DDNS Discrete Dynamics in Nature and Society 1607-887X 1026-0226 Hindawi Publishing Corporation 472746 10.1155/2014/472746 472746 Research Article Lyapunov Functions for a Class of Discrete SIRS Epidemic Models with Nonlinear Incidence Rate and Varying Population Sizes Wang Ying http://orcid.org/0000-0003-0567-4699 Teng Zhidong Rehim Mehbuba Guerrini Luca College of Mathematics and Model Sciences Xinjiang University Urumqi 830046 China xju.edu.cn 2014 2172014 2014 21 03 2014 21 04 2014 21 7 2014 2014 Copyright © 2014 Ying Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We investigate the dynamical behaviors of a class of discrete SIRS epidemic models with nonlinear incidence rate and varying population sizes. The model is required to possess different death rates for the susceptible, infectious, recovered, and constant recruitment into the susceptible class, infectious class, and recovered class, respectively. By using the inductive method, the positivity and boundedness of all solutions are obtained. Furthermore, by constructing new discrete type Lyapunov functions, the sufficient and necessary conditions on the global asymptotic stability of the disease-free equilibrium and endemic equilibrium are established.

1. Introduction

As well known in the theoretical study of epidemic models, the susceptible-infected-recovered (SIR) compartmental epidemic models are a kind of very important epidemic models and in recent years have been widely investigated. According to the assumptions of Kermack and McKendrick , the population of size N(t) at time t is divided into three distinct classes: the susceptible class of size S(t), the infectious class of size I(t), and the recovered class R(t) at time t. When a susceptible individual acquires the disease by contacting with an infectious individual, the susceptible individual moves into the infectious compartment and, subsequently, as a result of some measures such as medication or isolation the infector takes into the recovered class. If the recovered individuals retain their immunity permanently, then he/her remains in the recovered compartment. The model based on these assumptions is known as the SIR epidemic model. Furthermore, if the immunity is not permanent, that is, the recovered individual may lose his/her immunity after a period of time, then he/her returns to the susceptible class. Thus, we obtain the SIRS epidemic model.

Usually, there are two kinds of epidemic dynamical models: the continuous-time models described by differential equations and the discrete-time models described by difference equations. In this paper, we will focus our attention on discrete-time epidemic dynamical models. For an epidemic model, which is continuous-time model or discrete-time model, we all know that an important subject is to determine the global stability of the disease-free equilibrium and endemic equilibrium. Particularly, we expect to compute basic reproduction number R0 of the model and also to obtain the fact that the disease-free equilibrium is globally stable when R01, as well as the endemic equilibrium exists and is globally stable when R0>1.

Until now, the discrete-time SIR and SIRS epidemic models have been extensively studied in many articles; for example, see  and the reference therein. Many important results have been established. These results focus on the computation of the basic reproduction number, the local and global stability of the disease-free equilibrium and endemic equilibrium, the permanence, persistence, and extinction of the disease, and so forth. Particularly, in [2, 3], the authors studied a class of discrete-time SIRS epidemic models with time delays derived from corresponding continuous-time models by applying Mickens’ nonstandard finite difference scheme. The sufficient conditions on the global asymptotic stability of the disease-free equilibrium and the permanence of the disease are established. In , the authors studied a discrete-time SIRS epidemic model with bilinear incidence rate derived from corresponding continuous-time model by applying backward Euler finite difference scheme. The sufficient and necessary conditions on the global asymptotic stability of the disease-free equilibrium and endemic equilibrium are established. In , the authors discussed a class of discrete-time SIRS epidemic models with general nonlinear incidence rate derived from corresponding continuous-time model by applying forward Euler scheme. The sufficient conditions for the existence and local stability of the disease-free equilibrium and endemic equilibrium are obtained. In , the authors discussed a class of discrete-time SIRS epidemic models with standard incidence rate discretized from corresponding continuous-time model by applying forward Euler scheme. The sufficient condition for the global stability of the endemic equilibrium is established.

