical Sciences Shanxi University Taiyuan Shanxi 030006 China3Department of Mathematics North University of China Taiyuan Shanxi 030051 China4Department of Applied Mathematics Xidian University Xirsquoan Shaanxi 710071 China5 Institute of Information Economy Hangzhou Normal University Hangzhou 310036 China6Web Sciences Center University of Electronic Science and Technology of China Chengdu Sichuan 610054 China7Department of Physics University of Fribourg Chemin du Musee 3 1700 Fribourg Switzerland

Correspondence should be addressed to Gui-Quan Sun gquansun126com

Received 17 August 2013 Accepted 12 September 2013

Academic Editors P Leach O D Makinde and Y Xia

Copyright copy 2013 Gui-Quan Sun 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

An SEI autonomous model with logistic growth rate and its corresponding nonautonomous model are investigated For theautonomous case we give the attractive regions of equilibria and perform some numerical simulations Basic demographicreproduction number 119877119889 is obtained Moreover only the basic reproduction number 1198770 cannot ensure the existence of the positiveequilibrium which needs additional condition 119877119889 gt 1198771 For the nonautonomous case by introducing the basic reproductionnumber defined by the spectral radius we study the uniform persistence and extinction of the diseaseThe results show that for theperiodic system the basic reproduction number is more accurate than the average reproduction number

1 Introduction

Bernoulli was the first person to use mathematical methodto evaluate the effectiveness of inoculation for smallpox [1ndash6] Then in 1906 Hawer studied the regular occurrenceof measles by a discrete-time model Moreover Ross [34] adopted the continuous model to study the dynamicsof malaria between mosquitoes and humans in 1916 and1917 In 1927 Kermack and McKendrick [5 6] extendedthe above works and established the threshold theory Sofar mathematical models have gotten great developmentand have been used to study population dynamics ecologyand epidemic which can be classified in terms of differentaspects From the aspect of the incidence of infectiousdiseases there are bilinear incidence standard incidencesaturating incidence and so on According to the type ofdemographic import the constant import the exponentialimport and the logistic growth import are the most commonforms The simple exponential growth models can provide

an adequate approximation to population growth for theinitial period If no predation or intraspecific competitionfor populations is included the population can continue toincrease However it is impossible to grow immoderately dueto the intraspecific competition for environmental resourcessuch as food and habitat So for this case logistic model ismore reasonable and realistic which has been adopted andstudied [7ndash18]Moreover due to its rich dynamics the logisticmodels have been applied to many fields Fujikawa et al [9]applied the logistic model to show Escherichia coli growthInvernizzi and Terpin [14] used a generalized logistic modelto describe photosynthetic growth and predict biomass pro-duction Min et al [15] used logistic dynamics model todescribe coalmining citiesrsquo economic growthmechanism andsustainable developmentThere is a good fit in simulating thecoalmining citiesrsquo growth and development track based onresource development cycle Banaszak et al [17] investigatedlogisticmodels in flexiblemanufacturing and Brianzoni et al[18] studied a business-cycle model with logistic population

2 The Scientific World Journal

growth Muroya [13] investigated discrete models of non-autonomous logistic equations As a result this paper buildsan SEI ordinary differential model with the logistic growthrate and the standard incidence

For general epidemic models we mainly study theirthreshold dynamics that is the basic reproduction numberwhich determines whether the disease can invade the suscep-tible population successfully However for the system withlogistic growth rate besides the basic reproduction numberthe qualitative dynamics are controlled by a demographicthreshold 119877119889 which has a similar meaning and is called asthe basic demographic reproduction number If 119877119889 gt 1the population grows that is a critical mass of individualsfor the disease to spread may be supported If 119877119889 lt 1the population will not survive that is not enough mass ofindividuals may be supported for the disease to spread Forthis case the dynamical behavior of disease will be decidedby two thresholds 1198770 and 119877119889

It is well known that many diseases exhibit seasonalfluctuations such as whooping cough measles influenzapolio chickenpox mumps and rabies [19ndash22] Seasonallyeffective contact rate [22ndash26] periodic changing in the birthrate [27] and vaccination program [28] are often regardedas sources of periodicity Seasonally effective contact rate isrelated to the behavior of people and animals the temper-ature and the economy Due to the existence of differentseasons people have different activities which may lead to adifferent contact rate Because of various factors the economyin a different season has a very big difference Therefore thispaper studies the corresponding non-autonomous systemwhich is obtained by changing the constant transmission rateinto the periodic transmission rate Seasonal transmission isoften assumed to be sinusoidal (cosine function has the samemeaning) such that 120582(119905) = 120582(1 + 120578 sin(120587119905119887)) where 120578 isthe amplitude of seasonal variation in transmission (typicallyreferred to as the ldquostrength of seasonal forcingrdquo) and 2119887 isthe period which is a crude assumption for many infectiousdiseases [29ndash31] When 120578 = 0 there is no nonseasonalinfections Motivated by biological realism some recentpapers take the contact rate as 120582(119905) = 120582(1 + 120578 term(119905)) whereterm is a periodic function which is +1 during a period oftime and minus1 during other time More natural term can bewritten as 120582(119905) = 120582(1 + 120578)

term(119905) [29] Here we take the form120573(119905) = 119886[1 + 119887 sin(12058711990510)]

The paper is organized as follows In Section 2 weintroduce an autonomous model and analyze the equilibriaand their respective attractive region In Section 3 we studythe non-autonomous system in terms of global asymptoticstability of the disease-free equilibrium and the existence ofpositive periodic solutions Moreover numerical simulationsare also performed In Section 4 we give a brief discussion

2 Autonomous Model and Analysis

21 Model Formulation The model is a system of SEI ordi-nary differential equations where 119878 is the susceptible 119864 isthe exposed 119868 is the infected and 119873 = 119878 + 119864 + 119868 Thissystem considers the logistic growth rate and the standard

Table 1 Descriptions and values of parameters in model (1)

Parameter Interpretation Value119903 The intrinsic growth rate119896 The carrying capacity 100000120573 The transmission rate119898 The natural mortality rate 01120590 Clinical outcome rate 02120583 The disease-induced mortality rate 01119886 The baseline contact rate119887 Themagnitude of forcing

incidence which is fit for the long-term growth of many largepopulations The incubation period is considered for manydiseases which do not develop symptoms immediately andneed a period of time to accumulate a pathogen quantity forclinical outbreak such as rabies hand-foot-mouth diseasetuberculosis and AIDS [22 32] The model we employ is asfollows

119889119878

119889119905= 119903119873(1 minus

119873

119896) minus

120573119878119868

119873minus 119898119878

119889119864

119889119905=

120573119878119868

119873minus 120590119864 minus 119898119864

119889119868

119889119905= 120590119864 minus 119898119868 minus 120583119868

(1)

where all parameters are positive whose interpretations canbe seen in Table 1

Noticing the equations in model (1) we have

119889119873

119889119905= 119903119873(1 minus

119873

119896) minus 119898119873 minus 120583119868 (2)

When there exists no disease we have

119889119873

119889119905= 119903119873(1 minus

119873

119896) minus 119898119873 = [119903 (1 minus

119873

119896) minus 119898]119873 (3)

Let 119877119889 = 119903119898 if 119877119889 gt 1 119873 rarr 1198730

= (1 minus 119898119903)119896 for119873(0) gt 0 as 119905 rarr +infin that is the population will growand tend to a steady state 119873

0 If 119877119889 lt 1 then 119889119873119889119905 lt 0

which will cause the population to disappear Thus 119877119889 is thebasic demographic reproduction number From the aboveequation the feasible region can be obtained 119883 = (119878 119864 119868) |

119878 119864 119868 ge 0 0 le 119878 + 119864 + 119868 le (1 minus 119898119903)119896 where 119903 gt 119898

Theorem 1 The region119883 is positively invariant with respect tosystem (1)

22 Dynamical Analysis Let the right hand of system (1) tobe zero it is easy to see that system (1) has three equilibria

119874 = (0 0 0)

1198640 = ((1 minus119898

119903) 119896 0 0)

119864lowast = (119878lowast 119864lowast 119868lowast)

(4)

The Scientific World Journal 3

where 119874 is the origin 1198640 is the disease-free equilibrium and119864lowast is the endemic equilibrium Concretely one can have

119878lowast

=[119898120573 (119898 + 120590 + 120583) + 120583 (119898 + 120583) (119898 + 120590) (1198770 minus 1)]119873

lowast

120573 [(119898 + 120583) (119898 + 120590) (1198770 minus 1) + 119898 (119898 + 120590 + 120583)]

(5)

119864lowast

=[119903 (1 minus 119873

lowast119896) minus 119898] (119898 + 120583)119873

lowast

120590120583 (6)

119868lowast

=(119898 + 120583) (119898 + 120590) (1198770 minus 1)119873

lowast

120573 (119898 + 120590 + 120583) (7)

119873lowast

= (119896 [120573119898 (119898 + 120590 + 120583) (119877119889 minus 1)

+120583 (119898 + 120590) (119898 + 120583) (1 minus 1198770)])

times (119903120573 (119898 + 120590 + 120583))minus1

(8)

where 119877119889 = 119903119898 is the basic demographic reproductionnumber and1198770 = 120573120590(119898+120590)(119898+120583) is the basic reproductionnumber which can be obtained by the next-generationmatrixmethod [33ndash35] The introduction of the basic demographicreproduction number can be found in [36]

Moreover from (6) and (7) the conditions of the endemicequilibrium to exist are 1198770 gt 1 and 119877119889 gt 1198771 where 1198771 =

1+120583(119898+120583)(119898+120590)(1198770 minus1)119898120573(119898+120590+120583) So we can obtainthe following theorems

Theorem 2 The system (1) has three equilibria origin 119874disease-free equilibrium 1198640 and the endemic equilibrium 119864lowast119874 always exists if 119877119889 gt 1 1198640 exists if 1198770 gt 1 and 119877119889 gt 1198771 119864lowastexists

Theorem 3 When 119877119889 gt 1 and 1198770 lt 1 1198640 is globallyasymptotically stable

Proof By [33ndash35] we know that 1198640 is locally asymptoticallystable Now we define a Lyapunov function

119881 = 119864 +119898 + 120590

120590119868 ge 0 (9)

When 119877119889 gt 1 and 1198770 lt 1 the Lyapunov function satisfies

= +119898 + 120590

120590

119868

=120573119878119868

119873minus

(119898 + 120590) (119898 + 120583)

120590119868

le [120573 minus(119898 + 120590) (119898 + 120583)

120590] 119868

=(119898 + 120590) (119898 + 120583)

120590[1198770 minus 1] 119868

le 0

(10)

Moreover = 0 only hold when 119868 = 0 It is easy to verifythat the disease-free equilibrium point 1198640 is the only fixedpoint of the system Hence applying the Lyapunov-LaSalle

Rd

R0

E0

Elowast

O1 O2

O3

0

L

1

1

Figure 1 119877119889 in terms of 1198770 The region of 1198741 1198742 and 1198743 is thebasin of attraction of equilibrium 119874 the region of 1198640 is the basinof attraction of equilibrium 1198640 the region of 119864lowast is the basin ofattraction of equilibrium 119864lowast the line 119871 is 119877119889 = 1198771 = 1 + 120583(119898 +

120583)(119898 + 120590)(1198770 minus 1)119898120573(119898 + 120590 + 120583)

asymptotic stability theorem in [37 38] the disease-freeequilibrium point 1198640 is globally asymptotically stable

Since the proof of the stability of equilibria 119874 and 119864lowast ismore difficult we only give some numerical results

In sum we can show the respective basins of attractionof the three equilibria which can be seen in Figure 1 andconfirmed in Figure 2

(1) When 119877119889 lt 1 119874 is stable see Figures 2(b) and 2(c)(2) When119877119889 gt 1 and1198770 lt 1 1198640 is stable see Figure 2(d)(3) When 119877119889 gt 1 1198770 gt 1 and 119877119889 lt 1198771 119874 is stable see

Figure 2(a)(4) When 119877119889 gt 1 1198770 gt 1 and 119877119889 gt 1198771 119864lowast is stable see

Figure 2(e)

3 Nonautonomous Model and Analysis

31 The Basic Reproduction Number Now we consider thenon-autonomous case of themodel (1) when the transmissionrate is periodic which is given as follows

119889119878

119889119905= 119903119873(1 minus

119873

119896) minus

120573 (119905) 119878119868

119873minus 119898119878

119889119864

119889119905=

120573 (119905) 119878119868

119873minus 120590119864 minus 119898119864

119889119868

119889119905= 120590119864 minus 119898119868 minus 120583119868

(11)

where 120573(119905) is a periodic function which is proposed by [39]In the subsequent section we will discuss the dynamicalbehavior of the system (11)

For system (11) firstly we can give the basic reproductionnumber 1198770 According to the basic reproduction numberunder non-autonomous system we can refer to the methodof [40 41] From the last section we know that system (11)

4 The Scientific World Journal

80

70

60

50

40

30

20

10

00 50 100 150 200 250 300

R0 = 16667

Rd = 101

R1 = 12

I(t)

S(t)

(a)

R0 = 1667

Rd = 05

R1 = 10714

80

70

60

50

40

30

20

10

00 50 100 150 200 250 300

I(t)

S(t)

(b)

R0 = 06667

Rd = 08

R1 = 075

80

70

60

50

40

30

20

10

00 50 100 150 200 250 300

I(t)

S(t)

(c)

R0 = 06667

Rd = 13

R1 = 075

600

500

400

300

200

100

00 05 1 15 2 25 3 35 4

times104

I(t)

S(t)

(d)

R0 = 2333

Rd = 13

R1 = 12857

600

500

400

300

200

200

100

0100 300 400 500 600 700 800 900

I(t)

S(t)

(e)

Figure 2 The phase curves of the system under different initial conditions (a) In the region 1198743 with 119903 = 0101 and 120573 = 05 (b) in the region1198742 with 119903 = 005 and 120573 = 035 (c) in the region1198741 with 119903 = 008 and 120573 = 02 (d) in the region 1198640 with 119903 = 013 and 120573 = 02 (e) in the region119864lowastwith 119903 = 013 and 120573 = 07 The value of other parameters can be seen in Table 1

The Scientific World Journal 5

has one disease-free equilibrium 1198640 = (1198730 0 0) where119873

0=

(1 minus 119898119903)119896 By giving a new vector 119909 = (119864 119868) we have

119865 = (

120573 (119905) 119878119868

119873

0

) 119881 = (119898119864 + 120590119864

119898119868 + 120583119868 minus 120590119864)

119881minus

= (119898119864 + 120590119864

119898119868 + 120583119868) 119881

+= (

0

120590119864)

(12)

Taking the partial derivative of the above vectors aboutvariables 119864 119868 and substituting the disease-free equilibriumwe have

119865 (119905) = (0 120573 (119905)

0 0)

119881 (119905) = (119898 + 120590 0

minus120590 119898 + 120583)

(13)

According to [41] denote 119862120596 to be the ordered Banachspace of all 120596-periodic functions from R to R4 which isequippedwith themaximumnorm sdot and the positive cone119862+

120596= 120601 isin 119862120596 120601(119905) ge 0 forall119905 isin R+ Over the Banach space

we define a linear operator 119871 119862120596 rarr 119862120596 by

(119871120601) (119905)

= int

infin

0

119884 (119905 119905 minus 119886) 119865 (119905 minus 119886) 120601 (119905 minus 119886) 119889119886 forall119905 isin R+ 120601 isin 119862120596

(14)

where 119871 is called the next infection operator and the inter-pretation of 119884(119905 119905 minus 119886) 120601(119905 minus 119886) can be seen in [41] Thenthe spectral radius of 119871 is defined as the basic reproductionnumber

1198770 = 120588 (119871) (15)

In order to give the expression of the basic reproductionnumber we need to introduce the linear 120596-periodic system

119889119908

119889119905= [minus119881 (119905) +

119865 (119905)

120582]119908 119905 isin R+ (16)

with parameter 120582 isin R Let 119882(119905 119904 120582) 119905 ge 119904 be theevolution operator of system (16) on R2 In fact 119882(119905 119904 120582) =

Φ(119865120582)minus119881(119905) and Φ119865minus119881(119905) = 119882(119905 0 1) for all 119905 ge 0 ByTheorems 21 and 22 in [41] the basic reproduction numberalso can be defined as1205820 such that120588(Φ(1198651205820)minus119881(120596)) = 1 whichcan be straightforward to calculate

32 Global Stability of the Disease-Free Equilibrium

Theorem4 Thedisease-free equilibrium1198640 is globally asymp-totically stable when 1198770 lt 1 and 119877119889 gt 1

Proof Theorem 22 in [41] implies that 1198640 is locally asymp-totically stable when 1198770 lt 1 and 119877119889 gt 1 So we only need

to prove its global attractability It is easy to know that 119878(119905) le

1198730= (1 minus (119898119903))119896 Thus

119889119864

119889119905le 120573 (119905) 119868 minus (119898 + 120590) 119864

119889119868

119889119905= 120590119864 minus 119898119868 minus 120583119868

(17)

The right comparison system can be written as

119889119864

119889119905= 120573 (119905) 119868 minus (119898 + 120590) 119864

119889119868

119889119905= 120590119864 minus 119898119868 minus 120583119868

(18)

that is

119889ℎ

119889119905= (119865 (119905) minus 119881 (119905)) ℎ (119905) ℎ (119905) = (119864 (119905) 119868 (119905)) (19)

For (19) Lemma 21 in [42] shows that there is apositive 120596-periodic function ℎ(119905) = (119864(119905) 119868(119905))

119879 such thatℎ(119905) = 119890

119901119905ℎ(119905) is a solution of system (18) where 119901 =

(1120596) ln 120588(Φ119865minus119881(120596)) By Theorem 22 in [41] we know thatwhen 1198770 lt 1 and 119877119889 gt 1 120588(Φ119865minus119881(120596)) lt 1 and 119901 lt 0which implies ℎ(119905) rarr 0 as 119905 rarr infin Therefore the zerosolution of system (18) is globally asymptotically stable Bythe comparison principle [43] and the theory of asymptoticautonomous systems [44] when 1198770 lt 1 and 119877119889 gt 1 1198640is globally attractive Therefore the proposition that 1198640 isglobally asymptotically stable holds

33 Existence of Positive Periodic Solutions Before the proofof the existence of positive periodic solutions we firstlyintroduce some denotations Let 119906(119905 1199090) be the solution ofsystem (11) with the initial value 1199090 = (119878(0) 119864(0) 119868(0)) Bythe fundamental existence-uniqueness theorem [45] 119906(119905 1199090)is the unique solution of system (11) with 119906(0 1199090) = 1199090

Next we need to introduce the Poincare map119875 119883 rarr 119883

associated with system (11) that is

119875 (1199090) = 119906 (120596 1199090) forall1199090 isin 119883 (20)

where 120596 is the period Theorem 1 implies that 119883 is positivelyinvariant and 119875 is a dissipative point

Now we introduce two subsets of1198831198830 = (119878 119864 119868) isin 119883

119864 gt 0 119868 gt 0 and 1205971198830 = 119883 1198830

Lemma 5 (a) When 1198770 gt 1 and 119903 gt 119898 + 120583 there exists a120575 gt 0 such that when

1003817100381710038171003817(119878 (0) 119864 (0) 119868 (0)) minus 11986401003817100381710038171003817 le 120575 (21)

for any (119878(0) 119864(0) 119868(0)) isin 1198830 one has

lim sup119898rarrinfin

119889 [119875119898

(119878 (0) 119864 (0) 119868 (0)) 1198640] ge 120575 (22)

where 1198640 = (1198730 0 0)

6 The Scientific World Journal

(b) When 1198770 gt 1 and 119903 gt 119898 + 120583 there exists a 120575 gt 0 suchthat when

(119878 (0) 119864 (0) 119868 (0)) minus 119874 le 120575 (23)

for any (119878(0) 119864(0) 119868(0)) isin 1198830 one has

lim sup119898rarrinfin

119889 [119875119898

(119878 (0) 119864 (0) 119868 (0)) 119874] ge 120575 (24)

where 119874 = (0 0 0)

Proof (a) ByTheorem 22 in [41] we know that when 1198770 gt 1120588(Φ119865minus119881(120596)) gt 1 So there is a small enough positive number120598 such that 120588(Φ119865minus119881minus119872120598(120596)) gt 1 where

119872120598 = (0

120573 (119905) 120598119868

1198730

0 0

) (25)

If proposition (a) does not hold there is some(119878(0) 119864(0) 119868(0)) isin 1198830 such that

lim sup119898rarrinfin

119889 (119875119898

(119878 (0) 119864 (0) 119868 (0)) 1198640) lt 120575 (26)

We can assume that for all 119898 ge 0 119889(119875119898(119878(0) 119864(0)

119868(0)) 1198640) lt 120575 Applying the continuity of the solutions withrespect to the initial values

1003817100381710038171003817119906 (119905 119875119898

(119878 (0) 119864 (0) 119868 (0))) minus 119906 (119905 1198640)1003817100381710038171003817 le 120598

forall119898 ge 0 forall1199051 isin [0 120596]

(27)

Let 119905 = 119898120596 + 1199051 where 1199051 isin [0 120596] and 119898 = [119905120596] 119898 =

[119905120596] is the greatest integer which is not more than 119905120596Thenfor any 119905 ge 0

1003817100381710038171003817119906 (119905 (119878 (0) 119864 (0) 119868 (0))) minus 119906 (119905 1198640)1003817100381710038171003817

=1003817100381710038171003817119906 (1199051 119875

119898(119878 (0) 119864 (0) 119868 (0))) minus 119906 (1199051 1198640)

1003817100381710038171003817 le 120598

(28)

So 1198730

minus 120598 le 119878(119905) le 1198730

+ 120598 Then whenlim sup

119898rarrinfin119889(119875119898(119878(0) 119864(0) 119868(0)) 1198640) lt 120575

119889119864

119889119905ge 120573 (119905) 119868 minus

120573 (119905) 120598119868

1198730minus (119898 + 120590) 119864

119889119868

119889119905= 120590119864 minus 119898119868 minus 120583119868

(29)

Thus we can study the right linear system

119889119864

119889119905= 120573 (119905) 119868 minus

120573 (119905) 120598119868

1198730minus (119898 + 120590) 119864

119889119868

119889119905= 120590119864 minus 119898119868 minus 120583119868

(30)

For the system (30) there exists a positive 120596-periodicfunction 119892(119905) = (119864(119905) 119868(119905))

119879 such that 119892(119905) = 119890119901119905119892(119905) is a

solution of system (11) where 119901 = (1120596) ln 120588(Φ119865minus119881minus119872120598(120596))When 1198770 gt 1 120588(Φ119865minus119881minus119872120598(120596)) gt 1 which means that when

119892(0) gt 0119892(119905) rarr infin as 119905 rarr infin By the comparison principle[43] when 119864(0) gt 0 119868(0) gt 0 119864(119905) rarr infin 119868(119905) rarr infin as119905 rarr infinThere appears a contradictionThus the proposition(a) holds

(b) When 1198770 gt 1 and 119903 gt 119898 + 120583 we have

119889119873

119889119905= 119903119873(1 minus

119873

119896) minus 119898119873 minus 120583119868

ge [119903 (1 minus119873

119896) minus 119898 minus 120583]119873 gt 0

(31)

So if 119877119889 gt 1 119873 rarr 1198730 for 119873(0) gt 0 as 119905 rarr +infin that is

119882119878(119874) cap 1198830 = 0

Theorem 6 When 1198770 gt 1 and 119903 gt 119898 + 120583 there exists a 120575 gt 0

such that any solution (119878(119905) 119864(119905) 119868(119905)) of system (11)with initialvalue (119878(0) 119864(0) 119868(0)) isin (119878 119864 119868) isin 119883 119864 gt 0 119868 gt 0

satisfies

lim inf119905rarrinfin

119864 (119905) ge 120575 lim inf119905rarrinfin

119868 (119905) ge 120575 (32)

and system (11) has at least one positive periodic solution

Proof From system (11)

119878 (119905) = 119890minusint119905

0(120573(119904)119868(119904)119873+119898)119889119904

times [119878 (0) + int

119905

0

119903119873 (119904) (1 minus119873 (119904)

119896)

times119890int119904

0(120573(119888)119868(119888)119873(119888)+119898)119889119888

119889119904]

gt 0 forall119905 gt 0

(33)

119864 (119905) = 119890minus(119898+120590)119905

[119864 (0) + int

119905

0

120573 (119904) 119878 (119904) 119868 (119904)

119873 (119904)119890(119898+120590)119904

119889119904]

gt 0 forall119905 gt 0

(34)

119868 (119905) = 119890minus(119898+120583)119905

[119868 (0) + int

119905

0

120590119864 (119904) 119890(119898+120583)119904

119889119904]

gt 0 forall119905 gt 0

(35)

for any (119878(0) 119864(0) 119868(0)) isin 1198830 which shows that 1198830 ispositively invariant Moreover it is obvious to see that 1205971198830

is relatively closed in 119883 Denote

119872120597 = (119878 (0) 119864 (0) 119868 (0)) isin 1205971198830

119875119898

(119878 (0) 119864 (0) 119868 (0)) isin 1205971198830 forall119898 ge 0

(36)

Next we prove that

119872120597 = (119878 0 0) isin 119883 119878 ge 0 (37)

We only need to show that 119872120597 sube (119878 0 0) isin 119883 119878 ge 0which means that for any (119878(0) 119864(0) 119868(0)) isin 1205971198830 119864(119898120596) =

119868(119898120596) = 0 for all 119898 ge 0 If it does not hold there existsa 1198981 ge 0 such that (119864(1198981120596) 119868(1198981120596))

119879gt 0 Taking 1198981120596 as

The Scientific World Journal 7

25

2

15

1

05

01 15 2 25 3 35

times104

times104

S(0) = 25000 E(0) = 10000 I(0) = 24000

S(0) = 35000 E(0) = 20000 I(0) = 15000

S(0) = 15000 E(0) = 10000 I(0) = 20000

I(t)

S(t)

(a)

24

23

22

21

2

19

18

17

16

15

1415 2 25 3 35

times104

times104

S(0) = 25000 E(0) = 10000 I(0) = 24000

S(0) = 35000 E(0) = 20000 I(0) = 15000

S(0) = 15000 E(0) = 10000 I(0) = 20000

I(t)

S(t)

(b)

Figure 3 Phase plane of 119878(119905) and 119868(119905) (a) When the parameter values are 119903 = 013 119886 = 03 and 119887 = 02 1198770 = 09029 lt 1 119877119889 = 13 gt 1 (b)When the parameter values are 119903 = 03 119886 = 07 and 119887 = 02 1198770 = 21067 and 119903 gt 119898+120583 The value of other parameters can be seen in Table 1

the initial time and repeating the processes as in (33)ndash(35)we can have that (119878(119905) 119864(119905) 119868(119905))

119879gt 0 for all 119905 gt 1198981120596

Thus (119878(119905) 119864(119905) 119868(119905)) isin 1198830 for all 119905 gt 1198981120596 There appearsa contradiction which means that the equality (37) holdsTherefore 1198640 is acyclic in 1205971198830 Obviously when 1198770 gt 1 and119903 gt 119898 + 120583 119874 is acyclic in 1205971198830

Furthermore by Lemma 5 1198640 = (1198730 0 0) and 119874 =

(0 0 0) are isolated invariant sets in 119883 119882119878(1198640) cap 1198830 = 0and 119882

119878(119874) cap 1198830 = 0 By Theorem 131 and Remark 131

in [46] it can be obtained that 119875 is uniformly persistent withrespect to (1198830 1205971198830) that is there exists a 120575 gt 0 such that anysolution (119878(119905) 119864(119905) 119868(119905)) of system (11) with the initial value(119878(0) 119864(0) 119868(0)) isin (119878 119864 119868) isin 119883 119864 gt 0 119868 gt 0 satisfies

lim inf119905rarrinfin

119864 (119905) ge 120575 lim inf119905rarrinfin

119868 (119905) ge 120575 (38)

ApplyingTheorem 136 in [46] 119875 has a fixed point

(119878lowast(0) 119864

lowast(0) 119868lowast(0)) isin 1198830 (39)

From (33) we know 119878lowast

gt 0 for all 119905 isin [0 120596] 119878lowast(119905) is alsomore than zero for all 119905 gt 0 due to the periodicity Similarlyfor all 119905 ge 0119864lowast(119905) gt 0 119868lowast(119905) gt 0Therefore it can be obtainedthat one of the positive 120596-periodic solutions of system (11) is(119878lowast(119905) 119864lowast(119905) 119868lowast(119905))

34 Numerical Simulations Firstly we give some notationsIf 119892(119905) is a periodic function with period 120596 we define 119892 =

(1120596) int120596

0119892(119905)119889119905 119892119897 = min119905isin[0120596]119892(119905) 119892119906 = max119905isin[0120596]119892(119905) As

described in the previous section

1198771 (119905) = 1 +120583 (119898 + 120583) (119898 + 120590) (1198770 minus 1)

119898120573 (119905) (119898 + 120590 + 120583) (40)

So 119877119897

1= 1 + 120583(119898 + 120583)(119898 + 120590)(119877

119897

0minus 1)119898120573

119906(119898 + 120590 + 120583) 119877119906

1=

1 + 120583(119898 + 120583)(119898 + 120590)(119877119906

0minus 1)119898120573

119897(119898 + 120590 + 120583)

In this section we adopt 120573(119905) = 119886[1 + 119887 sin(12058711990510)]Then applying the numerical simulation to verify the abovesolution we give the following conclusion

(1) when 119877119889 lt 1 119874 is stable(2) when 119877119889 gt 1 and 1198770 lt 1 1198640 is stable see Figure 3(a)(3) when 1198770 gt 1 and 119903 gt 119898 + 120583 system (11) has at least

one positive periodic solution see Figure 3(b)

We can givemore results about the conditions of existenceof the positive periodic solution

(11015840) When 119877119889 gt 1 1198770 gt 1 and 119877119889 lt 119877

119906

1 119874 is stable see

Figures 4(a) and 4(b)(21015840) When 119877119889 gt 1 1198770 gt 1 and 119877119889 gt 119877

119906

1 system (11) has at

least one positive periodic solution see Figure 5

By numerical simulations we can give that the conditionswhich ensure the existence of positive periodic solution are1198770 gt 1 and 119903 gt 119898+120583 or 119877119889 gt 1 1198770 gt 1 and 119877119889 gt 119877

119906

1 In fact

119877119889 gt 1 1198770 gt 1 and 119877119889 gt 119877119906

1are the sufficient conditions for

1198770 gt 1 and 119903 gt 119898 + 120583 As a result the conditions 1198770 gt 1 and119903 gt 119898 + 120583 are broader

4 Discussion

This paper considers a logistic growth system whose birthprocess incorporates density-dependent effects This type ofmodel has a rich dynamical behavior and practical signif-icance By analyzing its equilibria and respective attractiveregion we find that the dynamical behavior of a disease will

8 The Scientific World Journal

25

2

15

1

05

00 05 1 15 2 25 3 35

times104

times104

S(0) = 25000 E(0) = 10000 I(0) = 24000

S(0) = 35000 E(0) = 20000 I(0) = 15000

S(0) = 15000 E(0) = 10000 I(0) = 20000

I(t)

S(t)

(a)

25

2

15

1

05

00 05 1 15 2 25 3 35

times104

times104

S(0) = 25000 E(0) = 10000 I(0) = 24000

S(0) = 35000 E(0) = 20000 I(0) = 15000

S(0) = 15000 E(0) = 10000 I(0) = 20000

I(t)

S(t)

(b)

Figure 4 Phase plane of 119878(119905) and 119868(119905) (a) When the parameter values are 119903 = 011 119886 = 09 119887 = 02 and 120583 = 03 1198770 = 14991 119877119889 = 11 gt 1119877119889 lt 119877

119906

1= 14159 and 119877119889 lt 119877

119897

1= 12773 (b) When the parameter values are 119903 = 0126 119886 = 07 119887 = 02 and 120583 = 03 1198770 = 21067

119877119889 = 126 gt 1 119877119889 lt 119877119906

1= 12964 and 119877119889 gt 119877

119897

1= 11976 The value of other parameters can be seen in Table 1

7000

6000

5000

4000

3000

2000

1000

005 1 15 2

times104

times104

S(0) = 22000 E(0) = 5000 I(0) = 1000

S(0) = 4000 E(0) = 2000 I(0) = 1000

S(0) = 10000 E(0) = 10000 I(0) = 6000

I(t)

S(t)

Figure 5 Phase plane of 119878(119905) and 119868(119905) When the parameter valuesare 119903 = 013 119886 = 036 and 119887 = 02 1198770 = 10834 119877119889 = 13 gt 1 119877119889 gt119877119906

1= 10435 and 119877119889 gt 119877

119897

1= 1029 The value of other parameters

can be seen in Table 1

be determined by two thresholds 1198770 and 119877119889 Only 1198770 gt 1

cannot promise the existence of the endemic equilibriumwhich also needs 119877119889 gt 1198771 When 1198770 gt 1 and 119877119889 lt 1198771the solutions of the system (1) will tend to the origin 119874 It

is caused by the phenomenon that the death number dueto disease cannot be supplemented by the birth numberpromptly Finally all people are infected and die outThe factinterpreted by this model is more reasonable Theoreticallywe prove the global asymptotic stability of the disease-free equilibrium and give respective attractive regions ofequilibria

Seasonally effective contact rate is themost common formwhich may be related to various factors and thus this paperstudies the corresponding non-autonomous system which isobtained by changing the constant transmission rate of theabove system into the periodic transmission rate For theperiodic systems their dynamical behaviors especially thebasic reproduction number have been investigated in depthby [41 47ndash55] which provide many methods that we canutilize For the obtained periodic model by analyzing theglobal asymptotic stability of the disease-free equilibrium andthe existence of positive periodic solution we have the similarresults as the autonomous system The dynamic behavior ofdisease will be decided by two conditions 1198770 gt 1 and 119903 gt

119898 + 120583 that show that when the disease is prevalent the birthrate should be larger than the death rate to guarantee thesustainable growth of population Otherwise the populationwill disappear In addition we will evaluate and comparethe basic reproduction number 1198770 and the average basicreproduction number 1198770 which has been adopted by [27 56ndash59] In this paper we can calculate the average reproductionnumber

1198770 =120573120590

(119898 + 120590) (119898 + 120583) (41)

The Scientific World Journal 9

where 120573 = (120) int20

0120573(119905)119889119905 When 119903 = 013 119886 = 03 and

119887 = 02 we know that 1198770 = 09029 and 1198770 = 1 When 119903 =

013 119896 = 100000 119886 = 036 119887 = 02 and 119898 = 01 120590 = 02120583 = 01 then 1198770 = 10834 and 1198770 = 12 In that sense it isconfirmed that the basic reproduction number 1198770 defined by[40] is more accurate than the average reproduction number1198770 which overestimates the risk of disease

