Research ArticleGlobal Stability of a Delayed SIRI Epidemic Model withNonlinear Incidence

Amine Bernoussi1 Abdelilah Kaddar2 and Said Asserda1

1Departement de Mathematiques Faculte des Sciences Universite Ibn Tofail Campus Universitaire BP 133 14000 Kenitra Morocco2Departement drsquoEconomie et de Gestion Faculte des Sciences Juridiques Economiques et Sociales Universite Mohammed V-SouissiRoute Outa Hsaine BP 5295 11100 Sala Al Jadida Morocco

Correspondence should be addressed to Abdelilah Kaddar akaddaryahoofr

Received 25 July 2014 Accepted 14 November 2014 Published 7 December 2014

Academic Editor Shouming Zhong

Copyright copy 2014 Amine Bernoussi et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

In this paper we propose the global dynamics of an SIRI epidemic model with latency and a general nonlinear incidence functionThemodel is based on the susceptible-infective-recovered (SIR) compartmental structure with relapse (SIRI) Sufficient conditionsfor the global stability of equilibria (the disease-free equilibrium and the endemic equilibrium) are obtained bymeans of Lyapunov-LaSalle theorem Also some numerical simulations are given to illustrate this result

1 Introduction

Epidemicmodels have long been an important tool for under-standing and controlling the spread of infectious diseasesMost of them are described by delayed differential equationsThe introduction of time delay is often used to modelthe latent period that is the time from the acquisition ofinfection to the time when the host becomes infectious [1 2]

Recently considerable attention has been paid to modelthe relapse phenomenon that is the return of signs andsymptoms of a disease after a remission Hence the recoveredindividual can return to the infectious class (see [3ndash6]) Forthe biological explanations of the relapse phenomenon wecite two examples

(i) For malaria Bignami [7] proposed that relapsesderived from persistence of small numbers of parasitein the blood Also it has been observed that theproportion of patients who have successive relapsesis relatively constant (see [8])

(ii) For tuberculosis relapse can be caused by incompletetreatment or by latent infection being observed thatHIV-positive patients are significantly more likelyto relapse than HIV-negative patients although it isoften difficult to differentiate relapse from reinfection(see [9])

In this paper we propose the following epidemic model withtime delay and relapse (delayed SIRI epidemic model) asfollows (see [10 11])

119889119878

119889119905= 119860 minus 120583119878 minus 119891 (119878 119868)

119889119868

119889119905= 119890minus120583120591

119891 (119878120591

119868120591

) minus (120583 + 120574 + 120572) 119868 + 120575119877

119889119877

119889119905= 120574119868 minus (120583 + 120575) 119877

(1)

The initial condition for the above system is

119878 (120579) = 1205931

(120579)

119868 (120579) = 1205932

(120579)

119877 (120579) = 1205933

(120579)

120579 isin [minus120591 0]

(2)

with 120593 = (1205931

1205932

1205933

) isin 119862+

times 119862+

times 119862+ such that 120593

119894

(120579) ge 0

(minus120591 le 120579 le 0 119894 = 1 2 3) Here 119862 denotes the Banach space119862([minus120591 0] 119877) of continuous functions mapping the interval[minus120591 0] into 119877 equipped with the supremum norm Thenonnegative cone of 119862 is defined as 119862+ = 119862([minus120591 0] 119877

+

)

Hindawi Publishing CorporationInternational Journal of Engineering MathematicsVolume 2014 Article ID 487589 6 pageshttpdxdoiorg1011552014487589

2 International Journal of Engineering Mathematics

Here 120595120591

= 120595(119905 minus 120591) for any given function 120595 119860 = 120583119873where 119873 = 119878 + 119868 + 119877 is the total number of population119878 is the number of susceptible individuals 119868 is the numberof infectious individuals 119877 is the number of recoveredindividuals 119860 is the recruitment rate of the population 120583 isthe natural death of the population 120572 is the death rate dueto disease 119891 is the nonlinear incidence function 120574 is therecovery rate of the infective individuals 120575 is the rate thatrecovered individuals relapse and regained infectious classand 120591 is the latent period

In model (1) the incidence function 119891(119878 119868) is a locallyLipschitz continuous function on 119877+ times119877+ satisfying 119891(0 119868) =119891(119878 0) = 0 for 119878 ge 0 119868 ge 0 and the following hold

(1198671

) 119891 is a strictly monotone increasing function of 119878 ge

0 for any fixed 119868 gt 0 and 119891 is a strictly monotoneincreasing function of 119868 ge 0 for any fixed 119878 ge 0

(1198672

) 120601(119878 119868) = 119891(119878 119868)119868 is a bounded and monotonedecreasing function of 119868 gt 0 for any fixed 119878 ge 0 and119870(119878) = lim

119868rarr0

+120601(119878 119868) is a continuous andmonotoneincreasing function on 119878 ge 0

This incidence function includes different forms pre-sented in literature (see eg [12ndash23])

System (1) always has a disease-free equilibrium 1198750

=

(119860120583 0 0) On the other hand under the hypothesis (1198672

) if

1198770

=119870 (119860120583) 119890

minus120583120591

120583 + 120574 + 120572 minus 120574120575 (120583 + 120575)gt 1 (3)

then system (1) also admits a unique endemic equilibrium119875lowast

= (119878lowast

119868lowast

119877lowast

) where 119878lowast 119868lowast and 119877lowast satisfy the followingsystem (see [24])

119860 minus 120583119878 minus 119891 (119878 119868) = 0

119890minus120583120591

119891 (119878 119868) minus (120583 + 120574 + 120572) 119868 + 120575119877 = 0

120574119868 minus (120583 + 120575) 119877 = 0

(4)

Hereafter we replace 120583 + 120574 + 120572 minus 120574120575(120583 + 120575) by 120578In [5] (1990) Tudor developed and analyzed qualitatively

one of the first SIRI epidemic models for the spread of aherpes-type infection in either human or animal populationsThis model consists of a system of nonlinear ordinary differ-ential equations with a bilinear incidence rate (ie 119891(119878 119868) =120573119878119868) and a constant total population (ie 119873 = 119878 + 119868 + 119877 =

constant)In [25] (1997) Moreira and Wang extended a Tudor-

model to include nonlinear incidence functions By usingan elementary analysis of Lienardrsquos equation and Lyapunovrsquosdirect method they derived sufficient conditions for theglobal asymptotic stability of the disease-free and endemicequilibria

In [23] (2000) Castillo-Garsow et al considered anSIRI model for drug use in a population of adolescentsThe authors assumed that 119873 = constant and 119891(119878 119868) =

120573119878119868119873 they estimated the parameters of the model and theydetermined a rough approximation of the basic reproductivenumber Based on these parameters they performed some

simulations that clearly showed the endemic character oftobacco use among adolescents

In [26] (2004) Blower developed a compartmentalmodelfor genital herpes assuming standard incidence rate (ie119891(119878 119868) = 120573119878119868119873) and constant recruitment rate

In [24] (2006) Korobeinikov proved that the endemicequilibrium of the model (1) with 120575 = 0 and 120591 = 0 is globallyasymptotically stable

In [27] (2007) van den Driessche and Zou proposedan integrodifferential equation to model a general relapsephenomenon in infectious diseases The resulting model inparticular case is a delay differential equation with a constantpopulation and standard incidence The basic reproductionnumber for this model is identified and some global resultsare obtained by employing the Lyapunov-Razumikhin tech-nique

In [10] (2007) Van DenDriessche and co-authors formu-lated a delay differential SIRImodel (System (1) with119891(119878 119868) =120573119878119868) For this system the endemic equilibrium is locallyasymptotically stable if 119877

0

gt 1 and the disease is shown to beuniformly persistent with the infective population size eitherapproaching or oscillating about the endemic level

In [22] (2011) Liu et al proposed a mathematical modelfor a disease with a general exposed distribution the pos-sibility of relapse and nonlinear incidence rate (119891(119878 119868) =

120573119878(119868119873)) By the method of Lyapunov functionals theyshowed that the disease dies out if 119877

0

= 1 and that the diseasebecomes endemic if 119877

0

gt 1 They also analyzed as a specialcase of this model the system (1) with 119891(119878 119868) = 119892(119878) sdot 119868the result confirms that the endemic equilibrium is globallyasymptotically stable

In [28] (2012) Abta et al considered a global asymptoticstability of a delayed SIR model (system (1) with 119891(119878 119868) =

120573119878119868(1 + 1205721

119878 + 1205722

119868) and 120575 = 0)In [29] (2013) Vargas-De-Leon presented the global sta-

bility conditions of an ordinary SIRI model with bilinear andstandard incidence rates respectively that includes recruit-ment rate of susceptible individuals into the communityand that the disease produces nonnegligible death in theinfectious class The author presented the construction ofLyapunov functions using suitable combinations of knownfunctions common quadratic and Volterra-type and a com-posite Volterra-type function

In [21] (2013) Georgescu and Zhang analyzed the dynam-ics of an ordinary SIRI model under the assumption that theincidence of infection is given by 119891(119878 119868) = 119862(119878)119892(119868) Theyobtained by means of Lyapunovrsquos second method sufficientconditions for the local stability of equilibria and theyshowed that global stability can be attained under suitablemonotonicity conditions

In [30] (2013) Shuai and van den Driessche presentedtwo systematic methods for the construction of Lyapunovfunctions for general infectious disease models (OrdinarySEIRI SIS etc) Specifically amatrix-theoreticmethod usingthe Perron eigenvector is applied to prove the global stabil-ity of the disease-free equilibrium while a graph-theoreticmethod based on Kirchhoff rsquos matrix tree theorem and twonew combinatorial identities are used to prove the globalstability of the endemic equilibrium

International Journal of Engineering Mathematics 3

In [31] (2014) Xu investigated a delayed SIRI model(system (1) with 119891(119878 119868) = 120573119878119868) The author established theglobal stability of a disease-free equilibrium and an endemicequilibrium by means of suitable Lyapunov functionals andLaSallersquos invariance principle

In this paper we extend the global stability results pre-sented in [31] to a delayed SIRI epidemic model (system (1))with a general nonlinear incidence function It is shown thatglobal stability can be attained under suitable monotonicityconditions and it is established that the basic reproductionnumber 119877

0

is a threshold parameter for the stability of adelayed SIRI model The rest of the paper is organized asfollows In Section 2 the global stability of disease-free andendemic equilibria are established In Section 3 numericalsimulations and concluding remarks are provided In theappendix some results on the global stability are stated

2 Global Stability Analysis of DelayedSIRI Model

In this section we discuss the global stability of a disease-freeequilibrium 119875

0

and an endemic equilibrium 119875lowast of system (1)

Since (119889119889119905)(119878+119868+119877) le 119860minus120583(119878+119868+119877) we have lim sup(119878+119868 + 119877) le 119860120583 Hence we discuss system (1) in the closed set

Ω

= (1205931

1205932

1205933

) isin 119862+

times 119862+

times 119862+

10038171003817100381710038171205931 + 1205932 + 1205933

1003817100381710038171003817 le119860

120583

(5)

It is easy to show thatΩ is positively invariant with respect tosystem (1) Next we consider the global asymptotic stability ofthe disease-free equilibrium 119875

0

and the endemic equilibrium119875lowast of (1) by Lyapunov functionals respectively

Proposition 1 If 1198770

le 1 then the disease-free equilibrium 1198750

is globally asymptotically stable

Proof Define a Lyapunov functional

1198810

(119905) = 119890minus120583120591

int

119878

120591

119860120583

(1 minus119870 (119860120583)

119870 (119906)) 119889119906 + 119868

+ 120578int

119905

119905minus120591

119868 (119906) 119889119906 +120575

120583 + 120575119877

(6)

We will show that 1198891198810

(119905)119889119905 le 0 for all 119905 ge 0 We have

1198891198810

(119905)

119889119905= 119890minus120583120591

(1 minus119870 (119860120583)

119870 (119878120591

)) 119878120591

+ 119890minus120583120591

119891 (119878120591

119868120591

) 119889120591

minus (120583 + 120574 + 120572) 119868 + 120575119877 + 120578 (119868 minus 119868120591

)

+120574120575

120583 + 120575119868 minus 120575119877

= 119890minus120583120591

(1 minus119870 (119860120583)

119870 (119878120591

)) (119860 minus 120583119878

120591

)

+119870 (119860120583)

119870 (119878120591

)119890minus120583120591

119891 (119878120591

119868120591

) minus 120578119868120591

= 120583119890minus120583120591

(1 minus119870 (119860120583)

119870 (119878120591

))(

119860

120583minus 119878120591

)

+ 120578119868120591

(120601 (119878120591

119868120591

)

120578

119870 (119860120583)

119870 (119878120591

)119890minus120583120591

minus 1)

(7)

By the hypothesis (1198671

) we obtain that

(1 minus119870 (119860120583)

119870 (119878120591

))(

119860

120583minus 119878120591

) le 0 (8)

where equality holds if and only if 119878 = 119860120583Furthermore It follows from the hypothesis (119867

2

) that

120601 (119878120591

119868120591

)

120578

119870 (119860120583) 119890minus120583120591

119870(119878120591

)le119870 (119878120591

)