However, we can see from the above literatures that the studies on the global stability for discrete-time SIRS epidemic models are not perfect. The necessary and sufficient conditions for the global stability of the disease-free equilibrium when basic reproduction number R01, as well as the global stability of the endemic equilibrium when R0>1, are established only for bilinear incidence rate (see ). Therefore, motivated by the above works, as an extension of the results given in , in this paper, we consider the following discrete-time SIRS epidemic model with nonlinear incidence rate and varying population sizes derived from corresponding continuous-time model by applying backward Euler scheme: (1)Sn+1-Sn=aA-βSn+1g(In+1)-d1Sn+1+δRn+1,In+1-In=bA+βSn+1g(In+1)-(d2+γ)In+1,Rn+1-Rn=cA+γIn+1-(d3+δ)Rn+1.

By constructing new discrete type Lyapunov functions and using the theory of stability of difference equations, we will establish the global asymptotic stability of equilibria only under basic hypothesis (H) (see Section 2). That is, the disease-free equilibrium is globally asymptotically stable if and only if basic reproduction number R01, and the endemic equilibrium is globally asymptotically stable if and only if R0>1.

The organization of this paper is as follows. In the second section, we give a model description and further obtain the results on the positivity and boundedness of solutions of model (1). In the third section, we discuss the existence and global asymptotic stability of equilibria of model (1) for case b=0. In the fourth section, we will study the global asymptotic stability of endemic equilibrium of model (1) for case b>0. Lastly, in the fifth section, we will give a conclusion.

2. Preliminaries

In model (1), Sn, In, and Rn denote the numbers of susceptible, infected, and recovered individuals at nth period, respectively. A is the recruitment rate of the total population, parameters a>0, b0, and c0 are the fraction constants of input to susceptible class Sn, infected class In, and recovered class Rn, respectively, and satisfying a+b+c=1. d1, d2, and d3 represent the death rate of susceptible, infected, and recovered individuals, respectively. Particularly, death rate d2 includes the natural death rate and the disease-related death rate of the infected individuals. δ is the rate at which recovered individuals lose immunity and return to the susceptible class. γ is the natural recovery rate of the infective individuals. β is the proportionality constant, and the transmission of the infection is governed by a nonlinear incidence rate βSg(I). In this paper, we always assume that A,d1,d2,d3,δ,β,γ are positive constants.

The initial condition for model (1) is given by (2)S0>0,I0>0,R0>0. Throughout this paper, we always assume that

g(I) is continuous and monotonically increasing on [0,+), g(0)=0, I/g(I) is also monotonically increasing on (0,+), and g(0) exists with g(0)>0.

Remark 1.

Hypothesis (H) is basic for model (1). Particularly, g(I)=I/(1+ωI) with constant ω0; then assumption (H1) naturally holds. Furthermore, if function g(I) satisfies that second-order derivative g(I) exists and g(I)0 for all I[0,), then we easily prove that I/g(I) is monotone increasing on I(0,+).

On the positivity and boundedness of all solutions of model (1) with initial condition (2), we have the following results.

Lemma 2.

For any solution (Sn,In,Rn) of model (1) with initial condition (2), it holds that (3)Sn>0,In>0,Rn>0n0.

Proof.

Model (1) is equivalent to the following form: (4)Sn+1=aA+δRn+1+Sn1+βg(In+1)+d1,In+1=bA+βSn+1g(In+1)+In1+d2+γ,Rn+1=cA+γIn+1+Rn1+d3+δ. When n=0, we have (5)S1=aA+δR1+S01+βg(I1)+d1,I1=bA+βS1g(I1)+I01+d2+γ,R1=cA+γI1+R01+d3+δ. Let C1=1+d3+δ and let C2=1+d2+γ; then obviously, C1C2-δγ>0. Substituting S1, R1 into I1, we obtain that I1 satisfies the following equation: (6)I1=βg(I1)h+δγβg(I1)I1+C1(bA+I0)(1+d1)(1+βg(I1)+d1)C1C2, where h=C1A(a+b)+δ(cA+R0)+C1(S0+I0). From this, we further obtain that I1 satisfies the following equation: (7)H(I1)(C1C2-δγ)βg(I1)+C1C2(1+d1)-C1(bA+I0)(1+d1)I1-hβg(I1)I1=0. From hypothesis (H), we obtain that H(I1) is monotonically increasing on (0,+), and, obviously, (8)limI10+H(I1)=-. If limI1g(I1)<, then we have (9)limI1H(I1)=C1C2(1+d1)+(C1C2-δγ)×βlimI1g(I1)>0 and if limI1g(I1)=+, then limI1H(I1)=+. Hence, there is a unique positive solution x*>0 such that H(x*)=0. Therefore, we have I1=x*>0. Further, from (5) we also obtain S1>0 and R1>0.