It should be noted that we live in a spatial world and it is anatural phenomenon that a substance goes from high densityregions to low density regions As a result epidemic modelsshould include spatial effects In a further study we need toinvestigate spatial epidemic models with seasonal factors

Acknowledgments

The research was partially supported by the National NaturalScience Foundation of China under Grant nos 1130149011301491 11331009 11147015 11171314 11101251 and 11105024Natural Science Foundation of ShanXi Province Grant no2012021002-1 and the Opening Foundation of Institute ofInformation Economy Hangzhou Normal University Grantno PD12001003002003

References

[1] K Dietz and J A P Heesterbeek ldquoBernoulli was ahead ofmodern epidemiologyrdquo Nature vol 408 no 6812 pp 513ndash5142000

[2] K Dietz and J A P Heesterbeek ldquoDaniel Bernoullirsquos epidemio-logical model revisitedrdquoMathematical Biosciences vol 180 pp1ndash21 2002

[3] R Ross ldquoAn application of the theory of probabilities to thestudy of a priori pathometry (part I)rdquo Proceedings of the RoyalSociety A vol 92 no 638 pp 204ndash230 1916

[4] R Ross and H P Hudson ldquoAn application of the theory ofprobabilities to the study of a priori pathometry (part II)rdquoProceedings of the Royal Society A vol 93 no 650 pp 212ndash2251917

[5] W O Kermack and A G McKendrick ldquoA contribution to themathematical theory of epidemics (part I)rdquo Proceedings of theRoyal Society A vol 115 no 772 pp 700ndash721 1927

[6] W O Kermack and A G McKendrick ldquoA contribution to themathematical theory of epidemics (part II)rdquo Proceedings of theRoyal Society A vol 138 no 834 pp 55ndash83 1932

[7] A Tsoularis and JWallace ldquoAnalysis of logistic growthmodelsrdquoMathematical Biosciences vol 179 no 1 pp 21ndash55 2002

[8] I Nasell ldquoOn the quasi-stationary distribution of the stochasticlogistic epidemicrdquoMathematical Biosciences vol 156 no 1-2 pp21ndash40 1999

[9] H Fujikawa A Kai and SMorozumi ldquoA new logisticmodel forEscherichia coli growth at constant and dynamic temperaturesrdquoFood Microbiology vol 21 no 5 pp 501ndash509 2004

[10] L Berezansky and E Braverman ldquoOscillation properties of alogistic equation with several delaysrdquo Journal of MathematicalAnalysis and Applications vol 247 no 1 pp 110ndash125 2000

[11] M Tabata N Eshima and I Takagi ldquoThe nonlinear integro-partial differential equation describing the logistic growth ofhuman population with migrationrdquo Applied Mathematics andComputation vol 98 pp 169ndash183 1999

[12] L Korobenko andE Braverman ldquoOn logisticmodels with a car-rying capacity dependent diffusion stability of equilibria andcoexistence with a regularly diffusing populationrdquo NonlinearAnalysis Real World Applications vol 13 no 6 pp 2648ndash26582012

[13] Y Muroya ldquoGlobal attractivity for discrete models of nonau-tonomous logistic equationsrdquo Computers andMathematics withApplications vol 53 no 7 pp 1059ndash1073 2007

[14] S Invernizzi and K Terpin ldquoA generalized logistic model forphotosynthetic growthrdquo Ecological Modelling vol 94 no 2-3pp 231ndash242 1997

[15] ZMinW Bang-Jun and J Feng ldquoCoalmining citiesrsquo economicgrowthmechanism and sustainable development analysis basedon logistic dynamics modelrdquo Procedia Earth and PlanetaryScience vol 1 no 1 pp 1737ndash1743 2009

[16] L I Anita S Anita and V Arnautu ldquoGlobal behavior foran age-dependent population model with logistic term andperiodic vital ratesrdquoAppliedMathematics andComputation vol206 no 1 pp 368ndash379 2008

[17] Z A Banaszak X Q Tang S C Wang and M B ZarembaldquoLogistics models in flexible manufacturingrdquo Computers inIndustry vol 43 no 3 pp 237ndash248 2000

[18] S Brianzoni C Mammana and E Michetti ldquoNonlineardynamics in a business-cycle model with logistic populationgrowthrdquo Chaos Solitons and Fractals vol 40 no 2 pp 717ndash7302009

[19] W P London and J A Yorke ldquoRecurrent outbreaks of measleschickenpox and mumps I Seasonal variation in contact ratesrdquoThe American Journal of Epidemiology vol 98 no 6 pp 453ndash468 1973

[20] S F Dowell ldquoSeasonal variation in host susceptibility and cyclesof certain infectious diseasesrdquo Emerging Infectious Diseases vol7 no 3 pp 369ndash374 2001

[21] ON Bjoslashrnstad B F Finkenstadt andB TGrenfell ldquoDynamicsof measles epidemics estimating scaling of transmission ratesusing a Time series SIR modelrdquo Ecological Monographs vol 72no 2 pp 169ndash184 2002

[22] J Zhang Z Jin G Q Sun X Sun and S Ruan ldquoModelingseasonal rabies epidemics in Chinardquo Bulletin of MathematicalBiology vol 74 no 5 pp 1226ndash1251 2012

[23] I B Schwartz ldquoSmall amplitude long period outbreaks inseasonally driven epidemicsrdquo Journal of Mathematical Biologyvol 30 no 5 pp 473ndash491 1992

[24] I B Schwartz and H L Smith ldquoInfinite subharmonic bifur-cation in an SEIR epidemic modelrdquo Journal of MathematicalBiology vol 18 no 3 pp 233ndash253 1983

[25] H L Smith ldquoMultiple stable subharmonics for a periodicepidemic modelrdquo Journal of Mathematical Biology vol 17 no2 pp 179ndash190 1983

[26] J Dushoff J B Plotkin S A Levin and D J D EarnldquoDynamical resonance can account for seasonality of influenzaepidemicsrdquo Proceedings of the National Academy of Sciences ofthe United States of America vol 101 no 48 pp 16915ndash169162004

[27] J L Ma and Z Ma ldquoEpidemic threshold conditions forseasonally forced SEIR modelsrdquo Mathematical Biosciences andEngineering vol 3 no 1 pp 161ndash172 2006

[28] D J D Earn P Rohani B M Bolker and B T Grenfell ldquoAsimple model for complex dynamical transitions in epidemicsrdquoScience vol 287 no 5453 pp 667ndash670 2000

10 The Scientific World Journal

[29] M J Keeling P Rohani and B T Grenfell ldquoSeasonally forceddisease dynamics explored as switching between attractorsrdquoPhysica D vol 148 no 3-4 pp 317ndash335 2001

[30] J L Aron and I B Schwartz ldquoSeasonality and period-doublingbifurcations in an epidemic modelrdquo Journal of TheoreticalBiology vol 110 no 4 pp 665ndash679 1984

[31] N C Grassly and C Fraser ldquoSeasonal infectious diseaseepidemiologyrdquo Proceedings of the Royal Society B vol 273 no1600 pp 2541ndash2550 2006

[32] J Zhang Z Jin G Q Sun T Zhou and S Ruan ldquoAnalysisof rabies in China transmission dynamics and controlrdquo PLoSONE vol 6 no 7 Article ID e20891 2011

[33] O Diekmann J A P Heesterbeek and J A Metz ldquoOnthe definition and the computation of the basic reproductionratio R0 in models for infectious diseases in heterogeneouspopulationsrdquo Journal of mathematical biology vol 28 no 4 pp365ndash382 1990

[34] O Diekmann J A P Heesterbeek and M G Roberts ldquoTheconstruction of next-generation matrices for compartmentalepidemic modelsrdquo Journal of the Royal Society Interface vol 7no 47 pp 873ndash885 2010

[35] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[36] F Berezovsky G Karev B J Song and C C Chavez ldquoA simpleepidemic model with surprising dynamicsrdquo Mathematical Bio-sciences and Engineering vol 2 pp 133ndash152 2005

[37] J LaSalle and S Lefschetz Stability by Liapunovrsquos DirectMethodAcademic Press New York NY USA 1961

[38] E A Barbashin Introduction to theTheory of Stability Wolters-Noordhoff Groningen The Netherlands 1970

[39] D Schenzle ldquoAn age-structured model of pre- and post-vaccination measles transmissionrdquoMathematical Medicine andBiology vol 1 no 2 pp 169ndash191 1984

[40] N Bacaer and S Guernaoui ldquoThe epidemic threshold ofvector-borne diseases with seasonality the case of cutaneousleishmaniasis in Chichaoua Moroccordquo Journal of MathematicalBiology vol 53 no 3 pp 421ndash436 2006

[41] W DWang and X Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008

[42] F Zhang and X Q Zhao ldquoA periodic epidemic model in apatchy environmentrdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 496ndash516 2007

[43] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press New York NY USA 1995

[44] H R Thieme ldquoConvergence results and a Poincare-Bendixsontrichotomy for asymptotically automous differential equationsrdquoJournal of Mathematical Biology vol 30 pp 755ndash763 1992

[45] L Perko Differential Equations and Dynamical SystemsSpringer New York NY USA 2000

[46] X Q Zhao Dynamical Systems in Population Biology SpringerNew York NY USA 2003

[47] Z G Bai and Y C Zhou ldquoThreshold dynamics of a bacillarydysentery model with seasonal fluctuationrdquo Discrete and Con-tinuous Dynamical Systems B vol 15 no 1 pp 1ndash14 2011

[48] Z G Bai Y C Zhou and T L Zhang ldquoExistence of multipleperiodic solutions for an SIRmodel with seasonalityrdquoNonlinear

Analysis Theory Methods and Applications vol 74 no 11 pp3548ndash3555 2011

[49] N Bacaer ldquoApproximation of the basic reproduction numberR0 for vector-borne diseases with a periodic vector populationrdquoBulletin of Mathematical Biology vol 69 no 3 pp 1067ndash10912007

[50] N Bacaer and X Abdurahman ldquoResonance of the epidemicthreshold in a periodic environmentrdquo Journal of MathematicalBiology vol 57 no 5 pp 649ndash673 2008

[51] N Bacaer ldquoPeriodic matrix population models growth ratebasic reproduction number and entropyrdquoBulletin ofMathemat-ical Biology vol 71 no 7 pp 1781ndash1792 2009

[52] N Bacaer and E H A Dads ldquoGenealogy with seasonalitythe basic reproduction number and the influenza pandemicrdquoJournal of Mathematical Biology vol 62 no 5 pp 741ndash762 2011

[53] L J Liu X Q Zhao and Y C Zhou ldquoA tuberculosis model withseasonalityrdquo Bulletin of Mathematical Biology vol 72 no 4 pp931ndash952 2010

[54] J L Liu ldquoThreshold dynamics for a HFMD epidemic modelwith periodic transmission raterdquo Nonlinear Dynamics vol 64no 1-2 pp 89ndash95 2011

[55] Y Nakata and T Kuniya ldquoGlobal dynamics of a class ofSEIRS epidemic models in a periodic environmentrdquo Journal ofMathematical Analysis and Applications vol 363 no 1 pp 230ndash237 2010

[56] B GWilliams andC Dye ldquoInfectious disease persistence whentransmission varies seasonallyrdquo Mathematical Biosciences vol145 no 1 pp 77ndash88 1997

[57] I A Moneim ldquoThe effect of using different types of periodiccontact rate on the behaviour of infectious diseases a simula-tion studyrdquo Computers in Biology and Medicine vol 37 no 11pp 1582ndash1590 2007

[58] C L Wesley and L J S Allen ldquoThe basic reproduction numberin epidemic models with periodic demographicsrdquo Journal ofBiological Dynamics vol 3 no 2-3 pp 116ndash129 2009

[59] D Greenhalgh and I A Moneim ldquoSIRS epidemic model andsimulations using different types of seasonal contact raterdquoSystems Analysis Modelling Simulation vol 43 no 5 pp 573ndash600 2003

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 The Scientific World Journal

growth Muroya [13] investigated discrete models of non-autonomous logistic equations As a result this paper buildsan SEI ordinary differential model with the logistic growthrate and the standard incidence

For general epidemic models we mainly study theirthreshold dynamics that is the basic reproduction numberwhich determines whether the disease can invade the suscep-tible population successfully However for the system withlogistic growth rate besides the basic reproduction numberthe qualitative dynamics are controlled by a demographicthreshold 119877119889 which has a similar meaning and is called asthe basic demographic reproduction number If 119877119889 gt 1the population grows that is a critical mass of individualsfor the disease to spread may be supported If 119877119889 lt 1the population will not survive that is not enough mass ofindividuals may be supported for the disease to spread Forthis case the dynamical behavior of disease will be decidedby two thresholds 1198770 and 119877119889

It is well known that many diseases exhibit seasonalfluctuations such as whooping cough measles influenzapolio chickenpox mumps and rabies [19ndash22] Seasonallyeffective contact rate [22ndash26] periodic changing in the birthrate [27] and vaccination program [28] are often regardedas sources of periodicity Seasonally effective contact rate isrelated to the behavior of people and animals the temper-ature and the economy Due to the existence of differentseasons people have different activities which may lead to adifferent contact rate Because of various factors the economyin a different season has a very big difference Therefore thispaper studies the corresponding non-autonomous systemwhich is obtained by changing the constant transmission rateinto the periodic transmission rate Seasonal transmission isoften assumed to be sinusoidal (cosine function has the samemeaning) such that 120582(119905) = 120582(1 + 120578 sin(120587119905119887)) where 120578 isthe amplitude of seasonal variation in transmission (typicallyreferred to as the ldquostrength of seasonal forcingrdquo) and 2119887 isthe period which is a crude assumption for many infectiousdiseases [29ndash31] When 120578 = 0 there is no nonseasonalinfections Motivated by biological realism some recentpapers take the contact rate as 120582(119905) = 120582(1 + 120578 term(119905)) whereterm is a periodic function which is +1 during a period oftime and minus1 during other time More natural term can bewritten as 120582(119905) = 120582(1 + 120578)

term(119905) [29] Here we take the form120573(119905) = 119886[1 + 119887 sin(12058711990510)]

The paper is organized as follows In Section 2 weintroduce an autonomous model and analyze the equilibriaand their respective attractive region In Section 3 we studythe non-autonomous system in terms of global asymptoticstability of the disease-free equilibrium and the existence ofpositive periodic solutions Moreover numerical simulationsare also performed In Section 4 we give a brief discussion

2 Autonomous Model and Analysis

21 Model Formulation The model is a system of SEI ordi-nary differential equations where 119878 is the susceptible 119864 isthe exposed 119868 is the infected and 119873 = 119878 + 119864 + 119868 Thissystem considers the logistic growth rate and the standard

Table 1 Descriptions and values of parameters in model (1)

Parameter Interpretation Value119903 The intrinsic growth rate119896 The carrying capacity 100000120573 The transmission rate119898 The natural mortality rate 01120590 Clinical outcome rate 02120583 The disease-induced mortality rate 01119886 The baseline contact rate119887 Themagnitude of forcing

incidence which is fit for the long-term growth of many largepopulations The incubation period is considered for manydiseases which do not develop symptoms immediately andneed a period of time to accumulate a pathogen quantity forclinical outbreak such as rabies hand-foot-mouth diseasetuberculosis and AIDS [22 32] The model we employ is asfollows

119889119878

119889119905= 119903119873(1 minus

119873

119896) minus

120573119878119868

119873minus 119898119878

119889119864

119889119905=

120573119878119868

119873minus 120590119864 minus 119898119864

119889119868

119889119905= 120590119864 minus 119898119868 minus 120583119868

(1)

where all parameters are positive whose interpretations canbe seen in Table 1

Noticing the equations in model (1) we have

119889119873

119889119905= 119903119873(1 minus

119873

119896) minus 119898119873 minus 120583119868 (2)

When there exists no disease we have

119889119873

119889119905= 119903119873(1 minus

119873

119896) minus 119898119873 = [119903 (1 minus

119873

119896) minus 119898]119873 (3)

Let 119877119889 = 119903119898 if 119877119889 gt 1 119873 rarr 1198730

= (1 minus 119898119903)119896 for119873(0) gt 0 as 119905 rarr +infin that is the population will growand tend to a steady state 119873

0 If 119877119889 lt 1 then 119889119873119889119905 lt 0

which will cause the population to disappear Thus 119877119889 is thebasic demographic reproduction number From the aboveequation the feasible region can be obtained 119883 = (119878 119864 119868) |

119878 119864 119868 ge 0 0 le 119878 + 119864 + 119868 le (1 minus 119898119903)119896 where 119903 gt 119898

Theorem 1 The region119883 is positively invariant with respect tosystem (1)

22 Dynamical Analysis Let the right hand of system (1) tobe zero it is easy to see that system (1) has three equilibria

119874 = (0 0 0)

1198640 = ((1 minus119898

119903) 119896 0 0)

119864lowast = (119878lowast 119864lowast 119868lowast)

(4)

The Scientific World Journal 3

where 119874 is the origin 1198640 is the disease-free equilibrium and119864lowast is the endemic equilibrium Concretely one can have

119878lowast

=[119898120573 (119898 + 120590 + 120583) + 120583 (119898 + 120583) (119898 + 120590) (1198770 minus 1)]119873

lowast

120573 [(119898 + 120583) (119898 + 120590) (1198770 minus 1) + 119898 (119898 + 120590 + 120583)]

(5)

119864lowast

=[119903 (1 minus 119873

lowast119896) minus 119898] (119898 + 120583)119873

lowast

120590120583 (6)

119868lowast

=(119898 + 120583) (119898 + 120590) (1198770 minus 1)119873

lowast

120573 (119898 + 120590 + 120583) (7)

119873lowast

= (119896 [120573119898 (119898 + 120590 + 120583) (119877119889 minus 1)

+120583 (119898 + 120590) (119898 + 120583) (1 minus 1198770)])

times (119903120573 (119898 + 120590 + 120583))minus1

(8)

where 119877119889 = 119903119898 is the basic demographic reproductionnumber and1198770 = 120573120590(119898+120590)(119898+120583) is the basic reproductionnumber which can be obtained by the next-generationmatrixmethod [33ndash35] The introduction of the basic demographicreproduction number can be found in [36]

Moreover from (6) and (7) the conditions of the endemicequilibrium to exist are 1198770 gt 1 and 119877119889 gt 1198771 where 1198771 =

1+120583(119898+120583)(119898+120590)(1198770 minus1)119898120573(119898+120590+120583) So we can obtainthe following theorems

Theorem 2 The system (1) has three equilibria origin 119874disease-free equilibrium 1198640 and the endemic equilibrium 119864lowast119874 always exists if 119877119889 gt 1 1198640 exists if 1198770 gt 1 and 119877119889 gt 1198771 119864lowastexists

Theorem 3 When 119877119889 gt 1 and 1198770 lt 1 1198640 is globallyasymptotically stable

Proof By [33ndash35] we know that 1198640 is locally asymptoticallystable Now we define a Lyapunov function

119881 = 119864 +119898 + 120590

120590119868 ge 0 (9)

When 119877119889 gt 1 and 1198770 lt 1 the Lyapunov function satisfies

= +119898 + 120590

120590

119868

=120573119878119868

119873minus

(119898 + 120590) (119898 + 120583)

120590119868

le [120573 minus(119898 + 120590) (119898 + 120583)

120590] 119868

=(119898 + 120590) (119898 + 120583)

120590[1198770 minus 1] 119868

le 0

(10)

Moreover = 0 only hold when 119868 = 0 It is easy to verifythat the disease-free equilibrium point 1198640 is the only fixedpoint of the system Hence applying the Lyapunov-LaSalle

Rd

R0

E0

Elowast

O1 O2

O3

0

L

1

1

Figure 1 119877119889 in terms of 1198770 The region of 1198741 1198742 and 1198743 is thebasin of attraction of equilibrium 119874 the region of 1198640 is the basinof attraction of equilibrium 1198640 the region of 119864lowast is the basin ofattraction of equilibrium 119864lowast the line 119871 is 119877119889 = 1198771 = 1 + 120583(119898 +

120583)(119898 + 120590)(1198770 minus 1)119898120573(119898 + 120590 + 120583)

asymptotic stability theorem in [37 38] the disease-freeequilibrium point 1198640 is globally asymptotically stable

Since the proof of the stability of equilibria 119874 and 119864lowast ismore difficult we only give some numerical results

In sum we can show the respective basins of attractionof the three equilibria which can be seen in Figure 1 andconfirmed in Figure 2

(1) When 119877119889 lt 1 119874 is stable see Figures 2(b) and 2(c)(2) When119877119889 gt 1 and1198770 lt 1 1198640 is stable see Figure 2(d)(3) When 119877119889 gt 1 1198770 gt 1 and 119877119889 lt 1198771 119874 is stable see

Figure 2(a)(4) When 119877119889 gt 1 1198770 gt 1 and 119877119889 gt 1198771 119864lowast is stable see

Figure 2(e)

3 Nonautonomous Model and Analysis

31 The Basic Reproduction Number Now we consider thenon-autonomous case of themodel (1) when the transmissionrate is periodic which is given as follows

119889119878

119889119905= 119903119873(1 minus

119873

119896) minus

120573 (119905) 119878119868

119873minus 119898119878

119889119864

119889119905=

120573 (119905) 119878119868

119873minus 120590119864 minus 119898119864

119889119868

119889119905= 120590119864 minus 119898119868 minus 120583119868

(11)

where 120573(119905) is a periodic function which is proposed by [39]In the subsequent section we will discuss the dynamicalbehavior of the system (11)

For system (11) firstly we can give the basic reproductionnumber 1198770 According to the basic reproduction numberunder non-autonomous system we can refer to the methodof [40 41] From the last section we know that system (11)

4 The Scientific World Journal

80

70

60

50

40

30

20

10

00 50 100 150 200 250 300

R0 = 16667

Rd = 101

R1 = 12

I(t)

S(t)

(a)

R0 = 1667

Rd = 05

R1 = 10714

80

70

60

50

40

30

20

10

00 50 100 150 200 250 300

I(t)

S(t)

(b)

R0 = 06667

Rd = 08

R1 = 075

80

70

60

50

40

30

20

10

00 50 100 150 200 250 300

I(t)

S(t)

(c)

R0 = 06667

Rd = 13

R1 = 075

600

500

400

300

200

100

00 05 1 15 2 25 3 35 4

times104

I(t)

S(t)

(d)

R0 = 2333

Rd = 13

R1 = 12857

600

500

400

300

200

200

100

0100 300 400 500 600 700 800 900

I(t)

S(t)

(e)

Figure 2 The phase curves of the system under different initial conditions (a) In the region 1198743 with 119903 = 0101 and 120573 = 05 (b) in the region1198742 with 119903 = 005 and 120573 = 035 (c) in the region1198741 with 119903 = 008 and 120573 = 02 (d) in the region 1198640 with 119903 = 013 and 120573 = 02 (e) in the region119864lowastwith 119903 = 013 and 120573 = 07 The value of other parameters can be seen in Table 1

The Scientific World Journal 5

has one disease-free equilibrium 1198640 = (1198730 0 0) where119873

0=

(1 minus 119898119903)119896 By giving a new vector 119909 = (119864 119868) we have

119865 = (

120573 (119905) 119878119868

119873

0

) 119881 = (119898119864 + 120590119864

119898119868 + 120583119868 minus 120590119864)

119881minus

= (119898119864 + 120590119864

119898119868 + 120583119868) 119881

+= (

0

120590119864)

(12)

Taking the partial derivative of the above vectors aboutvariables 119864 119868 and substituting the disease-free equilibriumwe have

119865 (119905) = (0 120573 (119905)

0 0)

119881 (119905) = (119898 + 120590 0

minus120590 119898 + 120583)

(13)

According to [41] denote 119862120596 to be the ordered Banachspace of all 120596-periodic functions from R to R4 which isequippedwith themaximumnorm sdot and the positive cone119862+

120596= 120601 isin 119862120596 120601(119905) ge 0 forall119905 isin R+ Over the Banach space

we define a linear operator 119871 119862120596 rarr 119862120596 by

(119871120601) (119905)

= int

infin

0

119884 (119905 119905 minus 119886) 119865 (119905 minus 119886) 120601 (119905 minus 119886) 119889119886 forall119905 isin R+ 120601 isin 119862120596

(14)

where 119871 is called the next infection operator and the inter-pretation of 119884(119905 119905 minus 119886) 120601(119905 minus 119886) can be seen in [41] Thenthe spectral radius of 119871 is defined as the basic reproductionnumber

1198770 = 120588 (119871) (15)

In order to give the expression of the basic reproductionnumber we need to introduce the linear 120596-periodic system

119889119908

119889119905= [minus119881 (119905) +

119865 (119905)

120582]119908 119905 isin R+ (16)

with parameter 120582 isin R Let 119882(119905 119904 120582) 119905 ge 119904 be theevolution operator of system (16) on R2 In fact 119882(119905 119904 120582) =

Φ(119865120582)minus119881(119905) and Φ119865minus119881(119905) = 119882(119905 0 1) for all 119905 ge 0 ByTheorems 21 and 22 in [41] the basic reproduction numberalso can be defined as1205820 such that120588(Φ(1198651205820)minus119881(120596)) = 1 whichcan be straightforward to calculate

32 Global Stability of the Disease-Free Equilibrium

Theorem4 Thedisease-free equilibrium1198640 is globally asymp-totically stable when 1198770 lt 1 and 119877119889 gt 1

Proof Theorem 22 in [41] implies that 1198640 is locally asymp-totically stable when 1198770 lt 1 and 119877119889 gt 1 So we only need

to prove its global attractability It is easy to know that 119878(119905) le

1198730= (1 minus (119898119903))119896 Thus

119889119864

119889119905le 120573 (119905) 119868 minus (119898 + 120590) 119864

119889119868

119889119905= 120590119864 minus 119898119868 minus 120583119868

(17)

The right comparison system can be written as

119889119864

119889119905= 120573 (119905) 119868 minus (119898 + 120590) 119864

119889119868

119889119905= 120590119864 minus 119898119868 minus 120583119868

(18)

that is

119889ℎ

119889119905= (119865 (119905) minus 119881 (119905)) ℎ (119905) ℎ (119905) = (119864 (119905) 119868 (119905)) (19)

For (19) Lemma 21 in [42] shows that there is apositive 120596-periodic function ℎ(119905) = (119864(119905) 119868(119905))

119879 such thatℎ(119905) = 119890

119901119905ℎ(119905) is a solution of system (18) where 119901 =

(1120596) ln 120588(Φ119865minus119881(120596)) By Theorem 22 in [41] we know thatwhen 1198770 lt 1 and 119877119889 gt 1 120588(Φ119865minus119881(120596)) lt 1 and 119901 lt 0which implies ℎ(119905) rarr 0 as 119905 rarr infin Therefore the zerosolution of system (18) is globally asymptotically stable Bythe comparison principle [43] and the theory of asymptoticautonomous systems [44] when 1198770 lt 1 and 119877119889 gt 1 1198640is globally attractive Therefore the proposition that 1198640 isglobally asymptotically stable holds

33 Existence of Positive Periodic Solutions Before the proofof the existence of positive periodic solutions we firstlyintroduce some denotations Let 119906(119905 1199090) be the solution ofsystem (11) with the initial value 1199090 = (119878(0) 119864(0) 119868(0)) Bythe fundamental existence-uniqueness theorem [45] 119906(119905 1199090)is the unique solution of system (11) with 119906(0 1199090) = 1199090

Next we need to introduce the Poincare map119875 119883 rarr 119883

associated with system (11) that is

119875 (1199090) = 119906 (120596 1199090) forall1199090 isin 119883 (20)

where 120596 is the period Theorem 1 implies that 119883 is positivelyinvariant and 119875 is a dissipative point

Now we introduce two subsets of1198831198830 = (119878 119864 119868) isin 119883

119864 gt 0 119868 gt 0 and 1205971198830 = 119883 1198830

Lemma 5 (a) When 1198770 gt 1 and 119903 gt 119898 + 120583 there exists a120575 gt 0 such that when

1003817100381710038171003817(119878 (0) 119864 (0) 119868 (0)) minus 11986401003817100381710038171003817 le 120575 (21)

for any (119878(0) 119864(0) 119868(0)) isin 1198830 one has

lim sup119898rarrinfin

119889 [119875119898

(119878 (0) 119864 (0) 119868 (0)) 1198640] ge 120575 (22)

where 1198640 = (1198730 0 0)

6 The Scientific World Journal

(b) When 1198770 gt 1 and 119903 gt 119898 + 120583 there exists a 120575 gt 0 suchthat when

(119878 (0) 119864 (0) 119868 (0)) minus 119874 le 120575 (23)

for any (119878(0) 119864(0) 119868(0)) isin 1198830 one has

lim sup119898rarrinfin

119889 [119875119898

(119878 (0) 119864 (0) 119868 (0)) 119874] ge 120575 (24)

where 119874 = (0 0 0)

Proof (a) ByTheorem 22 in [41] we know that when 1198770 gt 1120588(Φ119865minus119881(120596)) gt 1 So there is a small enough positive number120598 such that 120588(Φ119865minus119881minus119872120598(120596)) gt 1 where

119872120598 = (0

120573 (119905) 120598119868

1198730

0 0

) (25)

If proposition (a) does not hold there is some(119878(0) 119864(0) 119868(0)) isin 1198830 such that

lim sup119898rarrinfin

119889 (119875119898

(119878 (0) 119864 (0) 119868 (0)) 1198640) lt 120575 (26)

We can assume that for all 119898 ge 0 119889(119875119898(119878(0) 119864(0)

119868(0)) 1198640) lt 120575 Applying the continuity of the solutions withrespect to the initial values

1003817100381710038171003817119906 (119905 119875119898

(119878 (0) 119864 (0) 119868 (0))) minus 119906 (119905 1198640)1003817100381710038171003817 le 120598

forall119898 ge 0 forall1199051 isin [0 120596]

(27)

Let 119905 = 119898120596 + 1199051 where 1199051 isin [0 120596] and 119898 = [119905120596] 119898 =

[119905120596] is the greatest integer which is not more than 119905120596Thenfor any 119905 ge 0

1003817100381710038171003817119906 (119905 (119878 (0) 119864 (0) 119868 (0))) minus 119906 (119905 1198640)1003817100381710038171003817

=1003817100381710038171003817119906 (1199051 119875

119898(119878 (0) 119864 (0) 119868 (0))) minus 119906 (1199051 1198640)

1003817100381710038171003817 le 120598

(28)

So 1198730

minus 120598 le 119878(119905) le 1198730

+ 120598 Then whenlim sup

119898rarrinfin119889(119875119898(119878(0) 119864(0) 119868(0)) 1198640) lt 120575

119889119864

119889119905ge 120573 (119905) 119868 minus

120573 (119905) 120598119868

1198730minus (119898 + 120590) 119864

119889119868

119889119905= 120590119864 minus 119898119868 minus 120583119868

(29)

Thus we can study the right linear system

119889119864

119889119905= 120573 (119905) 119868 minus

120573 (119905) 120598119868

1198730minus (119898 + 120590) 119864

119889119868

119889119905= 120590119864 minus 119898119868 minus 120583119868

(30)

For the system (30) there exists a positive 120596-periodicfunction 119892(119905) = (119864(119905) 119868(119905))

119879 such that 119892(119905) = 119890119901119905119892(119905) is a

solution of system (11) where 119901 = (1120596) ln 120588(Φ119865minus119881minus119872120598(120596))When 1198770 gt 1 120588(Φ119865minus119881minus119872120598(120596)) gt 1 which means that when

119892(0) gt 0119892(119905) rarr infin as 119905 rarr infin By the comparison principle[43] when 119864(0) gt 0 119868(0) gt 0 119864(119905) rarr infin 119868(119905) rarr infin as119905 rarr infinThere appears a contradictionThus the proposition(a) holds

(b) When 1198770 gt 1 and 119903 gt 119898 + 120583 we have

119889119873

119889119905= 119903119873(1 minus

119873

119896) minus 119898119873 minus 120583119868

ge [119903 (1 minus119873

119896) minus 119898 minus 120583]119873 gt 0

(31)

So if 119877119889 gt 1 119873 rarr 1198730 for 119873(0) gt 0 as 119905 rarr +infin that is

119882119878(119874) cap 1198830 = 0

Theorem 6 When 1198770 gt 1 and 119903 gt 119898 + 120583 there exists a 120575 gt 0

such that any solution (119878(119905) 119864(119905) 119868(119905)) of system (11)with initialvalue (119878(0) 119864(0) 119868(0)) isin (119878 119864 119868) isin 119883 119864 gt 0 119868 gt 0

satisfies

lim inf119905rarrinfin

119864 (119905) ge 120575 lim inf119905rarrinfin

119868 (119905) ge 120575 (32)

and system (11) has at least one positive periodic solution

Proof From system (11)

119878 (119905) = 119890minusint119905

0(120573(119904)119868(119904)119873+119898)119889119904

times [119878 (0) + int

119905

0

119903119873 (119904) (1 minus119873 (119904)

119896)

times119890int119904

0(120573(119888)119868(119888)119873(119888)+119898)119889119888

119889119904]

gt 0 forall119905 gt 0

(33)

119864 (119905) = 119890minus(119898+120590)119905

[119864 (0) + int

119905

0

120573 (119904) 119878 (119904) 119868 (119904)

119873 (119904)119890(119898+120590)119904

119889119904]

gt 0 forall119905 gt 0

(34)

119868 (119905) = 119890minus(119898+120583)119905

[119868 (0) + int

119905

0

120590119864 (119904) 119890(119898+120583)119904

119889119904]

gt 0 forall119905 gt 0

(35)

for any (119878(0) 119864(0) 119868(0)) isin 1198830 which shows that 1198830 ispositively invariant Moreover it is obvious to see that 1205971198830

is relatively closed in 119883 Denote

119872120597 = (119878 (0) 119864 (0) 119868 (0)) isin 1205971198830

119875119898

(119878 (0) 119864 (0) 119868 (0)) isin 1205971198830 forall119898 ge 0

(36)

Next we prove that

119872120597 = (119878 0 0) isin 119883 119878 ge 0 (37)

We only need to show that 119872120597 sube (119878 0 0) isin 119883 119878 ge 0which means that for any (119878(0) 119864(0) 119868(0)) isin 1205971198830 119864(119898120596) =

119868(119898120596) = 0 for all 119898 ge 0 If it does not hold there existsa 1198981 ge 0 such that (119864(1198981120596) 119868(1198981120596))