120578

119870 (119860120583) 119890minus120583120591

119870(119878120591

)

le119870 (119860120583) 119890

minus120583120591

120578

le 1198770

(9)

Therefore 1198770

le 1 ensures that 1198891198810

(119905)119889119905 le 0 for all 119905 ge 0where 119889119881

0

(119905)119889119905 = 0 holds if (119878 119868 119877) = (119860120583 0 0) Henceit follows from system (1) that 119875

0

is the largest invariantset in (119878 119868 119877) | 119889119881

0

(119905)119889119905 = 0 From the Lyapunov-LaSalle asymptotic stability we obtain that 119875

0

is globallyasymptotically stable This completes the proof

Proposition 2 If 1198770

gt 1 then the endemic equilibrium 119875lowast

1

isglobally asymptotically stable

Proof To prove global stability of the endemic equilibriumwe define a Lyapunov functional119881(119905) = 119881

1

(119905)+1198812

(119905)+1198813

(119905)+

119881+

with

1198811

(119905) = 119878 minus 119878lowast

minus int

119878

119878

lowast

119891 (119878lowast

119868lowast

)

119891 (119906 119868lowast)119889119906

1198812

(119905) = 119890120583120591

(119868 minus 119868lowast

minus 119868lowast ln 119868

119868lowast)

1198813

(119905) =120575119890120583120591

120583 + 120575(119877 minus 119877

lowast

minus 119877lowast ln 119877

119877lowast)

119881+

= int

119905

119905minus120591

(119891 (119878 119868) minus 119891 (119878lowast

119868lowast

) minus 119891 (119878lowast

119868lowast

)

sdot ln119891 (119878 (119906) 119868 (119906))

119891 (119878lowast 119868lowast)) 119889119906

(10)

4 International Journal of Engineering Mathematics

The time derivative of the function 119881(119905) along the positivesolution of system (1) is

119889119881 (119905)

119889119905= (1 minus

119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)) (119860 minus 120583119878 minus 119891 (119878 119868))

+ 119890120583120591

(1 minus119868lowast

119868)

sdot (119890minus120583120591

119891 (119878120591

119868120591

+ 120575119877) minus (120583 + 120572 + 120574) 119868)

+120575119890120583120591

120583 + 120575(1 minus

119877lowast

119877) (120574119868 minus (120583 + 120575) 119877)

+ 119891 (119878 119868) minus 119891 (119878120591

119868120591

) + 119891 (119878lowast

119868lowast

)

sdot ln119891 (119878120591

119868120591

)

119891 (119878 119868)

(11)

Using the relation 119860 = 120583119878lowast

+ 119891(119878lowast

119868lowast

) simple calculationsgive

119889119881 (119905)

119889119905

= (1 minus119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)) (minus120583 (119878 minus 119878

lowast

) + 119891 (119878lowast

119868lowast

))

+119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)119891 (119878 119868) minus 119890

120583120591

(120583 + 120572 + 120574) 119868

minus119868lowast

119868119891 (119878120591

119868120591

) minus 120575119890120583120591

119868lowast

119868119877 + 119890120583120591

(120583 + 120572 + 120574) 119868lowast

+120575119890120583120591

120583 + 120575(120574119868 minus 120574119868

119877lowast

119877+ (120583 + 120575) 119877

lowast

)

+ 119891 (119878lowast

119868lowast

) ln119891 (119878120591

119868120591

)

119891 (119878 119868)

(12)

Here by using

119890120583120591

(120583 + 120572 + 120574) 119868lowast

minus120575119890120583120591

120583 + 120575120574119868lowast

= 119891 (119878lowast

119868lowast

)

(120583 + 120575) 119877lowast

= 120574119868lowast

ln119891 (119878120591

119868120591

)

119891 (119878 119868)

= ln119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)+ ln

119868lowast

119891 (119878120591

119868120591

)

119868119891 (119878lowast 119868lowast)+ ln

119868119891 (119878 119868lowast

)

119868lowast119891 (119878 119868)

(13)

straightforward calculations give

119889119881 (119905)

119889119905

= minus120583(1 minus119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)) (119878 minus 119878

lowast

)

minus 119891 (119878lowast

119868lowast

) (119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)minus 1 minus ln

119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast))

minus 119891 (119878lowast

119868lowast

) (119868lowast

119891 (119878120591

119868120591

)

119868119891 (119878lowast 119868lowast)minus 1 minus ln

119868lowast

119891 (119878120591

119868120591

)

119868119891 (119878lowast 119868lowast))

+ 119891 (119878lowast

119868lowast

) (119891 (119878 119868)

119891 (119878 119868lowast)minus

119868

119868lowast+119868119891 (119878 119868

lowast

)

119868lowast119891 (119878 119868)minus 1)

minus 119891 (119878lowast

119868lowast

) (119868119891 (119878 119868

lowast

)

119868lowast119891 (119878 119868)minus 1 minus ln

119868119891 (119878 119868lowast

)

119868lowast119891 (119878 119868))

+120575120574119868lowast

119890120583120591

120583 + 120575(2 minus

119868lowast

119877

119868119877lowastminus119868119877lowast

119868lowast119877)

= minus120583(1 minus119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)) (119878 minus 119878

lowast

)

minus 119891 (119878lowast

119868lowast

) (119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)minus 1 minus ln

119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast))

minus 119891 (119878lowast

119868lowast

) (119868lowast

119891 (119878120591

119868120591

)

119868119891 (119878 119868lowast)minus 1 minus ln

119868lowast

119891 (119878120591

119868120591

)

119868119891 (119878 119868lowast))

+119868

119868lowast119891 (119878lowast

119868lowast

) (1 minus119891 (119878 119868

lowast

)

119891 (119878 119868))(

120601 (119878 119868)

120601 (119878 119868lowast)minus 1)

minus 119891 (119878lowast

119868lowast

) (119868119891 (119878 119868

lowast

)

119868lowast119891 (119878 119868)minus 1 minus ln

119868119891 (119878 119868lowast

)

119868lowast119891 (119878 119868))

minus120575120574119890120583120591

120583 + 120575

119868119877lowast

119877(119868lowast

119877

119868119877lowastminus 1)

2

(14)

It follows from (1198671

) and (1198672

) that

minus120583(1 minus119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)) (119878 minus 119878

lowast

) le 0

119868

119868lowast119891 (119878lowast

119868lowast

) (1 minus119891 (119878 119868

lowast

)

119891 (119878 119868))(

120601 (119878 119868)

120601 (119878 119868lowast)minus 1)

le 0

(15)

Furthermore since the function 119892(119909) = 1minus119909+ ln(119909) is alwaysnonpositive for any 119909 gt 0 and 119892(119909) = 0 if and only if 119909 = 1then 119889119881(119905)119889119905 le 0 for all 119905 ge 0 where the equality holds onlyat the equilibrium point (119878lowast 119868lowast 119877lowast) Hence the functional119881satisfies all the conditions of Theorem A2 This proves that119875lowast

1

is globally asymptotically stable

International Journal of Engineering Mathematics 5

3 Numerical Simulations andConcluding Remarks

In this section we give a numerical simulation supporting thetheoretical analysis given in Section 2 Let

119891 (119878 119868) =120573119878119868

1 + 1205721

119878 + 1205722

119868 (16)

We take the parameters of the system (1) as follows

119860 = 10

1205721

= 09

1205722

= 09

120583 = 001

120574 = 002

120572 = 0005

120573 = 01

120575 = 0001

120591 = 10

(17)

By Proposition 2 the endemic equilibrium 119875lowast is globally

asymptotically stable see Figure 1In this paper we presented a mathematical analysis and

numerical simulations for an SIRI epidemiological modelapplied to the evolution of the spread of disease with relapsein a given population We denote 119877

0

the basic reproductionnumber It is defined as the average number of contagiouspersons infected by a typical infectious in a population of sus-ceptible We prove in this paper that the basic reproductionnumber 119877

0

depends on the incubation period and we showthat the disease-free equilibrium 119875

0

is globally asymptoticallystable if 119877

0

le 1 and that a unique endemic equilibrium 119875lowast is

globally asymptotically stable if 1198770

gt 1

Appendix

The Lyapunov-LaSalle Theorem

In the following we present the method of Lyapunov func-tionals in the context of a delay differential equations

119889119909

119889119905= 119891 (119909

119905

) (A1)

where 119891 119862 rarr 119877119899 is completely continuous and solutions

of (A1) are unique and continuously dependent on the initialdata We denote by 119909(120601) the solution of (A1) through (0 120601)For a continuous functional 119881 119862 rarr 119877 we define

= lim supℎrarr0

+

1

ℎ[(119881119909ℎ

(120601)) minus 119881 (120601)] (A2)

the derivative of 119881 along a solution of (A1) To statethe Lyapunov-LaSalle type theorem for (A1) we need thefollowing definition

0 100 200 300 400 5000

50

100

150

200

250

300

350

400

450

500

t

0 100 200 300 400 5000

50

100

150

200

250

300

350

400

450

t

I(t)

0 100 200 300 400 500100

200

300

400

500

600

700

800

900

1000

t

S(t)

R(t)

Figure 1 Solutions (119878 119868 119877) of the SIRI epidemic model (1) are glob-ally asymptotically stable and converge to the endemic equilibrium119875lowast

Definition A1 (see [32 page 30]) We say 119881 119862 rarr 119877

is a Lyapunov functional on a set 119866 in 119862 for (A1) if it iscontinuous on 119866 (the closure of 119866) and le 0 on 119866 We also

6 International Journal of Engineering Mathematics

define 119864 = 120601 isin 119866 (120601) = 0 and119872 is the largest set in 119864which is invariant with respect to (A1)

The following result is the Lyapunov-LaSalle type theo-rem for (A1)

Theorem A2 (see [32 page 30]) If 119881 is a Lyapunov func-tional on 119866 and 119909

119905

(120601) is a bounded solution of (A1) that staysin 119866 then 120596-limit set 120596(120601) sub 119872 that is 119909

119905

(120601) rarr 119872 as119905 rarr +infin

Conflict of Interests

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

References

[1] K L Cooke ldquoStability analysis for a vector disease modelrdquoRocky Mountain Journal of Mathematics vol 9 no 1 pp 31ndash421979

[2] H W Hethcote and P van den Driessche ldquoAn SIS epidemicmodel with variable population size and a delayrdquo Journal ofMathematical Biology vol 34 no 2 pp 177ndash194 1995

[3] J Chin Control of Communicable Diseases Manual AmericanPublic Health Association Washington DC USA 1999

[4] S W Martin Livestock Disease Eradication Evaluation of theCooperative State-Federal Bovine Tuberculosis Eradication Pro-gram National Academy Press Washington DC USA 1994

[5] D Tudor ldquoA deterministic model for herpes infections inhuman and animal populationsrdquo SIAM Review vol 32 no 1pp 136ndash139 1990

[6] K E VanLandingham H B Marsteller G W Ross and FG Hayden ldquoRelapse of herpes simplex encephalitis after con-ventional acyclovir therapyrdquo Journal of the American MedicalAssociation vol 259 no 7 pp 1051ndash1053 1988

[7] A Bignami ldquoConcerning the pathogenesis of relapses in malar-ial feversrdquo Southern Medical Journal vol 6 pp 79ndash89 1913

[8] N JWhite ldquoDeterminants of relapse periodicity inPlasmodiumvivaxmalariardquoMalaria Journal vol 10 article 297 2011

[9] H Cox Y Kebede S Allamuratova et al ldquoTuberculosis recur-rence and mortality after successful treatment impact of drugresistancerdquo PLoS Medicine vol 3 no 10 pp 1836ndash1843 2006

[10] P Van Den Driessche L I N Wang and X Zou ldquoModelingdiseases with latency and relapserdquoMathematical Biosciences andEngineering vol 4 no 2 pp 205ndash219 2007

[11] A Kaddar ldquoOn the dynamics of a delayed SIR epidemic modelwith a modified saturated incidence raterdquo Electronic Journal ofDifferential Equations no 133 pp 1ndash7 2009

[12] S Gao L Chen J J Nieto and A Torres ldquoAnalysis of adelayed epidemic model with pulse vaccination and saturationincidencerdquo Vaccine vol 24 no 35-36 pp 6037ndash6045 2006

[13] J A Yorke and W P London ldquoRecurrent outbreaks of measleschickenpox and mumps IIrdquo American Journal of Epidemiologyvol 98 no 6 pp 469ndash482 1973

[14] M GM Gomes L J White and G F Medley ldquoThe reinfectionthresholdrdquo Journal ofTheoretical Biology vol 236 no 1 pp 111ndash113 2005

[15] Y Zhou and H Liu ldquoStability of periodic solutions for an SISmodel with pulse vaccinationrdquo Mathematical and ComputerModelling vol 38 no 3-4 pp 299ndash308 2003

[16] R M Anderson and R M May ldquoRegulation and stability ofhost-parasite population interactions I Regulatory processesrdquoThe Journal of Animal Ecology vol 47 no 1 pp 219ndash267 1978

[17] F Zhang and Z-Z Li ldquoGlobal stability of an SIR epidemicmodel with constant infectious periodrdquo Applied Mathematicsand Computation vol 199 no 1 pp 285ndash291 2008