When n=1, in a similar way, we can obtain S2>0, I2>0 and R2>0. By the induction, we finally obtain that Sn>0, In>0 and Rn>0 for all n0. This completes the proof.

Lemma 3.

For any solution (Sn,In,Rn) of model (1) with initial condition (2), it holds that (10)limsupn(Sn+In+Rn)Ad, where d=min{d1,d2,d3}.

Proof.

Let Nn=Sn+In+Rn; then from model (1) we have (11)Nn=A+Nn-1-d1Sn-d2In-d3RnA+Nn-1-dNn. Hence, (12)NnA+Nn-11+d,n=1,2,. By using iteration method, we obtain (13)NnA+Nn-11+dA1+d+A(1+d)2+A(1+d)3++A(1+d)n+N0(1+d)nAd[1-1(1+d)n]+A(1+d)nN0. Therefore, it holds that (14)limsupn+NnAd. This completes the proof.

3. Case <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M99"><mml:mi>b</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>

If b=0, we have a+c=1, and a>0, c0; then model (1) becomes into the following form: (15)Sn+1-Sn=aA-βSn+1g(In+1)-d1Sn+1+δRn+1,In+1-In=βSn+1g(In+1)-(d2+γ)In+1,Rn+1-Rn=cA+γIn+1-(d3+δ)Rn+1. Particularly, when b=c=0, then a=1 and model (1) will become into the following well-known form: (16)Sn+1-Sn=A-βSn+1g(In+1)-d1Sn+1+δRn+1,In+1-In=βSn+1g(In+1)-(d2+γ)In+1,Rn+1-Rn=γIn+1-(d3+δ)Rn+1.

For model (15), under hypotheses (H), the basic reproduction number, that is an average number of secondary infectious cases produced by an infectious individual during his or her effective infectious period when introduced into an entirely susceptible population, can be defined by (17)R0=βA(δ+ad3)g(0)d1(d2+γ)(δ+d3). Here, β is the disease transmission rate, 1/(d2+γ) is the average infection period, and (18)limI0+βA(δ+ad3)d1(δ+d3)g(I)I=βA(δ+ad3)d1(δ+d3)g(0) implies that (βA(δ+ad3)/d1(δ+d3))g(0) denotes the number of new cases infected per unit time by one infective individual which is introduced into the susceptible compartment in the case that all the members of the population are susceptible. Particularly, for model (16) we have the basic reproduction number as follows: (19)R0=βAg(0)d1(d2+γ). On the existence of equilibria of model (15), we have the following result.

Theorem 4.

(1) If R01, then model (15) only has a unique disease-free equilibrium E0=(S0,0,R0), where S0=A(δ+ad3)/d1(δ+d3) and R0=cA/(δ+d3).

(2) If R0>1, then model (15) has a unique endemic equilibrium E*=(S*,I*,R*), except for the disease-free equilibrium E0.

Proof.

We know that an equilibrium E=(S,I,R) of model (15) satisfies (20)aA-βSg(I)-d1S+δR=0,βSg(I)-(d2+γ)I=0,cA+γI-(d3+δ)R=0. Firstly, when I=0, we have (21)aA-d1S+δR=0,cA-(d3+δ)R=0, from which we obtain the disease-free equilibrium E0=(S0,0,R0), where S0=A(δ+ad3)/d1(δ+d3) and R0=cA/(δ+d3).