119879gt 0 Taking 1198981120596 as

The Scientific World Journal 7

25

2

15

1

05

01 15 2 25 3 35

times104

times104

S(0) = 25000 E(0) = 10000 I(0) = 24000

S(0) = 35000 E(0) = 20000 I(0) = 15000

S(0) = 15000 E(0) = 10000 I(0) = 20000

I(t)

S(t)

(a)

24

23

22

21

2

19

18

17

16

15

1415 2 25 3 35

times104

times104

S(0) = 25000 E(0) = 10000 I(0) = 24000

S(0) = 35000 E(0) = 20000 I(0) = 15000

S(0) = 15000 E(0) = 10000 I(0) = 20000

I(t)

S(t)

(b)

Figure 3 Phase plane of 119878(119905) and 119868(119905) (a) When the parameter values are 119903 = 013 119886 = 03 and 119887 = 02 1198770 = 09029 lt 1 119877119889 = 13 gt 1 (b)When the parameter values are 119903 = 03 119886 = 07 and 119887 = 02 1198770 = 21067 and 119903 gt 119898+120583 The value of other parameters can be seen in Table 1

the initial time and repeating the processes as in (33)ndash(35)we can have that (119878(119905) 119864(119905) 119868(119905))

119879gt 0 for all 119905 gt 1198981120596

Thus (119878(119905) 119864(119905) 119868(119905)) isin 1198830 for all 119905 gt 1198981120596 There appearsa contradiction which means that the equality (37) holdsTherefore 1198640 is acyclic in 1205971198830 Obviously when 1198770 gt 1 and119903 gt 119898 + 120583 119874 is acyclic in 1205971198830

Furthermore by Lemma 5 1198640 = (1198730 0 0) and 119874 =

(0 0 0) are isolated invariant sets in 119883 119882119878(1198640) cap 1198830 = 0and 119882

119878(119874) cap 1198830 = 0 By Theorem 131 and Remark 131

in [46] it can be obtained that 119875 is uniformly persistent withrespect to (1198830 1205971198830) that is there exists a 120575 gt 0 such that anysolution (119878(119905) 119864(119905) 119868(119905)) of system (11) with the initial value(119878(0) 119864(0) 119868(0)) isin (119878 119864 119868) isin 119883 119864 gt 0 119868 gt 0 satisfies

lim inf119905rarrinfin

119864 (119905) ge 120575 lim inf119905rarrinfin

119868 (119905) ge 120575 (38)

ApplyingTheorem 136 in [46] 119875 has a fixed point

(119878lowast(0) 119864

lowast(0) 119868lowast(0)) isin 1198830 (39)

From (33) we know 119878lowast

gt 0 for all 119905 isin [0 120596] 119878lowast(119905) is alsomore than zero for all 119905 gt 0 due to the periodicity Similarlyfor all 119905 ge 0119864lowast(119905) gt 0 119868lowast(119905) gt 0Therefore it can be obtainedthat one of the positive 120596-periodic solutions of system (11) is(119878lowast(119905) 119864lowast(119905) 119868lowast(119905))

34 Numerical Simulations Firstly we give some notationsIf 119892(119905) is a periodic function with period 120596 we define 119892 =

(1120596) int120596

0119892(119905)119889119905 119892119897 = min119905isin[0120596]119892(119905) 119892119906 = max119905isin[0120596]119892(119905) As

described in the previous section

1198771 (119905) = 1 +120583 (119898 + 120583) (119898 + 120590) (1198770 minus 1)

119898120573 (119905) (119898 + 120590 + 120583) (40)

So 119877119897

1= 1 + 120583(119898 + 120583)(119898 + 120590)(119877

119897

0minus 1)119898120573

119906(119898 + 120590 + 120583) 119877119906

1=

1 + 120583(119898 + 120583)(119898 + 120590)(119877119906

0minus 1)119898120573

119897(119898 + 120590 + 120583)

In this section we adopt 120573(119905) = 119886[1 + 119887 sin(12058711990510)]Then applying the numerical simulation to verify the abovesolution we give the following conclusion

(1) when 119877119889 lt 1 119874 is stable(2) when 119877119889 gt 1 and 1198770 lt 1 1198640 is stable see Figure 3(a)(3) when 1198770 gt 1 and 119903 gt 119898 + 120583 system (11) has at least

one positive periodic solution see Figure 3(b)

We can givemore results about the conditions of existenceof the positive periodic solution

(11015840) When 119877119889 gt 1 1198770 gt 1 and 119877119889 lt 119877

119906

1 119874 is stable see

Figures 4(a) and 4(b)(21015840) When 119877119889 gt 1 1198770 gt 1 and 119877119889 gt 119877

119906

1 system (11) has at

least one positive periodic solution see Figure 5

By numerical simulations we can give that the conditionswhich ensure the existence of positive periodic solution are1198770 gt 1 and 119903 gt 119898+120583 or 119877119889 gt 1 1198770 gt 1 and 119877119889 gt 119877

119906

1 In fact

119877119889 gt 1 1198770 gt 1 and 119877119889 gt 119877119906

1are the sufficient conditions for

1198770 gt 1 and 119903 gt 119898 + 120583 As a result the conditions 1198770 gt 1 and119903 gt 119898 + 120583 are broader

4 Discussion

This paper considers a logistic growth system whose birthprocess incorporates density-dependent effects This type ofmodel has a rich dynamical behavior and practical signif-icance By analyzing its equilibria and respective attractiveregion we find that the dynamical behavior of a disease will

8 The Scientific World Journal

25

2

15

1

05

00 05 1 15 2 25 3 35

times104

times104

S(0) = 25000 E(0) = 10000 I(0) = 24000

S(0) = 35000 E(0) = 20000 I(0) = 15000

S(0) = 15000 E(0) = 10000 I(0) = 20000

I(t)

S(t)

(a)

25

2

15

1

05

00 05 1 15 2 25 3 35

times104

times104

S(0) = 25000 E(0) = 10000 I(0) = 24000

S(0) = 35000 E(0) = 20000 I(0) = 15000

S(0) = 15000 E(0) = 10000 I(0) = 20000

I(t)

S(t)

(b)

Figure 4 Phase plane of 119878(119905) and 119868(119905) (a) When the parameter values are 119903 = 011 119886 = 09 119887 = 02 and 120583 = 03 1198770 = 14991 119877119889 = 11 gt 1119877119889 lt 119877

119906

1= 14159 and 119877119889 lt 119877

119897

1= 12773 (b) When the parameter values are 119903 = 0126 119886 = 07 119887 = 02 and 120583 = 03 1198770 = 21067

119877119889 = 126 gt 1 119877119889 lt 119877119906

1= 12964 and 119877119889 gt 119877

119897

1= 11976 The value of other parameters can be seen in Table 1

7000

6000

5000

4000

3000

2000

1000

005 1 15 2

times104

times104

S(0) = 22000 E(0) = 5000 I(0) = 1000

S(0) = 4000 E(0) = 2000 I(0) = 1000

S(0) = 10000 E(0) = 10000 I(0) = 6000

I(t)

S(t)

Figure 5 Phase plane of 119878(119905) and 119868(119905) When the parameter valuesare 119903 = 013 119886 = 036 and 119887 = 02 1198770 = 10834 119877119889 = 13 gt 1 119877119889 gt119877119906

1= 10435 and 119877119889 gt 119877

119897

1= 1029 The value of other parameters

can be seen in Table 1

be determined by two thresholds 1198770 and 119877119889 Only 1198770 gt 1

cannot promise the existence of the endemic equilibriumwhich also needs 119877119889 gt 1198771 When 1198770 gt 1 and 119877119889 lt 1198771the solutions of the system (1) will tend to the origin 119874 It

is caused by the phenomenon that the death number dueto disease cannot be supplemented by the birth numberpromptly Finally all people are infected and die outThe factinterpreted by this model is more reasonable Theoreticallywe prove the global asymptotic stability of the disease-free equilibrium and give respective attractive regions ofequilibria

Seasonally effective contact rate is themost common formwhich may be related to various factors and thus this paperstudies the corresponding non-autonomous system which isobtained by changing the constant transmission rate of theabove system into the periodic transmission rate For theperiodic systems their dynamical behaviors especially thebasic reproduction number have been investigated in depthby [41 47ndash55] which provide many methods that we canutilize For the obtained periodic model by analyzing theglobal asymptotic stability of the disease-free equilibrium andthe existence of positive periodic solution we have the similarresults as the autonomous system The dynamic behavior ofdisease will be decided by two conditions 1198770 gt 1 and 119903 gt

119898 + 120583 that show that when the disease is prevalent the birthrate should be larger than the death rate to guarantee thesustainable growth of population Otherwise the populationwill disappear In addition we will evaluate and comparethe basic reproduction number 1198770 and the average basicreproduction number 1198770 which has been adopted by [27 56ndash59] In this paper we can calculate the average reproductionnumber

1198770 =120573120590

(119898 + 120590) (119898 + 120583) (41)

The Scientific World Journal 9

where 120573 = (120) int20

0120573(119905)119889119905 When 119903 = 013 119886 = 03 and

119887 = 02 we know that 1198770 = 09029 and 1198770 = 1 When 119903 =

013 119896 = 100000 119886 = 036 119887 = 02 and 119898 = 01 120590 = 02120583 = 01 then 1198770 = 10834 and 1198770 = 12 In that sense it isconfirmed that the basic reproduction number 1198770 defined by[40] is more accurate than the average reproduction number1198770 which overestimates the risk of disease

It should be noted that we live in a spatial world and it is anatural phenomenon that a substance goes from high densityregions to low density regions As a result epidemic modelsshould include spatial effects In a further study we need toinvestigate spatial epidemic models with seasonal factors

Acknowledgments

The research was partially supported by the National NaturalScience Foundation of China under Grant nos 1130149011301491 11331009 11147015 11171314 11101251 and 11105024Natural Science Foundation of ShanXi Province Grant no2012021002-1 and the Opening Foundation of Institute ofInformation Economy Hangzhou Normal University Grantno PD12001003002003

References

[1] K Dietz and J A P Heesterbeek ldquoBernoulli was ahead ofmodern epidemiologyrdquo Nature vol 408 no 6812 pp 513ndash5142000

[2] K Dietz and J A P Heesterbeek ldquoDaniel Bernoullirsquos epidemio-logical model revisitedrdquoMathematical Biosciences vol 180 pp1ndash21 2002

[3] R Ross ldquoAn application of the theory of probabilities to thestudy of a priori pathometry (part I)rdquo Proceedings of the RoyalSociety A vol 92 no 638 pp 204ndash230 1916

[4] R Ross and H P Hudson ldquoAn application of the theory ofprobabilities to the study of a priori pathometry (part II)rdquoProceedings of the Royal Society A vol 93 no 650 pp 212ndash2251917

[5] W O Kermack and A G McKendrick ldquoA contribution to themathematical theory of epidemics (part I)rdquo Proceedings of theRoyal Society A vol 115 no 772 pp 700ndash721 1927

[6] W O Kermack and A G McKendrick ldquoA contribution to themathematical theory of epidemics (part II)rdquo Proceedings of theRoyal Society A vol 138 no 834 pp 55ndash83 1932

[7] A Tsoularis and JWallace ldquoAnalysis of logistic growthmodelsrdquoMathematical Biosciences vol 179 no 1 pp 21ndash55 2002

[8] I Nasell ldquoOn the quasi-stationary distribution of the stochasticlogistic epidemicrdquoMathematical Biosciences vol 156 no 1-2 pp21ndash40 1999

[9] H Fujikawa A Kai and SMorozumi ldquoA new logisticmodel forEscherichia coli growth at constant and dynamic temperaturesrdquoFood Microbiology vol 21 no 5 pp 501ndash509 2004

[10] L Berezansky and E Braverman ldquoOscillation properties of alogistic equation with several delaysrdquo Journal of MathematicalAnalysis and Applications vol 247 no 1 pp 110ndash125 2000

[11] M Tabata N Eshima and I Takagi ldquoThe nonlinear integro-partial differential equation describing the logistic growth ofhuman population with migrationrdquo Applied Mathematics andComputation vol 98 pp 169ndash183 1999

[12] L Korobenko andE Braverman ldquoOn logisticmodels with a car-rying capacity dependent diffusion stability of equilibria andcoexistence with a regularly diffusing populationrdquo NonlinearAnalysis Real World Applications vol 13 no 6 pp 2648ndash26582012

[13] Y Muroya ldquoGlobal attractivity for discrete models of nonau-tonomous logistic equationsrdquo Computers andMathematics withApplications vol 53 no 7 pp 1059ndash1073 2007

[14] S Invernizzi and K Terpin ldquoA generalized logistic model forphotosynthetic growthrdquo Ecological Modelling vol 94 no 2-3pp 231ndash242 1997

[15] ZMinW Bang-Jun and J Feng ldquoCoalmining citiesrsquo economicgrowthmechanism and sustainable development analysis basedon logistic dynamics modelrdquo Procedia Earth and PlanetaryScience vol 1 no 1 pp 1737ndash1743 2009

[16] L I Anita S Anita and V Arnautu ldquoGlobal behavior foran age-dependent population model with logistic term andperiodic vital ratesrdquoAppliedMathematics andComputation vol206 no 1 pp 368ndash379 2008

[17] Z A Banaszak X Q Tang S C Wang and M B ZarembaldquoLogistics models in flexible manufacturingrdquo Computers inIndustry vol 43 no 3 pp 237ndash248 2000

[18] S Brianzoni C Mammana and E Michetti ldquoNonlineardynamics in a business-cycle model with logistic populationgrowthrdquo Chaos Solitons and Fractals vol 40 no 2 pp 717ndash7302009

[19] W P London and J A Yorke ldquoRecurrent outbreaks of measleschickenpox and mumps I Seasonal variation in contact ratesrdquoThe American Journal of Epidemiology vol 98 no 6 pp 453ndash468 1973

[20] S F Dowell ldquoSeasonal variation in host susceptibility and cyclesof certain infectious diseasesrdquo Emerging Infectious Diseases vol7 no 3 pp 369ndash374 2001

[21] ON Bjoslashrnstad B F Finkenstadt andB TGrenfell ldquoDynamicsof measles epidemics estimating scaling of transmission ratesusing a Time series SIR modelrdquo Ecological Monographs vol 72no 2 pp 169ndash184 2002

[22] J Zhang Z Jin G Q Sun X Sun and S Ruan ldquoModelingseasonal rabies epidemics in Chinardquo Bulletin of MathematicalBiology vol 74 no 5 pp 1226ndash1251 2012

[23] I B Schwartz ldquoSmall amplitude long period outbreaks inseasonally driven epidemicsrdquo Journal of Mathematical Biologyvol 30 no 5 pp 473ndash491 1992

[24] I B Schwartz and H L Smith ldquoInfinite subharmonic bifur-cation in an SEIR epidemic modelrdquo Journal of MathematicalBiology vol 18 no 3 pp 233ndash253 1983

[25] H L Smith ldquoMultiple stable subharmonics for a periodicepidemic modelrdquo Journal of Mathematical Biology vol 17 no2 pp 179ndash190 1983

[26] J Dushoff J B Plotkin S A Levin and D J D EarnldquoDynamical resonance can account for seasonality of influenzaepidemicsrdquo Proceedings of the National Academy of Sciences ofthe United States of America vol 101 no 48 pp 16915ndash169162004

[27] J L Ma and Z Ma ldquoEpidemic threshold conditions forseasonally forced SEIR modelsrdquo Mathematical Biosciences andEngineering vol 3 no 1 pp 161ndash172 2006

[28] D J D Earn P Rohani B M Bolker and B T Grenfell ldquoAsimple model for complex dynamical transitions in epidemicsrdquoScience vol 287 no 5453 pp 667ndash670 2000

10 The Scientific World Journal

[29] M J Keeling P Rohani and B T Grenfell ldquoSeasonally forceddisease dynamics explored as switching between attractorsrdquoPhysica D vol 148 no 3-4 pp 317ndash335 2001

[30] J L Aron and I B Schwartz ldquoSeasonality and period-doublingbifurcations in an epidemic modelrdquo Journal of TheoreticalBiology vol 110 no 4 pp 665ndash679 1984

[31] N C Grassly and C Fraser ldquoSeasonal infectious diseaseepidemiologyrdquo Proceedings of the Royal Society B vol 273 no1600 pp 2541ndash2550 2006

[32] J Zhang Z Jin G Q Sun T Zhou and S Ruan ldquoAnalysisof rabies in China transmission dynamics and controlrdquo PLoSONE vol 6 no 7 Article ID e20891 2011

[33] O Diekmann J A P Heesterbeek and J A Metz ldquoOnthe definition and the computation of the basic reproductionratio R0 in models for infectious diseases in heterogeneouspopulationsrdquo Journal of mathematical biology vol 28 no 4 pp365ndash382 1990

[34] O Diekmann J A P Heesterbeek and M G Roberts ldquoTheconstruction of next-generation matrices for compartmentalepidemic modelsrdquo Journal of the Royal Society Interface vol 7no 47 pp 873ndash885 2010

[35] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[36] F Berezovsky G Karev B J Song and C C Chavez ldquoA simpleepidemic model with surprising dynamicsrdquo Mathematical Bio-sciences and Engineering vol 2 pp 133ndash152 2005

[37] J LaSalle and S Lefschetz Stability by Liapunovrsquos DirectMethodAcademic Press New York NY USA 1961

[38] E A Barbashin Introduction to theTheory of Stability Wolters-Noordhoff Groningen The Netherlands 1970

[39] D Schenzle ldquoAn age-structured model of pre- and post-vaccination measles transmissionrdquoMathematical Medicine andBiology vol 1 no 2 pp 169ndash191 1984

[40] N Bacaer and S Guernaoui ldquoThe epidemic threshold ofvector-borne diseases with seasonality the case of cutaneousleishmaniasis in Chichaoua Moroccordquo Journal of MathematicalBiology vol 53 no 3 pp 421ndash436 2006

[41] W DWang and X Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008

[42] F Zhang and X Q Zhao ldquoA periodic epidemic model in apatchy environmentrdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 496ndash516 2007

[43] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press New York NY USA 1995

[44] H R Thieme ldquoConvergence results and a Poincare-Bendixsontrichotomy for asymptotically automous differential equationsrdquoJournal of Mathematical Biology vol 30 pp 755ndash763 1992

[45] L Perko Differential Equations and Dynamical SystemsSpringer New York NY USA 2000

[46] X Q Zhao Dynamical Systems in Population Biology SpringerNew York NY USA 2003

[47] Z G Bai and Y C Zhou ldquoThreshold dynamics of a bacillarydysentery model with seasonal fluctuationrdquo Discrete and Con-tinuous Dynamical Systems B vol 15 no 1 pp 1ndash14 2011

[48] Z G Bai Y C Zhou and T L Zhang ldquoExistence of multipleperiodic solutions for an SIRmodel with seasonalityrdquoNonlinear

Analysis Theory Methods and Applications vol 74 no 11 pp3548ndash3555 2011

[49] N Bacaer ldquoApproximation of the basic reproduction numberR0 for vector-borne diseases with a periodic vector populationrdquoBulletin of Mathematical Biology vol 69 no 3 pp 1067ndash10912007

[50] N Bacaer and X Abdurahman ldquoResonance of the epidemicthreshold in a periodic environmentrdquo Journal of MathematicalBiology vol 57 no 5 pp 649ndash673 2008

[51] N Bacaer ldquoPeriodic matrix population models growth ratebasic reproduction number and entropyrdquoBulletin ofMathemat-ical Biology vol 71 no 7 pp 1781ndash1792 2009

[52] N Bacaer and E H A Dads ldquoGenealogy with seasonalitythe basic reproduction number and the influenza pandemicrdquoJournal of Mathematical Biology vol 62 no 5 pp 741ndash762 2011

[53] L J Liu X Q Zhao and Y C Zhou ldquoA tuberculosis model withseasonalityrdquo Bulletin of Mathematical Biology vol 72 no 4 pp931ndash952 2010

[54] J L Liu ldquoThreshold dynamics for a HFMD epidemic modelwith periodic transmission raterdquo Nonlinear Dynamics vol 64no 1-2 pp 89ndash95 2011

[55] Y Nakata and T Kuniya ldquoGlobal dynamics of a class ofSEIRS epidemic models in a periodic environmentrdquo Journal ofMathematical Analysis and Applications vol 363 no 1 pp 230ndash237 2010

[56] B GWilliams andC Dye ldquoInfectious disease persistence whentransmission varies seasonallyrdquo Mathematical Biosciences vol145 no 1 pp 77ndash88 1997

[57] I A Moneim ldquoThe effect of using different types of periodiccontact rate on the behaviour of infectious diseases a simula-tion studyrdquo Computers in Biology and Medicine vol 37 no 11pp 1582ndash1590 2007

[58] C L Wesley and L J S Allen ldquoThe basic reproduction numberin epidemic models with periodic demographicsrdquo Journal ofBiological Dynamics vol 3 no 2-3 pp 116ndash129 2009

[59] D Greenhalgh and I A Moneim ldquoSIRS epidemic model andsimulations using different types of seasonal contact raterdquoSystems Analysis Modelling Simulation vol 43 no 5 pp 573ndash600 2003

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

The Scientific World Journal 3

where 119874 is the origin 1198640 is the disease-free equilibrium and119864lowast is the endemic equilibrium Concretely one can have

119878lowast

=[119898120573 (119898 + 120590 + 120583) + 120583 (119898 + 120583) (119898 + 120590) (1198770 minus 1)]119873

lowast

120573 [(119898 + 120583) (119898 + 120590) (1198770 minus 1) + 119898 (119898 + 120590 + 120583)]

(5)

119864lowast

=[119903 (1 minus 119873

lowast119896) minus 119898] (119898 + 120583)119873

lowast

120590120583 (6)

119868lowast

=(119898 + 120583) (119898 + 120590) (1198770 minus 1)119873

lowast

120573 (119898 + 120590 + 120583) (7)

119873lowast

= (119896 [120573119898 (119898 + 120590 + 120583) (119877119889 minus 1)

+120583 (119898 + 120590) (119898 + 120583) (1 minus 1198770)])

times (119903120573 (119898 + 120590 + 120583))minus1

(8)

where 119877119889 = 119903119898 is the basic demographic reproductionnumber and1198770 = 120573120590(119898+120590)(119898+120583) is the basic reproductionnumber which can be obtained by the next-generationmatrixmethod [33ndash35] The introduction of the basic demographicreproduction number can be found in [36]

Moreover from (6) and (7) the conditions of the endemicequilibrium to exist are 1198770 gt 1 and 119877119889 gt 1198771 where 1198771 =

1+120583(119898+120583)(119898+120590)(1198770 minus1)119898120573(119898+120590+120583) So we can obtainthe following theorems

Theorem 2 The system (1) has three equilibria origin 119874disease-free equilibrium 1198640 and the endemic equilibrium 119864lowast119874 always exists if 119877119889 gt 1 1198640 exists if 1198770 gt 1 and 119877119889 gt 1198771 119864lowastexists

Theorem 3 When 119877119889 gt 1 and 1198770 lt 1 1198640 is globallyasymptotically stable

Proof By [33ndash35] we know that 1198640 is locally asymptoticallystable Now we define a Lyapunov function

119881 = 119864 +119898 + 120590

120590119868 ge 0 (9)

When 119877119889 gt 1 and 1198770 lt 1 the Lyapunov function satisfies

= +119898 + 120590

120590

119868

=120573119878119868

119873minus

(119898 + 120590) (119898 + 120583)

120590119868

le [120573 minus(119898 + 120590) (119898 + 120583)

120590] 119868

=(119898 + 120590) (119898 + 120583)

120590[1198770 minus 1] 119868

le 0

(10)

Moreover = 0 only hold when 119868 = 0 It is easy to verifythat the disease-free equilibrium point 1198640 is the only fixedpoint of the system Hence applying the Lyapunov-LaSalle

Rd

R0

E0

Elowast

O1 O2

O3

0

L

1

1

Figure 1 119877119889 in terms of 1198770 The region of 1198741 1198742 and 1198743 is thebasin of attraction of equilibrium 119874 the region of 1198640 is the basinof attraction of equilibrium 1198640 the region of 119864lowast is the basin ofattraction of equilibrium 119864lowast the line 119871 is 119877119889 = 1198771 = 1 + 120583(119898 +

120583)(119898 + 120590)(1198770 minus 1)119898120573(119898 + 120590 + 120583)

asymptotic stability theorem in [37 38] the disease-freeequilibrium point 1198640 is globally asymptotically stable

Since the proof of the stability of equilibria 119874 and 119864lowast ismore difficult we only give some numerical results

In sum we can show the respective basins of attractionof the three equilibria which can be seen in Figure 1 andconfirmed in Figure 2

(1) When 119877119889 lt 1 119874 is stable see Figures 2(b) and 2(c)(2) When119877119889 gt 1 and1198770 lt 1 1198640 is stable see Figure 2(d)(3) When 119877119889 gt 1 1198770 gt 1 and 119877119889 lt 1198771 119874 is stable see

Figure 2(a)(4) When 119877119889 gt 1 1198770 gt 1 and 119877119889 gt 1198771 119864lowast is stable see

Figure 2(e)

3 Nonautonomous Model and Analysis

31 The Basic Reproduction Number Now we consider thenon-autonomous case of themodel (1) when the transmissionrate is periodic which is given as follows

119889119878

119889119905= 119903119873(1 minus

119873

119896) minus

120573 (119905) 119878119868

119873minus 119898119878

119889119864

119889119905=

120573 (119905) 119878119868

119873minus 120590119864 minus 119898119864

119889119868

119889119905= 120590119864 minus 119898119868 minus 120583119868

(11)

where 120573(119905) is a periodic function which is proposed by [39]In the subsequent section we will discuss the dynamicalbehavior of the system (11)

For system (11) firstly we can give the basic reproductionnumber 1198770 According to the basic reproduction numberunder non-autonomous system we can refer to the methodof [40 41] From the last section we know that system (11)

4 The Scientific World Journal

80

70

60

50

40

30

20

10

00 50 100 150 200 250 300

R0 = 16667

Rd = 101

R1 = 12

I(t)

S(t)

(a)

R0 = 1667

Rd = 05

R1 = 10714

80

70

60

50

40

30

20

10

00 50 100 150 200 250 300

I(t)

S(t)

(b)

R0 = 06667

Rd = 08

R1 = 075

80

70

60

50

40

30

20

10

00 50 100 150 200 250 300

I(t)

S(t)

(c)

R0 = 06667

Rd = 13

R1 = 075

600

500

400

300

200

100

00 05 1 15 2 25 3 35 4

times104

I(t)

S(t)

(d)

R0 = 2333

Rd = 13

R1 = 12857

600

500

400

300

200

200

100

0100 300 400 500 600 700 800 900

I(t)

S(t)

(e)

Figure 2 The phase curves of the system under different initial conditions (a) In the region 1198743 with 119903 = 0101 and 120573 = 05 (b) in the region1198742 with 119903 = 005 and 120573 = 035 (c) in the region1198741 with 119903 = 008 and 120573 = 02 (d) in the region 1198640 with 119903 = 013 and 120573 = 02 (e) in the region119864lowastwith 119903 = 013 and 120573 = 07 The value of other parameters can be seen in Table 1

The Scientific World Journal 5

has one disease-free equilibrium 1198640 = (1198730 0 0) where119873

0=

(1 minus 119898119903)119896 By giving a new vector 119909 = (119864 119868) we have

119865 = (

120573 (119905) 119878119868

119873

0

) 119881 = (119898119864 + 120590119864

119898119868 + 120583119868 minus 120590119864)

119881minus

= (119898119864 + 120590119864

119898119868 + 120583119868) 119881

+= (

0

120590119864)

(12)

Taking the partial derivative of the above vectors aboutvariables 119864 119868 and substituting the disease-free equilibriumwe have

119865 (119905) = (0 120573 (119905)

0 0)

119881 (119905) = (119898 + 120590 0

minus120590 119898 + 120583)

(13)

According to [41] denote 119862120596 to be the ordered Banachspace of all 120596-periodic functions from R to R4 which isequippedwith themaximumnorm sdot and the positive cone119862+

120596= 120601 isin 119862120596 120601(119905) ge 0 forall119905 isin R+ Over the Banach space

we define a linear operator 119871 119862120596 rarr 119862120596 by

(119871120601) (119905)

= int

infin

0

119884 (119905 119905 minus 119886) 119865 (119905 minus 119886) 120601 (119905 minus 119886) 119889119886 forall119905 isin R+ 120601 isin 119862120596

(14)

where 119871 is called the next infection operator and the inter-pretation of 119884(119905 119905 minus 119886) 120601(119905 minus 119886) can be seen in [41] Thenthe spectral radius of 119871 is defined as the basic reproductionnumber

1198770 = 120588 (119871) (15)

In order to give the expression of the basic reproductionnumber we need to introduce the linear 120596-periodic system

119889119908

119889119905= [minus119881 (119905) +

119865 (119905)

120582]119908 119905 isin R+ (16)

with parameter 120582 isin R Let 119882(119905 119904 120582) 119905 ge 119904 be theevolution operator of system (16) on R2 In fact 119882(119905 119904 120582) =

Φ(119865120582)minus119881(119905) and Φ119865minus119881(119905) = 119882(119905 0 1) for all 119905 ge 0 ByTheorems 21 and 22 in [41] the basic reproduction numberalso can be defined as1205820 such that120588(Φ(1198651205820)minus119881(120596)) = 1 whichcan be straightforward to calculate

32 Global Stability of the Disease-Free Equilibrium

Theorem4 Thedisease-free equilibrium1198640 is globally asymp-totically stable when 1198770 lt 1 and 119877119889 gt 1

Proof Theorem 22 in [41] implies that 1198640 is locally asymp-totically stable when 1198770 lt 1 and 119877119889 gt 1 So we only need

to prove its global attractability It is easy to know that 119878(119905) le

1198730= (1 minus (119898119903))119896 Thus

119889119864

119889119905le 120573 (119905) 119868 minus (119898 + 120590) 119864

119889119868

119889119905= 120590119864 minus 119898119868 minus 120583119868

(17)

The right comparison system can be written as

119889119864

119889119905= 120573 (119905) 119868 minus (119898 + 120590) 119864

119889119868

119889119905= 120590119864 minus 119898119868 minus 120583119868

(18)

that is

119889ℎ

119889119905= (119865 (119905) minus 119881 (119905)) ℎ (119905) ℎ (119905) = (119864 (119905) 119868 (119905)) (19)

For (19) Lemma 21 in [42] shows that there is apositive 120596-periodic function ℎ(119905) = (119864(119905) 119868(119905))

119879 such thatℎ(119905) = 119890

119901119905ℎ(119905) is a solution of system (18) where 119901 =

(1120596) ln 120588(Φ119865minus119881(120596)) By Theorem 22 in [41] we know thatwhen 1198770 lt 1 and 119877119889 gt 1 120588(Φ119865minus119881(120596)) lt 1 and 119901 lt 0which implies ℎ(119905) rarr 0 as 119905 rarr infin Therefore the zerosolution of system (18) is globally asymptotically stable Bythe comparison principle [43] and the theory of asymptoticautonomous systems [44] when 1198770 lt 1 and 119877119889 gt 1 1198640is globally attractive Therefore the proposition that 1198640 isglobally asymptotically stable holds

33 Existence of Positive Periodic Solutions Before the proofof the existence of positive periodic solutions we firstlyintroduce some denotations Let 119906(119905 1199090) be the solution ofsystem (11) with the initial value 1199090 = (119878(0) 119864(0) 119868(0)) Bythe fundamental existence-uniqueness theorem [45] 119906(119905 1199090)is the unique solution of system (11) with 119906(0 1199090) = 1199090

Next we need to introduce the Poincare map119875 119883 rarr 119883

associated with system (11) that is

119875 (1199090) = 119906 (120596 1199090) forall1199090 isin 119883 (20)

where 120596 is the period Theorem 1 implies that 119883 is positivelyinvariant and 119875 is a dissipative point

Now we introduce two subsets of1198831198830 = (119878 119864 119868) isin 119883

119864 gt 0 119868 gt 0 and 1205971198830 = 119883 1198830

Lemma 5 (a) When 1198770 gt 1 and 119903 gt 119898 + 120583 there exists a120575 gt 0 such that when

1003817100381710038171003817(119878 (0) 119864 (0) 119868 (0)) minus 11986401003817100381710038171003817 le 120575 (21)

for any (119878(0) 119864(0) 119868(0)) isin 1198830 one has

lim sup119898rarrinfin

119889 [119875119898

(119878 (0) 119864 (0) 119868 (0)) 1198640] ge 120575 (22)

where 1198640 = (1198730 0 0)

6 The Scientific World Journal

(b) When 1198770 gt 1 and 119903 gt 119898 + 120583 there exists a 120575 gt 0 suchthat when

(119878 (0) 119864 (0) 119868 (0)) minus 119874 le 120575 (23)

for any (119878(0) 119864(0) 119868(0)) isin 1198830 one has

lim sup119898rarrinfin

119889 [119875119898

(119878 (0) 119864 (0) 119868 (0)) 119874] ge 120575 (24)

where 119874 = (0 0 0)

Proof (a) ByTheorem 22 in [41] we know that when 1198770 gt 1120588(Φ119865minus119881(120596)) gt 1 So there is a small enough positive number120598 such that 120588(Φ119865minus119881minus119872120598(120596)) gt 1 where

119872120598 = (0

120573 (119905) 120598119868

1198730

0 0

) (25)

If proposition (a) does not hold there is some(119878(0) 119864(0) 119868(0)) isin 1198830 such that

lim sup119898rarrinfin

119889 (119875119898

(119878 (0) 119864 (0) 119868 (0)) 1198640) lt 120575 (26)

We can assume that for all 119898 ge 0 119889(119875119898(119878(0) 119864(0)

119868(0)) 1198640) lt 120575 Applying the continuity of the solutions withrespect to the initial values

1003817100381710038171003817119906 (119905 119875119898

(119878 (0) 119864 (0) 119868 (0))) minus 119906 (119905 1198640)1003817100381710038171003817 le 120598

forall119898 ge 0 forall1199051 isin [0 120596]

(27)

Let 119905 = 119898120596 + 1199051 where 1199051 isin [0 120596] and 119898 = [119905120596] 119898 =

[119905120596] is the greatest integer which is not more than 119905120596Thenfor any 119905 ge 0

1003817100381710038171003817119906 (119905 (119878 (0) 119864 (0) 119868 (0))) minus 119906 (119905 1198640)1003817100381710038171003817

=1003817100381710038171003817119906 (1199051 119875

119898(119878 (0) 119864 (0) 119868 (0))) minus 119906 (1199051 1198640)

1003817100381710038171003817 le 120598

(28)