[18] Z Jiang and J Wei ldquoStability and bifurcation analysis in adelayed SIR modelrdquo Chaos Solitons and Fractals vol 35 no 3pp 609ndash619 2008

[19] V Capasso and G Serio ldquoA generalization of the Kermack-McKendrick deterministic epidemic modelrdquoMathematical Bio-sciences vol 42 no 1-2 pp 43ndash61 1978

[20] R Xu and Z Ma ldquoStability of a delayed SIRS epidemic modelwith a nonlinear incidence raterdquo Chaos Solitons and Fractalsvol 41 no 5 pp 2319ndash2325 2009

[21] P Georgescu and H Zhang ldquoA Lyapunov functional for aSIRI model with nonlinear incidence of infection and relapserdquoApplied Mathematics and Computation vol 219 no 16 pp8496ndash8507 2013

[22] S Liu S Wang and L Wang ldquoGlobal dynamics of delayepidemic models with nonlinear incidence rate and relapserdquoNonlinear Analysis Real World Applications vol 12 no 1 pp119ndash127 2011

[23] C Castillo-Garsow G Jordan-Salivia and A Rodriguez Her-rera ldquoMathematical models for the dynamics of tobacco userecovery and relapserdquo Technical Report Series BU-1505-MCornell University 2000

[24] A Korobeinikov ldquoLyapunov functions and global stabilityfor SIR and SIRS epidemiological models with non-lineartransmissionrdquo Bulletin of Mathematical Biology vol 68 no 3pp 615ndash626 2006

[25] H N Moreira and Y Wang ldquoGlobal stability in an 119878 rarr 119868 rarr

119877 rarr 119868modelrdquo SIAM Review vol 39 no 3 pp 496ndash502 1997[26] S Blower ldquoModelling the genital herpes epidemicrdquoHerpes vol

11 supplement 3 pp 138Andash146A 2004[27] P van den Driessche and X Zou ldquoModeling relapse in infec-

tious diseasesrdquoMathematical Biosciences vol 207 no 1 pp 89ndash103 2007

[28] A Abta A Kaddar and H T Alaoui ldquoGlobal stability for delaySIR and SEIR epidemic models with saturated incidence ratesrdquoElectronic Journal of Differential Equations vol 2012 no 23 pp1ndash13 2012

[29] C Vargas-De-Leon ldquoOn the global stability of infectious dis-eases models with relapserdquoAbstraction ampApplication vol 9 pp50ndash61 2013

[30] Z Shuai and P van denDriessche ldquoGlobal stability of infectiousdisease models using Lyapunov functionsrdquo SIAM Journal onApplied Mathematics vol 73 no 4 pp 1513ndash1532 2013

[31] R Xu ldquoGlobal dynamics of an SEIRI epidemiological modelwith time delayrdquo Applied Mathematics and Computation vol232 pp 436ndash444 2014

[32] Y KuangDifferentiel EquationsWith Applications in PopulationDynamics Academic Press New York NY USA 1993

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 International Journal of Engineering Mathematics

Here 120595120591

= 120595(119905 minus 120591) for any given function 120595 119860 = 120583119873where 119873 = 119878 + 119868 + 119877 is the total number of population119878 is the number of susceptible individuals 119868 is the numberof infectious individuals 119877 is the number of recoveredindividuals 119860 is the recruitment rate of the population 120583 isthe natural death of the population 120572 is the death rate dueto disease 119891 is the nonlinear incidence function 120574 is therecovery rate of the infective individuals 120575 is the rate thatrecovered individuals relapse and regained infectious classand 120591 is the latent period

In model (1) the incidence function 119891(119878 119868) is a locallyLipschitz continuous function on 119877+ times119877+ satisfying 119891(0 119868) =119891(119878 0) = 0 for 119878 ge 0 119868 ge 0 and the following hold

(1198671

) 119891 is a strictly monotone increasing function of 119878 ge

0 for any fixed 119868 gt 0 and 119891 is a strictly monotoneincreasing function of 119868 ge 0 for any fixed 119878 ge 0

(1198672

) 120601(119878 119868) = 119891(119878 119868)119868 is a bounded and monotonedecreasing function of 119868 gt 0 for any fixed 119878 ge 0 and119870(119878) = lim

119868rarr0

+120601(119878 119868) is a continuous andmonotoneincreasing function on 119878 ge 0

This incidence function includes different forms pre-sented in literature (see eg [12ndash23])

System (1) always has a disease-free equilibrium 1198750

=

(119860120583 0 0) On the other hand under the hypothesis (1198672

) if

1198770

=119870 (119860120583) 119890

minus120583120591

120583 + 120574 + 120572 minus 120574120575 (120583 + 120575)gt 1 (3)

then system (1) also admits a unique endemic equilibrium119875lowast

= (119878lowast

119868lowast

119877lowast

) where 119878lowast 119868lowast and 119877lowast satisfy the followingsystem (see [24])

119860 minus 120583119878 minus 119891 (119878 119868) = 0

119890minus120583120591

119891 (119878 119868) minus (120583 + 120574 + 120572) 119868 + 120575119877 = 0

120574119868 minus (120583 + 120575) 119877 = 0

(4)

Hereafter we replace 120583 + 120574 + 120572 minus 120574120575(120583 + 120575) by 120578In [5] (1990) Tudor developed and analyzed qualitatively

one of the first SIRI epidemic models for the spread of aherpes-type infection in either human or animal populationsThis model consists of a system of nonlinear ordinary differ-ential equations with a bilinear incidence rate (ie 119891(119878 119868) =120573119878119868) and a constant total population (ie 119873 = 119878 + 119868 + 119877 =

constant)In [25] (1997) Moreira and Wang extended a Tudor-

model to include nonlinear incidence functions By usingan elementary analysis of Lienardrsquos equation and Lyapunovrsquosdirect method they derived sufficient conditions for theglobal asymptotic stability of the disease-free and endemicequilibria

In [23] (2000) Castillo-Garsow et al considered anSIRI model for drug use in a population of adolescentsThe authors assumed that 119873 = constant and 119891(119878 119868) =

120573119878119868119873 they estimated the parameters of the model and theydetermined a rough approximation of the basic reproductivenumber Based on these parameters they performed some

simulations that clearly showed the endemic character oftobacco use among adolescents

In [26] (2004) Blower developed a compartmentalmodelfor genital herpes assuming standard incidence rate (ie119891(119878 119868) = 120573119878119868119873) and constant recruitment rate

In [24] (2006) Korobeinikov proved that the endemicequilibrium of the model (1) with 120575 = 0 and 120591 = 0 is globallyasymptotically stable

In [27] (2007) van den Driessche and Zou proposedan integrodifferential equation to model a general relapsephenomenon in infectious diseases The resulting model inparticular case is a delay differential equation with a constantpopulation and standard incidence The basic reproductionnumber for this model is identified and some global resultsare obtained by employing the Lyapunov-Razumikhin tech-nique

In [10] (2007) Van DenDriessche and co-authors formu-lated a delay differential SIRImodel (System (1) with119891(119878 119868) =120573119878119868) For this system the endemic equilibrium is locallyasymptotically stable if 119877

0

gt 1 and the disease is shown to beuniformly persistent with the infective population size eitherapproaching or oscillating about the endemic level

In [22] (2011) Liu et al proposed a mathematical modelfor a disease with a general exposed distribution the pos-sibility of relapse and nonlinear incidence rate (119891(119878 119868) =

120573119878(119868119873)) By the method of Lyapunov functionals theyshowed that the disease dies out if 119877

0

= 1 and that the diseasebecomes endemic if 119877

0

gt 1 They also analyzed as a specialcase of this model the system (1) with 119891(119878 119868) = 119892(119878) sdot 119868the result confirms that the endemic equilibrium is globallyasymptotically stable

In [28] (2012) Abta et al considered a global asymptoticstability of a delayed SIR model (system (1) with 119891(119878 119868) =

120573119878119868(1 + 1205721

119878 + 1205722

119868) and 120575 = 0)In [29] (2013) Vargas-De-Leon presented the global sta-

bility conditions of an ordinary SIRI model with bilinear andstandard incidence rates respectively that includes recruit-ment rate of susceptible individuals into the communityand that the disease produces nonnegligible death in theinfectious class The author presented the construction ofLyapunov functions using suitable combinations of knownfunctions common quadratic and Volterra-type and a com-posite Volterra-type function

In [21] (2013) Georgescu and Zhang analyzed the dynam-ics of an ordinary SIRI model under the assumption that theincidence of infection is given by 119891(119878 119868) = 119862(119878)119892(119868) Theyobtained by means of Lyapunovrsquos second method sufficientconditions for the local stability of equilibria and theyshowed that global stability can be attained under suitablemonotonicity conditions

In [30] (2013) Shuai and van den Driessche presentedtwo systematic methods for the construction of Lyapunovfunctions for general infectious disease models (OrdinarySEIRI SIS etc) Specifically amatrix-theoreticmethod usingthe Perron eigenvector is applied to prove the global stabil-ity of the disease-free equilibrium while a graph-theoreticmethod based on Kirchhoff rsquos matrix tree theorem and twonew combinatorial identities are used to prove the globalstability of the endemic equilibrium

International Journal of Engineering Mathematics 3

In [31] (2014) Xu investigated a delayed SIRI model(system (1) with 119891(119878 119868) = 120573119878119868) The author established theglobal stability of a disease-free equilibrium and an endemicequilibrium by means of suitable Lyapunov functionals andLaSallersquos invariance principle

In this paper we extend the global stability results pre-sented in [31] to a delayed SIRI epidemic model (system (1))with a general nonlinear incidence function It is shown thatglobal stability can be attained under suitable monotonicityconditions and it is established that the basic reproductionnumber 119877

0

is a threshold parameter for the stability of adelayed SIRI model The rest of the paper is organized asfollows In Section 2 the global stability of disease-free andendemic equilibria are established In Section 3 numericalsimulations and concluding remarks are provided In theappendix some results on the global stability are stated

2 Global Stability Analysis of DelayedSIRI Model

In this section we discuss the global stability of a disease-freeequilibrium 119875

0

and an endemic equilibrium 119875lowast of system (1)

Since (119889119889119905)(119878+119868+119877) le 119860minus120583(119878+119868+119877) we have lim sup(119878+119868 + 119877) le 119860120583 Hence we discuss system (1) in the closed set

Ω

= (1205931

1205932

1205933

) isin 119862+

times 119862+

times 119862+

10038171003817100381710038171205931 + 1205932 + 1205933

1003817100381710038171003817 le119860

120583

(5)

It is easy to show thatΩ is positively invariant with respect tosystem (1) Next we consider the global asymptotic stability ofthe disease-free equilibrium 119875

0

and the endemic equilibrium119875lowast of (1) by Lyapunov functionals respectively

Proposition 1 If 1198770

le 1 then the disease-free equilibrium 1198750

is globally asymptotically stable

Proof Define a Lyapunov functional

1198810

(119905) = 119890minus120583120591

int

119878

120591

119860120583

(1 minus119870 (119860120583)

119870 (119906)) 119889119906 + 119868

+ 120578int

119905

119905minus120591

119868 (119906) 119889119906 +120575

120583 + 120575119877

(6)

We will show that 1198891198810

(119905)119889119905 le 0 for all 119905 ge 0 We have

1198891198810

(119905)

119889119905= 119890minus120583120591

(1 minus119870 (119860120583)

119870 (119878120591

)) 119878120591

+ 119890minus120583120591

119891 (119878120591

119868120591

) 119889120591

minus (120583 + 120574 + 120572) 119868 + 120575119877 + 120578 (119868 minus 119868120591

)

+120574120575

120583 + 120575119868 minus 120575119877

= 119890minus120583120591

(1 minus119870 (119860120583)

119870 (119878120591

)) (119860 minus 120583119878

120591

)

+119870 (119860120583)

119870 (119878120591

)119890minus120583120591

119891 (119878120591

119868120591

) minus 120578119868120591

= 120583119890minus120583120591

(1 minus119870 (119860120583)

119870 (119878120591

))(

119860

120583minus 119878120591

)

+ 120578119868120591

(120601 (119878120591

119868120591

)

120578

119870 (119860120583)

119870 (119878120591

)119890minus120583120591

minus 1)

(7)

By the hypothesis (1198671

) we obtain that

(1 minus119870 (119860120583)

119870 (119878120591

))(

119860

120583minus 119878120591

) le 0 (8)

where equality holds if and only if 119878 = 119860120583Furthermore It follows from the hypothesis (119867

2

) that

120601 (119878120591

119868120591

)

120578

119870 (119860120583) 119890minus120583120591

119870(119878120591

)le119870 (119878120591

)

120578

119870 (119860120583) 119890minus120583120591

119870(119878120591

)

le119870 (119860120583) 119890

minus120583120591

120578

le 1198770

(9)

Therefore 1198770

le 1 ensures that 1198891198810

(119905)119889119905 le 0 for all 119905 ge 0where 119889119881

0

(119905)119889119905 = 0 holds if (119878 119868 119877) = (119860120583 0 0) Henceit follows from system (1) that 119875