Secondly, when I>0, from the second and third equations of (20), we obtain (22)R=cA+γIδ+d3,S=(d2+γ)Iβg(I). Substituting R,S into the first equation of (20), we have (23)A(δ+ad3)δ+d3-d3(d2+γ)+δd2δ+d3I-d1(d2+γ)Iβg(I)=0. Denote (24)H(I)=A(δ+ad3)δ+d3-d3(d2+γ)+δd2δ+d3I-d1(d2+γ)Iβg(I). By hypothesis (H), H(I) is monotonically decreasing on (0,+) satisfying (25)limI0+H(I)=A(δ+ad3)δ+d3-d1(d2+γ)βg(0)=A(δ+ad3)δ+d3(1-1R0), and we also have (26)limI+H(I)=-.

When R01, we have limI0+H(I)0. Then, there is not any I*>0 such that H(I*)=0. Therefore, model (15) only has a unique disease-free equilibrium E0.

When R0>1, we have limI0+H(I)>0. Then, there exists a unique I*>0 such that H(I*)=0. Furthermore, we have S*=(d2+γ)I*/βg(I*)>0 and R*=(cA+γI*)/(δ+d3)>0. This implies that model (15) has a unique endemic equilibrium E*=(S*,I*,R*). This completes the proof.

Remark 5.

Particularly, for model (16), the disease-free equilibrium given in Theorem 4 will become into E0=(A/d1,0,0).

Now, we study the stability of equilibria of model (15). On the global stability of the disease-free equilibrium E0, we have the following result.

Theorem 6.

Disease-free equilibrium E0 of model (15) is globally asymptotically stable if and only if R01.

Proof.

The necessity is obvious; we only need to prove the sufficiency. Model (15) can be rewritten as the following form: (27)Sn+1-Sn=-(βg(In+1)+d1)(Sn+1-S0)+δ(Rn+1-R0)0000000000-βS0g(In+1),In+1-In=βg(In+1)(Sn+1-S0)-(d2+γ)In+1000000000+βS0g(In+1),Rn+1-Rn=γIn+1-(d3+δ)(Rn+1-R0). We consider the following Lyapunov function: (28)Wn=12(Sn-S0+In+Rn-R0)2+k12(Sn-S0)2+(k2+k3)In+k42(Rn-R0)2, where (29)k1=d1+d3δ,k2=k1S0,k3=d1+d2βg(0),k4=d2+d3+αγ. Calculating the difference of Wn along (27), we have (30)Wn+1-Wn=k12[(Sn+1-S0)2-(Sn-S0)2]+(k2+k3)(In+1-In)+k42[(Rn+1-R0)2-(Rn-R0)2]+12[(Sn+1-S0+In+1+Rn+1-R0)2iiiiiii-(Sn-S0+In+Rn-R0)2]=k12(Sn+1-Sn)(Sn-Sn+1+2(Sn+1-S0))+(k2+k3)(In+1-In)+k42(Rn+1-Rn)(Rn-Rn+1+2(Rn+1-R0))+12(Sn+1-Sn+In+1-In+Rn+1-Rn)×(Sn-Sn+1+2(Sn+1-S0)+In-In+1iiiii+2In+1+Rn-Rn+1+2(Rn+1-R0))k1(Sn+1-Sn)(Sn+1-S0)+(k2+k3)(In+1-In)+k4(Rn+1-Rn)(Rn+1-R0)+(Sn+1-Sn+In+1-In+Rn+1-Rn)×(Sn+1-S0+In+1+Rn+1-R0)=k1[-(βg(In+1)+d1)(Sn+1-S0)iiiii+δ(Rn+1-R0)-βS0g(In+1)](Sn+1-S0)+k2[βg(In+1)(Sn+1-S0)-(d2+γ)In+1iiiiiiiii+βS0g(In+1)]+k3[βSn+1g(In+1)-(d2+γ)In+1]+k4[γIn+1-(d3+δ)(Rn+1-R0)]×(Rn+1-R0)+(-d1Sn+1-d2In+1-d3Rn+1)×(Sn+1-S0+In+1+Rn+1-R0). Since R0=βAg(0)(δ+ad3)/d1(d2+γ)(δ+d3)1, we have βS0g(0)d2+γ. Hence, (31)Wn+1-Wnk1[-(βg(In+1)+d1)(Sn+1-S0)iiiiii+δ(Rn+1-R0)-βS0g(In+1)](Sn+1-S0)+k2[βg(In+1)(Sn+1-S0)-βS0g(0)In+1iiiiiiii+βS0g(In+1)]+k3[βSn+1g(In+1)-βS0g(0)In+1]+k4[γIn+1-(d3+δ)(Rn+1-R0)]×(Rn+1-R0)+[-d1(Sn+1-S0)-d2In+1-d3(Rn+1-R0)]×(Sn+1-S0+In+1+Rn+1-R0)=-[k1(βg(In+1)+d1)+d1](Sn+1-S0)2-d2In+12-[k4(d3+δ)+d3](Rn+1-R0)2+k2In+1βS0[g(In+1)In+1-g(0)]+k3βSn+1In+1[g(In+1)In+1-g(0)]+k3βIn+1g(0)(Sn+1-S0)-[k1(βg(In+1)+d1)+d1](Sn+1-S0)2-d2In+12-[k4(d3+δ)+d3](Rn+1-R0)2+βIn+1(k2S0+k3Sn+1)(g(In+1)In+1-g(0)). Under hypothesis (H), we have for any n0(32)g(In+1)In+1limI0+g(I)I=g(0). Hence, (33)Wn+1-Wn-[k1(βg(In+1)+d1)+d1](Sn+1-S0)2-d2In+12-[k4(d3+δ)+d3](Rn+1-R0)2. This implies that (34)Wn+1-Wn<0(Sn,In,Rn)(S0,0,R0). By Lyapunov’s theorems on the global asymptotical stability for difference equations, we directly obtained that the disease-free equilibrium E0 is globally asymptotically stable. This completes the proof.