So 1198730

minus 120598 le 119878(119905) le 1198730

+ 120598 Then whenlim sup

119898rarrinfin119889(119875119898(119878(0) 119864(0) 119868(0)) 1198640) lt 120575

119889119864

119889119905ge 120573 (119905) 119868 minus

120573 (119905) 120598119868

1198730minus (119898 + 120590) 119864

119889119868

119889119905= 120590119864 minus 119898119868 minus 120583119868

(29)

Thus we can study the right linear system

119889119864

119889119905= 120573 (119905) 119868 minus

120573 (119905) 120598119868

1198730minus (119898 + 120590) 119864

119889119868

119889119905= 120590119864 minus 119898119868 minus 120583119868

(30)

For the system (30) there exists a positive 120596-periodicfunction 119892(119905) = (119864(119905) 119868(119905))

119879 such that 119892(119905) = 119890119901119905119892(119905) is a

solution of system (11) where 119901 = (1120596) ln 120588(Φ119865minus119881minus119872120598(120596))When 1198770 gt 1 120588(Φ119865minus119881minus119872120598(120596)) gt 1 which means that when

119892(0) gt 0119892(119905) rarr infin as 119905 rarr infin By the comparison principle[43] when 119864(0) gt 0 119868(0) gt 0 119864(119905) rarr infin 119868(119905) rarr infin as119905 rarr infinThere appears a contradictionThus the proposition(a) holds

(b) When 1198770 gt 1 and 119903 gt 119898 + 120583 we have

119889119873

119889119905= 119903119873(1 minus

119873

119896) minus 119898119873 minus 120583119868

ge [119903 (1 minus119873

119896) minus 119898 minus 120583]119873 gt 0

(31)

So if 119877119889 gt 1 119873 rarr 1198730 for 119873(0) gt 0 as 119905 rarr +infin that is

119882119878(119874) cap 1198830 = 0

Theorem 6 When 1198770 gt 1 and 119903 gt 119898 + 120583 there exists a 120575 gt 0

such that any solution (119878(119905) 119864(119905) 119868(119905)) of system (11)with initialvalue (119878(0) 119864(0) 119868(0)) isin (119878 119864 119868) isin 119883 119864 gt 0 119868 gt 0

satisfies

lim inf119905rarrinfin

119864 (119905) ge 120575 lim inf119905rarrinfin

119868 (119905) ge 120575 (32)

and system (11) has at least one positive periodic solution

Proof From system (11)

119878 (119905) = 119890minusint119905

0(120573(119904)119868(119904)119873+119898)119889119904

times [119878 (0) + int

119905

0

119903119873 (119904) (1 minus119873 (119904)

119896)

times119890int119904

0(120573(119888)119868(119888)119873(119888)+119898)119889119888

119889119904]

gt 0 forall119905 gt 0

(33)

119864 (119905) = 119890minus(119898+120590)119905

[119864 (0) + int

119905

0

120573 (119904) 119878 (119904) 119868 (119904)

119873 (119904)119890(119898+120590)119904

119889119904]

gt 0 forall119905 gt 0

(34)

119868 (119905) = 119890minus(119898+120583)119905

[119868 (0) + int

119905

0

120590119864 (119904) 119890(119898+120583)119904

119889119904]

gt 0 forall119905 gt 0

(35)

for any (119878(0) 119864(0) 119868(0)) isin 1198830 which shows that 1198830 ispositively invariant Moreover it is obvious to see that 1205971198830

is relatively closed in 119883 Denote

119872120597 = (119878 (0) 119864 (0) 119868 (0)) isin 1205971198830

119875119898

(119878 (0) 119864 (0) 119868 (0)) isin 1205971198830 forall119898 ge 0

(36)

Next we prove that

119872120597 = (119878 0 0) isin 119883 119878 ge 0 (37)

We only need to show that 119872120597 sube (119878 0 0) isin 119883 119878 ge 0which means that for any (119878(0) 119864(0) 119868(0)) isin 1205971198830 119864(119898120596) =

119868(119898120596) = 0 for all 119898 ge 0 If it does not hold there existsa 1198981 ge 0 such that (119864(1198981120596) 119868(1198981120596))

119879gt 0 Taking 1198981120596 as

The Scientific World Journal 7

25

2

15

1

05

01 15 2 25 3 35

times104

times104

S(0) = 25000 E(0) = 10000 I(0) = 24000

S(0) = 35000 E(0) = 20000 I(0) = 15000

S(0) = 15000 E(0) = 10000 I(0) = 20000

I(t)

S(t)

(a)

24

23

22

21

2

19

18

17

16

15

1415 2 25 3 35

times104

times104

S(0) = 25000 E(0) = 10000 I(0) = 24000

S(0) = 35000 E(0) = 20000 I(0) = 15000

S(0) = 15000 E(0) = 10000 I(0) = 20000

I(t)

S(t)

(b)

Figure 3 Phase plane of 119878(119905) and 119868(119905) (a) When the parameter values are 119903 = 013 119886 = 03 and 119887 = 02 1198770 = 09029 lt 1 119877119889 = 13 gt 1 (b)When the parameter values are 119903 = 03 119886 = 07 and 119887 = 02 1198770 = 21067 and 119903 gt 119898+120583 The value of other parameters can be seen in Table 1

the initial time and repeating the processes as in (33)ndash(35)we can have that (119878(119905) 119864(119905) 119868(119905))

119879gt 0 for all 119905 gt 1198981120596

Thus (119878(119905) 119864(119905) 119868(119905)) isin 1198830 for all 119905 gt 1198981120596 There appearsa contradiction which means that the equality (37) holdsTherefore 1198640 is acyclic in 1205971198830 Obviously when 1198770 gt 1 and119903 gt 119898 + 120583 119874 is acyclic in 1205971198830

Furthermore by Lemma 5 1198640 = (1198730 0 0) and 119874 =

(0 0 0) are isolated invariant sets in 119883 119882119878(1198640) cap 1198830 = 0and 119882

119878(119874) cap 1198830 = 0 By Theorem 131 and Remark 131

in [46] it can be obtained that 119875 is uniformly persistent withrespect to (1198830 1205971198830) that is there exists a 120575 gt 0 such that anysolution (119878(119905) 119864(119905) 119868(119905)) of system (11) with the initial value(119878(0) 119864(0) 119868(0)) isin (119878 119864 119868) isin 119883 119864 gt 0 119868 gt 0 satisfies

lim inf119905rarrinfin

119864 (119905) ge 120575 lim inf119905rarrinfin

119868 (119905) ge 120575 (38)

ApplyingTheorem 136 in [46] 119875 has a fixed point

(119878lowast(0) 119864

lowast(0) 119868lowast(0)) isin 1198830 (39)

From (33) we know 119878lowast

gt 0 for all 119905 isin [0 120596] 119878lowast(119905) is alsomore than zero for all 119905 gt 0 due to the periodicity Similarlyfor all 119905 ge 0119864lowast(119905) gt 0 119868lowast(119905) gt 0Therefore it can be obtainedthat one of the positive 120596-periodic solutions of system (11) is(119878lowast(119905) 119864lowast(119905) 119868lowast(119905))

34 Numerical Simulations Firstly we give some notationsIf 119892(119905) is a periodic function with period 120596 we define 119892 =

(1120596) int120596

0119892(119905)119889119905 119892119897 = min119905isin[0120596]119892(119905) 119892119906 = max119905isin[0120596]119892(119905) As

described in the previous section

1198771 (119905) = 1 +120583 (119898 + 120583) (119898 + 120590) (1198770 minus 1)

119898120573 (119905) (119898 + 120590 + 120583) (40)

So 119877119897

1= 1 + 120583(119898 + 120583)(119898 + 120590)(119877

119897

0minus 1)119898120573

119906(119898 + 120590 + 120583) 119877119906

1=

1 + 120583(119898 + 120583)(119898 + 120590)(119877119906

0minus 1)119898120573

119897(119898 + 120590 + 120583)

In this section we adopt 120573(119905) = 119886[1 + 119887 sin(12058711990510)]Then applying the numerical simulation to verify the abovesolution we give the following conclusion

(1) when 119877119889 lt 1 119874 is stable(2) when 119877119889 gt 1 and 1198770 lt 1 1198640 is stable see Figure 3(a)(3) when 1198770 gt 1 and 119903 gt 119898 + 120583 system (11) has at least

one positive periodic solution see Figure 3(b)

We can givemore results about the conditions of existenceof the positive periodic solution

(11015840) When 119877119889 gt 1 1198770 gt 1 and 119877119889 lt 119877

119906

1 119874 is stable see

Figures 4(a) and 4(b)(21015840) When 119877119889 gt 1 1198770 gt 1 and 119877119889 gt 119877

119906

1 system (11) has at

least one positive periodic solution see Figure 5

By numerical simulations we can give that the conditionswhich ensure the existence of positive periodic solution are1198770 gt 1 and 119903 gt 119898+120583 or 119877119889 gt 1 1198770 gt 1 and 119877119889 gt 119877

119906

1 In fact

119877119889 gt 1 1198770 gt 1 and 119877119889 gt 119877119906

1are the sufficient conditions for

1198770 gt 1 and 119903 gt 119898 + 120583 As a result the conditions 1198770 gt 1 and119903 gt 119898 + 120583 are broader

4 Discussion

This paper considers a logistic growth system whose birthprocess incorporates density-dependent effects This type ofmodel has a rich dynamical behavior and practical signif-icance By analyzing its equilibria and respective attractiveregion we find that the dynamical behavior of a disease will

8 The Scientific World Journal

25

2

15

1

05

00 05 1 15 2 25 3 35

times104

times104

S(0) = 25000 E(0) = 10000 I(0) = 24000

S(0) = 35000 E(0) = 20000 I(0) = 15000

S(0) = 15000 E(0) = 10000 I(0) = 20000

I(t)

S(t)

(a)

25

2

15

1

05

00 05 1 15 2 25 3 35

times104

times104

S(0) = 25000 E(0) = 10000 I(0) = 24000

S(0) = 35000 E(0) = 20000 I(0) = 15000

S(0) = 15000 E(0) = 10000 I(0) = 20000

I(t)

S(t)

(b)

Figure 4 Phase plane of 119878(119905) and 119868(119905) (a) When the parameter values are 119903 = 011 119886 = 09 119887 = 02 and 120583 = 03 1198770 = 14991 119877119889 = 11 gt 1119877119889 lt 119877

119906

1= 14159 and 119877119889 lt 119877

119897

1= 12773 (b) When the parameter values are 119903 = 0126 119886 = 07 119887 = 02 and 120583 = 03 1198770 = 21067

119877119889 = 126 gt 1 119877119889 lt 119877119906

1= 12964 and 119877119889 gt 119877

119897

1= 11976 The value of other parameters can be seen in Table 1

7000

6000

5000

4000

3000

2000

1000

005 1 15 2

times104

times104

S(0) = 22000 E(0) = 5000 I(0) = 1000

S(0) = 4000 E(0) = 2000 I(0) = 1000

S(0) = 10000 E(0) = 10000 I(0) = 6000

I(t)

S(t)

Figure 5 Phase plane of 119878(119905) and 119868(119905) When the parameter valuesare 119903 = 013 119886 = 036 and 119887 = 02 1198770 = 10834 119877119889 = 13 gt 1 119877119889 gt119877119906

1= 10435 and 119877119889 gt 119877

119897

1= 1029 The value of other parameters

can be seen in Table 1

be determined by two thresholds 1198770 and 119877119889 Only 1198770 gt 1

cannot promise the existence of the endemic equilibriumwhich also needs 119877119889 gt 1198771 When 1198770 gt 1 and 119877119889 lt 1198771the solutions of the system (1) will tend to the origin 119874 It

is caused by the phenomenon that the death number dueto disease cannot be supplemented by the birth numberpromptly Finally all people are infected and die outThe factinterpreted by this model is more reasonable Theoreticallywe prove the global asymptotic stability of the disease-free equilibrium and give respective attractive regions ofequilibria

Seasonally effective contact rate is themost common formwhich may be related to various factors and thus this paperstudies the corresponding non-autonomous system which isobtained by changing the constant transmission rate of theabove system into the periodic transmission rate For theperiodic systems their dynamical behaviors especially thebasic reproduction number have been investigated in depthby [41 47ndash55] which provide many methods that we canutilize For the obtained periodic model by analyzing theglobal asymptotic stability of the disease-free equilibrium andthe existence of positive periodic solution we have the similarresults as the autonomous system The dynamic behavior ofdisease will be decided by two conditions 1198770 gt 1 and 119903 gt

119898 + 120583 that show that when the disease is prevalent the birthrate should be larger than the death rate to guarantee thesustainable growth of population Otherwise the populationwill disappear In addition we will evaluate and comparethe basic reproduction number 1198770 and the average basicreproduction number 1198770 which has been adopted by [27 56ndash59] In this paper we can calculate the average reproductionnumber

1198770 =120573120590

(119898 + 120590) (119898 + 120583) (41)

The Scientific World Journal 9

where 120573 = (120) int20

0120573(119905)119889119905 When 119903 = 013 119886 = 03 and

119887 = 02 we know that 1198770 = 09029 and 1198770 = 1 When 119903 =

013 119896 = 100000 119886 = 036 119887 = 02 and 119898 = 01 120590 = 02120583 = 01 then 1198770 = 10834 and 1198770 = 12 In that sense it isconfirmed that the basic reproduction number 1198770 defined by[40] is more accurate than the average reproduction number1198770 which overestimates the risk of disease

It should be noted that we live in a spatial world and it is anatural phenomenon that a substance goes from high densityregions to low density regions As a result epidemic modelsshould include spatial effects In a further study we need toinvestigate spatial epidemic models with seasonal factors

Acknowledgments

The research was partially supported by the National NaturalScience Foundation of China under Grant nos 1130149011301491 11331009 11147015 11171314 11101251 and 11105024Natural Science Foundation of ShanXi Province Grant no2012021002-1 and the Opening Foundation of Institute ofInformation Economy Hangzhou Normal University Grantno PD12001003002003

References

[1] K Dietz and J A P Heesterbeek ldquoBernoulli was ahead ofmodern epidemiologyrdquo Nature vol 408 no 6812 pp 513ndash5142000

[2] K Dietz and J A P Heesterbeek ldquoDaniel Bernoullirsquos epidemio-logical model revisitedrdquoMathematical Biosciences vol 180 pp1ndash21 2002

[3] R Ross ldquoAn application of the theory of probabilities to thestudy of a priori pathometry (part I)rdquo Proceedings of the RoyalSociety A vol 92 no 638 pp 204ndash230 1916

[4] R Ross and H P Hudson ldquoAn application of the theory ofprobabilities to the study of a priori pathometry (part II)rdquoProceedings of the Royal Society A vol 93 no 650 pp 212ndash2251917

[5] W O Kermack and A G McKendrick ldquoA contribution to themathematical theory of epidemics (part I)rdquo Proceedings of theRoyal Society A vol 115 no 772 pp 700ndash721 1927

[6] W O Kermack and A G McKendrick ldquoA contribution to themathematical theory of epidemics (part II)rdquo Proceedings of theRoyal Society A vol 138 no 834 pp 55ndash83 1932

[7] A Tsoularis and JWallace ldquoAnalysis of logistic growthmodelsrdquoMathematical Biosciences vol 179 no 1 pp 21ndash55 2002

[8] I Nasell ldquoOn the quasi-stationary distribution of the stochasticlogistic epidemicrdquoMathematical Biosciences vol 156 no 1-2 pp21ndash40 1999

[9] H Fujikawa A Kai and SMorozumi ldquoA new logisticmodel forEscherichia coli growth at constant and dynamic temperaturesrdquoFood Microbiology vol 21 no 5 pp 501ndash509 2004

[10] L Berezansky and E Braverman ldquoOscillation properties of alogistic equation with several delaysrdquo Journal of MathematicalAnalysis and Applications vol 247 no 1 pp 110ndash125 2000

[11] M Tabata N Eshima and I Takagi ldquoThe nonlinear integro-partial differential equation describing the logistic growth ofhuman population with migrationrdquo Applied Mathematics andComputation vol 98 pp 169ndash183 1999

[12] L Korobenko andE Braverman ldquoOn logisticmodels with a car-rying capacity dependent diffusion stability of equilibria andcoexistence with a regularly diffusing populationrdquo NonlinearAnalysis Real World Applications vol 13 no 6 pp 2648ndash26582012

[13] Y Muroya ldquoGlobal attractivity for discrete models of nonau-tonomous logistic equationsrdquo Computers andMathematics withApplications vol 53 no 7 pp 1059ndash1073 2007

[14] S Invernizzi and K Terpin ldquoA generalized logistic model forphotosynthetic growthrdquo Ecological Modelling vol 94 no 2-3pp 231ndash242 1997

[15] ZMinW Bang-Jun and J Feng ldquoCoalmining citiesrsquo economicgrowthmechanism and sustainable development analysis basedon logistic dynamics modelrdquo Procedia Earth and PlanetaryScience vol 1 no 1 pp 1737ndash1743 2009

[16] L I Anita S Anita and V Arnautu ldquoGlobal behavior foran age-dependent population model with logistic term andperiodic vital ratesrdquoAppliedMathematics andComputation vol206 no 1 pp 368ndash379 2008

[17] Z A Banaszak X Q Tang S C Wang and M B ZarembaldquoLogistics models in flexible manufacturingrdquo Computers inIndustry vol 43 no 3 pp 237ndash248 2000

[18] S Brianzoni C Mammana and E Michetti ldquoNonlineardynamics in a business-cycle model with logistic populationgrowthrdquo Chaos Solitons and Fractals vol 40 no 2 pp 717ndash7302009

[19] W P London and J A Yorke ldquoRecurrent outbreaks of measleschickenpox and mumps I Seasonal variation in contact ratesrdquoThe American Journal of Epidemiology vol 98 no 6 pp 453ndash468 1973

[20] S F Dowell ldquoSeasonal variation in host susceptibility and cyclesof certain infectious diseasesrdquo Emerging Infectious Diseases vol7 no 3 pp 369ndash374 2001

[21] ON Bjoslashrnstad B F Finkenstadt andB TGrenfell ldquoDynamicsof measles epidemics estimating scaling of transmission ratesusing a Time series SIR modelrdquo Ecological Monographs vol 72no 2 pp 169ndash184 2002

[22] J Zhang Z Jin G Q Sun X Sun and S Ruan ldquoModelingseasonal rabies epidemics in Chinardquo Bulletin of MathematicalBiology vol 74 no 5 pp 1226ndash1251 2012

[23] I B Schwartz ldquoSmall amplitude long period outbreaks inseasonally driven epidemicsrdquo Journal of Mathematical Biologyvol 30 no 5 pp 473ndash491 1992

[24] I B Schwartz and H L Smith ldquoInfinite subharmonic bifur-cation in an SEIR epidemic modelrdquo Journal of MathematicalBiology vol 18 no 3 pp 233ndash253 1983

[25] H L Smith ldquoMultiple stable subharmonics for a periodicepidemic modelrdquo Journal of Mathematical Biology vol 17 no2 pp 179ndash190 1983

[26] J Dushoff J B Plotkin S A Levin and D J D EarnldquoDynamical resonance can account for seasonality of influenzaepidemicsrdquo Proceedings of the National Academy of Sciences ofthe United States of America vol 101 no 48 pp 16915ndash169162004

[27] J L Ma and Z Ma ldquoEpidemic threshold conditions forseasonally forced SEIR modelsrdquo Mathematical Biosciences andEngineering vol 3 no 1 pp 161ndash172 2006

[28] D J D Earn P Rohani B M Bolker and B T Grenfell ldquoAsimple model for complex dynamical transitions in epidemicsrdquoScience vol 287 no 5453 pp 667ndash670 2000

10 The Scientific World Journal

[29] M J Keeling P Rohani and B T Grenfell ldquoSeasonally forceddisease dynamics explored as switching between attractorsrdquoPhysica D vol 148 no 3-4 pp 317ndash335 2001

[30] J L Aron and I B Schwartz ldquoSeasonality and period-doublingbifurcations in an epidemic modelrdquo Journal of TheoreticalBiology vol 110 no 4 pp 665ndash679 1984

[31] N C Grassly and C Fraser ldquoSeasonal infectious diseaseepidemiologyrdquo Proceedings of the Royal Society B vol 273 no1600 pp 2541ndash2550 2006

[32] J Zhang Z Jin G Q Sun T Zhou and S Ruan ldquoAnalysisof rabies in China transmission dynamics and controlrdquo PLoSONE vol 6 no 7 Article ID e20891 2011

[33] O Diekmann J A P Heesterbeek and J A Metz ldquoOnthe definition and the computation of the basic reproductionratio R0 in models for infectious diseases in heterogeneouspopulationsrdquo Journal of mathematical biology vol 28 no 4 pp365ndash382 1990

[34] O Diekmann J A P Heesterbeek and M G Roberts ldquoTheconstruction of next-generation matrices for compartmentalepidemic modelsrdquo Journal of the Royal Society Interface vol 7no 47 pp 873ndash885 2010

[35] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[36] F Berezovsky G Karev B J Song and C C Chavez ldquoA simpleepidemic model with surprising dynamicsrdquo Mathematical Bio-sciences and Engineering vol 2 pp 133ndash152 2005

[37] J LaSalle and S Lefschetz Stability by Liapunovrsquos DirectMethodAcademic Press New York NY USA 1961

[38] E A Barbashin Introduction to theTheory of Stability Wolters-Noordhoff Groningen The Netherlands 1970

[39] D Schenzle ldquoAn age-structured model of pre- and post-vaccination measles transmissionrdquoMathematical Medicine andBiology vol 1 no 2 pp 169ndash191 1984

[40] N Bacaer and S Guernaoui ldquoThe epidemic threshold ofvector-borne diseases with seasonality the case of cutaneousleishmaniasis in Chichaoua Moroccordquo Journal of MathematicalBiology vol 53 no 3 pp 421ndash436 2006

[41] W DWang and X Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008

[42] F Zhang and X Q Zhao ldquoA periodic epidemic model in apatchy environmentrdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 496ndash516 2007

[43] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press New York NY USA 1995

[44] H R Thieme ldquoConvergence results and a Poincare-Bendixsontrichotomy for asymptotically automous differential equationsrdquoJournal of Mathematical Biology vol 30 pp 755ndash763 1992

[45] L Perko Differential Equations and Dynamical SystemsSpringer New York NY USA 2000

[46] X Q Zhao Dynamical Systems in Population Biology SpringerNew York NY USA 2003

[47] Z G Bai and Y C Zhou ldquoThreshold dynamics of a bacillarydysentery model with seasonal fluctuationrdquo Discrete and Con-tinuous Dynamical Systems B vol 15 no 1 pp 1ndash14 2011

[48] Z G Bai Y C Zhou and T L Zhang ldquoExistence of multipleperiodic solutions for an SIRmodel with seasonalityrdquoNonlinear

Analysis Theory Methods and Applications vol 74 no 11 pp3548ndash3555 2011

[49] N Bacaer ldquoApproximation of the basic reproduction numberR0 for vector-borne diseases with a periodic vector populationrdquoBulletin of Mathematical Biology vol 69 no 3 pp 1067ndash10912007

[50] N Bacaer and X Abdurahman ldquoResonance of the epidemicthreshold in a periodic environmentrdquo Journal of MathematicalBiology vol 57 no 5 pp 649ndash673 2008

[51] N Bacaer ldquoPeriodic matrix population models growth ratebasic reproduction number and entropyrdquoBulletin ofMathemat-ical Biology vol 71 no 7 pp 1781ndash1792 2009

[52] N Bacaer and E H A Dads ldquoGenealogy with seasonalitythe basic reproduction number and the influenza pandemicrdquoJournal of Mathematical Biology vol 62 no 5 pp 741ndash762 2011

[53] L J Liu X Q Zhao and Y C Zhou ldquoA tuberculosis model withseasonalityrdquo Bulletin of Mathematical Biology vol 72 no 4 pp931ndash952 2010

[54] J L Liu ldquoThreshold dynamics for a HFMD epidemic modelwith periodic transmission raterdquo Nonlinear Dynamics vol 64no 1-2 pp 89ndash95 2011

[55] Y Nakata and T Kuniya ldquoGlobal dynamics of a class ofSEIRS epidemic models in a periodic environmentrdquo Journal ofMathematical Analysis and Applications vol 363 no 1 pp 230ndash237 2010

[56] B GWilliams andC Dye ldquoInfectious disease persistence whentransmission varies seasonallyrdquo Mathematical Biosciences vol145 no 1 pp 77ndash88 1997

[57] I A Moneim ldquoThe effect of using different types of periodiccontact rate on the behaviour of infectious diseases a simula-tion studyrdquo Computers in Biology and Medicine vol 37 no 11pp 1582ndash1590 2007

[58] C L Wesley and L J S Allen ldquoThe basic reproduction numberin epidemic models with periodic demographicsrdquo Journal ofBiological Dynamics vol 3 no 2-3 pp 116ndash129 2009

[59] D Greenhalgh and I A Moneim ldquoSIRS epidemic model andsimulations using different types of seasonal contact raterdquoSystems Analysis Modelling Simulation vol 43 no 5 pp 573ndash600 2003

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 The Scientific World Journal

80

70

60

50

40

30

20

10

00 50 100 150 200 250 300

R0 = 16667

Rd = 101

R1 = 12

I(t)

S(t)

(a)

R0 = 1667

Rd = 05

R1 = 10714

80

70

60

50

40

30

20

10

00 50 100 150 200 250 300

I(t)

S(t)

(b)

R0 = 06667

Rd = 08

R1 = 075

80

70

60

50

40

30

20

10

00 50 100 150 200 250 300

I(t)

S(t)

(c)

R0 = 06667

Rd = 13

R1 = 075

600

500

400

300

200

100

00 05 1 15 2 25 3 35 4

times104

I(t)

S(t)

(d)

R0 = 2333

Rd = 13

R1 = 12857

600

500

400

300

200

200

100

0100 300 400 500 600 700 800 900

I(t)

S(t)

(e)

Figure 2 The phase curves of the system under different initial conditions (a) In the region 1198743 with 119903 = 0101 and 120573 = 05 (b) in the region1198742 with 119903 = 005 and 120573 = 035 (c) in the region1198741 with 119903 = 008 and 120573 = 02 (d) in the region 1198640 with 119903 = 013 and 120573 = 02 (e) in the region119864lowastwith 119903 = 013 and 120573 = 07 The value of other parameters can be seen in Table 1

The Scientific World Journal 5

has one disease-free equilibrium 1198640 = (1198730 0 0) where119873

0=

(1 minus 119898119903)119896 By giving a new vector 119909 = (119864 119868) we have

119865 = (

120573 (119905) 119878119868

119873

0

) 119881 = (119898119864 + 120590119864

119898119868 + 120583119868 minus 120590119864)

119881minus

= (119898119864 + 120590119864

119898119868 + 120583119868) 119881

+= (

0

120590119864)

(12)

Taking the partial derivative of the above vectors aboutvariables 119864 119868 and substituting the disease-free equilibriumwe have

119865 (119905) = (0 120573 (119905)

0 0)

119881 (119905) = (119898 + 120590 0

minus120590 119898 + 120583)

(13)

According to [41] denote 119862120596 to be the ordered Banachspace of all 120596-periodic functions from R to R4 which isequippedwith themaximumnorm sdot and the positive cone119862+

120596= 120601 isin 119862120596 120601(119905) ge 0 forall119905 isin R+ Over the Banach space

we define a linear operator 119871 119862120596 rarr 119862120596 by

(119871120601) (119905)

= int

infin

0

119884 (119905 119905 minus 119886) 119865 (119905 minus 119886) 120601 (119905 minus 119886) 119889119886 forall119905 isin R+ 120601 isin 119862120596

(14)

where 119871 is called the next infection operator and the inter-pretation of 119884(119905 119905 minus 119886) 120601(119905 minus 119886) can be seen in [41] Thenthe spectral radius of 119871 is defined as the basic reproductionnumber

1198770 = 120588 (119871) (15)

In order to give the expression of the basic reproductionnumber we need to introduce the linear 120596-periodic system

119889119908

119889119905= [minus119881 (119905) +

119865 (119905)

120582]119908 119905 isin R+ (16)

with parameter 120582 isin R Let 119882(119905 119904 120582) 119905 ge 119904 be theevolution operator of system (16) on R2 In fact 119882(119905 119904 120582) =

Φ(119865120582)minus119881(119905) and Φ119865minus119881(119905) = 119882(119905 0 1) for all 119905 ge 0 ByTheorems 21 and 22 in [41] the basic reproduction numberalso can be defined as1205820 such that120588(Φ(1198651205820)minus119881(120596)) = 1 whichcan be straightforward to calculate

32 Global Stability of the Disease-Free Equilibrium

Theorem4 Thedisease-free equilibrium1198640 is globally asymp-totically stable when 1198770 lt 1 and 119877119889 gt 1

Proof Theorem 22 in [41] implies that 1198640 is locally asymp-totically stable when 1198770 lt 1 and 119877119889 gt 1 So we only need

to prove its global attractability It is easy to know that 119878(119905) le

1198730= (1 minus (119898119903))119896 Thus

119889119864

119889119905le 120573 (119905) 119868 minus (119898 + 120590) 119864

119889119868

119889119905= 120590119864 minus 119898119868 minus 120583119868

(17)

The right comparison system can be written as

119889119864

119889119905= 120573 (119905) 119868 minus (119898 + 120590) 119864

119889119868

119889119905= 120590119864 minus 119898119868 minus 120583119868

(18)

that is

119889ℎ

119889119905= (119865 (119905) minus 119881 (119905)) ℎ (119905) ℎ (119905) = (119864 (119905) 119868 (119905)) (19)

For (19) Lemma 21 in [42] shows that there is apositive 120596-periodic function ℎ(119905) = (119864(119905) 119868(119905))

119879 such thatℎ(119905) = 119890

119901119905ℎ(119905) is a solution of system (18) where 119901 =

(1120596) ln 120588(Φ119865minus119881(120596)) By Theorem 22 in [41] we know thatwhen 1198770 lt 1 and 119877119889 gt 1 120588(Φ119865minus119881(120596)) lt 1 and 119901 lt 0which implies ℎ(119905) rarr 0 as 119905 rarr infin Therefore the zerosolution of system (18) is globally asymptotically stable Bythe comparison principle [43] and the theory of asymptoticautonomous systems [44] when 1198770 lt 1 and 119877119889 gt 1 1198640is globally attractive Therefore the proposition that 1198640 isglobally asymptotically stable holds

33 Existence of Positive Periodic Solutions Before the proofof the existence of positive periodic solutions we firstlyintroduce some denotations Let 119906(119905 1199090) be the solution ofsystem (11) with the initial value 1199090 = (119878(0) 119864(0) 119868(0)) Bythe fundamental existence-uniqueness theorem [45] 119906(119905 1199090)is the unique solution of system (11) with 119906(0 1199090) = 1199090

Next we need to introduce the Poincare map119875 119883 rarr 119883

associated with system (11) that is

119875 (1199090) = 119906 (120596 1199090) forall1199090 isin 119883 (20)

where 120596 is the period Theorem 1 implies that 119883 is positivelyinvariant and 119875 is a dissipative point

Now we introduce two subsets of1198831198830 = (119878 119864 119868) isin 119883

119864 gt 0 119868 gt 0 and 1205971198830 = 119883 1198830

Lemma 5 (a) When 1198770 gt 1 and 119903 gt 119898 + 120583 there exists a120575 gt 0 such that when

1003817100381710038171003817(119878 (0) 119864 (0) 119868 (0)) minus 11986401003817100381710038171003817 le 120575 (21)

for any (119878(0) 119864(0) 119868(0)) isin 1198830 one has

lim sup119898rarrinfin

119889 [119875119898

(119878 (0) 119864 (0) 119868 (0)) 1198640] ge 120575 (22)

where 1198640 = (1198730 0 0)

6 The Scientific World Journal

(b) When 1198770 gt 1 and 119903 gt 119898 + 120583 there exists a 120575 gt 0 suchthat when

(119878 (0) 119864 (0) 119868 (0)) minus 119874 le 120575 (23)

for any (119878(0) 119864(0) 119868(0)) isin 1198830 one has

lim sup119898rarrinfin

119889 [119875119898

(119878 (0) 119864 (0) 119868 (0)) 119874] ge 120575 (24)

where 119874 = (0 0 0)

Proof (a) ByTheorem 22 in [41] we know that when 1198770 gt 1120588(Φ119865minus119881(120596)) gt 1 So there is a small enough positive number120598 such that 120588(Φ119865minus119881minus119872120598(120596)) gt 1 where

119872120598 = (0

120573 (119905) 120598119868

1198730

0 0

) (25)

If proposition (a) does not hold there is some(119878(0) 119864(0) 119868(0)) isin 1198830 such that

lim sup119898rarrinfin

119889 (119875119898

(119878 (0) 119864 (0) 119868 (0)) 1198640) lt 120575 (26)

We can assume that for all 119898 ge 0 119889(119875119898(119878(0) 119864(0)

119868(0)) 1198640) lt 120575 Applying the continuity of the solutions withrespect to the initial values

1003817100381710038171003817119906 (119905 119875119898

(119878 (0) 119864 (0) 119868 (0))) minus 119906 (119905 1198640)1003817100381710038171003817 le 120598

forall119898 ge 0 forall1199051 isin [0 120596]

(27)

Let 119905 = 119898120596 + 1199051 where 1199051 isin [0 120596] and 119898 = [119905120596] 119898 =

[119905120596] is the greatest integer which is not more than 119905120596Thenfor any 119905 ge 0

1003817100381710038171003817119906 (119905 (119878 (0) 119864 (0) 119868 (0))) minus 119906 (119905 1198640)1003817100381710038171003817

=1003817100381710038171003817119906 (1199051 119875

119898(119878 (0) 119864 (0) 119868 (0))) minus 119906 (1199051 1198640)

1003817100381710038171003817 le 120598

(28)

So 1198730

minus 120598 le 119878(119905) le 1198730

+ 120598 Then whenlim sup

119898rarrinfin119889(119875119898(119878(0) 119864(0) 119868(0)) 1198640) lt 120575

119889119864

119889119905ge 120573 (119905) 119868 minus

120573 (119905) 120598119868

1198730minus (119898 + 120590) 119864

119889119868

119889119905= 120590119864 minus 119898119868 minus 120583119868

(29)

Thus we can study the right linear system

119889119864

119889119905= 120573 (119905) 119868 minus

120573 (119905) 120598119868

1198730minus (119898 + 120590) 119864

119889119868

119889119905= 120590119864 minus 119898119868 minus 120583119868

(30)

For the system (30) there exists a positive 120596-periodicfunction 119892(119905) = (119864(119905) 119868(119905))