0

is the largest invariantset in (119878 119868 119877) | 119889119881

0

(119905)119889119905 = 0 From the Lyapunov-LaSalle asymptotic stability we obtain that 119875

0

is globallyasymptotically stable This completes the proof

Proposition 2 If 1198770

gt 1 then the endemic equilibrium 119875lowast

1

isglobally asymptotically stable

Proof To prove global stability of the endemic equilibriumwe define a Lyapunov functional119881(119905) = 119881

1

(119905)+1198812

(119905)+1198813

(119905)+

119881+

with

1198811

(119905) = 119878 minus 119878lowast

minus int

119878

119878

lowast

119891 (119878lowast

119868lowast

)

119891 (119906 119868lowast)119889119906

1198812

(119905) = 119890120583120591

(119868 minus 119868lowast

minus 119868lowast ln 119868

119868lowast)

1198813

(119905) =120575119890120583120591

120583 + 120575(119877 minus 119877

lowast

minus 119877lowast ln 119877

119877lowast)

119881+

= int

119905

119905minus120591

(119891 (119878 119868) minus 119891 (119878lowast

119868lowast

) minus 119891 (119878lowast

119868lowast

)

sdot ln119891 (119878 (119906) 119868 (119906))

119891 (119878lowast 119868lowast)) 119889119906

(10)

4 International Journal of Engineering Mathematics

The time derivative of the function 119881(119905) along the positivesolution of system (1) is

119889119881 (119905)

119889119905= (1 minus

119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)) (119860 minus 120583119878 minus 119891 (119878 119868))

+ 119890120583120591

(1 minus119868lowast

119868)

sdot (119890minus120583120591

119891 (119878120591

119868120591

+ 120575119877) minus (120583 + 120572 + 120574) 119868)

+120575119890120583120591

120583 + 120575(1 minus

119877lowast

119877) (120574119868 minus (120583 + 120575) 119877)

+ 119891 (119878 119868) minus 119891 (119878120591

119868120591

) + 119891 (119878lowast

119868lowast

)

sdot ln119891 (119878120591

119868120591

)

119891 (119878 119868)

(11)

Using the relation 119860 = 120583119878lowast

+ 119891(119878lowast

119868lowast

) simple calculationsgive

119889119881 (119905)

119889119905

= (1 minus119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)) (minus120583 (119878 minus 119878

lowast

) + 119891 (119878lowast

119868lowast

))

+119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)119891 (119878 119868) minus 119890

120583120591

(120583 + 120572 + 120574) 119868

minus119868lowast

119868119891 (119878120591

119868120591

) minus 120575119890120583120591

119868lowast

119868119877 + 119890120583120591

(120583 + 120572 + 120574) 119868lowast

+120575119890120583120591

120583 + 120575(120574119868 minus 120574119868

119877lowast

119877+ (120583 + 120575) 119877

lowast

)

+ 119891 (119878lowast

119868lowast

) ln119891 (119878120591

119868120591

)

119891 (119878 119868)

(12)

Here by using

119890120583120591

(120583 + 120572 + 120574) 119868lowast

minus120575119890120583120591

120583 + 120575120574119868lowast

= 119891 (119878lowast

119868lowast

)

(120583 + 120575) 119877lowast

= 120574119868lowast

ln119891 (119878120591

119868120591

)

119891 (119878 119868)

= ln119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)+ ln

119868lowast

119891 (119878120591

119868120591

)

119868119891 (119878lowast 119868lowast)+ ln

119868119891 (119878 119868lowast

)

119868lowast119891 (119878 119868)

(13)

straightforward calculations give

119889119881 (119905)

119889119905

= minus120583(1 minus119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)) (119878 minus 119878

lowast

)

minus 119891 (119878lowast

119868lowast

) (119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)minus 1 minus ln

119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast))

minus 119891 (119878lowast

119868lowast

) (119868lowast

119891 (119878120591

119868120591

)

119868119891 (119878lowast 119868lowast)minus 1 minus ln

119868lowast

119891 (119878120591

119868120591

)

119868119891 (119878lowast 119868lowast))

+ 119891 (119878lowast

119868lowast

) (119891 (119878 119868)

119891 (119878 119868lowast)minus

119868

119868lowast+119868119891 (119878 119868

lowast

)

119868lowast119891 (119878 119868)minus 1)

minus 119891 (119878lowast

119868lowast

) (119868119891 (119878 119868

lowast

)

119868lowast119891 (119878 119868)minus 1 minus ln

119868119891 (119878 119868lowast

)

119868lowast119891 (119878 119868))

+120575120574119868lowast

119890120583120591

120583 + 120575(2 minus

119868lowast

119877

119868119877lowastminus119868119877lowast

119868lowast119877)

= minus120583(1 minus119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)) (119878 minus 119878

lowast

)

minus 119891 (119878lowast

119868lowast

) (119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)minus 1 minus ln

119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast))

minus 119891 (119878lowast

119868lowast

) (119868lowast

119891 (119878120591

119868120591

)

119868119891 (119878 119868lowast)minus 1 minus ln

119868lowast

119891 (119878120591

119868120591

)

119868119891 (119878 119868lowast))

+119868

119868lowast119891 (119878lowast

119868lowast

) (1 minus119891 (119878 119868

lowast

)

119891 (119878 119868))(

120601 (119878 119868)

120601 (119878 119868lowast)minus 1)

minus 119891 (119878lowast

119868lowast

) (119868119891 (119878 119868

lowast

)

119868lowast119891 (119878 119868)minus 1 minus ln

119868119891 (119878 119868lowast

)

119868lowast119891 (119878 119868))

minus120575120574119890120583120591

120583 + 120575

119868119877lowast

119877(119868lowast

119877

119868119877lowastminus 1)

2

(14)

It follows from (1198671

) and (1198672

) that

minus120583(1 minus119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)) (119878 minus 119878

lowast

) le 0

119868

119868lowast119891 (119878lowast

119868lowast

) (1 minus119891 (119878 119868

lowast

)

119891 (119878 119868))(

120601 (119878 119868)

120601 (119878 119868lowast)minus 1)

le 0

(15)

Furthermore since the function 119892(119909) = 1minus119909+ ln(119909) is alwaysnonpositive for any 119909 gt 0 and 119892(119909) = 0 if and only if 119909 = 1then 119889119881(119905)119889119905 le 0 for all 119905 ge 0 where the equality holds onlyat the equilibrium point (119878lowast 119868lowast 119877lowast) Hence the functional119881satisfies all the conditions of Theorem A2 This proves that119875lowast

1

is globally asymptotically stable

International Journal of Engineering Mathematics 5

3 Numerical Simulations andConcluding Remarks

In this section we give a numerical simulation supporting thetheoretical analysis given in Section 2 Let

119891 (119878 119868) =120573119878119868

1 + 1205721

119878 + 1205722

119868 (16)

We take the parameters of the system (1) as follows

119860 = 10

1205721

= 09

1205722

= 09

120583 = 001

120574 = 002

120572 = 0005

120573 = 01

120575 = 0001

120591 = 10

(17)

By Proposition 2 the endemic equilibrium 119875lowast is globally

asymptotically stable see Figure 1In this paper we presented a mathematical analysis and

numerical simulations for an SIRI epidemiological modelapplied to the evolution of the spread of disease with relapsein a given population We denote 119877

0

the basic reproductionnumber It is defined as the average number of contagiouspersons infected by a typical infectious in a population of sus-ceptible We prove in this paper that the basic reproductionnumber 119877

0

depends on the incubation period and we showthat the disease-free equilibrium 119875

0

is globally asymptoticallystable if 119877

0

le 1 and that a unique endemic equilibrium 119875lowast is

globally asymptotically stable if 1198770

gt 1

Appendix

The Lyapunov-LaSalle Theorem

In the following we present the method of Lyapunov func-tionals in the context of a delay differential equations

119889119909

119889119905= 119891 (119909

119905

) (A1)

where 119891 119862 rarr 119877119899 is completely continuous and solutions

of (A1) are unique and continuously dependent on the initialdata We denote by 119909(120601) the solution of (A1) through (0 120601)For a continuous functional 119881 119862 rarr 119877 we define

= lim supℎrarr0

+

1

ℎ[(119881119909ℎ

(120601)) minus 119881 (120601)] (A2)

the derivative of 119881 along a solution of (A1) To statethe Lyapunov-LaSalle type theorem for (A1) we need thefollowing definition

0 100 200 300 400 5000

50

100

150

200

250

300

350

400

450

500

t

0 100 200 300 400 5000

50

100

150

200

250

300

350

400

450

t

I(t)

0 100 200 300 400 500100

200

300

400

500

600

700

800

900

1000

t

S(t)

R(t)

Figure 1 Solutions (119878 119868 119877) of the SIRI epidemic model (1) are glob-ally asymptotically stable and converge to the endemic equilibrium119875lowast

Definition A1 (see [32 page 30]) We say 119881 119862 rarr 119877

is a Lyapunov functional on a set 119866 in 119862 for (A1) if it iscontinuous on 119866 (the closure of 119866) and le 0 on 119866 We also

6 International Journal of Engineering Mathematics

define 119864 = 120601 isin 119866 (120601) = 0 and119872 is the largest set in 119864which is invariant with respect to (A1)

The following result is the Lyapunov-LaSalle type theo-rem for (A1)

Theorem A2 (see [32 page 30]) If 119881 is a Lyapunov func-tional on 119866 and 119909

119905

(120601) is a bounded solution of (A1) that staysin 119866 then 120596-limit set 120596(120601) sub 119872 that is 119909

119905

(120601) rarr 119872 as119905 rarr +infin

Conflict of Interests

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

References

[1] K L Cooke ldquoStability analysis for a vector disease modelrdquoRocky Mountain Journal of Mathematics vol 9 no 1 pp 31ndash421979

[2] H W Hethcote and P van den Driessche ldquoAn SIS epidemicmodel with variable population size and a delayrdquo Journal ofMathematical Biology vol 34 no 2 pp 177ndash194 1995

[3] J Chin Control of Communicable Diseases Manual AmericanPublic Health Association Washington DC USA 1999

[4] S W Martin Livestock Disease Eradication Evaluation of theCooperative State-Federal Bovine Tuberculosis Eradication Pro-gram National Academy Press Washington DC USA 1994

[5] D Tudor ldquoA deterministic model for herpes infections inhuman and animal populationsrdquo SIAM Review vol 32 no 1pp 136ndash139 1990

[6] K E VanLandingham H B Marsteller G W Ross and FG Hayden ldquoRelapse of herpes simplex encephalitis after con-ventional acyclovir therapyrdquo Journal of the American MedicalAssociation vol 259 no 7 pp 1051ndash1053 1988

[7] A Bignami ldquoConcerning the pathogenesis of relapses in malar-ial feversrdquo Southern Medical Journal vol 6 pp 79ndash89 1913

[8] N JWhite ldquoDeterminants of relapse periodicity inPlasmodiumvivaxmalariardquoMalaria Journal vol 10 article 297 2011

[9] H Cox Y Kebede S Allamuratova et al ldquoTuberculosis recur-rence and mortality after successful treatment impact of drugresistancerdquo PLoS Medicine vol 3 no 10 pp 1836ndash1843 2006

[10] P Van Den Driessche L I N Wang and X Zou ldquoModelingdiseases with latency and relapserdquoMathematical Biosciences andEngineering vol 4 no 2 pp 205ndash219 2007

[11] A Kaddar ldquoOn the dynamics of a delayed SIR epidemic modelwith a modified saturated incidence raterdquo Electronic Journal ofDifferential Equations no 133 pp 1ndash7 2009

[12] S Gao L Chen J J Nieto and A Torres ldquoAnalysis of adelayed epidemic model with pulse vaccination and saturationincidencerdquo Vaccine vol 24 no 35-36 pp 6037ndash6045 2006

[13] J A Yorke and W P London ldquoRecurrent outbreaks of measleschickenpox and mumps IIrdquo American Journal of Epidemiologyvol 98 no 6 pp 469ndash482 1973

[14] M GM Gomes L J White and G F Medley ldquoThe reinfectionthresholdrdquo Journal ofTheoretical Biology vol 236 no 1 pp 111ndash113 2005

[15] Y Zhou and H Liu ldquoStability of periodic solutions for an SISmodel with pulse vaccinationrdquo Mathematical and ComputerModelling vol 38 no 3-4 pp 299ndash308 2003

[16] R M Anderson and R M May ldquoRegulation and stability ofhost-parasite population interactions I Regulatory processesrdquoThe Journal of Animal Ecology vol 47 no 1 pp 219ndash267 1978

[17] F Zhang and Z-Z Li ldquoGlobal stability of an SIR epidemicmodel with constant infectious periodrdquo Applied Mathematicsand Computation vol 199 no 1 pp 285ndash291 2008

[18] Z Jiang and J Wei ldquoStability and bifurcation analysis in adelayed SIR modelrdquo Chaos Solitons and Fractals vol 35 no 3pp 609ndash619 2008

[19] V Capasso and G Serio ldquoA generalization of the Kermack-McKendrick deterministic epidemic modelrdquoMathematical Bio-sciences vol 42 no 1-2 pp 43ndash61 1978