On the global stability of the endemic equilibrium E*, we have the following result.

Theorem 7.

Endemic equilibrium E* of model (15) is globally asymptotically stable if and only if R0>1.

Proof.

The necessity is obvious, we only need to prove the sufficiency. Model (15) can be rewritten as the following form: (35)Sn+1-Sn=-(βg(In+1)+d1)(Sn+1-S*)+δ(Rn+1-R*)0000000000-βS*(g(In+1)-g(I*)),In+1-In=βg(In+1)(Sn+1-S*)-(d2+γ)(In+1-I*)000000000+βS*(g(In+1)-g(I*)),Rn+1-Rn=γ(In+1-I*)-(d3+δ)(Rn+1-R*). We also have (36)In+1-In=βSn+1g(In+1)-(d2+γ)In+1=In+1[βSn+1g(In+1)In+1-(d2+γ)]=In+1[βSn+1g(In+1)In+1-βS*g(I*)I*]=In+1[βSn+1g(In+1)In+1-βS*g(I*)I*+βSn+1g(I*)I*iiiiiiiiiiiii-βSn+1g(I*)I*]=In+1[βSn+1(g(In+1)In+1-g(I*)I*)iiiiiiiiiiiii+βg(I*)I*(Sn+1-S*)].

We consider the following Lyapunov function: (37)Vn=12(Sn-S*+In-I*+Rn-R*)2+k12(Sn-S*)2+k2I*Ing(τ)-g(I*)g(τ)dτ+k32(Rn-R*)2+k4(InI*-1-lnInI*), where (38)k1=d1+d3δ,k2=k1S*,k3=d2+d3+αγ,k4=(d1+d2)(I*)2βg(I*). Calculating the difference of Wn along (35), we have (39)Vn+1-Vn=k12[(Sn+1-S*)2-(Sn-S*)2]+k2InIn+1g(τ)-g(I*)g(τ)dτ+k32[(Rn+1-R*)2-(Rn-R*)2]+k4(In+1-InI*-lnIn+1In)+12[(Sn+1-S*+In+1-I*+Rn+1-R*)2iiiiiiii-(Sn-S*+In-I*+Rn-R*)2]k1(Sn+1-Sn)(Sn+1-S*)+k2(In+1-In)[g(In+1)-g(I*)g(In+1)]+k3(Rn+1-Rn)(Rn+1-R*)+k4(In+1-In)[In+1-I*I*In+1]+(Sn+1-Sn+In+1-In+Rn+1-Rn)×(Sn+1-S*+In+1-I*+Rn+1-R*)=k1[-(βg(In+1)+d1)(Sn+1-S*)iiiiii+δ(Rn+1-R*)-βS*(g(In+1)-g(I*))]×(Sn+1-S*)+k2[βg(In+1)(Sn+1-S*)iiiiiiiiii-(d2+γ)(In+1-I*)iiiiiiiiii+βS*(g(In+1)-g(I*))]×[g(In+1)-g(I*)g(In+1)]+k3[γ(In+1-I*)-(d3+δ)(Rn+1-R*)]×(Rn+1-R*)+k4In+1[βSn+1(g(In+1)In+1-g(I*)I*)iiiiiiiiiiiiiii+βg(I*)I*(Sn+1-S*)][In+1-I*I*In+1]+(-d1Sn+1-d2In+1-d3Rn+1)×(Sn+1-S*+In+1-I*+Rn+1-R*)-[k1(βg(In+1)+d1)+d1](Sn+1-S*)2-d2(In+1-I*)2-[k3(d3+δ)+d3](Rn+1-R*)2+k2g(In+1)-g(I*)g(In+1)×[βS*(g(In+1)-g(I*))iiiiiiiii-(d2+γ)(In+1-I*)]+k4I*βSn+1(In+1-I*)(g(In+1)In+1-g(I*)I*). Further, from hypothesis (H) and d2+γ=βS*(g(I*)/I*), we have (40)k2g(In+1)-g(I*)g(In+1)×[βS*(g(In+1)-g(I*))-(d2+γ)(In+1-I*)]=k2g(In+1)-g(I*)g(In+1)[βS*g(In+1)-(d2+γ)In+1]=k2In+1g(In+1)(g(In+1)-g(I*))×[βS*g(In+1)In+1-(d2+γ)]=k2In+1g(In+1)(g(In+1)-g(I*))×[βS*g(In+1)In+1-βS*g(I*)I*]=k2βS*In+1g(In+1)(g(In+1)-g(I*))[g(In+1)In+1-g(I*)I*]0,k4I*βSn+1(In+1-I*)(g(In+1)In+1-g(I*)I*)0. Hence, (41)Vn+1-Vn-[k1(βg(In+1)+d1)+d1](Sn+1-S*)2-d2(In+1-I*)2-[k3(d3+δ)+d3](Rn+1-R*)2. This