119879 such that 119892(119905) = 119890119901119905119892(119905) is a

solution of system (11) where 119901 = (1120596) ln 120588(Φ119865minus119881minus119872120598(120596))When 1198770 gt 1 120588(Φ119865minus119881minus119872120598(120596)) gt 1 which means that when

119892(0) gt 0119892(119905) rarr infin as 119905 rarr infin By the comparison principle[43] when 119864(0) gt 0 119868(0) gt 0 119864(119905) rarr infin 119868(119905) rarr infin as119905 rarr infinThere appears a contradictionThus the proposition(a) holds

(b) When 1198770 gt 1 and 119903 gt 119898 + 120583 we have

119889119873

119889119905= 119903119873(1 minus

119873

119896) minus 119898119873 minus 120583119868

ge [119903 (1 minus119873

119896) minus 119898 minus 120583]119873 gt 0

(31)

So if 119877119889 gt 1 119873 rarr 1198730 for 119873(0) gt 0 as 119905 rarr +infin that is

119882119878(119874) cap 1198830 = 0

Theorem 6 When 1198770 gt 1 and 119903 gt 119898 + 120583 there exists a 120575 gt 0

such that any solution (119878(119905) 119864(119905) 119868(119905)) of system (11)with initialvalue (119878(0) 119864(0) 119868(0)) isin (119878 119864 119868) isin 119883 119864 gt 0 119868 gt 0

satisfies

lim inf119905rarrinfin

119864 (119905) ge 120575 lim inf119905rarrinfin

119868 (119905) ge 120575 (32)

and system (11) has at least one positive periodic solution

Proof From system (11)

119878 (119905) = 119890minusint119905

0(120573(119904)119868(119904)119873+119898)119889119904

times [119878 (0) + int

119905

0

119903119873 (119904) (1 minus119873 (119904)

119896)

times119890int119904

0(120573(119888)119868(119888)119873(119888)+119898)119889119888

119889119904]

gt 0 forall119905 gt 0

(33)

119864 (119905) = 119890minus(119898+120590)119905

[119864 (0) + int

119905

0

120573 (119904) 119878 (119904) 119868 (119904)

119873 (119904)119890(119898+120590)119904

119889119904]

gt 0 forall119905 gt 0

(34)

119868 (119905) = 119890minus(119898+120583)119905

[119868 (0) + int

119905

0

120590119864 (119904) 119890(119898+120583)119904

119889119904]

gt 0 forall119905 gt 0

(35)

for any (119878(0) 119864(0) 119868(0)) isin 1198830 which shows that 1198830 ispositively invariant Moreover it is obvious to see that 1205971198830

is relatively closed in 119883 Denote

119872120597 = (119878 (0) 119864 (0) 119868 (0)) isin 1205971198830

119875119898

(119878 (0) 119864 (0) 119868 (0)) isin 1205971198830 forall119898 ge 0

(36)

Next we prove that

119872120597 = (119878 0 0) isin 119883 119878 ge 0 (37)

We only need to show that 119872120597 sube (119878 0 0) isin 119883 119878 ge 0which means that for any (119878(0) 119864(0) 119868(0)) isin 1205971198830 119864(119898120596) =

119868(119898120596) = 0 for all 119898 ge 0 If it does not hold there existsa 1198981 ge 0 such that (119864(1198981120596) 119868(1198981120596))

119879gt 0 Taking 1198981120596 as

The Scientific World Journal 7

25

2

15

1

05

01 15 2 25 3 35

times104

times104

S(0) = 25000 E(0) = 10000 I(0) = 24000

S(0) = 35000 E(0) = 20000 I(0) = 15000

S(0) = 15000 E(0) = 10000 I(0) = 20000

I(t)

S(t)

(a)

24

23

22

21

2

19

18

17

16

15

1415 2 25 3 35

times104

times104

S(0) = 25000 E(0) = 10000 I(0) = 24000

S(0) = 35000 E(0) = 20000 I(0) = 15000

S(0) = 15000 E(0) = 10000 I(0) = 20000

I(t)

S(t)

(b)

Figure 3 Phase plane of 119878(119905) and 119868(119905) (a) When the parameter values are 119903 = 013 119886 = 03 and 119887 = 02 1198770 = 09029 lt 1 119877119889 = 13 gt 1 (b)When the parameter values are 119903 = 03 119886 = 07 and 119887 = 02 1198770 = 21067 and 119903 gt 119898+120583 The value of other parameters can be seen in Table 1

the initial time and repeating the processes as in (33)ndash(35)we can have that (119878(119905) 119864(119905) 119868(119905))

119879gt 0 for all 119905 gt 1198981120596

Thus (119878(119905) 119864(119905) 119868(119905)) isin 1198830 for all 119905 gt 1198981120596 There appearsa contradiction which means that the equality (37) holdsTherefore 1198640 is acyclic in 1205971198830 Obviously when 1198770 gt 1 and119903 gt 119898 + 120583 119874 is acyclic in 1205971198830

Furthermore by Lemma 5 1198640 = (1198730 0 0) and 119874 =

(0 0 0) are isolated invariant sets in 119883 119882119878(1198640) cap 1198830 = 0and 119882

119878(119874) cap 1198830 = 0 By Theorem 131 and Remark 131

in [46] it can be obtained that 119875 is uniformly persistent withrespect to (1198830 1205971198830) that is there exists a 120575 gt 0 such that anysolution (119878(119905) 119864(119905) 119868(119905)) of system (11) with the initial value(119878(0) 119864(0) 119868(0)) isin (119878 119864 119868) isin 119883 119864 gt 0 119868 gt 0 satisfies

lim inf119905rarrinfin

119864 (119905) ge 120575 lim inf119905rarrinfin

119868 (119905) ge 120575 (38)

ApplyingTheorem 136 in [46] 119875 has a fixed point

(119878lowast(0) 119864

lowast(0) 119868lowast(0)) isin 1198830 (39)

From (33) we know 119878lowast

gt 0 for all 119905 isin [0 120596] 119878lowast(119905) is alsomore than zero for all 119905 gt 0 due to the periodicity Similarlyfor all 119905 ge 0119864lowast(119905) gt 0 119868lowast(119905) gt 0Therefore it can be obtainedthat one of the positive 120596-periodic solutions of system (11) is(119878lowast(119905) 119864lowast(119905) 119868lowast(119905))

34 Numerical Simulations Firstly we give some notationsIf 119892(119905) is a periodic function with period 120596 we define 119892 =

(1120596) int120596

0119892(119905)119889119905 119892119897 = min119905isin[0120596]119892(119905) 119892119906 = max119905isin[0120596]119892(119905) As

described in the previous section

1198771 (119905) = 1 +120583 (119898 + 120583) (119898 + 120590) (1198770 minus 1)

119898120573 (119905) (119898 + 120590 + 120583) (40)

So 119877119897

1= 1 + 120583(119898 + 120583)(119898 + 120590)(119877

119897

0minus 1)119898120573

119906(119898 + 120590 + 120583) 119877119906

1=

1 + 120583(119898 + 120583)(119898 + 120590)(119877119906

0minus 1)119898120573

119897(119898 + 120590 + 120583)

In this section we adopt 120573(119905) = 119886[1 + 119887 sin(12058711990510)]Then applying the numerical simulation to verify the abovesolution we give the following conclusion

(1) when 119877119889 lt 1 119874 is stable(2) when 119877119889 gt 1 and 1198770 lt 1 1198640 is stable see Figure 3(a)(3) when 1198770 gt 1 and 119903 gt 119898 + 120583 system (11) has at least

one positive periodic solution see Figure 3(b)

We can givemore results about the conditions of existenceof the positive periodic solution

(11015840) When 119877119889 gt 1 1198770 gt 1 and 119877119889 lt 119877

119906

1 119874 is stable see

Figures 4(a) and 4(b)(21015840) When 119877119889 gt 1 1198770 gt 1 and 119877119889 gt 119877

119906

1 system (11) has at

least one positive periodic solution see Figure 5

By numerical simulations we can give that the conditionswhich ensure the existence of positive periodic solution are1198770 gt 1 and 119903 gt 119898+120583 or 119877119889 gt 1 1198770 gt 1 and 119877119889 gt 119877

119906

1 In fact

119877119889 gt 1 1198770 gt 1 and 119877119889 gt 119877119906

1are the sufficient conditions for

1198770 gt 1 and 119903 gt 119898 + 120583 As a result the conditions 1198770 gt 1 and119903 gt 119898 + 120583 are broader

4 Discussion

This paper considers a logistic growth system whose birthprocess incorporates density-dependent effects This type ofmodel has a rich dynamical behavior and practical signif-icance By analyzing its equilibria and respective attractiveregion we find that the dynamical behavior of a disease will

8 The Scientific World Journal

25

2

15

1

05

00 05 1 15 2 25 3 35

times104

times104

S(0) = 25000 E(0) = 10000 I(0) = 24000

S(0) = 35000 E(0) = 20000 I(0) = 15000

S(0) = 15000 E(0) = 10000 I(0) = 20000

I(t)

S(t)

(a)

25

2

15

1

05

00 05 1 15 2 25 3 35

times104

times104

S(0) = 25000 E(0) = 10000 I(0) = 24000

S(0) = 35000 E(0) = 20000 I(0) = 15000

S(0) = 15000 E(0) = 10000 I(0) = 20000

I(t)

S(t)

(b)

Figure 4 Phase plane of 119878(119905) and 119868(119905) (a) When the parameter values are 119903 = 011 119886 = 09 119887 = 02 and 120583 = 03 1198770 = 14991 119877119889 = 11 gt 1119877119889 lt 119877

119906

1= 14159 and 119877119889 lt 119877

119897

1= 12773 (b) When the parameter values are 119903 = 0126 119886 = 07 119887 = 02 and 120583 = 03 1198770 = 21067

119877119889 = 126 gt 1 119877119889 lt 119877119906

1= 12964 and 119877119889 gt 119877

119897

1= 11976 The value of other parameters can be seen in Table 1

7000

6000

5000

4000

3000

2000

1000

005 1 15 2

times104

times104

S(0) = 22000 E(0) = 5000 I(0) = 1000

S(0) = 4000 E(0) = 2000 I(0) = 1000

S(0) = 10000 E(0) = 10000 I(0) = 6000

I(t)

S(t)

Figure 5 Phase plane of 119878(119905) and 119868(119905) When the parameter valuesare 119903 = 013 119886 = 036 and 119887 = 02 1198770 = 10834 119877119889 = 13 gt 1 119877119889 gt119877119906

1= 10435 and 119877119889 gt 119877

119897

1= 1029 The value of other parameters

can be seen in Table 1

be determined by two thresholds 1198770 and 119877119889 Only 1198770 gt 1

cannot promise the existence of the endemic equilibriumwhich also needs 119877119889 gt 1198771 When 1198770 gt 1 and 119877119889 lt 1198771the solutions of the system (1) will tend to the origin 119874 It

is caused by the phenomenon that the death number dueto disease cannot be supplemented by the birth numberpromptly Finally all people are infected and die outThe factinterpreted by this model is more reasonable Theoreticallywe prove the global asymptotic stability of the disease-free equilibrium and give respective attractive regions ofequilibria

Seasonally effective contact rate is themost common formwhich may be related to various factors and thus this paperstudies the corresponding non-autonomous system which isobtained by changing the constant transmission rate of theabove system into the periodic transmission rate For theperiodic systems their dynamical behaviors especially thebasic reproduction number have been investigated in depthby [41 47ndash55] which provide many methods that we canutilize For the obtained periodic model by analyzing theglobal asymptotic stability of the disease-free equilibrium andthe existence of positive periodic solution we have the similarresults as the autonomous system The dynamic behavior ofdisease will be decided by two conditions 1198770 gt 1 and 119903 gt

119898 + 120583 that show that when the disease is prevalent the birthrate should be larger than the death rate to guarantee thesustainable growth of population Otherwise the populationwill disappear In addition we will evaluate and comparethe basic reproduction number 1198770 and the average basicreproduction number 1198770 which has been adopted by [27 56ndash59] In this paper we can calculate the average reproductionnumber

1198770 =120573120590

(119898 + 120590) (119898 + 120583) (41)

The Scientific World Journal 9

where 120573 = (120) int20

0120573(119905)119889119905 When 119903 = 013 119886 = 03 and

119887 = 02 we know that 1198770 = 09029 and 1198770 = 1 When 119903 =

013 119896 = 100000 119886 = 036 119887 = 02 and 119898 = 01 120590 = 02120583 = 01 then 1198770 = 10834 and 1198770 = 12 In that sense it isconfirmed that the basic reproduction number 1198770 defined by[40] is more accurate than the average reproduction number1198770 which overestimates the risk of disease

It should be noted that we live in a spatial world and it is anatural phenomenon that a substance goes from high densityregions to low density regions As a result epidemic modelsshould include spatial effects In a further study we need toinvestigate spatial epidemic models with seasonal factors

Acknowledgments

The research was partially supported by the National NaturalScience Foundation of China under Grant nos 1130149011301491 11331009 11147015 11171314 11101251 and 11105024Natural Science Foundation of ShanXi Province Grant no2012021002-1 and the Opening Foundation of Institute ofInformation Economy Hangzhou Normal University Grantno PD12001003002003

References

[1] K Dietz and J A P Heesterbeek ldquoBernoulli was ahead ofmodern epidemiologyrdquo Nature vol 408 no 6812 pp 513ndash5142000

[2] K Dietz and J A P Heesterbeek ldquoDaniel Bernoullirsquos epidemio-logical model revisitedrdquoMathematical Biosciences vol 180 pp1ndash21 2002

[3] R Ross ldquoAn application of the theory of probabilities to thestudy of a priori pathometry (part I)rdquo Proceedings of the RoyalSociety A vol 92 no 638 pp 204ndash230 1916

[4] R Ross and H P Hudson ldquoAn application of the theory ofprobabilities to the study of a priori pathometry (part II)rdquoProceedings of the Royal Society A vol 93 no 650 pp 212ndash2251917

[5] W O Kermack and A G McKendrick ldquoA contribution to themathematical theory of epidemics (part I)rdquo Proceedings of theRoyal Society A vol 115 no 772 pp 700ndash721 1927

[6] W O Kermack and A G McKendrick ldquoA contribution to themathematical theory of epidemics (part II)rdquo Proceedings of theRoyal Society A vol 138 no 834 pp 55ndash83 1932

[7] A Tsoularis and JWallace ldquoAnalysis of logistic growthmodelsrdquoMathematical Biosciences vol 179 no 1 pp 21ndash55 2002

[8] I Nasell ldquoOn the quasi-stationary distribution of the stochasticlogistic epidemicrdquoMathematical Biosciences vol 156 no 1-2 pp21ndash40 1999

[9] H Fujikawa A Kai and SMorozumi ldquoA new logisticmodel forEscherichia coli growth at constant and dynamic temperaturesrdquoFood Microbiology vol 21 no 5 pp 501ndash509 2004

[10] L Berezansky and E Braverman ldquoOscillation properties of alogistic equation with several delaysrdquo Journal of MathematicalAnalysis and Applications vol 247 no 1 pp 110ndash125 2000

[11] M Tabata N Eshima and I Takagi ldquoThe nonlinear integro-partial differential equation describing the logistic growth ofhuman population with migrationrdquo Applied Mathematics andComputation vol 98 pp 169ndash183 1999

[12] L Korobenko andE Braverman ldquoOn logisticmodels with a car-rying capacity dependent diffusion stability of equilibria andcoexistence with a regularly diffusing populationrdquo NonlinearAnalysis Real World Applications vol 13 no 6 pp 2648ndash26582012

[13] Y Muroya ldquoGlobal attractivity for discrete models of nonau-tonomous logistic equationsrdquo Computers andMathematics withApplications vol 53 no 7 pp 1059ndash1073 2007

[14] S Invernizzi and K Terpin ldquoA generalized logistic model forphotosynthetic growthrdquo Ecological Modelling vol 94 no 2-3pp 231ndash242 1997

[15] ZMinW Bang-Jun and J Feng ldquoCoalmining citiesrsquo economicgrowthmechanism and sustainable development analysis basedon logistic dynamics modelrdquo Procedia Earth and PlanetaryScience vol 1 no 1 pp 1737ndash1743 2009

[16] L I Anita S Anita and V Arnautu ldquoGlobal behavior foran age-dependent population model with logistic term andperiodic vital ratesrdquoAppliedMathematics andComputation vol206 no 1 pp 368ndash379 2008

[17] Z A Banaszak X Q Tang S C Wang and M B ZarembaldquoLogistics models in flexible manufacturingrdquo Computers inIndustry vol 43 no 3 pp 237ndash248 2000

[18] S Brianzoni C Mammana and E Michetti ldquoNonlineardynamics in a business-cycle model with logistic populationgrowthrdquo Chaos Solitons and Fractals vol 40 no 2 pp 717ndash7302009

[19] W P London and J A Yorke ldquoRecurrent outbreaks of measleschickenpox and mumps I Seasonal variation in contact ratesrdquoThe American Journal of Epidemiology vol 98 no 6 pp 453ndash468 1973

[20] S F Dowell ldquoSeasonal variation in host susceptibility and cyclesof certain infectious diseasesrdquo Emerging Infectious Diseases vol7 no 3 pp 369ndash374 2001

[21] ON Bjoslashrnstad B F Finkenstadt andB TGrenfell ldquoDynamicsof measles epidemics estimating scaling of transmission ratesusing a Time series SIR modelrdquo Ecological Monographs vol 72no 2 pp 169ndash184 2002

[22] J Zhang Z Jin G Q Sun X Sun and S Ruan ldquoModelingseasonal rabies epidemics in Chinardquo Bulletin of MathematicalBiology vol 74 no 5 pp 1226ndash1251 2012

[23] I B Schwartz ldquoSmall amplitude long period outbreaks inseasonally driven epidemicsrdquo Journal of Mathematical Biologyvol 30 no 5 pp 473ndash491 1992

[24] I B Schwartz and H L Smith ldquoInfinite subharmonic bifur-cation in an SEIR epidemic modelrdquo Journal of MathematicalBiology vol 18 no 3 pp 233ndash253 1983

[25] H L Smith ldquoMultiple stable subharmonics for a periodicepidemic modelrdquo Journal of Mathematical Biology vol 17 no2 pp 179ndash190 1983

[26] J Dushoff J B Plotkin S A Levin and D J D EarnldquoDynamical resonance can account for seasonality of influenzaepidemicsrdquo Proceedings of the National Academy of Sciences ofthe United States of America vol 101 no 48 pp 16915ndash169162004

[27] J L Ma and Z Ma ldquoEpidemic threshold conditions forseasonally forced SEIR modelsrdquo Mathematical Biosciences andEngineering vol 3 no 1 pp 161ndash172 2006

[28] D J D Earn P Rohani B M Bolker and B T Grenfell ldquoAsimple model for complex dynamical transitions in epidemicsrdquoScience vol 287 no 5453 pp 667ndash670 2000

10 The Scientific World Journal

[29] M J Keeling P Rohani and B T Grenfell ldquoSeasonally forceddisease dynamics explored as switching between attractorsrdquoPhysica D vol 148 no 3-4 pp 317ndash335 2001

[30] J L Aron and I B Schwartz ldquoSeasonality and period-doublingbifurcations in an epidemic modelrdquo Journal of TheoreticalBiology vol 110 no 4 pp 665ndash679 1984

[31] N C Grassly and C Fraser ldquoSeasonal infectious diseaseepidemiologyrdquo Proceedings of the Royal Society B vol 273 no1600 pp 2541ndash2550 2006

[32] J Zhang Z Jin G Q Sun T Zhou and S Ruan ldquoAnalysisof rabies in China transmission dynamics and controlrdquo PLoSONE vol 6 no 7 Article ID e20891 2011

[33] O Diekmann J A P Heesterbeek and J A Metz ldquoOnthe definition and the computation of the basic reproductionratio R0 in models for infectious diseases in heterogeneouspopulationsrdquo Journal of mathematical biology vol 28 no 4 pp365ndash382 1990

[34] O Diekmann J A P Heesterbeek and M G Roberts ldquoTheconstruction of next-generation matrices for compartmentalepidemic modelsrdquo Journal of the Royal Society Interface vol 7no 47 pp 873ndash885 2010

[35] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[36] F Berezovsky G Karev B J Song and C C Chavez ldquoA simpleepidemic model with surprising dynamicsrdquo Mathematical Bio-sciences and Engineering vol 2 pp 133ndash152 2005

[37] J LaSalle and S Lefschetz Stability by Liapunovrsquos DirectMethodAcademic Press New York NY USA 1961

[38] E A Barbashin Introduction to theTheory of Stability Wolters-Noordhoff Groningen The Netherlands 1970

[39] D Schenzle ldquoAn age-structured model of pre- and post-vaccination measles transmissionrdquoMathematical Medicine andBiology vol 1 no 2 pp 169ndash191 1984

[40] N Bacaer and S Guernaoui ldquoThe epidemic threshold ofvector-borne diseases with seasonality the case of cutaneousleishmaniasis in Chichaoua Moroccordquo Journal of MathematicalBiology vol 53 no 3 pp 421ndash436 2006

[41] W DWang and X Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008

[42] F Zhang and X Q Zhao ldquoA periodic epidemic model in apatchy environmentrdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 496ndash516 2007

[43] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press New York NY USA 1995

[44] H R Thieme ldquoConvergence results and a Poincare-Bendixsontrichotomy for asymptotically automous differential equationsrdquoJournal of Mathematical Biology vol 30 pp 755ndash763 1992

[45] L Perko Differential Equations and Dynamical SystemsSpringer New York NY USA 2000

[46] X Q Zhao Dynamical Systems in Population Biology SpringerNew York NY USA 2003

[47] Z G Bai and Y C Zhou ldquoThreshold dynamics of a bacillarydysentery model with seasonal fluctuationrdquo Discrete and Con-tinuous Dynamical Systems B vol 15 no 1 pp 1ndash14 2011

[48] Z G Bai Y C Zhou and T L Zhang ldquoExistence of multipleperiodic solutions for an SIRmodel with seasonalityrdquoNonlinear

Analysis Theory Methods and Applications vol 74 no 11 pp3548ndash3555 2011

[49] N Bacaer ldquoApproximation of the basic reproduction numberR0 for vector-borne diseases with a periodic vector populationrdquoBulletin of Mathematical Biology vol 69 no 3 pp 1067ndash10912007

[50] N Bacaer and X Abdurahman ldquoResonance of the epidemicthreshold in a periodic environmentrdquo Journal of MathematicalBiology vol 57 no 5 pp 649ndash673 2008

[51] N Bacaer ldquoPeriodic matrix population models growth ratebasic reproduction number and entropyrdquoBulletin ofMathemat-ical Biology vol 71 no 7 pp 1781ndash1792 2009

[52] N Bacaer and E H A Dads ldquoGenealogy with seasonalitythe basic reproduction number and the influenza pandemicrdquoJournal of Mathematical Biology vol 62 no 5 pp 741ndash762 2011

[53] L J Liu X Q Zhao and Y C Zhou ldquoA tuberculosis model withseasonalityrdquo Bulletin of Mathematical Biology vol 72 no 4 pp931ndash952 2010

[54] J L Liu ldquoThreshold dynamics for a HFMD epidemic modelwith periodic transmission raterdquo Nonlinear Dynamics vol 64no 1-2 pp 89ndash95 2011

[55] Y Nakata and T Kuniya ldquoGlobal dynamics of a class ofSEIRS epidemic models in a periodic environmentrdquo Journal ofMathematical Analysis and Applications vol 363 no 1 pp 230ndash237 2010

[56] B GWilliams andC Dye ldquoInfectious disease persistence whentransmission varies seasonallyrdquo Mathematical Biosciences vol145 no 1 pp 77ndash88 1997

[57] I A Moneim ldquoThe effect of using different types of periodiccontact rate on the behaviour of infectious diseases a simula-tion studyrdquo Computers in Biology and Medicine vol 37 no 11pp 1582ndash1590 2007

[58] C L Wesley and L J S Allen ldquoThe basic reproduction numberin epidemic models with periodic demographicsrdquo Journal ofBiological Dynamics vol 3 no 2-3 pp 116ndash129 2009

[59] D Greenhalgh and I A Moneim ldquoSIRS epidemic model andsimulations using different types of seasonal contact raterdquoSystems Analysis Modelling Simulation vol 43 no 5 pp 573ndash600 2003

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

The Scientific World Journal 5

has one disease-free equilibrium 1198640 = (1198730 0 0) where119873

0=

(1 minus 119898119903)119896 By giving a new vector 119909 = (119864 119868) we have

119865 = (

120573 (119905) 119878119868

119873

0

) 119881 = (119898119864 + 120590119864

119898119868 + 120583119868 minus 120590119864)

119881minus

= (119898119864 + 120590119864

119898119868 + 120583119868) 119881

+= (

0

120590119864)

(12)

Taking the partial derivative of the above vectors aboutvariables 119864 119868 and substituting the disease-free equilibriumwe have

119865 (119905) = (0 120573 (119905)

0 0)

119881 (119905) = (119898 + 120590 0

minus120590 119898 + 120583)

(13)

According to [41] denote 119862120596 to be the ordered Banachspace of all 120596-periodic functions from R to R4 which isequippedwith themaximumnorm sdot and the positive cone119862+

120596= 120601 isin 119862120596 120601(119905) ge 0 forall119905 isin R+ Over the Banach space

we define a linear operator 119871 119862120596 rarr 119862120596 by

(119871120601) (119905)

= int

infin

0

119884 (119905 119905 minus 119886) 119865 (119905 minus 119886) 120601 (119905 minus 119886) 119889119886 forall119905 isin R+ 120601 isin 119862120596

(14)

where 119871 is called the next infection operator and the inter-pretation of 119884(119905 119905 minus 119886) 120601(119905 minus 119886) can be seen in [41] Thenthe spectral radius of 119871 is defined as the basic reproductionnumber

1198770 = 120588 (119871) (15)

In order to give the expression of the basic reproductionnumber we need to introduce the linear 120596-periodic system

119889119908

119889119905= [minus119881 (119905) +

119865 (119905)

120582]119908 119905 isin R+ (16)

with parameter 120582 isin R Let 119882(119905 119904 120582) 119905 ge 119904 be theevolution operator of system (16) on R2 In fact 119882(119905 119904 120582) =

Φ(119865120582)minus119881(119905) and Φ119865minus119881(119905) = 119882(119905 0 1) for all 119905 ge 0 ByTheorems 21 and 22 in [41] the basic reproduction numberalso can be defined as1205820 such that120588(Φ(1198651205820)minus119881(120596)) = 1 whichcan be straightforward to calculate

32 Global Stability of the Disease-Free Equilibrium

Theorem4 Thedisease-free equilibrium1198640 is globally asymp-totically stable when 1198770 lt 1 and 119877119889 gt 1

Proof Theorem 22 in [41] implies that 1198640 is locally asymp-totically stable when 1198770 lt 1 and 119877119889 gt 1 So we only need

to prove its global attractability It is easy to know that 119878(119905) le

1198730= (1 minus (119898119903))119896 Thus

119889119864

119889119905le 120573 (119905) 119868 minus (119898 + 120590) 119864

119889119868

119889119905= 120590119864 minus 119898119868 minus 120583119868

(17)

The right comparison system can be written as

119889119864

119889119905= 120573 (119905) 119868 minus (119898 + 120590) 119864

119889119868

119889119905= 120590119864 minus 119898119868 minus 120583119868

(18)

that is

119889ℎ

119889119905= (119865 (119905) minus 119881 (119905)) ℎ (119905) ℎ (119905) = (119864 (119905) 119868 (119905)) (19)

For (19) Lemma 21 in [42] shows that there is apositive 120596-periodic function ℎ(119905) = (119864(119905) 119868(119905))

119879 such thatℎ(119905) = 119890

119901119905ℎ(119905) is a solution of system (18) where 119901 =

(1120596) ln 120588(Φ119865minus119881(120596)) By Theorem 22 in [41] we know thatwhen 1198770 lt 1 and 119877119889 gt 1 120588(Φ119865minus119881(120596)) lt 1 and 119901 lt 0which implies ℎ(119905) rarr 0 as 119905 rarr infin Therefore the zerosolution of system (18) is globally asymptotically stable Bythe comparison principle [43] and the theory of asymptoticautonomous systems [44] when 1198770 lt 1 and 119877119889 gt 1 1198640is globally attractive Therefore the proposition that 1198640 isglobally asymptotically stable holds

33 Existence of Positive Periodic Solutions Before the proofof the existence of positive periodic solutions we firstlyintroduce some denotations Let 119906(119905 1199090) be the solution ofsystem (11) with the initial value 1199090 = (119878(0) 119864(0) 119868(0)) Bythe fundamental existence-uniqueness theorem [45] 119906(119905 1199090)is the unique solution of system (11) with 119906(0 1199090) = 1199090

Next we need to introduce the Poincare map119875 119883 rarr 119883

associated with system (11) that is

119875 (1199090) = 119906 (120596 1199090) forall1199090 isin 119883 (20)

where 120596 is the period Theorem 1 implies that 119883 is positivelyinvariant and 119875 is a dissipative point

Now we introduce two subsets of1198831198830 = (119878 119864 119868) isin 119883

119864 gt 0 119868 gt 0 and 1205971198830 = 119883 1198830

Lemma 5 (a) When 1198770 gt 1 and 119903 gt 119898 + 120583 there exists a120575 gt 0 such that when

1003817100381710038171003817(119878 (0) 119864 (0) 119868 (0)) minus 11986401003817100381710038171003817 le 120575 (21)

for any (119878(0) 119864(0) 119868(0)) isin 1198830 one has

lim sup119898rarrinfin

119889 [119875119898

(119878 (0) 119864 (0) 119868 (0)) 1198640] ge 120575 (22)

where 1198640 = (1198730 0 0)

6 The Scientific World Journal

(b) When 1198770 gt 1 and 119903 gt 119898 + 120583 there exists a 120575 gt 0 suchthat when

(119878 (0) 119864 (0) 119868 (0)) minus 119874 le 120575 (23)

for any (119878(0) 119864(0) 119868(0)) isin 1198830 one has

lim sup119898rarrinfin

119889 [119875119898

(119878 (0) 119864 (0) 119868 (0)) 119874] ge 120575 (24)

where 119874 = (0 0 0)

Proof (a) ByTheorem 22 in [41] we know that when 1198770 gt 1120588(Φ119865minus119881(120596)) gt 1 So there is a small enough positive number120598 such that 120588(Φ119865minus119881minus119872120598(120596)) gt 1 where

119872120598 = (0

120573 (119905) 120598119868

1198730

0 0

) (25)

If proposition (a) does not hold there is some(119878(0) 119864(0) 119868(0)) isin 1198830 such that

lim sup119898rarrinfin

119889 (119875119898

(119878 (0) 119864 (0) 119868 (0)) 1198640) lt 120575 (26)

We can assume that for all 119898 ge 0 119889(119875119898(119878(0) 119864(0)

119868(0)) 1198640) lt 120575 Applying the continuity of the solutions withrespect to the initial values

1003817100381710038171003817119906 (119905 119875119898

(119878 (0) 119864 (0) 119868 (0))) minus 119906 (119905 1198640)1003817100381710038171003817 le 120598

forall119898 ge 0 forall1199051 isin [0 120596]

(27)

Let 119905 = 119898120596 + 1199051 where 1199051 isin [0 120596] and 119898 = [119905120596] 119898 =

[119905120596] is the greatest integer which is not more than 119905120596Thenfor any 119905 ge 0

1003817100381710038171003817119906 (119905 (119878 (0) 119864 (0) 119868 (0))) minus 119906 (119905 1198640)1003817100381710038171003817

=1003817100381710038171003817119906 (1199051 119875

119898(119878 (0) 119864 (0) 119868 (0))) minus 119906 (1199051 1198640)

1003817100381710038171003817 le 120598

(28)

So 1198730

minus 120598 le 119878(119905) le 1198730

+ 120598 Then whenlim sup

119898rarrinfin119889(119875119898(119878(0) 119864(0) 119868(0)) 1198640) lt 120575

119889119864

119889119905ge 120573 (119905) 119868 minus

120573 (119905) 120598119868

1198730minus (119898 + 120590) 119864

119889119868

119889119905= 120590119864 minus 119898119868 minus 120583119868

(29)

Thus we can study the right linear system

119889119864

119889119905= 120573 (119905) 119868 minus

120573 (119905) 120598119868

1198730minus (119898 + 120590) 119864

119889119868

119889119905= 120590119864 minus 119898119868 minus 120583119868

(30)

For the system (30) there exists a positive 120596-periodicfunction 119892(119905) = (119864(119905) 119868(119905))

119879 such that 119892(119905) = 119890119901119905119892(119905) is a

solution of system (11) where 119901 = (1120596) ln 120588(Φ119865minus119881minus119872120598(120596))When 1198770 gt 1 120588(Φ119865minus119881minus119872120598(120596)) gt 1 which means that when

119892(0) gt 0119892(119905) rarr infin as 119905 rarr infin By the comparison principle[43] when 119864(0) gt 0 119868(0) gt 0 119864(119905) rarr infin 119868(119905) rarr infin as119905 rarr infinThere appears a contradictionThus the proposition(a) holds

(b) When 1198770 gt 1 and 119903 gt 119898 + 120583 we have

119889119873

119889119905= 119903119873(1 minus

119873

119896) minus 119898119873 minus 120583119868

ge [119903 (1 minus119873

119896) minus 119898 minus 120583]119873 gt 0

(31)

So if 119877119889 gt 1 119873 rarr 1198730 for 119873(0) gt 0 as 119905 rarr +infin that is

119882119878(119874) cap 1198830 = 0

Theorem 6 When 1198770 gt 1 and 119903 gt 119898 + 120583 there exists a 120575 gt 0

such that any solution (119878(119905) 119864(119905) 119868(119905)) of system (11)with initialvalue (119878(0) 119864(0) 119868(0)) isin (119878 119864 119868) isin 119883 119864 gt 0 119868 gt 0

satisfies

lim inf119905rarrinfin

119864 (119905) ge 120575 lim inf119905rarrinfin

119868 (119905) ge 120575 (32)

and system (11) has at least one positive periodic solution

Proof From system (11)

119878 (119905) = 119890minusint119905

0(120573(119904)119868(119904)119873+119898)119889119904

times [119878 (0) + int

119905

0

119903119873 (119904) (1 minus119873 (119904)

119896)

times119890int119904

0(120573(119888)119868(119888)119873(119888)+119898)119889119888

119889119904]

gt 0 forall119905 gt 0

(33)

119864 (119905) = 119890minus(119898+120590)119905

[119864 (0) + int

119905

0

120573 (119904) 119878 (119904) 119868 (119904)