[20] R Xu and Z Ma ldquoStability of a delayed SIRS epidemic modelwith a nonlinear incidence raterdquo Chaos Solitons and Fractalsvol 41 no 5 pp 2319ndash2325 2009

[21] P Georgescu and H Zhang ldquoA Lyapunov functional for aSIRI model with nonlinear incidence of infection and relapserdquoApplied Mathematics and Computation vol 219 no 16 pp8496ndash8507 2013

[22] S Liu S Wang and L Wang ldquoGlobal dynamics of delayepidemic models with nonlinear incidence rate and relapserdquoNonlinear Analysis Real World Applications vol 12 no 1 pp119ndash127 2011

[23] C Castillo-Garsow G Jordan-Salivia and A Rodriguez Her-rera ldquoMathematical models for the dynamics of tobacco userecovery and relapserdquo Technical Report Series BU-1505-MCornell University 2000

[24] A Korobeinikov ldquoLyapunov functions and global stabilityfor SIR and SIRS epidemiological models with non-lineartransmissionrdquo Bulletin of Mathematical Biology vol 68 no 3pp 615ndash626 2006

[25] H N Moreira and Y Wang ldquoGlobal stability in an 119878 rarr 119868 rarr

119877 rarr 119868modelrdquo SIAM Review vol 39 no 3 pp 496ndash502 1997[26] S Blower ldquoModelling the genital herpes epidemicrdquoHerpes vol

11 supplement 3 pp 138Andash146A 2004[27] P van den Driessche and X Zou ldquoModeling relapse in infec-

tious diseasesrdquoMathematical Biosciences vol 207 no 1 pp 89ndash103 2007

[28] A Abta A Kaddar and H T Alaoui ldquoGlobal stability for delaySIR and SEIR epidemic models with saturated incidence ratesrdquoElectronic Journal of Differential Equations vol 2012 no 23 pp1ndash13 2012

[29] C Vargas-De-Leon ldquoOn the global stability of infectious dis-eases models with relapserdquoAbstraction ampApplication vol 9 pp50ndash61 2013

[30] Z Shuai and P van denDriessche ldquoGlobal stability of infectiousdisease models using Lyapunov functionsrdquo SIAM Journal onApplied Mathematics vol 73 no 4 pp 1513ndash1532 2013

[31] R Xu ldquoGlobal dynamics of an SEIRI epidemiological modelwith time delayrdquo Applied Mathematics and Computation vol232 pp 436ndash444 2014

[32] Y KuangDifferentiel EquationsWith Applications in PopulationDynamics Academic Press New York NY USA 1993

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

International Journal of Engineering Mathematics 3

In [31] (2014) Xu investigated a delayed SIRI model(system (1) with 119891(119878 119868) = 120573119878119868) The author established theglobal stability of a disease-free equilibrium and an endemicequilibrium by means of suitable Lyapunov functionals andLaSallersquos invariance principle

In this paper we extend the global stability results pre-sented in [31] to a delayed SIRI epidemic model (system (1))with a general nonlinear incidence function It is shown thatglobal stability can be attained under suitable monotonicityconditions and it is established that the basic reproductionnumber 119877

0

is a threshold parameter for the stability of adelayed SIRI model The rest of the paper is organized asfollows In Section 2 the global stability of disease-free andendemic equilibria are established In Section 3 numericalsimulations and concluding remarks are provided In theappendix some results on the global stability are stated

2 Global Stability Analysis of DelayedSIRI Model

In this section we discuss the global stability of a disease-freeequilibrium 119875

0

and an endemic equilibrium 119875lowast of system (1)

Since (119889119889119905)(119878+119868+119877) le 119860minus120583(119878+119868+119877) we have lim sup(119878+119868 + 119877) le 119860120583 Hence we discuss system (1) in the closed set

Ω

= (1205931

1205932

1205933

) isin 119862+

times 119862+

times 119862+

10038171003817100381710038171205931 + 1205932 + 1205933

1003817100381710038171003817 le119860

120583

(5)

It is easy to show thatΩ is positively invariant with respect tosystem (1) Next we consider the global asymptotic stability ofthe disease-free equilibrium 119875

0

and the endemic equilibrium119875lowast of (1) by Lyapunov functionals respectively

Proposition 1 If 1198770

le 1 then the disease-free equilibrium 1198750

is globally asymptotically stable

Proof Define a Lyapunov functional

1198810

(119905) = 119890minus120583120591

int

119878

120591

119860120583

(1 minus119870 (119860120583)

119870 (119906)) 119889119906 + 119868

+ 120578int

119905

119905minus120591

119868 (119906) 119889119906 +120575

120583 + 120575119877

(6)

We will show that 1198891198810

(119905)119889119905 le 0 for all 119905 ge 0 We have

1198891198810

(119905)

119889119905= 119890minus120583120591

(1 minus119870 (119860120583)

119870 (119878120591

)) 119878120591

+ 119890minus120583120591

119891 (119878120591

119868120591

) 119889120591

minus (120583 + 120574 + 120572) 119868 + 120575119877 + 120578 (119868 minus 119868120591

)

+120574120575

120583 + 120575119868 minus 120575119877

= 119890minus120583120591

(1 minus119870 (119860120583)

119870 (119878120591

)) (119860 minus 120583119878

120591

)

+119870 (119860120583)

119870 (119878120591

)119890minus120583120591

119891 (119878120591

119868120591

) minus 120578119868120591

= 120583119890minus120583120591

(1 minus119870 (119860120583)

119870 (119878120591

))(

119860

120583minus 119878120591

)

+ 120578119868120591

(120601 (119878120591

119868120591

)

120578

119870 (119860120583)

119870 (119878120591

)119890minus120583120591

minus 1)

(7)

By the hypothesis (1198671

) we obtain that

(1 minus119870 (119860120583)

119870 (119878120591

))(

119860

120583minus 119878120591

) le 0 (8)

where equality holds if and only if 119878 = 119860120583Furthermore It follows from the hypothesis (119867

2

) that

120601 (119878120591

119868120591

)

120578

119870 (119860120583) 119890minus120583120591

119870(119878120591

)le119870 (119878120591

)

120578

119870 (119860120583) 119890minus120583120591

119870(119878120591

)

le119870 (119860120583) 119890

minus120583120591

120578

le 1198770

(9)

Therefore 1198770

le 1 ensures that 1198891198810

(119905)119889119905 le 0 for all 119905 ge 0where 119889119881

0

(119905)119889119905 = 0 holds if (119878 119868 119877) = (119860120583 0 0) Henceit follows from system (1) that 119875

0

is the largest invariantset in (119878 119868 119877) | 119889119881

0

(119905)119889119905 = 0 From the Lyapunov-LaSalle asymptotic stability we obtain that 119875

0

is globallyasymptotically stable This completes the proof

Proposition 2 If 1198770

gt 1 then the endemic equilibrium 119875lowast

1

isglobally asymptotically stable

Proof To prove global stability of the endemic equilibriumwe define a Lyapunov functional119881(119905) = 119881

1

(119905)+1198812

(119905)+1198813

(119905)+

119881+

with

1198811

(119905) = 119878 minus 119878lowast

minus int

119878

119878

lowast

119891 (119878lowast

119868lowast

)

119891 (119906 119868lowast)119889119906

1198812

(119905) = 119890120583120591

(119868 minus 119868lowast

minus 119868lowast ln 119868

119868lowast)

1198813

(119905) =120575119890120583120591

120583 + 120575(119877 minus 119877

lowast

minus 119877lowast ln 119877

119877lowast)

119881+

= int

119905

119905minus120591

(119891 (119878 119868) minus 119891 (119878lowast

119868lowast

) minus 119891 (119878lowast

119868lowast

)

sdot ln119891 (119878 (119906) 119868 (119906))

119891 (119878lowast 119868lowast)) 119889119906

(10)

4 International Journal of Engineering Mathematics

The time derivative of the function 119881(119905) along the positivesolution of system (1) is

119889119881 (119905)

119889119905= (1 minus

119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)) (119860 minus 120583119878 minus 119891 (119878 119868))

+ 119890120583120591

(1 minus119868lowast

119868)

sdot (119890minus120583120591

119891 (119878120591

119868120591

+ 120575119877) minus (120583 + 120572 + 120574) 119868)

+120575119890120583120591

120583 + 120575(1 minus

119877lowast

119877) (120574119868 minus (120583 + 120575) 119877)

+ 119891 (119878 119868) minus 119891 (119878120591

119868120591

) + 119891 (119878lowast

119868lowast

)

sdot ln119891 (119878120591

119868120591

)

119891 (119878 119868)

(11)

Using the relation 119860 = 120583119878lowast

+ 119891(119878lowast

119868lowast

) simple calculationsgive

119889119881 (119905)

119889119905

= (1 minus119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)) (minus120583 (119878 minus 119878

lowast

) + 119891 (119878lowast

119868lowast

))

+119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)119891 (119878 119868) minus 119890

120583120591

(120583 + 120572 + 120574) 119868

minus119868lowast

119868119891 (119878120591

119868120591

) minus 120575119890120583120591

119868lowast

119868119877 + 119890120583120591

(120583 + 120572 + 120574) 119868lowast

+120575119890120583120591

120583 + 120575(120574119868 minus 120574119868

119877lowast

119877+ (120583 + 120575) 119877

lowast

)

+ 119891 (119878lowast

119868lowast

) ln119891 (119878120591

119868120591

)

119891 (119878 119868)

(12)

Here by using

119890120583120591

(120583 + 120572 + 120574) 119868lowast

minus120575119890120583120591

120583 + 120575120574119868lowast

= 119891 (119878lowast

119868lowast

)

(120583 + 120575) 119877lowast

= 120574119868lowast

ln119891 (119878120591

119868120591

)

119891 (119878 119868)

= ln119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)+ ln

119868lowast

119891 (119878120591

119868120591

)

119868119891 (119878lowast 119868lowast)+ ln

119868119891 (119878 119868lowast

)

119868lowast119891 (119878 119868)

(13)

straightforward calculations give

119889119881 (119905)

119889119905

= minus120583(1 minus119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)) (119878 minus 119878

lowast

)

minus 119891 (119878lowast

119868lowast

) (119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)minus 1 minus ln

119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast))

minus 119891 (119878lowast

119868lowast

) (119868lowast

119891 (119878120591

119868120591

)

119868119891 (119878lowast 119868lowast)minus 1 minus ln

119868lowast

119891 (119878120591

119868120591

)

119868119891 (119878lowast 119868lowast))

+ 119891 (119878lowast

119868lowast

) (119891 (119878 119868)

119891 (119878 119868lowast)minus

119868

119868lowast+119868119891 (119878 119868

lowast

)

119868lowast119891 (119878 119868)minus 1)

minus 119891 (119878lowast

119868lowast

) (119868119891 (119878 119868

lowast

)

119868lowast119891 (119878 119868)minus 1 minus ln

119868119891 (119878 119868lowast

)

119868lowast119891 (119878 119868))

+120575120574119868lowast

119890120583120591

120583 + 120575(2 minus

119868lowast

119877

119868119877lowastminus119868119877lowast

119868lowast119877)

= minus120583(1 minus119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)) (119878 minus 119878

lowast

)

minus 119891 (119878lowast

119868lowast

) (119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)minus 1 minus ln

119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast))

minus 119891 (119878lowast

119868lowast

) (119868lowast

119891 (119878120591

119868120591

)

119868119891 (119878 119868lowast)minus 1 minus ln

119868lowast

119891 (119878120591

119868120591

)

119868119891 (119878 119868lowast))

+119868

119868lowast119891 (119878lowast

119868lowast

) (1 minus119891 (119878 119868

lowast

)

119891 (119878 119868))(

120601 (119878 119868)

120601 (119878 119868lowast)minus 1)

minus 119891 (119878lowast

119868lowast

) (119868119891 (119878 119868

lowast

)

119868lowast119891 (119878 119868)minus 1 minus ln

119868119891 (119878 119868lowast

)

119868lowast119891 (119878 119868))

minus120575120574119890120583120591

120583 + 120575

119868119877lowast

119877(119868lowast

119877

119868119877lowastminus 1)

2

(14)

It follows from (1198671

) and (1198672

) that

minus120583(1 minus119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)) (119878 minus 119878

lowast

) le 0

119868

119868lowast119891 (119878lowast

119868lowast

) (1 minus119891 (119878 119868

lowast

)

119891 (119878 119868))(

120601 (119878 119868)

120601 (119878 119868lowast)minus 1)

le 0

(15)

Furthermore since the function 119892(119909) = 1minus119909+ ln(119909) is alwaysnonpositive for any 119909 gt 0 and 119892(119909) = 0 if and only if 119909 = 1then 119889119881(119905)119889119905 le 0 for all 119905 ge 0 where the equality holds onlyat the equilibrium point (119878lowast 119868lowast 119877lowast) Hence the functional119881satisfies all the conditions of Theorem A2 This proves that119875lowast

1

is globally asymptotically stable

International Journal of Engineering Mathematics 5

3 Numerical Simulations andConcluding Remarks

In this section we give a numerical simulation supporting thetheoretical analysis given in Section 2 Let