119873 (119904)119890(119898+120590)119904

119889119904]

gt 0 forall119905 gt 0

(34)

119868 (119905) = 119890minus(119898+120583)119905

[119868 (0) + int

119905

0

120590119864 (119904) 119890(119898+120583)119904

119889119904]

gt 0 forall119905 gt 0

(35)

for any (119878(0) 119864(0) 119868(0)) isin 1198830 which shows that 1198830 ispositively invariant Moreover it is obvious to see that 1205971198830

is relatively closed in 119883 Denote

119872120597 = (119878 (0) 119864 (0) 119868 (0)) isin 1205971198830

119875119898

(119878 (0) 119864 (0) 119868 (0)) isin 1205971198830 forall119898 ge 0

(36)

Next we prove that

119872120597 = (119878 0 0) isin 119883 119878 ge 0 (37)

We only need to show that 119872120597 sube (119878 0 0) isin 119883 119878 ge 0which means that for any (119878(0) 119864(0) 119868(0)) isin 1205971198830 119864(119898120596) =

119868(119898120596) = 0 for all 119898 ge 0 If it does not hold there existsa 1198981 ge 0 such that (119864(1198981120596) 119868(1198981120596))

119879gt 0 Taking 1198981120596 as

The Scientific World Journal 7

25

2

15

1

05

01 15 2 25 3 35

times104

times104

S(0) = 25000 E(0) = 10000 I(0) = 24000

S(0) = 35000 E(0) = 20000 I(0) = 15000

S(0) = 15000 E(0) = 10000 I(0) = 20000

I(t)

S(t)

(a)

24

23

22

21

2

19

18

17

16

15

1415 2 25 3 35

times104

times104

S(0) = 25000 E(0) = 10000 I(0) = 24000

S(0) = 35000 E(0) = 20000 I(0) = 15000

S(0) = 15000 E(0) = 10000 I(0) = 20000

I(t)

S(t)

(b)

Figure 3 Phase plane of 119878(119905) and 119868(119905) (a) When the parameter values are 119903 = 013 119886 = 03 and 119887 = 02 1198770 = 09029 lt 1 119877119889 = 13 gt 1 (b)When the parameter values are 119903 = 03 119886 = 07 and 119887 = 02 1198770 = 21067 and 119903 gt 119898+120583 The value of other parameters can be seen in Table 1

the initial time and repeating the processes as in (33)ndash(35)we can have that (119878(119905) 119864(119905) 119868(119905))

119879gt 0 for all 119905 gt 1198981120596

Thus (119878(119905) 119864(119905) 119868(119905)) isin 1198830 for all 119905 gt 1198981120596 There appearsa contradiction which means that the equality (37) holdsTherefore 1198640 is acyclic in 1205971198830 Obviously when 1198770 gt 1 and119903 gt 119898 + 120583 119874 is acyclic in 1205971198830

Furthermore by Lemma 5 1198640 = (1198730 0 0) and 119874 =

(0 0 0) are isolated invariant sets in 119883 119882119878(1198640) cap 1198830 = 0and 119882

119878(119874) cap 1198830 = 0 By Theorem 131 and Remark 131

in [46] it can be obtained that 119875 is uniformly persistent withrespect to (1198830 1205971198830) that is there exists a 120575 gt 0 such that anysolution (119878(119905) 119864(119905) 119868(119905)) of system (11) with the initial value(119878(0) 119864(0) 119868(0)) isin (119878 119864 119868) isin 119883 119864 gt 0 119868 gt 0 satisfies

lim inf119905rarrinfin

119864 (119905) ge 120575 lim inf119905rarrinfin

119868 (119905) ge 120575 (38)

ApplyingTheorem 136 in [46] 119875 has a fixed point

(119878lowast(0) 119864

lowast(0) 119868lowast(0)) isin 1198830 (39)

From (33) we know 119878lowast

gt 0 for all 119905 isin [0 120596] 119878lowast(119905) is alsomore than zero for all 119905 gt 0 due to the periodicity Similarlyfor all 119905 ge 0119864lowast(119905) gt 0 119868lowast(119905) gt 0Therefore it can be obtainedthat one of the positive 120596-periodic solutions of system (11) is(119878lowast(119905) 119864lowast(119905) 119868lowast(119905))

34 Numerical Simulations Firstly we give some notationsIf 119892(119905) is a periodic function with period 120596 we define 119892 =

(1120596) int120596

0119892(119905)119889119905 119892119897 = min119905isin[0120596]119892(119905) 119892119906 = max119905isin[0120596]119892(119905) As

described in the previous section

1198771 (119905) = 1 +120583 (119898 + 120583) (119898 + 120590) (1198770 minus 1)

119898120573 (119905) (119898 + 120590 + 120583) (40)

So 119877119897

1= 1 + 120583(119898 + 120583)(119898 + 120590)(119877

119897

0minus 1)119898120573

119906(119898 + 120590 + 120583) 119877119906

1=

1 + 120583(119898 + 120583)(119898 + 120590)(119877119906

0minus 1)119898120573

119897(119898 + 120590 + 120583)

In this section we adopt 120573(119905) = 119886[1 + 119887 sin(12058711990510)]Then applying the numerical simulation to verify the abovesolution we give the following conclusion

(1) when 119877119889 lt 1 119874 is stable(2) when 119877119889 gt 1 and 1198770 lt 1 1198640 is stable see Figure 3(a)(3) when 1198770 gt 1 and 119903 gt 119898 + 120583 system (11) has at least

one positive periodic solution see Figure 3(b)

We can givemore results about the conditions of existenceof the positive periodic solution

(11015840) When 119877119889 gt 1 1198770 gt 1 and 119877119889 lt 119877

119906

1 119874 is stable see

Figures 4(a) and 4(b)(21015840) When 119877119889 gt 1 1198770 gt 1 and 119877119889 gt 119877

119906

1 system (11) has at

least one positive periodic solution see Figure 5

By numerical simulations we can give that the conditionswhich ensure the existence of positive periodic solution are1198770 gt 1 and 119903 gt 119898+120583 or 119877119889 gt 1 1198770 gt 1 and 119877119889 gt 119877

119906

1 In fact

119877119889 gt 1 1198770 gt 1 and 119877119889 gt 119877119906

1are the sufficient conditions for

1198770 gt 1 and 119903 gt 119898 + 120583 As a result the conditions 1198770 gt 1 and119903 gt 119898 + 120583 are broader

4 Discussion

This paper considers a logistic growth system whose birthprocess incorporates density-dependent effects This type ofmodel has a rich dynamical behavior and practical signif-icance By analyzing its equilibria and respective attractiveregion we find that the dynamical behavior of a disease will

8 The Scientific World Journal

25

2

15

1

05

00 05 1 15 2 25 3 35

times104

times104

S(0) = 25000 E(0) = 10000 I(0) = 24000

S(0) = 35000 E(0) = 20000 I(0) = 15000

S(0) = 15000 E(0) = 10000 I(0) = 20000

I(t)

S(t)

(a)

25

2

15

1

05

00 05 1 15 2 25 3 35

times104

times104

S(0) = 25000 E(0) = 10000 I(0) = 24000

S(0) = 35000 E(0) = 20000 I(0) = 15000

S(0) = 15000 E(0) = 10000 I(0) = 20000

I(t)

S(t)

(b)

Figure 4 Phase plane of 119878(119905) and 119868(119905) (a) When the parameter values are 119903 = 011 119886 = 09 119887 = 02 and 120583 = 03 1198770 = 14991 119877119889 = 11 gt 1119877119889 lt 119877

119906

1= 14159 and 119877119889 lt 119877

119897

1= 12773 (b) When the parameter values are 119903 = 0126 119886 = 07 119887 = 02 and 120583 = 03 1198770 = 21067

119877119889 = 126 gt 1 119877119889 lt 119877119906

1= 12964 and 119877119889 gt 119877

119897

1= 11976 The value of other parameters can be seen in Table 1

7000

6000

5000

4000

3000

2000

1000

005 1 15 2

times104

times104

S(0) = 22000 E(0) = 5000 I(0) = 1000

S(0) = 4000 E(0) = 2000 I(0) = 1000

S(0) = 10000 E(0) = 10000 I(0) = 6000

I(t)

S(t)

Figure 5 Phase plane of 119878(119905) and 119868(119905) When the parameter valuesare 119903 = 013 119886 = 036 and 119887 = 02 1198770 = 10834 119877119889 = 13 gt 1 119877119889 gt119877119906

1= 10435 and 119877119889 gt 119877

119897

1= 1029 The value of other parameters

can be seen in Table 1

be determined by two thresholds 1198770 and 119877119889 Only 1198770 gt 1

cannot promise the existence of the endemic equilibriumwhich also needs 119877119889 gt 1198771 When 1198770 gt 1 and 119877119889 lt 1198771the solutions of the system (1) will tend to the origin 119874 It

is caused by the phenomenon that the death number dueto disease cannot be supplemented by the birth numberpromptly Finally all people are infected and die outThe factinterpreted by this model is more reasonable Theoreticallywe prove the global asymptotic stability of the disease-free equilibrium and give respective attractive regions ofequilibria

Seasonally effective contact rate is themost common formwhich may be related to various factors and thus this paperstudies the corresponding non-autonomous system which isobtained by changing the constant transmission rate of theabove system into the periodic transmission rate For theperiodic systems their dynamical behaviors especially thebasic reproduction number have been investigated in depthby [41 47ndash55] which provide many methods that we canutilize For the obtained periodic model by analyzing theglobal asymptotic stability of the disease-free equilibrium andthe existence of positive periodic solution we have the similarresults as the autonomous system The dynamic behavior ofdisease will be decided by two conditions 1198770 gt 1 and 119903 gt

119898 + 120583 that show that when the disease is prevalent the birthrate should be larger than the death rate to guarantee thesustainable growth of population Otherwise the populationwill disappear In addition we will evaluate and comparethe basic reproduction number 1198770 and the average basicreproduction number 1198770 which has been adopted by [27 56ndash59] In this paper we can calculate the average reproductionnumber

1198770 =120573120590

(119898 + 120590) (119898 + 120583) (41)

The Scientific World Journal 9

where 120573 = (120) int20

0120573(119905)119889119905 When 119903 = 013 119886 = 03 and

119887 = 02 we know that 1198770 = 09029 and 1198770 = 1 When 119903 =

013 119896 = 100000 119886 = 036 119887 = 02 and 119898 = 01 120590 = 02120583 = 01 then 1198770 = 10834 and 1198770 = 12 In that sense it isconfirmed that the basic reproduction number 1198770 defined by[40] is more accurate than the average reproduction number1198770 which overestimates the risk of disease

It should be noted that we live in a spatial world and it is anatural phenomenon that a substance goes from high densityregions to low density regions As a result epidemic modelsshould include spatial effects In a further study we need toinvestigate spatial epidemic models with seasonal factors

Acknowledgments

The research was partially supported by the National NaturalScience Foundation of China under Grant nos 1130149011301491 11331009 11147015 11171314 11101251 and 11105024Natural Science Foundation of ShanXi Province Grant no2012021002-1 and the Opening Foundation of Institute ofInformation Economy Hangzhou Normal University Grantno PD12001003002003

References

[1] K Dietz and J A P Heesterbeek ldquoBernoulli was ahead ofmodern epidemiologyrdquo Nature vol 408 no 6812 pp 513ndash5142000

[2] K Dietz and J A P Heesterbeek ldquoDaniel Bernoullirsquos epidemio-logical model revisitedrdquoMathematical Biosciences vol 180 pp1ndash21 2002

[3] R Ross ldquoAn application of the theory of probabilities to thestudy of a priori pathometry (part I)rdquo Proceedings of the RoyalSociety A vol 92 no 638 pp 204ndash230 1916

[4] R Ross and H P Hudson ldquoAn application of the theory ofprobabilities to the study of a priori pathometry (part II)rdquoProceedings of the Royal Society A vol 93 no 650 pp 212ndash2251917

[5] W O Kermack and A G McKendrick ldquoA contribution to themathematical theory of epidemics (part I)rdquo Proceedings of theRoyal Society A vol 115 no 772 pp 700ndash721 1927

[6] W O Kermack and A G McKendrick ldquoA contribution to themathematical theory of epidemics (part II)rdquo Proceedings of theRoyal Society A vol 138 no 834 pp 55ndash83 1932

[7] A Tsoularis and JWallace ldquoAnalysis of logistic growthmodelsrdquoMathematical Biosciences vol 179 no 1 pp 21ndash55 2002

[8] I Nasell ldquoOn the quasi-stationary distribution of the stochasticlogistic epidemicrdquoMathematical Biosciences vol 156 no 1-2 pp21ndash40 1999

[9] H Fujikawa A Kai and SMorozumi ldquoA new logisticmodel forEscherichia coli growth at constant and dynamic temperaturesrdquoFood Microbiology vol 21 no 5 pp 501ndash509 2004

[10] L Berezansky and E Braverman ldquoOscillation properties of alogistic equation with several delaysrdquo Journal of MathematicalAnalysis and Applications vol 247 no 1 pp 110ndash125 2000

[11] M Tabata N Eshima and I Takagi ldquoThe nonlinear integro-partial differential equation describing the logistic growth ofhuman population with migrationrdquo Applied Mathematics andComputation vol 98 pp 169ndash183 1999

[12] L Korobenko andE Braverman ldquoOn logisticmodels with a car-rying capacity dependent diffusion stability of equilibria andcoexistence with a regularly diffusing populationrdquo NonlinearAnalysis Real World Applications vol 13 no 6 pp 2648ndash26582012

[13] Y Muroya ldquoGlobal attractivity for discrete models of nonau-tonomous logistic equationsrdquo Computers andMathematics withApplications vol 53 no 7 pp 1059ndash1073 2007

[14] S Invernizzi and K Terpin ldquoA generalized logistic model forphotosynthetic growthrdquo Ecological Modelling vol 94 no 2-3pp 231ndash242 1997

[15] ZMinW Bang-Jun and J Feng ldquoCoalmining citiesrsquo economicgrowthmechanism and sustainable development analysis basedon logistic dynamics modelrdquo Procedia Earth and PlanetaryScience vol 1 no 1 pp 1737ndash1743 2009

[16] L I Anita S Anita and V Arnautu ldquoGlobal behavior foran age-dependent population model with logistic term andperiodic vital ratesrdquoAppliedMathematics andComputation vol206 no 1 pp 368ndash379 2008

[17] Z A Banaszak X Q Tang S C Wang and M B ZarembaldquoLogistics models in flexible manufacturingrdquo Computers inIndustry vol 43 no 3 pp 237ndash248 2000

[18] S Brianzoni C Mammana and E Michetti ldquoNonlineardynamics in a business-cycle model with logistic populationgrowthrdquo Chaos Solitons and Fractals vol 40 no 2 pp 717ndash7302009

[19] W P London and J A Yorke ldquoRecurrent outbreaks of measleschickenpox and mumps I Seasonal variation in contact ratesrdquoThe American Journal of Epidemiology vol 98 no 6 pp 453ndash468 1973

[20] S F Dowell ldquoSeasonal variation in host susceptibility and cyclesof certain infectious diseasesrdquo Emerging Infectious Diseases vol7 no 3 pp 369ndash374 2001

[21] ON Bjoslashrnstad B F Finkenstadt andB TGrenfell ldquoDynamicsof measles epidemics estimating scaling of transmission ratesusing a Time series SIR modelrdquo Ecological Monographs vol 72no 2 pp 169ndash184 2002

[22] J Zhang Z Jin G Q Sun X Sun and S Ruan ldquoModelingseasonal rabies epidemics in Chinardquo Bulletin of MathematicalBiology vol 74 no 5 pp 1226ndash1251 2012

[23] I B Schwartz ldquoSmall amplitude long period outbreaks inseasonally driven epidemicsrdquo Journal of Mathematical Biologyvol 30 no 5 pp 473ndash491 1992

[24] I B Schwartz and H L Smith ldquoInfinite subharmonic bifur-cation in an SEIR epidemic modelrdquo Journal of MathematicalBiology vol 18 no 3 pp 233ndash253 1983

[25] H L Smith ldquoMultiple stable subharmonics for a periodicepidemic modelrdquo Journal of Mathematical Biology vol 17 no2 pp 179ndash190 1983

[26] J Dushoff J B Plotkin S A Levin and D J D EarnldquoDynamical resonance can account for seasonality of influenzaepidemicsrdquo Proceedings of the National Academy of Sciences ofthe United States of America vol 101 no 48 pp 16915ndash169162004

[27] J L Ma and Z Ma ldquoEpidemic threshold conditions forseasonally forced SEIR modelsrdquo Mathematical Biosciences andEngineering vol 3 no 1 pp 161ndash172 2006

[28] D J D Earn P Rohani B M Bolker and B T Grenfell ldquoAsimple model for complex dynamical transitions in epidemicsrdquoScience vol 287 no 5453 pp 667ndash670 2000

10 The Scientific World Journal

[29] M J Keeling P Rohani and B T Grenfell ldquoSeasonally forceddisease dynamics explored as switching between attractorsrdquoPhysica D vol 148 no 3-4 pp 317ndash335 2001

[30] J L Aron and I B Schwartz ldquoSeasonality and period-doublingbifurcations in an epidemic modelrdquo Journal of TheoreticalBiology vol 110 no 4 pp 665ndash679 1984

[31] N C Grassly and C Fraser ldquoSeasonal infectious diseaseepidemiologyrdquo Proceedings of the Royal Society B vol 273 no1600 pp 2541ndash2550 2006

[32] J Zhang Z Jin G Q Sun T Zhou and S Ruan ldquoAnalysisof rabies in China transmission dynamics and controlrdquo PLoSONE vol 6 no 7 Article ID e20891 2011

[33] O Diekmann J A P Heesterbeek and J A Metz ldquoOnthe definition and the computation of the basic reproductionratio R0 in models for infectious diseases in heterogeneouspopulationsrdquo Journal of mathematical biology vol 28 no 4 pp365ndash382 1990

[34] O Diekmann J A P Heesterbeek and M G Roberts ldquoTheconstruction of next-generation matrices for compartmentalepidemic modelsrdquo Journal of the Royal Society Interface vol 7no 47 pp 873ndash885 2010

[35] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[36] F Berezovsky G Karev B J Song and C C Chavez ldquoA simpleepidemic model with surprising dynamicsrdquo Mathematical Bio-sciences and Engineering vol 2 pp 133ndash152 2005

[37] J LaSalle and S Lefschetz Stability by Liapunovrsquos DirectMethodAcademic Press New York NY USA 1961

[38] E A Barbashin Introduction to theTheory of Stability Wolters-Noordhoff Groningen The Netherlands 1970

[39] D Schenzle ldquoAn age-structured model of pre- and post-vaccination measles transmissionrdquoMathematical Medicine andBiology vol 1 no 2 pp 169ndash191 1984

[40] N Bacaer and S Guernaoui ldquoThe epidemic threshold ofvector-borne diseases with seasonality the case of cutaneousleishmaniasis in Chichaoua Moroccordquo Journal of MathematicalBiology vol 53 no 3 pp 421ndash436 2006

[41] W DWang and X Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008

[42] F Zhang and X Q Zhao ldquoA periodic epidemic model in apatchy environmentrdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 496ndash516 2007

[43] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press New York NY USA 1995

[44] H R Thieme ldquoConvergence results and a Poincare-Bendixsontrichotomy for asymptotically automous differential equationsrdquoJournal of Mathematical Biology vol 30 pp 755ndash763 1992

[45] L Perko Differential Equations and Dynamical SystemsSpringer New York NY USA 2000

[46] X Q Zhao Dynamical Systems in Population Biology SpringerNew York NY USA 2003

[47] Z G Bai and Y C Zhou ldquoThreshold dynamics of a bacillarydysentery model with seasonal fluctuationrdquo Discrete and Con-tinuous Dynamical Systems B vol 15 no 1 pp 1ndash14 2011

[48] Z G Bai Y C Zhou and T L Zhang ldquoExistence of multipleperiodic solutions for an SIRmodel with seasonalityrdquoNonlinear

Analysis Theory Methods and Applications vol 74 no 11 pp3548ndash3555 2011

[49] N Bacaer ldquoApproximation of the basic reproduction numberR0 for vector-borne diseases with a periodic vector populationrdquoBulletin of Mathematical Biology vol 69 no 3 pp 1067ndash10912007

[50] N Bacaer and X Abdurahman ldquoResonance of the epidemicthreshold in a periodic environmentrdquo Journal of MathematicalBiology vol 57 no 5 pp 649ndash673 2008

[51] N Bacaer ldquoPeriodic matrix population models growth ratebasic reproduction number and entropyrdquoBulletin ofMathemat-ical Biology vol 71 no 7 pp 1781ndash1792 2009

[52] N Bacaer and E H A Dads ldquoGenealogy with seasonalitythe basic reproduction number and the influenza pandemicrdquoJournal of Mathematical Biology vol 62 no 5 pp 741ndash762 2011

[53] L J Liu X Q Zhao and Y C Zhou ldquoA tuberculosis model withseasonalityrdquo Bulletin of Mathematical Biology vol 72 no 4 pp931ndash952 2010

[54] J L Liu ldquoThreshold dynamics for a HFMD epidemic modelwith periodic transmission raterdquo Nonlinear Dynamics vol 64no 1-2 pp 89ndash95 2011

[55] Y Nakata and T Kuniya ldquoGlobal dynamics of a class ofSEIRS epidemic models in a periodic environmentrdquo Journal ofMathematical Analysis and Applications vol 363 no 1 pp 230ndash237 2010

[56] B GWilliams andC Dye ldquoInfectious disease persistence whentransmission varies seasonallyrdquo Mathematical Biosciences vol145 no 1 pp 77ndash88 1997

[57] I A Moneim ldquoThe effect of using different types of periodiccontact rate on the behaviour of infectious diseases a simula-tion studyrdquo Computers in Biology and Medicine vol 37 no 11pp 1582ndash1590 2007

[58] C L Wesley and L J S Allen ldquoThe basic reproduction numberin epidemic models with periodic demographicsrdquo Journal ofBiological Dynamics vol 3 no 2-3 pp 116ndash129 2009

[59] D Greenhalgh and I A Moneim ldquoSIRS epidemic model andsimulations using different types of seasonal contact raterdquoSystems Analysis Modelling Simulation vol 43 no 5 pp 573ndash600 2003

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 The Scientific World Journal

(b) When 1198770 gt 1 and 119903 gt 119898 + 120583 there exists a 120575 gt 0 suchthat when

(119878 (0) 119864 (0) 119868 (0)) minus 119874 le 120575 (23)

for any (119878(0) 119864(0) 119868(0)) isin 1198830 one has

lim sup119898rarrinfin

119889 [119875119898

(119878 (0) 119864 (0) 119868 (0)) 119874] ge 120575 (24)

where 119874 = (0 0 0)

Proof (a) ByTheorem 22 in [41] we know that when 1198770 gt 1120588(Φ119865minus119881(120596)) gt 1 So there is a small enough positive number120598 such that 120588(Φ119865minus119881minus119872120598(120596)) gt 1 where

119872120598 = (0

120573 (119905) 120598119868

1198730

0 0

) (25)

If proposition (a) does not hold there is some(119878(0) 119864(0) 119868(0)) isin 1198830 such that

lim sup119898rarrinfin

119889 (119875119898

(119878 (0) 119864 (0) 119868 (0)) 1198640) lt 120575 (26)

We can assume that for all 119898 ge 0 119889(119875119898(119878(0) 119864(0)

119868(0)) 1198640) lt 120575 Applying the continuity of the solutions withrespect to the initial values

1003817100381710038171003817119906 (119905 119875119898

(119878 (0) 119864 (0) 119868 (0))) minus 119906 (119905 1198640)1003817100381710038171003817 le 120598

forall119898 ge 0 forall1199051 isin [0 120596]

(27)

Let 119905 = 119898120596 + 1199051 where 1199051 isin [0 120596] and 119898 = [119905120596] 119898 =

[119905120596] is the greatest integer which is not more than 119905120596Thenfor any 119905 ge 0

1003817100381710038171003817119906 (119905 (119878 (0) 119864 (0) 119868 (0))) minus 119906 (119905 1198640)1003817100381710038171003817

=1003817100381710038171003817119906 (1199051 119875

119898(119878 (0) 119864 (0) 119868 (0))) minus 119906 (1199051 1198640)

1003817100381710038171003817 le 120598

(28)

So 1198730

minus 120598 le 119878(119905) le 1198730

+ 120598 Then whenlim sup

119898rarrinfin119889(119875119898(119878(0) 119864(0) 119868(0)) 1198640) lt 120575

119889119864

119889119905ge 120573 (119905) 119868 minus

120573 (119905) 120598119868

1198730minus (119898 + 120590) 119864

119889119868

119889119905= 120590119864 minus 119898119868 minus 120583119868

(29)

Thus we can study the right linear system

119889119864

119889119905= 120573 (119905) 119868 minus

120573 (119905) 120598119868

1198730minus (119898 + 120590) 119864

119889119868

119889119905= 120590119864 minus 119898119868 minus 120583119868

(30)

For the system (30) there exists a positive 120596-periodicfunction 119892(119905) = (119864(119905) 119868(119905))

119879 such that 119892(119905) = 119890119901119905119892(119905) is a

solution of system (11) where 119901 = (1120596) ln 120588(Φ119865minus119881minus119872120598(120596))When 1198770 gt 1 120588(Φ119865minus119881minus119872120598(120596)) gt 1 which means that when

119892(0) gt 0119892(119905) rarr infin as 119905 rarr infin By the comparison principle[43] when 119864(0) gt 0 119868(0) gt 0 119864(119905) rarr infin 119868(119905) rarr infin as119905 rarr infinThere appears a contradictionThus the proposition(a) holds

(b) When 1198770 gt 1 and 119903 gt 119898 + 120583 we have

119889119873

119889119905= 119903119873(1 minus

119873

119896) minus 119898119873 minus 120583119868

ge [119903 (1 minus119873

119896) minus 119898 minus 120583]119873 gt 0

(31)

So if 119877119889 gt 1 119873 rarr 1198730 for 119873(0) gt 0 as 119905 rarr +infin that is

119882119878(119874) cap 1198830 = 0

Theorem 6 When 1198770 gt 1 and 119903 gt 119898 + 120583 there exists a 120575 gt 0

such that any solution (119878(119905) 119864(119905) 119868(119905)) of system (11)with initialvalue (119878(0) 119864(0) 119868(0)) isin (119878 119864 119868) isin 119883 119864 gt 0 119868 gt 0

satisfies

lim inf119905rarrinfin

119864 (119905) ge 120575 lim inf119905rarrinfin

119868 (119905) ge 120575 (32)

and system (11) has at least one positive periodic solution

Proof From system (11)

119878 (119905) = 119890minusint119905

0(120573(119904)119868(119904)119873+119898)119889119904

times [119878 (0) + int

119905

0

119903119873 (119904) (1 minus119873 (119904)

119896)

times119890int119904

0(120573(119888)119868(119888)119873(119888)+119898)119889119888

119889119904]

gt 0 forall119905 gt 0

(33)

119864 (119905) = 119890minus(119898+120590)119905

[119864 (0) + int

119905

0

120573 (119904) 119878 (119904) 119868 (119904)

119873 (119904)119890(119898+120590)119904

119889119904]

gt 0 forall119905 gt 0

(34)

119868 (119905) = 119890minus(119898+120583)119905

[119868 (0) + int

119905

0

120590119864 (119904) 119890(119898+120583)119904

119889119904]

gt 0 forall119905 gt 0

(35)

for any (119878(0) 119864(0) 119868(0)) isin 1198830 which shows that 1198830 ispositively invariant Moreover it is obvious to see that 1205971198830

is relatively closed in 119883 Denote

119872120597 = (119878 (0) 119864 (0) 119868 (0)) isin 1205971198830

119875119898

(119878 (0) 119864 (0) 119868 (0)) isin 1205971198830 forall119898 ge 0

(36)

Next we prove that

119872120597 = (119878 0 0) isin 119883 119878 ge 0 (37)

We only need to show that 119872120597 sube (119878 0 0) isin 119883 119878 ge 0which means that for any (119878(0) 119864(0) 119868(0)) isin 1205971198830 119864(119898120596) =

119868(119898120596) = 0 for all 119898 ge 0 If it does not hold there existsa 1198981 ge 0 such that (119864(1198981120596) 119868(1198981120596))

119879gt 0 Taking 1198981120596 as

The Scientific World Journal 7

25

2

15

1

05

01 15 2 25 3 35

times104

times104

S(0) = 25000 E(0) = 10000 I(0) = 24000

S(0) = 35000 E(0) = 20000 I(0) = 15000

S(0) = 15000 E(0) = 10000 I(0) = 20000

I(t)

S(t)

(a)

24

23

22

21

2

19

18

17

16

15

1415 2 25 3 35

times104

times104

S(0) = 25000 E(0) = 10000 I(0) = 24000

S(0) = 35000 E(0) = 20000 I(0) = 15000

S(0) = 15000 E(0) = 10000 I(0) = 20000

I(t)

S(t)

(b)

Figure 3 Phase plane of 119878(119905) and 119868(119905) (a) When the parameter values are 119903 = 013 119886 = 03 and 119887 = 02 1198770 = 09029 lt 1 119877119889 = 13 gt 1 (b)When the parameter values are 119903 = 03 119886 = 07 and 119887 = 02 1198770 = 21067 and 119903 gt 119898+120583 The value of other parameters can be seen in Table 1

the initial time and repeating the processes as in (33)ndash(35)we can have that (119878(119905) 119864(119905) 119868(119905))

119879gt 0 for all 119905 gt 1198981120596

Thus (119878(119905) 119864(119905) 119868(119905)) isin 1198830 for all 119905 gt 1198981120596 There appearsa contradiction which means that the equality (37) holdsTherefore 1198640 is acyclic in 1205971198830 Obviously when 1198770 gt 1 and119903 gt 119898 + 120583 119874 is acyclic in 1205971198830

Furthermore by Lemma 5 1198640 = (1198730 0 0) and 119874 =

(0 0 0) are isolated invariant sets in 119883 119882119878(1198640) cap 1198830 = 0and 119882

119878(119874) cap 1198830 = 0 By Theorem 131 and Remark 131

in [46] it can be obtained that 119875 is uniformly persistent withrespect to (1198830 1205971198830) that is there exists a 120575 gt 0 such that anysolution (119878(119905) 119864(119905) 119868(119905)) of system (11) with the initial value(119878(0) 119864(0) 119868(0)) isin (119878 119864 119868) isin 119883 119864 gt 0 119868 gt 0 satisfies

lim inf119905rarrinfin

119864 (119905) ge 120575 lim inf119905rarrinfin

119868 (119905) ge 120575 (38)

ApplyingTheorem 136 in [46] 119875 has a fixed point

(119878lowast(0) 119864

lowast(0) 119868lowast(0)) isin 1198830 (39)

From (33) we know 119878lowast

gt 0 for all 119905 isin [0 120596] 119878lowast(119905) is alsomore than zero for all 119905 gt 0 due to the periodicity Similarlyfor all 119905 ge 0119864lowast(119905) gt 0 119868lowast(119905) gt 0Therefore it can be obtainedthat one of the positive 120596-periodic solutions of system (11) is(119878lowast(119905) 119864lowast(119905) 119868lowast(119905))

34 Numerical Simulations Firstly we give some notationsIf 119892(119905) is a periodic function with period 120596 we define 119892 =

(1120596) int120596

0119892(119905)119889119905 119892119897 = min119905isin[0120596]119892(119905) 119892119906 = max119905isin[0120596]119892(119905) As

described in the previous section

1198771 (119905) = 1 +120583 (119898 + 120583) (119898 + 120590) (1198770 minus 1)

119898120573 (119905) (119898 + 120590 + 120583) (40)

So 119877119897

1= 1 + 120583(119898 + 120583)(119898 + 120590)(119877

119897

0minus 1)119898120573

119906(119898 + 120590 + 120583) 119877119906

1=

1 + 120583(119898 + 120583)(119898 + 120590)(119877119906

0minus 1)119898120573

119897(119898 + 120590 + 120583)

In this section we adopt 120573(119905) = 119886[1 + 119887 sin(12058711990510)]Then applying the numerical simulation to verify the abovesolution we give the following conclusion

(1) when 119877119889 lt 1 119874 is stable(2) when 119877119889 gt 1 and 1198770 lt 1 1198640 is stable see Figure 3(a)(3) when 1198770 gt 1 and 119903 gt 119898 + 120583 system (11) has at least

one positive periodic solution see Figure 3(b)

We can givemore results about the conditions of existenceof the positive periodic solution

(11015840) When 119877119889 gt 1 1198770 gt 1 and 119877119889 lt 119877

119906

1 119874 is stable see

Figures 4(a) and 4(b)(21015840) When 119877119889 gt 1 1198770 gt 1 and 119877119889 gt 119877

119906

1 system (11) has at

least one positive periodic solution see Figure 5

By numerical simulations we can give that the conditionswhich ensure the existence of positive periodic solution are1198770 gt 1 and 119903 gt 119898+120583 or 119877119889 gt 1 1198770 gt 1 and 119877119889 gt 119877

119906

1 In fact

119877119889 gt 1 1198770 gt 1 and 119877119889 gt 119877119906

1are the sufficient conditions for

1198770 gt 1 and 119903 gt 119898 + 120583 As a result the conditions 1198770 gt 1 and119903 gt 119898 + 120583 are broader

4 Discussion

This paper considers a logistic growth system whose birthprocess incorporates density-dependent effects This type ofmodel has a rich dynamical behavior and practical signif-icance By analyzing its equilibria and respective attractiveregion we find that the dynamical behavior of a disease will

8 The Scientific World Journal

25

2

15

1

05

00 05 1 15 2 25 3 35

times104

times104

S(0) = 25000 E(0) = 10000 I(0) = 24000

S(0) = 35000 E(0) = 20000 I(0) = 15000

S(0) = 15000 E(0) = 10000 I(0) = 20000

I(t)

S(t)

(a)

25

2

15

1

05

00 05 1 15 2 25 3 35

times104

times104

S(0) = 25000 E(0) = 10000 I(0) = 24000

S(0) = 35000 E(0) = 20000 I(0) = 15000

S(0) = 15000 E(0) = 10000 I(0) = 20000

I(t)

S(t)

(b)

Figure 4 Phase plane of 119878(119905) and 119868(119905) (a) When the parameter values are 119903 = 011 119886 = 09 119887 = 02 and 120583 = 03 1198770 = 14991 119877119889 = 11 gt 1119877119889 lt 119877

119906

1= 14159 and 119877119889 lt 119877

119897

1= 12773 (b) When the parameter values are 119903 = 0126 119886 = 07 119887 = 02 and 120583 = 03 1198770 = 21067

119877119889 = 126 gt 1 119877119889 lt 119877119906

1= 12964 and 119877119889 gt 119877

119897

1= 11976 The value of other parameters can be seen in Table 1

7000

6000

5000

4000

3000

2000

1000

005 1 15 2

times104

times104

S(0) = 22000 E(0) = 5000 I(0) = 1000

S(0) = 4000 E(0) = 2000 I(0) = 1000

S(0) = 10000 E(0) = 10000 I(0) = 6000

I(t)

S(t)

Figure 5 Phase plane of 119878(119905) and 119868(119905) When the parameter valuesare 119903 = 013 119886 = 036 and 119887 = 02 1198770 = 10834 119877119889 = 13 gt 1 119877119889 gt119877119906

1= 10435 and 119877119889 gt 119877

119897

1= 1029 The value of other parameters

can be seen in Table 1

be determined by two thresholds 1198770 and 119877119889 Only 1198770 gt 1

cannot promise the existence of the endemic equilibriumwhich also needs 119877119889 gt 1198771 When 1198770 gt 1 and 119877119889 lt 1198771the solutions of the system (1) will tend to the origin 119874 It

is caused by the phenomenon that the death number dueto disease cannot be supplemented by the birth numberpromptly Finally all people are infected and die outThe factinterpreted by this model is more reasonable Theoreticallywe prove the global asymptotic stability of the disease-free equilibrium and give respective attractive regions ofequilibria

Seasonally effective contact rate is themost common formwhich may be related to various factors and thus this paperstudies the corresponding non-autonomous system which isobtained by changing the constant transmission rate of theabove system into the periodic transmission rate For theperiodic systems their dynamical behaviors especially thebasic reproduction number have been investigated in depthby [41 47ndash55] which provide many methods that we canutilize For the obtained periodic model by analyzing theglobal asymptotic stability of the disease-free equilibrium andthe existence of positive periodic solution we have the similarresults as the autonomous system The dynamic behavior ofdisease will be decided by two conditions 1198770 gt 1 and 119903 gt

119898 + 120583 that show that when the disease is prevalent the birthrate should be larger than the death rate to guarantee thesustainable growth of population Otherwise the populationwill disappear In addition we will evaluate and comparethe basic reproduction number 1198770 and the average basicreproduction number 1198770 which has been adopted by [27 56ndash59] In this paper we can calculate the average reproductionnumber

1198770 =120573120590

(119898 + 120590) (119898 + 120583) (41)

The Scientific World Journal 9

where 120573 = (120) int20

0120573(119905)119889119905 When 119903 = 013 119886 = 03 and

119887 = 02 we know that 1198770 = 09029 and 1198770 = 1 When 119903 =

013 119896 = 100000 119886 = 036 119887 = 02 and 119898 = 01 120590 = 02120583 = 01 then 1198770 = 10834 and 1198770 = 12 In that sense it isconfirmed that the basic reproduction number 1198770 defined by[40] is more accurate than the average reproduction number1198770 which overestimates the risk of disease

It should be noted that we live in a spatial world and it is anatural phenomenon that a substance goes from high densityregions to low density regions As a result epidemic modelsshould include spatial effects In a further study we need toinvestigate spatial epidemic models with seasonal factors

Acknowledgments

The research was partially supported by the National NaturalScience Foundation of China under Grant nos 1130149011301491 11331009 11147015 11171314 11101251 and 11105024Natural Science Foundation of ShanXi Province Grant no2012021002-1 and the Opening Foundation of Institute ofInformation Economy Hangzhou Normal University Grantno PD12001003002003

References

[1] K Dietz and J A P Heesterbeek ldquoBernoulli was ahead ofmodern epidemiologyrdquo Nature vol 408 no 6812 pp 513ndash5142000

[2] K Dietz and J A P Heesterbeek ldquoDaniel Bernoullirsquos epidemio-logical model revisitedrdquoMathematical Biosciences vol 180 pp1ndash21 2002

[3] R Ross ldquoAn application of the theory of probabilities to thestudy of a priori pathometry (part I)rdquo Proceedings of the RoyalSociety A vol 92 no 638 pp 204ndash230 1916

[4] R Ross and H P Hudson ldquoAn application of the theory ofprobabilities to the study of a priori pathometry (part II)rdquoProceedings of the Royal Society A vol 93 no 650 pp 212ndash2251917

[5] W O Kermack and A G McKendrick ldquoA contribution to themathematical theory of epidemics (part I)rdquo Proceedings of theRoyal Society A vol 115 no 772 pp 700ndash721 1927

[6] W O Kermack and A G McKendrick ldquoA contribution to themathematical theory of epidemics (part II)rdquo Proceedings of theRoyal Society A vol 138 no 834 pp 55ndash83 1932

[7] A Tsoularis and JWallace ldquoAnalysis of logistic growthmodelsrdquoMathematical Biosciences vol 179 no 1 pp 21ndash55 2002

[8] I Nasell ldquoOn the quasi-stationary distribution of the stochasticlogistic epidemicrdquoMathematical Biosciences vol 156 no 1-2 pp21ndash40 1999

[9] H Fujikawa A Kai and SMorozumi ldquoA new logisticmodel forEscherichia coli growth at constant and dynamic temperaturesrdquoFood Microbiology vol 21 no 5 pp 501ndash509 2004

[10] L Berezansky and E Braverman ldquoOscillation properties of alogistic equation with several delaysrdquo Journal of MathematicalAnalysis and Applications vol 247 no 1 pp 110ndash125 2000

[11] M Tabata N Eshima and I Takagi ldquoThe nonlinear integro-partial differential equation describing the logistic growth ofhuman population with migrationrdquo Applied Mathematics andComputation vol 98 pp 169ndash183 1999

[12] L Korobenko andE Braverman ldquoOn logisticmodels with a car-rying capacity dependent diffusion stability of equilibria andcoexistence with a regularly diffusing populationrdquo NonlinearAnalysis Real World Applications vol 13 no 6 pp 2648ndash26582012

[13] Y Muroya ldquoGlobal attractivity for discrete models of nonau-tonomous logistic equationsrdquo Computers andMathematics withApplications vol 53 no 7 pp 1059ndash1073 2007

[14] S Invernizzi and K Terpin ldquoA generalized logistic model forphotosynthetic growthrdquo Ecological Modelling vol 94 no 2-3pp 231ndash242 1997

[15] ZMinW Bang-Jun and J Feng ldquoCoalmining citiesrsquo economicgrowthmechanism and sustainable development analysis basedon logistic dynamics modelrdquo Procedia Earth and PlanetaryScience vol 1 no 1 pp 1737ndash1743 2009

[16] L I Anita S Anita and V Arnautu ldquoGlobal behavior foran age-dependent population model with logistic term andperiodic vital ratesrdquoAppliedMathematics andComputation vol206 no 1 pp 368ndash379 2008

[17] Z A Banaszak X Q Tang S C Wang and M B ZarembaldquoLogistics models in flexible manufacturingrdquo Computers inIndustry vol 43 no 3 pp 237ndash248 2000

[18] S Brianzoni C Mammana and E Michetti ldquoNonlineardynamics in a business-cycle model with logistic populationgrowthrdquo Chaos Solitons and Fractals vol 40 no 2 pp 717ndash7302009

[19] W P London and J A Yorke ldquoRecurrent outbreaks of measleschickenpox and mumps I Seasonal variation in contact ratesrdquoThe American Journal of Epidemiology vol 98 no 6 pp 453ndash468 1973

[20] S F Dowell ldquoSeasonal variation in host susceptibility and cyclesof certain infectious diseasesrdquo Emerging Infectious Diseases vol7 no 3 pp 369ndash374 2001

[21] ON Bjoslashrnstad B F Finkenstadt andB TGrenfell ldquoDynamicsof measles epidemics estimating scaling of transmission ratesusing a Time series SIR modelrdquo Ecological Monographs vol 72no 2 pp 169ndash184 2002

[22] J Zhang Z Jin G Q Sun X Sun and S Ruan ldquoModelingseasonal rabies epidemics in Chinardquo Bulletin of MathematicalBiology vol 74 no 5 pp 1226ndash1251 2012

[23] I B Schwartz ldquoSmall amplitude long period outbreaks inseasonally driven epidemicsrdquo Journal of Mathematical Biologyvol 30 no 5 pp 473ndash491 1992

[24] I B Schwartz and H L Smith ldquoInfinite subharmonic bifur-cation in an SEIR epidemic modelrdquo Journal of MathematicalBiology vol 18 no 3 pp 233ndash253 1983

[25] H L Smith ldquoMultiple stable subharmonics for a periodicepidemic modelrdquo Journal of Mathematical Biology vol 17 no2 pp 179ndash190 1983

[26] J Dushoff J B Plotkin S A Levin and D J D EarnldquoDynamical resonance can account for seasonality of influenzaepidemicsrdquo Proceedings of the National Academy of Sciences ofthe United States of America vol 101 no 48 pp 16915ndash169162004

[27] J L Ma and Z Ma ldquoEpidemic threshold conditions forseasonally forced SEIR modelsrdquo Mathematical Biosciences andEngineering vol 3 no 1 pp 161ndash172 2006

[28] D J D Earn P Rohani B M Bolker and B T Grenfell ldquoAsimple model for complex dynamical transitions in epidemicsrdquoScience vol 287 no 5453 pp 667ndash670 2000

10 The Scientific World Journal

[29] M J Keeling P Rohani and B T Grenfell ldquoSeasonally forceddisease dynamics explored as switching between attractorsrdquoPhysica D vol 148 no 3-4 pp 317ndash335 2001

[30] J L Aron and I B Schwartz ldquoSeasonality and period-doublingbifurcations in an epidemic modelrdquo Journal of TheoreticalBiology vol 110 no 4 pp 665ndash679 1984

[31] N C Grassly and C Fraser ldquoSeasonal infectious diseaseepidemiologyrdquo Proceedings of the Royal Society B vol 273 no1600 pp 2541ndash2550 2006

[32] J Zhang Z Jin G Q Sun T Zhou and S Ruan ldquoAnalysisof rabies in China transmission dynamics and controlrdquo PLoSONE vol 6 no 7 Article ID e20891 2011

[33] O Diekmann J A P Heesterbeek and J A Metz ldquoOnthe definition and the computation of the basic reproductionratio R0 in models for infectious diseases in heterogeneouspopulationsrdquo Journal of mathematical biology vol 28 no 4 pp365ndash382 1990

[34] O Diekmann J A P Heesterbeek and M G Roberts ldquoTheconstruction of next-generation matrices for compartmentalepidemic modelsrdquo Journal of the Royal Society Interface vol 7no 47 pp 873ndash885 2010

[35] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[36] F Berezovsky G Karev B J Song and C C Chavez ldquoA simpleepidemic model with surprising dynamicsrdquo Mathematical Bio-sciences and Engineering vol 2 pp 133ndash152 2005

[37] J LaSalle and S Lefschetz Stability by Liapunovrsquos DirectMethodAcademic Press New York NY USA 1961

[38] E A Barbashin Introduction to theTheory of Stability Wolters-Noordhoff Groningen The Netherlands 1970

[39] D Schenzle ldquoAn age-structured model of pre- and post-vaccination measles transmissionrdquoMathematical Medicine andBiology vol 1 no 2 pp 169ndash191 1984

[40] N Bacaer and S Guernaoui ldquoThe epidemic threshold ofvector-borne diseases with seasonality the case of cutaneousleishmaniasis in Chichaoua Moroccordquo Journal of MathematicalBiology vol 53 no 3 pp 421ndash436 2006

[41] W DWang and X Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008

[42] F Zhang and X Q Zhao ldquoA periodic epidemic model in apatchy environmentrdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 496ndash516 2007

[43] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press New York NY USA 1995

[44] H R Thieme ldquoConvergence results and a Poincare-Bendixsontrichotomy for asymptotically automous differential equationsrdquoJournal of Mathematical Biology vol 30 pp 755ndash763 1992

[45] L Perko Differential Equations and Dynamical SystemsSpringer New York NY USA 2000

[46] X Q Zhao Dynamical Systems in Population Biology SpringerNew York NY USA 2003

[47] Z G Bai and Y C Zhou ldquoThreshold dynamics of a bacillarydysentery model with seasonal fluctuationrdquo Discrete and Con-tinuous Dynamical Systems B vol 15 no 1 pp 1ndash14 2011

[48] Z G Bai Y C Zhou and T L Zhang ldquoExistence of multipleperiodic solutions for an SIRmodel with seasonalityrdquoNonlinear

Analysis Theory Methods and Applications vol 74 no 11 pp3548ndash3555 2011

[49] N Bacaer ldquoApproximation of the basic reproduction numberR0 for vector-borne diseases with a periodic vector populationrdquoBulletin of Mathematical Biology vol 69 no 3 pp 1067ndash10912007

[50] N Bacaer and X Abdurahman ldquoResonance of the epidemicthreshold in a periodic environmentrdquo Journal of MathematicalBiology vol 57 no 5 pp 649ndash673 2008

[51] N Bacaer ldquoPeriodic matrix population models growth ratebasic reproduction number and entropyrdquoBulletin ofMathemat-ical Biology vol 71 no 7 pp 1781ndash1792 2009

[52] N Bacaer and E H A Dads ldquoGenealogy with seasonalitythe basic reproduction number and the influenza pandemicrdquoJournal of Mathematical Biology vol 62 no 5 pp 741ndash762 2011

[53] L J Liu X Q Zhao and Y C Zhou ldquoA tuberculosis model withseasonalityrdquo Bulletin of Mathematical Biology vol 72 no 4 pp931ndash952 2010

[54] J L Liu ldquoThreshold dynamics for a HFMD epidemic modelwith periodic transmission raterdquo Nonlinear Dynamics vol 64no 1-2 pp 89ndash95 2011

[55] Y Nakata and T Kuniya ldquoGlobal dynamics of a class ofSEIRS epidemic models in a periodic environmentrdquo Journal ofMathematical Analysis and Applications vol 363 no 1 pp 230ndash237 2010

[56] B GWilliams andC Dye ldquoInfectious disease persistence whentransmission varies seasonallyrdquo Mathematical Biosciences vol145 no 1 pp 77ndash88 1997

[57] I A Moneim ldquoThe effect of using different types of periodiccontact rate on the behaviour of infectious diseases a simula-tion studyrdquo Computers in Biology and Medicine vol 37 no 11pp 1582ndash1590 2007

[58] C L Wesley and L J S Allen ldquoThe basic reproduction numberin epidemic models with periodic demographicsrdquo Journal ofBiological Dynamics vol 3 no 2-3 pp 116ndash129 2009

[59] D Greenhalgh and I A Moneim ldquoSIRS epidemic model andsimulations using different types of seasonal contact raterdquoSystems Analysis Modelling Simulation vol 43 no 5 pp 573ndash600 2003

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

The Scientific World Journal 7

25

2

15

1

05

01 15 2 25 3 35

times104

times104

S(0) = 25000 E(0) = 10000 I(0) = 24000

S(0) = 35000 E(0) = 20000 I(0) = 15000

S(0) = 15000 E(0) = 10000 I(0) = 20000

I(t)

S(t)

(a)

24

23

22

21

2

19

18

17

16

15

1415 2 25 3 35

times104

times104

S(0) = 25000 E(0) = 10000 I(0) = 24000

S(0) = 35000 E(0) = 20000 I(0) = 15000

S(0) = 15000 E(0) = 10000 I(0) = 20000

I(t)

S(t)

(b)

Figure 3 Phase plane of 119878(119905) and 119868(119905) (a) When the parameter values are 119903 = 013 119886 = 03 and 119887 = 02 1198770 = 09029 lt 1 119877119889 = 13 gt 1 (b)When the parameter values are 119903 = 03 119886 = 07 and 119887 = 02 1198770 = 21067 and 119903 gt 119898+120583 The value of other parameters can be seen in Table 1

the initial time and repeating the processes as in (33)ndash(35)we can have that (119878(119905) 119864(119905) 119868(119905))

119879gt 0 for all 119905 gt 1198981120596

Thus (119878(119905) 119864(119905) 119868(119905)) isin 1198830 for all 119905 gt 1198981120596 There appearsa contradiction which means that the equality (37) holdsTherefore 1198640 is acyclic in 1205971198830 Obviously when 1198770 gt 1 and119903 gt 119898 + 120583 119874 is acyclic in 1205971198830

Furthermore by Lemma 5 1198640 = (1198730 0 0) and 119874 =

(0 0 0) are isolated invariant sets in 119883 119882119878(1198640) cap 1198830 = 0and 119882

119878(119874) cap 1198830 = 0 By Theorem 131 and Remark 131

in [46] it can be obtained that 119875 is uniformly persistent withrespect to (1198830 1205971198830) that is there exists a 120575 gt 0 such that anysolution (119878(119905) 119864(119905) 119868(119905)) of system (11) with the initial value(119878(0) 119864(0) 119868(0)) isin (119878 119864 119868) isin 119883 119864 gt 0 119868 gt 0 satisfies

lim inf119905rarrinfin

119864 (119905) ge 120575 lim inf119905rarrinfin

119868 (119905) ge 120575 (38)

ApplyingTheorem 136 in [46] 119875 has a fixed point

(119878lowast(0) 119864

lowast(0) 119868lowast(0)) isin 1198830 (39)

From (33) we know 119878lowast

gt 0 for all 119905 isin [0 120596] 119878lowast(119905) is alsomore than zero for all 119905 gt 0 due to the periodicity Similarlyfor all 119905 ge 0119864lowast(119905) gt 0 119868lowast(119905) gt 0Therefore it can be obtainedthat one of the positive 120596-periodic solutions of system (11) is(119878lowast(119905) 119864lowast(119905) 119868lowast(119905))

34 Numerical Simulations Firstly we give some notationsIf 119892(119905) is a periodic function with period 120596 we define 119892 =

(1120596) int120596

0119892(119905)119889119905 119892119897 = min119905isin[0120596]119892(119905) 119892119906 = max119905isin[0120596]119892(119905) As

described in the previous section

1198771 (119905) = 1 +120583 (119898 + 120583) (119898 + 120590) (1198770 minus 1)

119898120573 (119905) (119898 + 120590 + 120583) (40)

So 119877119897

1= 1 + 120583(119898 + 120583)(119898 + 120590)(119877

119897

0minus 1)119898120573

119906(119898 + 120590 + 120583) 119877119906

1=

1 + 120583(119898 + 120583)(119898 + 120590)(119877119906

0minus 1)119898120573

119897(119898 + 120590 + 120583)

In this section we adopt 120573(119905) = 119886[1 + 119887 sin(12058711990510)]Then applying the numerical simulation to verify the abovesolution we give the following conclusion

(1) when 119877119889 lt 1 119874 is stable(2) when 119877119889 gt 1 and 1198770 lt 1 1198640 is stable see Figure 3(a)(3) when 1198770 gt 1 and 119903 gt 119898 + 120583 system (11) has at least

one positive periodic solution see Figure 3(b)

We can givemore results about the conditions of existenceof the positive periodic solution

(11015840) When 119877119889 gt 1 1198770 gt 1 and 119877119889 lt 119877

119906

1 119874 is stable see

Figures 4(a) and 4(b)(21015840) When 119877119889 gt 1 1198770 gt 1 and 119877119889 gt 119877

119906

1 system (11) has at

least one positive periodic solution see Figure 5

By numerical simulations we can give that the conditionswhich ensure the existence of positive periodic solution are1198770 gt 1 and 119903 gt 119898+120583 or 119877119889 gt 1 1198770 gt 1 and 119877119889 gt 119877

119906

1 In fact

119877119889 gt 1 1198770 gt 1 and 119877119889 gt 119877119906

1are the sufficient conditions for

1198770 gt 1 and 119903 gt 119898 + 120583 As a result the conditions 1198770 gt 1 and119903 gt 119898 + 120583 are broader

4 Discussion

This paper considers a logistic growth system whose birthprocess incorporates density-dependent effects This type ofmodel has a rich dynamical behavior and practical signif-icance By analyzing its equilibria and respective attractiveregion we find that the dynamical behavior of a disease will

8 The Scientific World Journal

25

2

15

1

05

00 05 1 15 2 25 3 35

times104

times104

S(0) = 25000 E(0) = 10000 I(0) = 24000

S(0) = 35000 E(0) = 20000 I(0) = 15000

S(0) = 15000 E(0) = 10000 I(0) = 20000

I(t)

S(t)

(a)

25

2

15

1

05

00 05 1 15 2 25 3 35

times104

times104

S(0) = 25000 E(0) = 10000 I(0) = 24000

S(0) = 35000 E(0) = 20000 I(0) = 15000

S(0) = 15000 E(0) = 10000 I(0) = 20000

I(t)

S(t)

(b)

Figure 4 Phase plane of 119878(119905) and 119868(119905) (a) When the parameter values are 119903 = 011 119886 = 09 119887 = 02 and 120583 = 03 1198770 = 14991 119877119889 = 11 gt 1119877119889 lt 119877

119906

1= 14159 and 119877119889 lt 119877

119897

1= 12773 (b) When the parameter values are 119903 = 0126 119886 = 07 119887 = 02 and 120583 = 03 1198770 = 21067

119877119889 = 126 gt 1 119877119889 lt 119877119906

1= 12964 and 119877119889 gt 119877

119897

1= 11976 The value of other parameters can be seen in Table 1

7000

6000

5000

4000

3000

2000

1000

005 1 15 2

times104

times104

S(0) = 22000 E(0) = 5000 I(0) = 1000

S(0) = 4000 E(0) = 2000 I(0) = 1000

S(0) = 10000 E(0) = 10000 I(0) = 6000

I(t)

S(t)

Figure 5 Phase plane of 119878(119905) and 119868(119905) When the parameter valuesare 119903 = 013 119886 = 036 and 119887 = 02 1198770 = 10834 119877119889 = 13 gt 1 119877119889 gt119877119906

1= 10435 and 119877119889 gt 119877

119897

1= 1029 The value of other parameters

can be seen in Table 1

be determined by two thresholds 1198770 and 119877119889 Only 1198770 gt 1

cannot promise the existence of the endemic equilibriumwhich also needs 119877119889 gt 1198771 When 1198770 gt 1 and 119877119889 lt 1198771the solutions of the system (1) will tend to the origin 119874 It

is caused by the phenomenon that the death number dueto disease cannot be supplemented by the birth numberpromptly Finally all people are infected and die outThe factinterpreted by this model is more reasonable Theoreticallywe prove the global asymptotic stability of the disease-free equilibrium and give respective attractive regions ofequilibria

Seasonally effective contact rate is themost common formwhich may be related to various factors and thus this paperstudies the corresponding non-autonomous system which isobtained by changing the constant transmission rate of theabove system into the periodic transmission rate For theperiodic systems their dynamical behaviors especially thebasic reproduction number have been investigated in depthby [41 47ndash55] which provide many methods that we canutilize For the obtained periodic model by analyzing theglobal asymptotic stability of the disease-free equilibrium andthe existence of positive periodic solution we have the similarresults as the autonomous system The dynamic behavior ofdisease will be decided by two conditions 1198770 gt 1 and 119903 gt

119898 + 120583 that show that when the disease is prevalent the birthrate should be larger than the death rate to guarantee thesustainable growth of population Otherwise the populationwill disappear In addition we will evaluate and comparethe basic reproduction number 1198770 and the average basicreproduction number 1198770 which has been adopted by [27 56ndash59] In this paper we can calculate the average reproductionnumber

1198770 =120573120590

(119898 + 120590) (119898 + 120583) (41)

The Scientific World Journal 9

where 120573 = (120) int20

0120573(119905)119889119905 When 119903 = 013 119886 = 03 and

119887 = 02 we know that 1198770 = 09029 and 1198770 = 1 When 119903 =

013 119896 = 100000 119886 = 036 119887 = 02 and 119898 = 01 120590 = 02120583 = 01 then 1198770 = 10834 and 1198770 = 12 In that sense it isconfirmed that the basic reproduction number 1198770 defined by[40] is more accurate than the average reproduction number1198770 which overestimates the risk of disease

It should be noted that we live in a spatial world and it is anatural phenomenon that a substance goes from high densityregions to low density regions As a result epidemic modelsshould include spatial effects In a further study we need toinvestigate spatial epidemic models with seasonal factors

Acknowledgments

The research was partially supported by the National NaturalScience Foundation of China under Grant nos 1130149011301491 11331009 11147015 11171314 11101251 and 11105024Natural Science Foundation of ShanXi Province Grant no2012021002-1 and the Opening Foundation of Institute ofInformation Economy Hangzhou Normal University Grantno PD12001003002003

References

[1] K Dietz and J A P Heesterbeek ldquoBernoulli was ahead ofmodern epidemiologyrdquo Nature vol 408 no 6812 pp 513ndash5142000

[2] K Dietz and J A P Heesterbeek ldquoDaniel Bernoullirsquos epidemio-logical model revisitedrdquoMathematical Biosciences vol 180 pp1ndash21 2002

[3] R Ross ldquoAn application of the theory of probabilities to thestudy of a priori pathometry (part I)rdquo Proceedings of the RoyalSociety A vol 92 no 638 pp 204ndash230 1916

[4] R Ross and H P Hudson ldquoAn application of the theory ofprobabilities to the study of a priori pathometry (part II)rdquoProceedings of the Royal Society A vol 93 no 650 pp 212ndash2251917

[5] W O Kermack and A G McKendrick ldquoA contribution to themathematical theory of epidemics (part I)rdquo Proceedings of theRoyal Society A vol 115 no 772 pp 700ndash721 1927

[6] W O Kermack and A G McKendrick ldquoA contribution to themathematical theory of epidemics (part II)rdquo Proceedings of theRoyal Society A vol 138 no 834 pp 55ndash83 1932

[7] A Tsoularis and JWallace ldquoAnalysis of logistic growthmodelsrdquoMathematical Biosciences vol 179 no 1 pp 21ndash55 2002

[8] I Nasell ldquoOn the quasi-stationary distribution of the stochasticlogistic epidemicrdquoMathematical Biosciences vol 156 no 1-2 pp21ndash40 1999

[9] H Fujikawa A Kai and SMorozumi ldquoA new logisticmodel forEscherichia coli growth at constant and dynamic temperaturesrdquoFood Microbiology vol 21 no 5 pp 501ndash509 2004

[10] L Berezansky and E Braverman ldquoOscillation properties of alogistic equation with several delaysrdquo Journal of MathematicalAnalysis and Applications vol 247 no 1 pp 110ndash125 2000

[11] M Tabata N Eshima and I Takagi ldquoThe nonlinear integro-partial differential equation describing the logistic growth ofhuman population with migrationrdquo Applied Mathematics andComputation vol 98 pp 169ndash183 1999

[12] L Korobenko andE Braverman ldquoOn logisticmodels with a car-rying capacity dependent diffusion stability of equilibria andcoexistence with a regularly diffusing populationrdquo NonlinearAnalysis Real World Applications vol 13 no 6 pp 2648ndash26582012

[13] Y Muroya ldquoGlobal attractivity for discrete models of nonau-tonomous logistic equationsrdquo Computers andMathematics withApplications vol 53 no 7 pp 1059ndash1073 2007

[14] S Invernizzi and K Terpin ldquoA generalized logistic model forphotosynthetic growthrdquo Ecological Modelling vol 94 no 2-3pp 231ndash242 1997

[15] ZMinW Bang-Jun and J Feng ldquoCoalmining citiesrsquo economicgrowthmechanism and sustainable development analysis basedon logistic dynamics modelrdquo Procedia Earth and PlanetaryScience vol 1 no 1 pp 1737ndash1743 2009

[16] L I Anita S Anita and V Arnautu ldquoGlobal behavior foran age-dependent population model with logistic term andperiodic vital ratesrdquoAppliedMathematics andComputation vol206 no 1 pp 368ndash379 2008

[17] Z A Banaszak X Q Tang S C Wang and M B ZarembaldquoLogistics models in flexible manufacturingrdquo Computers inIndustry vol 43 no 3 pp 237ndash248 2000

[18] S Brianzoni C Mammana and E Michetti ldquoNonlineardynamics in a business-cycle model with logistic populationgrowthrdquo Chaos Solitons and Fractals vol 40 no 2 pp 717ndash7302009

[19] W P London and J A Yorke ldquoRecurrent outbreaks of measleschickenpox and mumps I Seasonal variation in contact ratesrdquoThe American Journal of Epidemiology vol 98 no 6 pp 453ndash468 1973

[20] S F Dowell ldquoSeasonal variation in host susceptibility and cyclesof certain infectious diseasesrdquo Emerging Infectious Diseases vol7 no 3 pp 369ndash374 2001

[21] ON Bjoslashrnstad B F Finkenstadt andB TGrenfell ldquoDynamicsof measles epidemics estimating scaling of transmission ratesusing a Time series SIR modelrdquo Ecological Monographs vol 72no 2 pp 169ndash184 2002

[22] J Zhang Z Jin G Q Sun X Sun and S Ruan ldquoModelingseasonal rabies epidemics in Chinardquo Bulletin of MathematicalBiology vol 74 no 5 pp 1226ndash1251 2012

[23] I B Schwartz ldquoSmall amplitude long period outbreaks inseasonally driven epidemicsrdquo Journal of Mathematical Biologyvol 30 no 5 pp 473ndash491 1992

[24] I B Schwartz and H L Smith ldquoInfinite subharmonic bifur-cation in an SEIR epidemic modelrdquo Journal of MathematicalBiology vol 18 no 3 pp 233ndash253 1983

[25] H L Smith ldquoMultiple stable subharmonics for a periodicepidemic modelrdquo Journal of Mathematical Biology vol 17 no2 pp 179ndash190 1983

[26] J Dushoff J B Plotkin S A Levin and D J D EarnldquoDynamical resonance can account for seasonality of influenzaepidemicsrdquo Proceedings of the National Academy of Sciences ofthe United States of America vol 101 no 48 pp 16915ndash169162004

[27] J L Ma and Z Ma ldquoEpidemic threshold conditions forseasonally forced SEIR modelsrdquo Mathematical Biosciences andEngineering vol 3 no 1 pp 161ndash172 2006

[28] D J D Earn P Rohani B M Bolker and B T Grenfell ldquoAsimple model for complex dynamical transitions in epidemicsrdquoScience vol 287 no 5453 pp 667ndash670 2000

10 The Scientific World Journal

[29] M J Keeling P Rohani and B T Grenfell ldquoSeasonally forceddisease dynamics explored as switching between attractorsrdquoPhysica D vol 148 no 3-4 pp 317ndash335 2001

[30] J L Aron and I B Schwartz ldquoSeasonality and period-doublingbifurcations in an epidemic modelrdquo Journal of TheoreticalBiology vol 110 no 4 pp 665ndash679 1984

[31] N C Grassly and C Fraser ldquoSeasonal infectious diseaseepidemiologyrdquo Proceedings of the Royal Society B vol 273 no1600 pp 2541ndash2550 2006

[32] J Zhang Z Jin G Q Sun T Zhou and S Ruan ldquoAnalysisof rabies in China transmission dynamics and controlrdquo PLoSONE vol 6 no 7 Article ID e20891 2011

[33] O Diekmann J A P Heesterbeek and J A Metz ldquoOnthe definition and the computation of the basic reproductionratio R0 in models for infectious diseases in heterogeneouspopulationsrdquo Journal of mathematical biology vol 28 no 4 pp365ndash382 1990

[34] O Diekmann J A P Heesterbeek and M G Roberts ldquoTheconstruction of next-generation matrices for compartmentalepidemic modelsrdquo Journal of the Royal Society Interface vol 7no 47 pp 873ndash885 2010

[35] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[36] F Berezovsky G Karev B J Song and C C Chavez ldquoA simpleepidemic model with surprising dynamicsrdquo Mathematical Bio-sciences and Engineering vol 2 pp 133ndash152 2005

[37] J LaSalle and S Lefschetz Stability by Liapunovrsquos DirectMethodAcademic Press New York NY USA 1961

[38] E A Barbashin Introduction to theTheory of Stability Wolters-Noordhoff Groningen The Netherlands 1970

[39] D Schenzle ldquoAn age-structured model of pre- and post-vaccination measles transmissionrdquoMathematical Medicine andBiology vol 1 no 2 pp 169ndash191 1984

[40] N Bacaer and S Guernaoui ldquoThe epidemic threshold ofvector-borne diseases with seasonality the case of cutaneousleishmaniasis in Chichaoua Moroccordquo Journal of MathematicalBiology vol 53 no 3 pp 421ndash436 2006

[41] W DWang and X Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008

[42] F Zhang and X Q Zhao ldquoA periodic epidemic model in apatchy environmentrdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 496ndash516 2007

[43] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press New York NY USA 1995

[44] H R Thieme ldquoConvergence results and a Poincare-Bendixsontrichotomy for asymptotically automous differential equationsrdquoJournal of Mathematical Biology vol 30 pp 755ndash763 1992

[45] L Perko Differential Equations and Dynamical SystemsSpringer New York NY USA 2000

[46] X Q Zhao Dynamical Systems in Population Biology SpringerNew York NY USA 2003

[47] Z G Bai and Y C Zhou ldquoThreshold dynamics of a bacillarydysentery model with seasonal fluctuationrdquo Discrete and Con-tinuous Dynamical Systems B vol 15 no 1 pp 1ndash14 2011

[48] Z G Bai Y C Zhou and T L Zhang ldquoExistence of multipleperiodic solutions for an SIRmodel with seasonalityrdquoNonlinear

Analysis Theory Methods and Applications vol 74 no 11 pp3548ndash3555 2011

[49] N Bacaer ldquoApproximation of the basic reproduction numberR0 for vector-borne diseases with a periodic vector populationrdquoBulletin of Mathematical Biology vol 69 no 3 pp 1067ndash10912007

[50] N Bacaer and X Abdurahman ldquoResonance of the epidemicthreshold in a periodic environmentrdquo Journal of MathematicalBiology vol 57 no 5 pp 649ndash673 2008

[51] N Bacaer ldquoPeriodic matrix population models growth ratebasic reproduction number and entropyrdquoBulletin ofMathemat-ical Biology vol 71 no 7 pp 1781ndash1792 2009

[52] N Bacaer and E H A Dads ldquoGenealogy with seasonalitythe basic reproduction number and the influenza pandemicrdquoJournal of Mathematical Biology vol 62 no 5 pp 741ndash762 2011

[53] L J Liu X Q Zhao and Y C Zhou ldquoA tuberculosis model withseasonalityrdquo Bulletin of Mathematical Biology vol 72 no 4 pp931ndash952 2010