119891 (119878 119868) =120573119878119868

1 + 1205721

119878 + 1205722

119868 (16)

We take the parameters of the system (1) as follows

119860 = 10

1205721

= 09

1205722

= 09

120583 = 001

120574 = 002

120572 = 0005

120573 = 01

120575 = 0001

120591 = 10

(17)

By Proposition 2 the endemic equilibrium 119875lowast is globally

asymptotically stable see Figure 1In this paper we presented a mathematical analysis and

numerical simulations for an SIRI epidemiological modelapplied to the evolution of the spread of disease with relapsein a given population We denote 119877

0

the basic reproductionnumber It is defined as the average number of contagiouspersons infected by a typical infectious in a population of sus-ceptible We prove in this paper that the basic reproductionnumber 119877

0

depends on the incubation period and we showthat the disease-free equilibrium 119875

0

is globally asymptoticallystable if 119877

0

le 1 and that a unique endemic equilibrium 119875lowast is

globally asymptotically stable if 1198770

gt 1

Appendix

The Lyapunov-LaSalle Theorem

In the following we present the method of Lyapunov func-tionals in the context of a delay differential equations

119889119909

119889119905= 119891 (119909

119905

) (A1)

where 119891 119862 rarr 119877119899 is completely continuous and solutions

of (A1) are unique and continuously dependent on the initialdata We denote by 119909(120601) the solution of (A1) through (0 120601)For a continuous functional 119881 119862 rarr 119877 we define

= lim supℎrarr0

+

1

ℎ[(119881119909ℎ

(120601)) minus 119881 (120601)] (A2)

the derivative of 119881 along a solution of (A1) To statethe Lyapunov-LaSalle type theorem for (A1) we need thefollowing definition

0 100 200 300 400 5000

50

100

150

200

250

300

350

400

450

500

t

0 100 200 300 400 5000

50

100

150

200

250

300

350

400

450

t

I(t)

0 100 200 300 400 500100

200

300

400

500

600

700

800

900

1000

t

S(t)

R(t)

Figure 1 Solutions (119878 119868 119877) of the SIRI epidemic model (1) are glob-ally asymptotically stable and converge to the endemic equilibrium119875lowast

Definition A1 (see [32 page 30]) We say 119881 119862 rarr 119877

is a Lyapunov functional on a set 119866 in 119862 for (A1) if it iscontinuous on 119866 (the closure of 119866) and le 0 on 119866 We also

6 International Journal of Engineering Mathematics

define 119864 = 120601 isin 119866 (120601) = 0 and119872 is the largest set in 119864which is invariant with respect to (A1)

The following result is the Lyapunov-LaSalle type theo-rem for (A1)

Theorem A2 (see [32 page 30]) If 119881 is a Lyapunov func-tional on 119866 and 119909

119905

(120601) is a bounded solution of (A1) that staysin 119866 then 120596-limit set 120596(120601) sub 119872 that is 119909

119905

(120601) rarr 119872 as119905 rarr +infin

Conflict of Interests

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

References

[1] K L Cooke ldquoStability analysis for a vector disease modelrdquoRocky Mountain Journal of Mathematics vol 9 no 1 pp 31ndash421979

[2] H W Hethcote and P van den Driessche ldquoAn SIS epidemicmodel with variable population size and a delayrdquo Journal ofMathematical Biology vol 34 no 2 pp 177ndash194 1995

[3] J Chin Control of Communicable Diseases Manual AmericanPublic Health Association Washington DC USA 1999

[4] S W Martin Livestock Disease Eradication Evaluation of theCooperative State-Federal Bovine Tuberculosis Eradication Pro-gram National Academy Press Washington DC USA 1994

[5] D Tudor ldquoA deterministic model for herpes infections inhuman and animal populationsrdquo SIAM Review vol 32 no 1pp 136ndash139 1990

[6] K E VanLandingham H B Marsteller G W Ross and FG Hayden ldquoRelapse of herpes simplex encephalitis after con-ventional acyclovir therapyrdquo Journal of the American MedicalAssociation vol 259 no 7 pp 1051ndash1053 1988

[7] A Bignami ldquoConcerning the pathogenesis of relapses in malar-ial feversrdquo Southern Medical Journal vol 6 pp 79ndash89 1913

[8] N JWhite ldquoDeterminants of relapse periodicity inPlasmodiumvivaxmalariardquoMalaria Journal vol 10 article 297 2011

[9] H Cox Y Kebede S Allamuratova et al ldquoTuberculosis recur-rence and mortality after successful treatment impact of drugresistancerdquo PLoS Medicine vol 3 no 10 pp 1836ndash1843 2006

[10] P Van Den Driessche L I N Wang and X Zou ldquoModelingdiseases with latency and relapserdquoMathematical Biosciences andEngineering vol 4 no 2 pp 205ndash219 2007

[11] A Kaddar ldquoOn the dynamics of a delayed SIR epidemic modelwith a modified saturated incidence raterdquo Electronic Journal ofDifferential Equations no 133 pp 1ndash7 2009

[12] S Gao L Chen J J Nieto and A Torres ldquoAnalysis of adelayed epidemic model with pulse vaccination and saturationincidencerdquo Vaccine vol 24 no 35-36 pp 6037ndash6045 2006

[13] J A Yorke and W P London ldquoRecurrent outbreaks of measleschickenpox and mumps IIrdquo American Journal of Epidemiologyvol 98 no 6 pp 469ndash482 1973

[14] M GM Gomes L J White and G F Medley ldquoThe reinfectionthresholdrdquo Journal ofTheoretical Biology vol 236 no 1 pp 111ndash113 2005

[15] Y Zhou and H Liu ldquoStability of periodic solutions for an SISmodel with pulse vaccinationrdquo Mathematical and ComputerModelling vol 38 no 3-4 pp 299ndash308 2003

[16] R M Anderson and R M May ldquoRegulation and stability ofhost-parasite population interactions I Regulatory processesrdquoThe Journal of Animal Ecology vol 47 no 1 pp 219ndash267 1978

[17] F Zhang and Z-Z Li ldquoGlobal stability of an SIR epidemicmodel with constant infectious periodrdquo Applied Mathematicsand Computation vol 199 no 1 pp 285ndash291 2008

[18] Z Jiang and J Wei ldquoStability and bifurcation analysis in adelayed SIR modelrdquo Chaos Solitons and Fractals vol 35 no 3pp 609ndash619 2008

[19] V Capasso and G Serio ldquoA generalization of the Kermack-McKendrick deterministic epidemic modelrdquoMathematical Bio-sciences vol 42 no 1-2 pp 43ndash61 1978

[20] R Xu and Z Ma ldquoStability of a delayed SIRS epidemic modelwith a nonlinear incidence raterdquo Chaos Solitons and Fractalsvol 41 no 5 pp 2319ndash2325 2009

[21] P Georgescu and H Zhang ldquoA Lyapunov functional for aSIRI model with nonlinear incidence of infection and relapserdquoApplied Mathematics and Computation vol 219 no 16 pp8496ndash8507 2013

[22] S Liu S Wang and L Wang ldquoGlobal dynamics of delayepidemic models with nonlinear incidence rate and relapserdquoNonlinear Analysis Real World Applications vol 12 no 1 pp119ndash127 2011

[23] C Castillo-Garsow G Jordan-Salivia and A Rodriguez Her-rera ldquoMathematical models for the dynamics of tobacco userecovery and relapserdquo Technical Report Series BU-1505-MCornell University 2000

[24] A Korobeinikov ldquoLyapunov functions and global stabilityfor SIR and SIRS epidemiological models with non-lineartransmissionrdquo Bulletin of Mathematical Biology vol 68 no 3pp 615ndash626 2006

[25] H N Moreira and Y Wang ldquoGlobal stability in an 119878 rarr 119868 rarr

119877 rarr 119868modelrdquo SIAM Review vol 39 no 3 pp 496ndash502 1997[26] S Blower ldquoModelling the genital herpes epidemicrdquoHerpes vol

11 supplement 3 pp 138Andash146A 2004[27] P van den Driessche and X Zou ldquoModeling relapse in infec-

tious diseasesrdquoMathematical Biosciences vol 207 no 1 pp 89ndash103 2007

[28] A Abta A Kaddar and H T Alaoui ldquoGlobal stability for delaySIR and SEIR epidemic models with saturated incidence ratesrdquoElectronic Journal of Differential Equations vol 2012 no 23 pp1ndash13 2012

[29] C Vargas-De-Leon ldquoOn the global stability of infectious dis-eases models with relapserdquoAbstraction ampApplication vol 9 pp50ndash61 2013

[30] Z Shuai and P van denDriessche ldquoGlobal stability of infectiousdisease models using Lyapunov functionsrdquo SIAM Journal onApplied Mathematics vol 73 no 4 pp 1513ndash1532 2013

[31] R Xu ldquoGlobal dynamics of an SEIRI epidemiological modelwith time delayrdquo Applied Mathematics and Computation vol232 pp 436ndash444 2014

[32] Y KuangDifferentiel EquationsWith Applications in PopulationDynamics Academic Press New York NY USA 1993

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 International Journal of Engineering Mathematics

The time derivative of the function 119881(119905) along the positivesolution of system (1) is

119889119881 (119905)

119889119905= (1 minus

119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)) (119860 minus 120583119878 minus 119891 (119878 119868))

+ 119890120583120591

(1 minus119868lowast

119868)

sdot (119890minus120583120591

119891 (119878120591

119868120591

+ 120575119877) minus (120583 + 120572 + 120574) 119868)

+120575119890120583120591

120583 + 120575(1 minus

119877lowast

119877) (120574119868 minus (120583 + 120575) 119877)

+ 119891 (119878 119868) minus 119891 (119878120591

119868120591

) + 119891 (119878lowast

119868lowast

)

sdot ln119891 (119878120591

119868120591

)

119891 (119878 119868)

(11)

Using the relation 119860 = 120583119878lowast

+ 119891(119878lowast

119868lowast

) simple calculationsgive

119889119881 (119905)

119889119905

= (1 minus119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)) (minus120583 (119878 minus 119878

lowast

) + 119891 (119878lowast

119868lowast

))

+119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)119891 (119878 119868) minus 119890

120583120591

(120583 + 120572 + 120574) 119868

minus119868lowast

119868119891 (119878120591

119868120591

) minus 120575119890120583120591

119868lowast

119868119877 + 119890120583120591

(120583 + 120572 + 120574) 119868lowast

+120575119890120583120591

120583 + 120575(120574119868 minus 120574119868

119877lowast

119877+ (120583 + 120575) 119877

lowast

)

+ 119891 (119878lowast

119868lowast

) ln119891 (119878120591

119868120591

)

119891 (119878 119868)

(12)

Here by using

119890120583120591

(120583 + 120572 + 120574) 119868lowast

minus120575119890120583120591

120583 + 120575120574119868lowast

= 119891 (119878lowast

119868lowast

)

(120583 + 120575) 119877lowast

= 120574119868lowast

ln119891 (119878120591

119868120591

)

119891 (119878 119868)

= ln119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)+ ln

119868lowast

119891 (119878120591

119868120591

)

119868119891 (119878lowast 119868lowast)+ ln

119868119891 (119878 119868lowast

)

119868lowast119891 (119878 119868)

(13)

straightforward calculations give

119889119881 (119905)

119889119905

= minus120583(1 minus119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)) (119878 minus 119878

lowast

)

minus 119891 (119878lowast

119868lowast

) (119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)minus 1 minus ln

119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast))

minus 119891 (119878lowast

119868lowast

) (119868lowast

119891 (119878120591

119868120591

)

119868119891 (119878lowast 119868lowast)minus 1 minus ln

119868lowast

119891 (119878120591

119868120591

)

119868119891 (119878lowast 119868lowast))

+ 119891 (119878lowast

119868lowast

) (119891 (119878 119868)

119891 (119878 119868lowast)minus

119868

119868lowast+119868119891 (119878 119868

lowast

)

119868lowast119891 (119878 119868)minus 1)

minus 119891 (119878lowast

119868lowast

) (119868119891 (119878 119868

lowast

)

119868lowast119891 (119878 119868)minus 1 minus ln

119868119891 (119878 119868lowast

)

119868lowast119891 (119878 119868))

+120575120574119868lowast

119890120583120591

120583 + 120575(2 minus

119868lowast

119877

119868119877lowastminus119868119877lowast

119868lowast119877)

= minus120583(1 minus119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)) (119878 minus 119878

lowast

)

minus 119891 (119878lowast

119868lowast

) (119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)minus 1 minus ln

119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast))

minus 119891 (119878lowast

119868lowast

) (119868lowast

119891 (119878120591

119868120591

)

119868119891 (119878 119868lowast)minus 1 minus ln

119868lowast

119891 (119878120591

119868120591

)

119868119891 (119878 119868lowast))

+119868

119868lowast119891 (119878lowast

119868lowast

) (1 minus119891 (119878 119868

lowast

)

119891 (119878 119868))(

120601 (119878 119868)

120601 (119878 119868lowast)minus 1)

minus 119891 (119878lowast

119868lowast

) (119868119891 (119878 119868

lowast

)

119868lowast119891 (119878 119868)minus 1 minus ln

119868119891 (119878 119868lowast

)

119868lowast119891 (119878 119868))

minus120575120574119890120583120591

120583 + 120575

119868119877lowast

119877(119868lowast

119877

119868119877lowastminus 1)

2

(14)

It follows from (1198671

) and (1198672

) that

minus120583(1 minus119891 (119878lowast

119868lowast

)

119891 (119878 119868lowast)) (119878 minus 119878

lowast

) le 0

119868

119868lowast119891 (119878lowast

119868lowast

) (1 minus119891 (119878 119868

lowast

)

119891 (119878 119868))(

120601 (119878 119868)

120601 (119878 119868lowast)minus 1)

le 0

(15)

Furthermore since the function 119892(119909) = 1minus119909+ ln(119909) is alwaysnonpositive for any 119909 gt 0 and 119892(119909) = 0 if and only if 119909 = 1then 119889119881(119905)119889119905 le 0 for all 119905 ge 0 where the equality holds onlyat the equilibrium point (119878lowast 119868lowast 119877lowast) Hence the functional119881satisfies all the conditions of Theorem A2 This proves that119875lowast

1

is globally asymptotically stable

International Journal of Engineering Mathematics 5

3 Numerical Simulations andConcluding Remarks

In this section we give a numerical simulation supporting thetheoretical analysis given in Section 2 Let

119891 (119878 119868) =120573119878119868

1 + 1205721

119878 + 1205722

119868 (16)

We take the parameters of the system (1) as follows

119860 = 10

1205721

= 09

1205722

= 09

120583 = 001

120574 = 002

120572 = 0005

120573 = 01

120575 = 0001

120591 = 10

(17)

By Proposition 2 the endemic equilibrium 119875lowast is globally

asymptotically stable see Figure 1In this paper we presented a mathematical analysis and

numerical simulations for an SIRI epidemiological modelapplied to the evolution of the spread of disease with relapsein a given population We denote 119877

0

the basic reproductionnumber It is defined as the average number of contagiouspersons infected by a typical infectious in a population of sus-ceptible We prove in this paper that the basic reproductionnumber 119877

0

depends on the incubation period and we showthat the disease-free equilibrium 119875

0

is globally asymptoticallystable if 119877

0

le 1 and that a unique endemic equilibrium 119875lowast is

globally asymptotically stable if 1198770

gt 1

Appendix

The Lyapunov-LaSalle Theorem

In the following we present the method of Lyapunov func-tionals in the context of a delay differential equations

119889119909

119889119905= 119891 (119909

119905

) (A1)

where 119891 119862 rarr 119877119899 is completely continuous and solutions

of (A1) are unique and continuously dependent on the initialdata We denote by 119909(120601) the solution of (A1) through (0 120601)For a continuous functional 119881 119862 rarr 119877 we define

= lim supℎrarr0

+

1

ℎ[(119881119909ℎ

(120601)) minus 119881 (120601)] (A2)

the derivative of 119881 along a solution of (A1) To statethe Lyapunov-LaSalle type theorem for (A1) we need thefollowing definition

0 100 200 300 400 5000

50

100

150

200

250

300

350

400

450

500

t

0 100 200 300 400 5000

50

100

150

200

250

300

350

400

450

t

I(t)

0 100 200 300 400 500100

200

300

400

500

600

700

800

900

1000

t

S(t)

R(t)

Figure 1 Solutions (119878 119868 119877) of the SIRI epidemic model (1) are glob-ally asymptotically stable and converge to the endemic equilibrium119875lowast

Definition A1 (see [32 page 30]) We say 119881 119862 rarr 119877

is a Lyapunov functional on a set 119866 in 119862 for (A1) if it iscontinuous on 119866 (the closure of 119866) and le 0 on 119866 We also

6 International Journal of Engineering Mathematics

define 119864 = 120601 isin 119866 (120601) = 0 and119872 is the largest set in 119864which is invariant with respect to (A1)

The following result is the Lyapunov-LaSalle type theo-rem for (A1)

Theorem A2 (see [32 page 30]) If 119881 is a Lyapunov func-tional on 119866 and 119909

119905

(120601) is a bounded solution of (A1) that staysin 119866 then 120596-limit set 120596(120601) sub 119872 that is 119909

119905

(120601) rarr 119872 as119905 rarr +infin

Conflict of Interests

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

References

[1] K L Cooke ldquoStability analysis for a vector disease modelrdquoRocky Mountain Journal of Mathematics vol 9 no 1 pp 31ndash421979

[2] H W Hethcote and P van den Driessche ldquoAn SIS epidemicmodel with variable population size and a delayrdquo Journal ofMathematical Biology vol 34 no 2 pp 177ndash194 1995

[3] J Chin Control of Communicable Diseases Manual AmericanPublic Health Association Washington DC USA 1999

[4] S W Martin Livestock Disease Eradication Evaluation of theCooperative State-Federal Bovine Tuberculosis Eradication Pro-gram National Academy Press Washington DC USA 1994

[5] D Tudor ldquoA deterministic model for herpes infections inhuman and animal populationsrdquo SIAM Review vol 32 no 1pp 136ndash139 1990

[6] K E VanLandingham H B Marsteller G W Ross and FG Hayden ldquoRelapse of herpes simplex encephalitis after con-ventional acyclovir therapyrdquo Journal of the American MedicalAssociation vol 259 no 7 pp 1051ndash1053 1988

[7] A Bignami ldquoConcerning the pathogenesis of relapses in malar-ial feversrdquo Southern Medical Journal vol 6 pp 79ndash89 1913

[8] N JWhite ldquoDeterminants of relapse periodicity inPlasmodiumvivaxmalariardquoMalaria Journal vol 10 article 297 2011

[9] H Cox Y Kebede S Allamuratova et al ldquoTuberculosis recur-rence and mortality after successful treatment impact of drugresistancerdquo PLoS Medicine vol 3 no 10 pp 1836ndash1843 2006

[10] P Van Den Driessche L I N Wang and X Zou ldquoModelingdiseases with latency and relapserdquoMathematical Biosciences andEngineering vol 4 no 2 pp 205ndash219 2007

[11] A Kaddar ldquoOn the dynamics of a delayed SIR epidemic modelwith a modified saturated incidence raterdquo Electronic Journal ofDifferential Equations no 133 pp 1ndash7 2009

[12] S Gao L Chen J J Nieto and A Torres ldquoAnalysis of adelayed epidemic model with pulse vaccination and saturationincidencerdquo Vaccine vol 24 no 35-36 pp 6037ndash6045 2006

[13] J A Yorke and W P London ldquoRecurrent outbreaks of measleschickenpox and mumps IIrdquo American Journal of Epidemiologyvol 98 no 6 pp 469ndash482 1973

[14] M GM Gomes L J White and G F Medley ldquoThe reinfectionthresholdrdquo Journal ofTheoretical Biology vol 236 no 1 pp 111ndash113 2005

[15] Y Zhou and H Liu ldquoStability of periodic solutions for an SISmodel with pulse vaccinationrdquo Mathematical and ComputerModelling vol 38 no 3-4 pp 299ndash308 2003

[16] R M Anderson and R M May ldquoRegulation and stability ofhost-parasite population interactions I Regulatory processesrdquoThe Journal of Animal Ecology vol 47 no 1 pp 219ndash267 1978

[17] F Zhang and Z-Z Li ldquoGlobal stability of an SIR epidemicmodel with constant infectious periodrdquo Applied Mathematicsand Computation vol 199 no 1 pp 285ndash291 2008

[18] Z Jiang and J Wei ldquoStability and bifurcation analysis in adelayed SIR modelrdquo Chaos Solitons and Fractals vol 35 no 3pp 609ndash619 2008

[19] V Capasso and G Serio ldquoA generalization of the Kermack-McKendrick deterministic epidemic modelrdquoMathematical Bio-sciences vol 42 no 1-2 pp 43ndash61 1978

[20] R Xu and Z Ma ldquoStability of a delayed SIRS epidemic modelwith a nonlinear incidence raterdquo Chaos Solitons and Fractalsvol 41 no 5 pp 2319ndash2325 2009

[21] P Georgescu and H Zhang ldquoA Lyapunov functional for aSIRI model with nonlinear incidence of infection and relapserdquoApplied Mathematics and Computation vol 219 no 16 pp8496ndash8507 2013

[22] S Liu S Wang and L Wang ldquoGlobal dynamics of delayepidemic models with nonlinear incidence rate and relapserdquoNonlinear Analysis Real World Applications vol 12 no 1 pp119ndash127 2011

[23] C Castillo-Garsow G Jordan-Salivia and A Rodriguez Her-rera ldquoMathematical models for the dynamics of tobacco userecovery and relapserdquo Technical Report Series BU-1505-MCornell University 2000

[24] A Korobeinikov ldquoLyapunov functions and global stabilityfor SIR and SIRS epidemiological models with non-lineartransmissionrdquo Bulletin of Mathematical Biology vol 68 no 3pp 615ndash626 2006

[25] H N Moreira and Y Wang ldquoGlobal stability in an 119878 rarr 119868 rarr

119877 rarr 119868modelrdquo SIAM Review vol 39 no 3 pp 496ndash502 1997[26] S Blower ldquoModelling the genital herpes epidemicrdquoHerpes vol

11 supplement 3 pp 138Andash146A 2004[27] P van den Driessche and X Zou ldquoModeling relapse in infec-

tious diseasesrdquoMathematical Biosciences vol 207 no 1 pp 89ndash103 2007

[28] A Abta A Kaddar and H T Alaoui ldquoGlobal stability for delaySIR and SEIR epidemic models with saturated incidence ratesrdquoElectronic Journal of Differential Equations vol 2012 no 23 pp1ndash13 2012

[29] C Vargas-De-Leon ldquoOn the global stability of infectious dis-eases models with relapserdquoAbstraction ampApplication vol 9 pp50ndash61 2013

[30] Z Shuai and P van denDriessche ldquoGlobal stability of infectiousdisease models using Lyapunov functionsrdquo SIAM Journal onApplied Mathematics vol 73 no 4 pp 1513ndash1532 2013

[31] R Xu ldquoGlobal dynamics of an SEIRI epidemiological modelwith time delayrdquo Applied Mathematics and Computation vol232 pp 436ndash444 2014

[32] Y KuangDifferentiel EquationsWith Applications in PopulationDynamics Academic Press New York NY USA 1993

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

International Journal of Engineering Mathematics 5

3 Numerical Simulations andConcluding Remarks

In this section we give a numerical simulation supporting thetheoretical analysis given in Section 2 Let

119891 (119878 119868) =120573119878119868

1 + 1205721

119878 + 1205722

119868 (16)

We take the parameters of the system (1) as follows

119860 = 10

1205721

= 09

1205722

= 09

120583 = 001

120574 = 002

120572 = 0005

120573 = 01

120575 = 0001

120591 = 10

(17)

By Proposition 2 the endemic equilibrium 119875lowast is globally

asymptotically stable see Figure 1In this paper we presented a mathematical analysis and

numerical simulations for an SIRI epidemiological modelapplied to the evolution of the spread of disease with relapsein a given population We denote 119877

0

the basic reproductionnumber It is defined as the average number of contagiouspersons infected by a typical infectious in a population of sus-ceptible We prove in this paper that the basic reproductionnumber 119877

0

depends on the incubation period and we showthat the disease-free equilibrium 119875

0

is globally asymptoticallystable if 119877

0

le 1 and that a unique endemic equilibrium 119875lowast is

globally asymptotically stable if 1198770

gt 1

Appendix

The Lyapunov-LaSalle Theorem

In the following we present the method of Lyapunov func-tionals in the context of a delay differential equations

119889119909

119889119905= 119891 (119909

119905

) (A1)

where 119891 119862 rarr 119877119899 is completely continuous and solutions

of (A1) are unique and continuously dependent on the initialdata We denote by 119909(120601) the solution of (A1) through (0 120601)For a continuous functional 119881 119862 rarr 119877 we define

= lim supℎrarr0

+

1

ℎ[(119881119909ℎ

(120601)) minus 119881 (120601)] (A2)

the derivative of 119881 along a solution of (A1) To statethe Lyapunov-LaSalle type theorem for (A1) we need thefollowing definition

0 100 200 300 400 5000

50

100

150

200

250

300

350

400

450

500

t

0 100 200 300 400 5000

50

100

150

200

250

300

350

400

450

t

I(t)

0 100 200 300 400 500100

200

300

400

500

600

700

800

900

1000

t

S(t)

R(t)

Figure 1 Solutions (119878 119868 119877) of the SIRI epidemic model (1) are glob-ally asymptotically stable and converge to the endemic equilibrium119875lowast

Definition A1 (see [32 page 30]) We say 119881 119862 rarr 119877

is a Lyapunov functional on a set 119866 in 119862 for (A1) if it iscontinuous on 119866 (the closure of 119866) and le 0 on 119866 We also

6 International Journal of Engineering Mathematics

define 119864 = 120601 isin 119866 (120601) = 0 and119872 is the largest set in 119864which is invariant with respect to (A1)