[54] J L Liu ldquoThreshold dynamics for a HFMD epidemic modelwith periodic transmission raterdquo Nonlinear Dynamics vol 64no 1-2 pp 89ndash95 2011

[55] Y Nakata and T Kuniya ldquoGlobal dynamics of a class ofSEIRS epidemic models in a periodic environmentrdquo Journal ofMathematical Analysis and Applications vol 363 no 1 pp 230ndash237 2010

[56] B GWilliams andC Dye ldquoInfectious disease persistence whentransmission varies seasonallyrdquo Mathematical Biosciences vol145 no 1 pp 77ndash88 1997

[57] I A Moneim ldquoThe effect of using different types of periodiccontact rate on the behaviour of infectious diseases a simula-tion studyrdquo Computers in Biology and Medicine vol 37 no 11pp 1582ndash1590 2007

[58] C L Wesley and L J S Allen ldquoThe basic reproduction numberin epidemic models with periodic demographicsrdquo Journal ofBiological Dynamics vol 3 no 2-3 pp 116ndash129 2009

[59] D Greenhalgh and I A Moneim ldquoSIRS epidemic model andsimulations using different types of seasonal contact raterdquoSystems Analysis Modelling Simulation vol 43 no 5 pp 573ndash600 2003

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 The Scientific World Journal

25

2

15

1

05

00 05 1 15 2 25 3 35

times104

times104

S(0) = 25000 E(0) = 10000 I(0) = 24000

S(0) = 35000 E(0) = 20000 I(0) = 15000

S(0) = 15000 E(0) = 10000 I(0) = 20000

I(t)

S(t)

(a)

25

2

15

1

05

00 05 1 15 2 25 3 35

times104

times104

S(0) = 25000 E(0) = 10000 I(0) = 24000

S(0) = 35000 E(0) = 20000 I(0) = 15000

S(0) = 15000 E(0) = 10000 I(0) = 20000

I(t)

S(t)

(b)

Figure 4 Phase plane of 119878(119905) and 119868(119905) (a) When the parameter values are 119903 = 011 119886 = 09 119887 = 02 and 120583 = 03 1198770 = 14991 119877119889 = 11 gt 1119877119889 lt 119877

119906

1= 14159 and 119877119889 lt 119877

119897

1= 12773 (b) When the parameter values are 119903 = 0126 119886 = 07 119887 = 02 and 120583 = 03 1198770 = 21067

119877119889 = 126 gt 1 119877119889 lt 119877119906

1= 12964 and 119877119889 gt 119877

119897

1= 11976 The value of other parameters can be seen in Table 1

7000

6000

5000

4000

3000

2000

1000

005 1 15 2

times104

times104

S(0) = 22000 E(0) = 5000 I(0) = 1000

S(0) = 4000 E(0) = 2000 I(0) = 1000

S(0) = 10000 E(0) = 10000 I(0) = 6000

I(t)

S(t)

Figure 5 Phase plane of 119878(119905) and 119868(119905) When the parameter valuesare 119903 = 013 119886 = 036 and 119887 = 02 1198770 = 10834 119877119889 = 13 gt 1 119877119889 gt119877119906

1= 10435 and 119877119889 gt 119877

119897

1= 1029 The value of other parameters

can be seen in Table 1

be determined by two thresholds 1198770 and 119877119889 Only 1198770 gt 1

cannot promise the existence of the endemic equilibriumwhich also needs 119877119889 gt 1198771 When 1198770 gt 1 and 119877119889 lt 1198771the solutions of the system (1) will tend to the origin 119874 It

is caused by the phenomenon that the death number dueto disease cannot be supplemented by the birth numberpromptly Finally all people are infected and die outThe factinterpreted by this model is more reasonable Theoreticallywe prove the global asymptotic stability of the disease-free equilibrium and give respective attractive regions ofequilibria

Seasonally effective contact rate is themost common formwhich may be related to various factors and thus this paperstudies the corresponding non-autonomous system which isobtained by changing the constant transmission rate of theabove system into the periodic transmission rate For theperiodic systems their dynamical behaviors especially thebasic reproduction number have been investigated in depthby [41 47ndash55] which provide many methods that we canutilize For the obtained periodic model by analyzing theglobal asymptotic stability of the disease-free equilibrium andthe existence of positive periodic solution we have the similarresults as the autonomous system The dynamic behavior ofdisease will be decided by two conditions 1198770 gt 1 and 119903 gt

119898 + 120583 that show that when the disease is prevalent the birthrate should be larger than the death rate to guarantee thesustainable growth of population Otherwise the populationwill disappear In addition we will evaluate and comparethe basic reproduction number 1198770 and the average basicreproduction number 1198770 which has been adopted by [27 56ndash59] In this paper we can calculate the average reproductionnumber

1198770 =120573120590

(119898 + 120590) (119898 + 120583) (41)

The Scientific World Journal 9

where 120573 = (120) int20

0120573(119905)119889119905 When 119903 = 013 119886 = 03 and

119887 = 02 we know that 1198770 = 09029 and 1198770 = 1 When 119903 =

013 119896 = 100000 119886 = 036 119887 = 02 and 119898 = 01 120590 = 02120583 = 01 then 1198770 = 10834 and 1198770 = 12 In that sense it isconfirmed that the basic reproduction number 1198770 defined by[40] is more accurate than the average reproduction number1198770 which overestimates the risk of disease

It should be noted that we live in a spatial world and it is anatural phenomenon that a substance goes from high densityregions to low density regions As a result epidemic modelsshould include spatial effects In a further study we need toinvestigate spatial epidemic models with seasonal factors

Acknowledgments

The research was partially supported by the National NaturalScience Foundation of China under Grant nos 1130149011301491 11331009 11147015 11171314 11101251 and 11105024Natural Science Foundation of ShanXi Province Grant no2012021002-1 and the Opening Foundation of Institute ofInformation Economy Hangzhou Normal University Grantno PD12001003002003

References

[1] K Dietz and J A P Heesterbeek ldquoBernoulli was ahead ofmodern epidemiologyrdquo Nature vol 408 no 6812 pp 513ndash5142000

[2] K Dietz and J A P Heesterbeek ldquoDaniel Bernoullirsquos epidemio-logical model revisitedrdquoMathematical Biosciences vol 180 pp1ndash21 2002

[3] R Ross ldquoAn application of the theory of probabilities to thestudy of a priori pathometry (part I)rdquo Proceedings of the RoyalSociety A vol 92 no 638 pp 204ndash230 1916

[4] R Ross and H P Hudson ldquoAn application of the theory ofprobabilities to the study of a priori pathometry (part II)rdquoProceedings of the Royal Society A vol 93 no 650 pp 212ndash2251917

[5] W O Kermack and A G McKendrick ldquoA contribution to themathematical theory of epidemics (part I)rdquo Proceedings of theRoyal Society A vol 115 no 772 pp 700ndash721 1927

[6] W O Kermack and A G McKendrick ldquoA contribution to themathematical theory of epidemics (part II)rdquo Proceedings of theRoyal Society A vol 138 no 834 pp 55ndash83 1932

[7] A Tsoularis and JWallace ldquoAnalysis of logistic growthmodelsrdquoMathematical Biosciences vol 179 no 1 pp 21ndash55 2002

[8] I Nasell ldquoOn the quasi-stationary distribution of the stochasticlogistic epidemicrdquoMathematical Biosciences vol 156 no 1-2 pp21ndash40 1999

[9] H Fujikawa A Kai and SMorozumi ldquoA new logisticmodel forEscherichia coli growth at constant and dynamic temperaturesrdquoFood Microbiology vol 21 no 5 pp 501ndash509 2004

[10] L Berezansky and E Braverman ldquoOscillation properties of alogistic equation with several delaysrdquo Journal of MathematicalAnalysis and Applications vol 247 no 1 pp 110ndash125 2000

[11] M Tabata N Eshima and I Takagi ldquoThe nonlinear integro-partial differential equation describing the logistic growth ofhuman population with migrationrdquo Applied Mathematics andComputation vol 98 pp 169ndash183 1999

[12] L Korobenko andE Braverman ldquoOn logisticmodels with a car-rying capacity dependent diffusion stability of equilibria andcoexistence with a regularly diffusing populationrdquo NonlinearAnalysis Real World Applications vol 13 no 6 pp 2648ndash26582012

[13] Y Muroya ldquoGlobal attractivity for discrete models of nonau-tonomous logistic equationsrdquo Computers andMathematics withApplications vol 53 no 7 pp 1059ndash1073 2007

[14] S Invernizzi and K Terpin ldquoA generalized logistic model forphotosynthetic growthrdquo Ecological Modelling vol 94 no 2-3pp 231ndash242 1997

[15] ZMinW Bang-Jun and J Feng ldquoCoalmining citiesrsquo economicgrowthmechanism and sustainable development analysis basedon logistic dynamics modelrdquo Procedia Earth and PlanetaryScience vol 1 no 1 pp 1737ndash1743 2009

[16] L I Anita S Anita and V Arnautu ldquoGlobal behavior foran age-dependent population model with logistic term andperiodic vital ratesrdquoAppliedMathematics andComputation vol206 no 1 pp 368ndash379 2008

[17] Z A Banaszak X Q Tang S C Wang and M B ZarembaldquoLogistics models in flexible manufacturingrdquo Computers inIndustry vol 43 no 3 pp 237ndash248 2000

[18] S Brianzoni C Mammana and E Michetti ldquoNonlineardynamics in a business-cycle model with logistic populationgrowthrdquo Chaos Solitons and Fractals vol 40 no 2 pp 717ndash7302009

[19] W P London and J A Yorke ldquoRecurrent outbreaks of measleschickenpox and mumps I Seasonal variation in contact ratesrdquoThe American Journal of Epidemiology vol 98 no 6 pp 453ndash468 1973

[20] S F Dowell ldquoSeasonal variation in host susceptibility and cyclesof certain infectious diseasesrdquo Emerging Infectious Diseases vol7 no 3 pp 369ndash374 2001

[21] ON Bjoslashrnstad B F Finkenstadt andB TGrenfell ldquoDynamicsof measles epidemics estimating scaling of transmission ratesusing a Time series SIR modelrdquo Ecological Monographs vol 72no 2 pp 169ndash184 2002

[22] J Zhang Z Jin G Q Sun X Sun and S Ruan ldquoModelingseasonal rabies epidemics in Chinardquo Bulletin of MathematicalBiology vol 74 no 5 pp 1226ndash1251 2012

[23] I B Schwartz ldquoSmall amplitude long period outbreaks inseasonally driven epidemicsrdquo Journal of Mathematical Biologyvol 30 no 5 pp 473ndash491 1992

[24] I B Schwartz and H L Smith ldquoInfinite subharmonic bifur-cation in an SEIR epidemic modelrdquo Journal of MathematicalBiology vol 18 no 3 pp 233ndash253 1983

[25] H L Smith ldquoMultiple stable subharmonics for a periodicepidemic modelrdquo Journal of Mathematical Biology vol 17 no2 pp 179ndash190 1983

[26] J Dushoff J B Plotkin S A Levin and D J D EarnldquoDynamical resonance can account for seasonality of influenzaepidemicsrdquo Proceedings of the National Academy of Sciences ofthe United States of America vol 101 no 48 pp 16915ndash169162004

[27] J L Ma and Z Ma ldquoEpidemic threshold conditions forseasonally forced SEIR modelsrdquo Mathematical Biosciences andEngineering vol 3 no 1 pp 161ndash172 2006

[28] D J D Earn P Rohani B M Bolker and B T Grenfell ldquoAsimple model for complex dynamical transitions in epidemicsrdquoScience vol 287 no 5453 pp 667ndash670 2000

10 The Scientific World Journal

[29] M J Keeling P Rohani and B T Grenfell ldquoSeasonally forceddisease dynamics explored as switching between attractorsrdquoPhysica D vol 148 no 3-4 pp 317ndash335 2001

[30] J L Aron and I B Schwartz ldquoSeasonality and period-doublingbifurcations in an epidemic modelrdquo Journal of TheoreticalBiology vol 110 no 4 pp 665ndash679 1984

[31] N C Grassly and C Fraser ldquoSeasonal infectious diseaseepidemiologyrdquo Proceedings of the Royal Society B vol 273 no1600 pp 2541ndash2550 2006

[32] J Zhang Z Jin G Q Sun T Zhou and S Ruan ldquoAnalysisof rabies in China transmission dynamics and controlrdquo PLoSONE vol 6 no 7 Article ID e20891 2011

[33] O Diekmann J A P Heesterbeek and J A Metz ldquoOnthe definition and the computation of the basic reproductionratio R0 in models for infectious diseases in heterogeneouspopulationsrdquo Journal of mathematical biology vol 28 no 4 pp365ndash382 1990

[34] O Diekmann J A P Heesterbeek and M G Roberts ldquoTheconstruction of next-generation matrices for compartmentalepidemic modelsrdquo Journal of the Royal Society Interface vol 7no 47 pp 873ndash885 2010

[35] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[36] F Berezovsky G Karev B J Song and C C Chavez ldquoA simpleepidemic model with surprising dynamicsrdquo Mathematical Bio-sciences and Engineering vol 2 pp 133ndash152 2005

[37] J LaSalle and S Lefschetz Stability by Liapunovrsquos DirectMethodAcademic Press New York NY USA 1961

[38] E A Barbashin Introduction to theTheory of Stability Wolters-Noordhoff Groningen The Netherlands 1970

[39] D Schenzle ldquoAn age-structured model of pre- and post-vaccination measles transmissionrdquoMathematical Medicine andBiology vol 1 no 2 pp 169ndash191 1984

[40] N Bacaer and S Guernaoui ldquoThe epidemic threshold ofvector-borne diseases with seasonality the case of cutaneousleishmaniasis in Chichaoua Moroccordquo Journal of MathematicalBiology vol 53 no 3 pp 421ndash436 2006

[41] W DWang and X Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008

[42] F Zhang and X Q Zhao ldquoA periodic epidemic model in apatchy environmentrdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 496ndash516 2007

[43] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press New York NY USA 1995

[44] H R Thieme ldquoConvergence results and a Poincare-Bendixsontrichotomy for asymptotically automous differential equationsrdquoJournal of Mathematical Biology vol 30 pp 755ndash763 1992

[45] L Perko Differential Equations and Dynamical SystemsSpringer New York NY USA 2000

[46] X Q Zhao Dynamical Systems in Population Biology SpringerNew York NY USA 2003

[47] Z G Bai and Y C Zhou ldquoThreshold dynamics of a bacillarydysentery model with seasonal fluctuationrdquo Discrete and Con-tinuous Dynamical Systems B vol 15 no 1 pp 1ndash14 2011

[48] Z G Bai Y C Zhou and T L Zhang ldquoExistence of multipleperiodic solutions for an SIRmodel with seasonalityrdquoNonlinear

Analysis Theory Methods and Applications vol 74 no 11 pp3548ndash3555 2011

[49] N Bacaer ldquoApproximation of the basic reproduction numberR0 for vector-borne diseases with a periodic vector populationrdquoBulletin of Mathematical Biology vol 69 no 3 pp 1067ndash10912007

[50] N Bacaer and X Abdurahman ldquoResonance of the epidemicthreshold in a periodic environmentrdquo Journal of MathematicalBiology vol 57 no 5 pp 649ndash673 2008

[51] N Bacaer ldquoPeriodic matrix population models growth ratebasic reproduction number and entropyrdquoBulletin ofMathemat-ical Biology vol 71 no 7 pp 1781ndash1792 2009

[52] N Bacaer and E H A Dads ldquoGenealogy with seasonalitythe basic reproduction number and the influenza pandemicrdquoJournal of Mathematical Biology vol 62 no 5 pp 741ndash762 2011

[53] L J Liu X Q Zhao and Y C Zhou ldquoA tuberculosis model withseasonalityrdquo Bulletin of Mathematical Biology vol 72 no 4 pp931ndash952 2010

[54] J L Liu ldquoThreshold dynamics for a HFMD epidemic modelwith periodic transmission raterdquo Nonlinear Dynamics vol 64no 1-2 pp 89ndash95 2011

[55] Y Nakata and T Kuniya ldquoGlobal dynamics of a class ofSEIRS epidemic models in a periodic environmentrdquo Journal ofMathematical Analysis and Applications vol 363 no 1 pp 230ndash237 2010

[56] B GWilliams andC Dye ldquoInfectious disease persistence whentransmission varies seasonallyrdquo Mathematical Biosciences vol145 no 1 pp 77ndash88 1997

[57] I A Moneim ldquoThe effect of using different types of periodiccontact rate on the behaviour of infectious diseases a simula-tion studyrdquo Computers in Biology and Medicine vol 37 no 11pp 1582ndash1590 2007

[58] C L Wesley and L J S Allen ldquoThe basic reproduction numberin epidemic models with periodic demographicsrdquo Journal ofBiological Dynamics vol 3 no 2-3 pp 116ndash129 2009

[59] D Greenhalgh and I A Moneim ldquoSIRS epidemic model andsimulations using different types of seasonal contact raterdquoSystems Analysis Modelling Simulation vol 43 no 5 pp 573ndash600 2003

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

The Scientific World Journal 9

where 120573 = (120) int20

0120573(119905)119889119905 When 119903 = 013 119886 = 03 and

119887 = 02 we know that 1198770 = 09029 and 1198770 = 1 When 119903 =

013 119896 = 100000 119886 = 036 119887 = 02 and 119898 = 01 120590 = 02120583 = 01 then 1198770 = 10834 and 1198770 = 12 In that sense it isconfirmed that the basic reproduction number 1198770 defined by[40] is more accurate than the average reproduction number1198770 which overestimates the risk of disease

It should be noted that we live in a spatial world and it is anatural phenomenon that a substance goes from high densityregions to low density regions As a result epidemic modelsshould include spatial effects In a further study we need toinvestigate spatial epidemic models with seasonal factors

Acknowledgments

The research was partially supported by the National NaturalScience Foundation of China under Grant nos 1130149011301491 11331009 11147015 11171314 11101251 and 11105024Natural Science Foundation of ShanXi Province Grant no2012021002-1 and the Opening Foundation of Institute ofInformation Economy Hangzhou Normal University Grantno PD12001003002003

References

[1] K Dietz and J A P Heesterbeek ldquoBernoulli was ahead ofmodern epidemiologyrdquo Nature vol 408 no 6812 pp 513ndash5142000

[2] K Dietz and J A P Heesterbeek ldquoDaniel Bernoullirsquos epidemio-logical model revisitedrdquoMathematical Biosciences vol 180 pp1ndash21 2002

[3] R Ross ldquoAn application of the theory of probabilities to thestudy of a priori pathometry (part I)rdquo Proceedings of the RoyalSociety A vol 92 no 638 pp 204ndash230 1916

[4] R Ross and H P Hudson ldquoAn application of the theory ofprobabilities to the study of a priori pathometry (part II)rdquoProceedings of the Royal Society A vol 93 no 650 pp 212ndash2251917

[5] W O Kermack and A G McKendrick ldquoA contribution to themathematical theory of epidemics (part I)rdquo Proceedings of theRoyal Society A vol 115 no 772 pp 700ndash721 1927

[6] W O Kermack and A G McKendrick ldquoA contribution to themathematical theory of epidemics (part II)rdquo Proceedings of theRoyal Society A vol 138 no 834 pp 55ndash83 1932

[7] A Tsoularis and JWallace ldquoAnalysis of logistic growthmodelsrdquoMathematical Biosciences vol 179 no 1 pp 21ndash55 2002

[8] I Nasell ldquoOn the quasi-stationary distribution of the stochasticlogistic epidemicrdquoMathematical Biosciences vol 156 no 1-2 pp21ndash40 1999

[9] H Fujikawa A Kai and SMorozumi ldquoA new logisticmodel forEscherichia coli growth at constant and dynamic temperaturesrdquoFood Microbiology vol 21 no 5 pp 501ndash509 2004

[10] L Berezansky and E Braverman ldquoOscillation properties of alogistic equation with several delaysrdquo Journal of MathematicalAnalysis and Applications vol 247 no 1 pp 110ndash125 2000

[11] M Tabata N Eshima and I Takagi ldquoThe nonlinear integro-partial differential equation describing the logistic growth ofhuman population with migrationrdquo Applied Mathematics andComputation vol 98 pp 169ndash183 1999

[12] L Korobenko andE Braverman ldquoOn logisticmodels with a car-rying capacity dependent diffusion stability of equilibria andcoexistence with a regularly diffusing populationrdquo NonlinearAnalysis Real World Applications vol 13 no 6 pp 2648ndash26582012

[13] Y Muroya ldquoGlobal attractivity for discrete models of nonau-tonomous logistic equationsrdquo Computers andMathematics withApplications vol 53 no 7 pp 1059ndash1073 2007

[14] S Invernizzi and K Terpin ldquoA generalized logistic model forphotosynthetic growthrdquo Ecological Modelling vol 94 no 2-3pp 231ndash242 1997

[15] ZMinW Bang-Jun and J Feng ldquoCoalmining citiesrsquo economicgrowthmechanism and sustainable development analysis basedon logistic dynamics modelrdquo Procedia Earth and PlanetaryScience vol 1 no 1 pp 1737ndash1743 2009

[16] L I Anita S Anita and V Arnautu ldquoGlobal behavior foran age-dependent population model with logistic term andperiodic vital ratesrdquoAppliedMathematics andComputation vol206 no 1 pp 368ndash379 2008

[17] Z A Banaszak X Q Tang S C Wang and M B ZarembaldquoLogistics models in flexible manufacturingrdquo Computers inIndustry vol 43 no 3 pp 237ndash248 2000

[18] S Brianzoni C Mammana and E Michetti ldquoNonlineardynamics in a business-cycle model with logistic populationgrowthrdquo Chaos Solitons and Fractals vol 40 no 2 pp 717ndash7302009

[19] W P London and J A Yorke ldquoRecurrent outbreaks of measleschickenpox and mumps I Seasonal variation in contact ratesrdquoThe American Journal of Epidemiology vol 98 no 6 pp 453ndash468 1973

[20] S F Dowell ldquoSeasonal variation in host susceptibility and cyclesof certain infectious diseasesrdquo Emerging Infectious Diseases vol7 no 3 pp 369ndash374 2001

[21] ON Bjoslashrnstad B F Finkenstadt andB TGrenfell ldquoDynamicsof measles epidemics estimating scaling of transmission ratesusing a Time series SIR modelrdquo Ecological Monographs vol 72no 2 pp 169ndash184 2002

[22] J Zhang Z Jin G Q Sun X Sun and S Ruan ldquoModelingseasonal rabies epidemics in Chinardquo Bulletin of MathematicalBiology vol 74 no 5 pp 1226ndash1251 2012

[23] I B Schwartz ldquoSmall amplitude long period outbreaks inseasonally driven epidemicsrdquo Journal of Mathematical Biologyvol 30 no 5 pp 473ndash491 1992

[24] I B Schwartz and H L Smith ldquoInfinite subharmonic bifur-cation in an SEIR epidemic modelrdquo Journal of MathematicalBiology vol 18 no 3 pp 233ndash253 1983

[25] H L Smith ldquoMultiple stable subharmonics for a periodicepidemic modelrdquo Journal of Mathematical Biology vol 17 no2 pp 179ndash190 1983

[26] J Dushoff J B Plotkin S A Levin and D J D EarnldquoDynamical resonance can account for seasonality of influenzaepidemicsrdquo Proceedings of the National Academy of Sciences ofthe United States of America vol 101 no 48 pp 16915ndash169162004

[27] J L Ma and Z Ma ldquoEpidemic threshold conditions forseasonally forced SEIR modelsrdquo Mathematical Biosciences andEngineering vol 3 no 1 pp 161ndash172 2006

[28] D J D Earn P Rohani B M Bolker and B T Grenfell ldquoAsimple model for complex dynamical transitions in epidemicsrdquoScience vol 287 no 5453 pp 667ndash670 2000

10 The Scientific World Journal

[29] M J Keeling P Rohani and B T Grenfell ldquoSeasonally forceddisease dynamics explored as switching between attractorsrdquoPhysica D vol 148 no 3-4 pp 317ndash335 2001

[30] J L Aron and I B Schwartz ldquoSeasonality and period-doublingbifurcations in an epidemic modelrdquo Journal of TheoreticalBiology vol 110 no 4 pp 665ndash679 1984

[31] N C Grassly and C Fraser ldquoSeasonal infectious diseaseepidemiologyrdquo Proceedings of the Royal Society B vol 273 no1600 pp 2541ndash2550 2006

[32] J Zhang Z Jin G Q Sun T Zhou and S Ruan ldquoAnalysisof rabies in China transmission dynamics and controlrdquo PLoSONE vol 6 no 7 Article ID e20891 2011

[33] O Diekmann J A P Heesterbeek and J A Metz ldquoOnthe definition and the computation of the basic reproductionratio R0 in models for infectious diseases in heterogeneouspopulationsrdquo Journal of mathematical biology vol 28 no 4 pp365ndash382 1990

[34] O Diekmann J A P Heesterbeek and M G Roberts ldquoTheconstruction of next-generation matrices for compartmentalepidemic modelsrdquo Journal of the Royal Society Interface vol 7no 47 pp 873ndash885 2010

[35] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[36] F Berezovsky G Karev B J Song and C C Chavez ldquoA simpleepidemic model with surprising dynamicsrdquo Mathematical Bio-sciences and Engineering vol 2 pp 133ndash152 2005

[37] J LaSalle and S Lefschetz Stability by Liapunovrsquos DirectMethodAcademic Press New York NY USA 1961

[38] E A Barbashin Introduction to theTheory of Stability Wolters-Noordhoff Groningen The Netherlands 1970

[39] D Schenzle ldquoAn age-structured model of pre- and post-vaccination measles transmissionrdquoMathematical Medicine andBiology vol 1 no 2 pp 169ndash191 1984

[40] N Bacaer and S Guernaoui ldquoThe epidemic threshold ofvector-borne diseases with seasonality the case of cutaneousleishmaniasis in Chichaoua Moroccordquo Journal of MathematicalBiology vol 53 no 3 pp 421ndash436 2006

[41] W DWang and X Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008

[42] F Zhang and X Q Zhao ldquoA periodic epidemic model in apatchy environmentrdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 496ndash516 2007

[43] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press New York NY USA 1995

[44] H R Thieme ldquoConvergence results and a Poincare-Bendixsontrichotomy for asymptotically automous differential equationsrdquoJournal of Mathematical Biology vol 30 pp 755ndash763 1992

[45] L Perko Differential Equations and Dynamical SystemsSpringer New York NY USA 2000

[46] X Q Zhao Dynamical Systems in Population Biology SpringerNew York NY USA 2003

[47] Z G Bai and Y C Zhou ldquoThreshold dynamics of a bacillarydysentery model with seasonal fluctuationrdquo Discrete and Con-tinuous Dynamical Systems B vol 15 no 1 pp 1ndash14 2011

[48] Z G Bai Y C Zhou and T L Zhang ldquoExistence of multipleperiodic solutions for an SIRmodel with seasonalityrdquoNonlinear

Analysis Theory Methods and Applications vol 74 no 11 pp3548ndash3555 2011

[49] N Bacaer ldquoApproximation of the basic reproduction numberR0 for vector-borne diseases with a periodic vector populationrdquoBulletin of Mathematical Biology vol 69 no 3 pp 1067ndash10912007

[50] N Bacaer and X Abdurahman ldquoResonance of the epidemicthreshold in a periodic environmentrdquo Journal of MathematicalBiology vol 57 no 5 pp 649ndash673 2008

[51] N Bacaer ldquoPeriodic matrix population models growth ratebasic reproduction number and entropyrdquoBulletin ofMathemat-ical Biology vol 71 no 7 pp 1781ndash1792 2009

[52] N Bacaer and E H A Dads ldquoGenealogy with seasonalitythe basic reproduction number and the influenza pandemicrdquoJournal of Mathematical Biology vol 62 no 5 pp 741ndash762 2011

[53] L J Liu X Q Zhao and Y C Zhou ldquoA tuberculosis model withseasonalityrdquo Bulletin of Mathematical Biology vol 72 no 4 pp931ndash952 2010

[54] J L Liu ldquoThreshold dynamics for a HFMD epidemic modelwith periodic transmission raterdquo Nonlinear Dynamics vol 64no 1-2 pp 89ndash95 2011

[55] Y Nakata and T Kuniya ldquoGlobal dynamics of a class ofSEIRS epidemic models in a periodic environmentrdquo Journal ofMathematical Analysis and Applications vol 363 no 1 pp 230ndash237 2010

[56] B GWilliams andC Dye ldquoInfectious disease persistence whentransmission varies seasonallyrdquo Mathematical Biosciences vol145 no 1 pp 77ndash88 1997

[57] I A Moneim ldquoThe effect of using different types of periodiccontact rate on the behaviour of infectious diseases a simula-tion studyrdquo Computers in Biology and Medicine vol 37 no 11pp 1582ndash1590 2007

[58] C L Wesley and L J S Allen ldquoThe basic reproduction numberin epidemic models with periodic demographicsrdquo Journal ofBiological Dynamics vol 3 no 2-3 pp 116ndash129 2009

[59] D Greenhalgh and I A Moneim ldquoSIRS epidemic model andsimulations using different types of seasonal contact raterdquoSystems Analysis Modelling Simulation vol 43 no 5 pp 573ndash600 2003

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 The Scientific World Journal

[29] M J Keeling P Rohani and B T Grenfell ldquoSeasonally forceddisease dynamics explored as switching between attractorsrdquoPhysica D vol 148 no 3-4 pp 317ndash335 2001

[30] J L Aron and I B Schwartz ldquoSeasonality and period-doublingbifurcations in an epidemic modelrdquo Journal of TheoreticalBiology vol 110 no 4 pp 665ndash679 1984

[31] N C Grassly and C Fraser ldquoSeasonal infectious diseaseepidemiologyrdquo Proceedings of the Royal Society B vol 273 no1600 pp 2541ndash2550 2006

[32] J Zhang Z Jin G Q Sun T Zhou and S Ruan ldquoAnalysisof rabies in China transmission dynamics and controlrdquo PLoSONE vol 6 no 7 Article ID e20891 2011

[33] O Diekmann J A P Heesterbeek and J A Metz ldquoOnthe definition and the computation of the basic reproductionratio R0 in models for infectious diseases in heterogeneouspopulationsrdquo Journal of mathematical biology vol 28 no 4 pp365ndash382 1990

[34] O Diekmann J A P Heesterbeek and M G Roberts ldquoTheconstruction of next-generation matrices for compartmentalepidemic modelsrdquo Journal of the Royal Society Interface vol 7no 47 pp 873ndash885 2010

[35] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[36] F Berezovsky G Karev B J Song and C C Chavez ldquoA simpleepidemic model with surprising dynamicsrdquo Mathematical Bio-sciences and Engineering vol 2 pp 133ndash152 2005

[37] J LaSalle and S Lefschetz Stability by Liapunovrsquos DirectMethodAcademic Press New York NY USA 1961

[38] E A Barbashin Introduction to theTheory of Stability Wolters-Noordhoff Groningen The Netherlands 1970

[39] D Schenzle ldquoAn age-structured model of pre- and post-vaccination measles transmissionrdquoMathematical Medicine andBiology vol 1 no 2 pp 169ndash191 1984

[40] N Bacaer and S Guernaoui ldquoThe epidemic threshold ofvector-borne diseases with seasonality the case of cutaneousleishmaniasis in Chichaoua Moroccordquo Journal of MathematicalBiology vol 53 no 3 pp 421ndash436 2006

[41] W DWang and X Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008

[42] F Zhang and X Q Zhao ldquoA periodic epidemic model in apatchy environmentrdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 496ndash516 2007

[43] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press New York NY USA 1995

[44] H R Thieme ldquoConvergence results and a Poincare-Bendixsontrichotomy for asymptotically automous differential equationsrdquoJournal of Mathematical Biology vol 30 pp 755ndash763 1992

[45] L Perko Differential Equations and Dynamical SystemsSpringer New York NY USA 2000

[46] X Q Zhao Dynamical Systems in Population Biology SpringerNew York NY USA 2003

[47] Z G Bai and Y C Zhou ldquoThreshold dynamics of a bacillarydysentery model with seasonal fluctuationrdquo Discrete and Con-tinuous Dynamical Systems B vol 15 no 1 pp 1ndash14 2011

[48] Z G Bai Y C Zhou and T L Zhang ldquoExistence of multipleperiodic solutions for an SIRmodel with seasonalityrdquoNonlinear

Analysis Theory Methods and Applications vol 74 no 11 pp3548ndash3555 2011

[49] N Bacaer ldquoApproximation of the basic reproduction numberR0 for vector-borne diseases with a periodic vector populationrdquoBulletin of Mathematical Biology vol 69 no 3 pp 1067ndash10912007

[50] N Bacaer and X Abdurahman ldquoResonance of the epidemicthreshold in a periodic environmentrdquo Journal of MathematicalBiology vol 57 no 5 pp 649ndash673 2008

[51] N Bacaer ldquoPeriodic matrix population models growth ratebasic reproduction number and entropyrdquoBulletin ofMathemat-ical Biology vol 71 no 7 pp 1781ndash1792 2009

[52] N Bacaer and E H A Dads ldquoGenealogy with seasonalitythe basic reproduction number and the influenza pandemicrdquoJournal of Mathematical Biology vol 62 no 5 pp 741ndash762 2011

[53] L J Liu X Q Zhao and Y C Zhou ldquoA tuberculosis model withseasonalityrdquo Bulletin of Mathematical Biology vol 72 no 4 pp931ndash952 2010

[54] J L Liu ldquoThreshold dynamics for a HFMD epidemic modelwith periodic transmission raterdquo Nonlinear Dynamics vol 64no 1-2 pp 89ndash95 2011

[55] Y Nakata and T Kuniya ldquoGlobal dynamics of a class ofSEIRS epidemic models in a periodic environmentrdquo Journal ofMathematical Analysis and Applications vol 363 no 1 pp 230ndash237 2010

[56] B GWilliams andC Dye ldquoInfectious disease persistence whentransmission varies seasonallyrdquo Mathematical Biosciences vol145 no 1 pp 77ndash88 1997

[57] I A Moneim ldquoThe effect of using different types of periodiccontact rate on the behaviour of infectious diseases a simula-tion studyrdquo Computers in Biology and Medicine vol 37 no 11pp 1582ndash1590 2007

[58] C L Wesley and L J S Allen ldquoThe basic reproduction numberin epidemic models with periodic demographicsrdquo Journal ofBiological Dynamics vol 3 no 2-3 pp 116ndash129 2009

[59] D Greenhalgh and I A Moneim ldquoSIRS epidemic model andsimulations using different types of seasonal contact raterdquoSystems Analysis Modelling Simulation vol 43 no 5 pp 573ndash600 2003

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