The following result is the Lyapunov-LaSalle type theo-rem for (A1)

Theorem A2 (see [32 page 30]) If 119881 is a Lyapunov func-tional on 119866 and 119909

119905

(120601) is a bounded solution of (A1) that staysin 119866 then 120596-limit set 120596(120601) sub 119872 that is 119909

119905

(120601) rarr 119872 as119905 rarr +infin

Conflict of Interests

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

References

[1] K L Cooke ldquoStability analysis for a vector disease modelrdquoRocky Mountain Journal of Mathematics vol 9 no 1 pp 31ndash421979

[2] H W Hethcote and P van den Driessche ldquoAn SIS epidemicmodel with variable population size and a delayrdquo Journal ofMathematical Biology vol 34 no 2 pp 177ndash194 1995

[3] J Chin Control of Communicable Diseases Manual AmericanPublic Health Association Washington DC USA 1999

[4] S W Martin Livestock Disease Eradication Evaluation of theCooperative State-Federal Bovine Tuberculosis Eradication Pro-gram National Academy Press Washington DC USA 1994

[5] D Tudor ldquoA deterministic model for herpes infections inhuman and animal populationsrdquo SIAM Review vol 32 no 1pp 136ndash139 1990

[6] K E VanLandingham H B Marsteller G W Ross and FG Hayden ldquoRelapse of herpes simplex encephalitis after con-ventional acyclovir therapyrdquo Journal of the American MedicalAssociation vol 259 no 7 pp 1051ndash1053 1988

[7] A Bignami ldquoConcerning the pathogenesis of relapses in malar-ial feversrdquo Southern Medical Journal vol 6 pp 79ndash89 1913

[8] N JWhite ldquoDeterminants of relapse periodicity inPlasmodiumvivaxmalariardquoMalaria Journal vol 10 article 297 2011

[9] H Cox Y Kebede S Allamuratova et al ldquoTuberculosis recur-rence and mortality after successful treatment impact of drugresistancerdquo PLoS Medicine vol 3 no 10 pp 1836ndash1843 2006

[10] P Van Den Driessche L I N Wang and X Zou ldquoModelingdiseases with latency and relapserdquoMathematical Biosciences andEngineering vol 4 no 2 pp 205ndash219 2007

[11] A Kaddar ldquoOn the dynamics of a delayed SIR epidemic modelwith a modified saturated incidence raterdquo Electronic Journal ofDifferential Equations no 133 pp 1ndash7 2009

[12] S Gao L Chen J J Nieto and A Torres ldquoAnalysis of adelayed epidemic model with pulse vaccination and saturationincidencerdquo Vaccine vol 24 no 35-36 pp 6037ndash6045 2006

[13] J A Yorke and W P London ldquoRecurrent outbreaks of measleschickenpox and mumps IIrdquo American Journal of Epidemiologyvol 98 no 6 pp 469ndash482 1973

[14] M GM Gomes L J White and G F Medley ldquoThe reinfectionthresholdrdquo Journal ofTheoretical Biology vol 236 no 1 pp 111ndash113 2005

[15] Y Zhou and H Liu ldquoStability of periodic solutions for an SISmodel with pulse vaccinationrdquo Mathematical and ComputerModelling vol 38 no 3-4 pp 299ndash308 2003

[16] R M Anderson and R M May ldquoRegulation and stability ofhost-parasite population interactions I Regulatory processesrdquoThe Journal of Animal Ecology vol 47 no 1 pp 219ndash267 1978

[17] F Zhang and Z-Z Li ldquoGlobal stability of an SIR epidemicmodel with constant infectious periodrdquo Applied Mathematicsand Computation vol 199 no 1 pp 285ndash291 2008

[18] Z Jiang and J Wei ldquoStability and bifurcation analysis in adelayed SIR modelrdquo Chaos Solitons and Fractals vol 35 no 3pp 609ndash619 2008

[19] V Capasso and G Serio ldquoA generalization of the Kermack-McKendrick deterministic epidemic modelrdquoMathematical Bio-sciences vol 42 no 1-2 pp 43ndash61 1978

[20] R Xu and Z Ma ldquoStability of a delayed SIRS epidemic modelwith a nonlinear incidence raterdquo Chaos Solitons and Fractalsvol 41 no 5 pp 2319ndash2325 2009

[21] P Georgescu and H Zhang ldquoA Lyapunov functional for aSIRI model with nonlinear incidence of infection and relapserdquoApplied Mathematics and Computation vol 219 no 16 pp8496ndash8507 2013

[22] S Liu S Wang and L Wang ldquoGlobal dynamics of delayepidemic models with nonlinear incidence rate and relapserdquoNonlinear Analysis Real World Applications vol 12 no 1 pp119ndash127 2011

[23] C Castillo-Garsow G Jordan-Salivia and A Rodriguez Her-rera ldquoMathematical models for the dynamics of tobacco userecovery and relapserdquo Technical Report Series BU-1505-MCornell University 2000

[24] A Korobeinikov ldquoLyapunov functions and global stabilityfor SIR and SIRS epidemiological models with non-lineartransmissionrdquo Bulletin of Mathematical Biology vol 68 no 3pp 615ndash626 2006

[25] H N Moreira and Y Wang ldquoGlobal stability in an 119878 rarr 119868 rarr

119877 rarr 119868modelrdquo SIAM Review vol 39 no 3 pp 496ndash502 1997[26] S Blower ldquoModelling the genital herpes epidemicrdquoHerpes vol

11 supplement 3 pp 138Andash146A 2004[27] P van den Driessche and X Zou ldquoModeling relapse in infec-

tious diseasesrdquoMathematical Biosciences vol 207 no 1 pp 89ndash103 2007

[28] A Abta A Kaddar and H T Alaoui ldquoGlobal stability for delaySIR and SEIR epidemic models with saturated incidence ratesrdquoElectronic Journal of Differential Equations vol 2012 no 23 pp1ndash13 2012

[29] C Vargas-De-Leon ldquoOn the global stability of infectious dis-eases models with relapserdquoAbstraction ampApplication vol 9 pp50ndash61 2013

[30] Z Shuai and P van denDriessche ldquoGlobal stability of infectiousdisease models using Lyapunov functionsrdquo SIAM Journal onApplied Mathematics vol 73 no 4 pp 1513ndash1532 2013

[31] R Xu ldquoGlobal dynamics of an SEIRI epidemiological modelwith time delayrdquo Applied Mathematics and Computation vol232 pp 436ndash444 2014

[32] Y KuangDifferentiel EquationsWith Applications in PopulationDynamics Academic Press New York NY USA 1993

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 International Journal of Engineering Mathematics

define 119864 = 120601 isin 119866 (120601) = 0 and119872 is the largest set in 119864which is invariant with respect to (A1)

The following result is the Lyapunov-LaSalle type theo-rem for (A1)

Theorem A2 (see [32 page 30]) If 119881 is a Lyapunov func-tional on 119866 and 119909

119905

(120601) is a bounded solution of (A1) that staysin 119866 then 120596-limit set 120596(120601) sub 119872 that is 119909

119905

(120601) rarr 119872 as119905 rarr +infin

Conflict of Interests

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

References

[1] K L Cooke ldquoStability analysis for a vector disease modelrdquoRocky Mountain Journal of Mathematics vol 9 no 1 pp 31ndash421979

[2] H W Hethcote and P van den Driessche ldquoAn SIS epidemicmodel with variable population size and a delayrdquo Journal ofMathematical Biology vol 34 no 2 pp 177ndash194 1995

[3] J Chin Control of Communicable Diseases Manual AmericanPublic Health Association Washington DC USA 1999

[4] S W Martin Livestock Disease Eradication Evaluation of theCooperative State-Federal Bovine Tuberculosis Eradication Pro-gram National Academy Press Washington DC USA 1994

[5] D Tudor ldquoA deterministic model for herpes infections inhuman and animal populationsrdquo SIAM Review vol 32 no 1pp 136ndash139 1990

[6] K E VanLandingham H B Marsteller G W Ross and FG Hayden ldquoRelapse of herpes simplex encephalitis after con-ventional acyclovir therapyrdquo Journal of the American MedicalAssociation vol 259 no 7 pp 1051ndash1053 1988

[7] A Bignami ldquoConcerning the pathogenesis of relapses in malar-ial feversrdquo Southern Medical Journal vol 6 pp 79ndash89 1913

[8] N JWhite ldquoDeterminants of relapse periodicity inPlasmodiumvivaxmalariardquoMalaria Journal vol 10 article 297 2011

[9] H Cox Y Kebede S Allamuratova et al ldquoTuberculosis recur-rence and mortality after successful treatment impact of drugresistancerdquo PLoS Medicine vol 3 no 10 pp 1836ndash1843 2006

[10] P Van Den Driessche L I N Wang and X Zou ldquoModelingdiseases with latency and relapserdquoMathematical Biosciences andEngineering vol 4 no 2 pp 205ndash219 2007

[11] A Kaddar ldquoOn the dynamics of a delayed SIR epidemic modelwith a modified saturated incidence raterdquo Electronic Journal ofDifferential Equations no 133 pp 1ndash7 2009

[12] S Gao L Chen J J Nieto and A Torres ldquoAnalysis of adelayed epidemic model with pulse vaccination and saturationincidencerdquo Vaccine vol 24 no 35-36 pp 6037ndash6045 2006

[13] J A Yorke and W P London ldquoRecurrent outbreaks of measleschickenpox and mumps IIrdquo American Journal of Epidemiologyvol 98 no 6 pp 469ndash482 1973

[14] M GM Gomes L J White and G F Medley ldquoThe reinfectionthresholdrdquo Journal ofTheoretical Biology vol 236 no 1 pp 111ndash113 2005

[15] Y Zhou and H Liu ldquoStability of periodic solutions for an SISmodel with pulse vaccinationrdquo Mathematical and ComputerModelling vol 38 no 3-4 pp 299ndash308 2003

[16] R M Anderson and R M May ldquoRegulation and stability ofhost-parasite population interactions I Regulatory processesrdquoThe Journal of Animal Ecology vol 47 no 1 pp 219ndash267 1978

[17] F Zhang and Z-Z Li ldquoGlobal stability of an SIR epidemicmodel with constant infectious periodrdquo Applied Mathematicsand Computation vol 199 no 1 pp 285ndash291 2008

[18] Z Jiang and J Wei ldquoStability and bifurcation analysis in adelayed SIR modelrdquo Chaos Solitons and Fractals vol 35 no 3pp 609ndash619 2008

[19] V Capasso and G Serio ldquoA generalization of the Kermack-McKendrick deterministic epidemic modelrdquoMathematical Bio-sciences vol 42 no 1-2 pp 43ndash61 1978

[20] R Xu and Z Ma ldquoStability of a delayed SIRS epidemic modelwith a nonlinear incidence raterdquo Chaos Solitons and Fractalsvol 41 no 5 pp 2319ndash2325 2009

[21] P Georgescu and H Zhang ldquoA Lyapunov functional for aSIRI model with nonlinear incidence of infection and relapserdquoApplied Mathematics and Computation vol 219 no 16 pp8496ndash8507 2013

[22] S Liu S Wang and L Wang ldquoGlobal dynamics of delayepidemic models with nonlinear incidence rate and relapserdquoNonlinear Analysis Real World Applications vol 12 no 1 pp119ndash127 2011

[23] C Castillo-Garsow G Jordan-Salivia and A Rodriguez Her-rera ldquoMathematical models for the dynamics of tobacco userecovery and relapserdquo Technical Report Series BU-1505-MCornell University 2000

[24] A Korobeinikov ldquoLyapunov functions and global stabilityfor SIR and SIRS epidemiological models with non-lineartransmissionrdquo Bulletin of Mathematical Biology vol 68 no 3pp 615ndash626 2006

[25] H N Moreira and Y Wang ldquoGlobal stability in an 119878 rarr 119868 rarr

119877 rarr 119868modelrdquo SIAM Review vol 39 no 3 pp 496ndash502 1997[26] S Blower ldquoModelling the genital herpes epidemicrdquoHerpes vol

11 supplement 3 pp 138Andash146A 2004[27] P van den Driessche and X Zou ldquoModeling relapse in infec-

tious diseasesrdquoMathematical Biosciences vol 207 no 1 pp 89ndash103 2007

[28] A Abta A Kaddar and H T Alaoui ldquoGlobal stability for delaySIR and SEIR epidemic models with saturated incidence ratesrdquoElectronic Journal of Differential Equations vol 2012 no 23 pp1ndash13 2012

[29] C Vargas-De-Leon ldquoOn the global stability of infectious dis-eases models with relapserdquoAbstraction ampApplication vol 9 pp50ndash61 2013

[30] Z Shuai and P van denDriessche ldquoGlobal stability of infectiousdisease models using Lyapunov functionsrdquo SIAM Journal onApplied Mathematics vol 73 no 4 pp 1513ndash1532 2013

[31] R Xu ldquoGlobal dynamics of an SEIRI epidemiological modelwith time delayrdquo Applied Mathematics and Computation vol232 pp 436ndash444 2014

[32] Y KuangDifferentiel EquationsWith Applications in PopulationDynamics Academic Press New York NY USA 1993

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