implies that (42)Vn+1-Vn<0(Sn,In,Rn)(S*,I*,R*). By Lyapunov’s theorems on the globally asymptotical stability for difference equations, we directly obtained that the endemic equilibrium E* is globally asymptotically stable. This completes the proof.

Remark 8.

From the above discussion we immediately see that the basic reproduction number R0 can completely determine the global asymptotic stability of model (15).

As a consequence of Theorems 6 and 7, for model (16) we have the following corollary.

Corollary 9.

For model (16) one has the following.

Disease-free equilibrium E0 is globally asymptotically stable if and only if R01.

Endemic equilibrium E* is globally asymptotically stable if and only if R0>1.

Remark 10.

From Corollary 9, we see that the corresponding results on the global asymptotic stability obtained in  for discrete-time SIRS epidemic models with bilinear incidence rate are extended to the models with nonlinear incidence rate. Furthermore, comparing with Lyapunov functions established in , we see that, in order to study the global asymptotic stability of model (15), a new Lyapunov function is constructed in this paper.

4. Case <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M185"><mml:mi>b</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>

We firstly discuss the existence of equilibria of model (1). It is easy to see that if b>0, model (1) has no disease-free equilibrium. We have the following result about the existence of endemic equilibrium of model (1).

Theorem 11.

Model (1) always has a unique endemic equilibrium E*=(S*,I*,R*).

Proof.

From model (1) we know that the endemic equilibrium E*=(S*,I*,R*) satisfies (43)aA-βSg(I)-d1S+δR=0,bA+βSg(I)-(d2+γ)I=0,cA+γI-(d3+δ)R=0. By the second and third equations of (43), we can obtain (44)R=cA+γIδ+d3,S=(d2+γ)I-bAβg(I). Substituting (44) into the first equation of (43), we have (45)-[d3(d2+γ)+δd2]I+A[δ+(a+b)d3]-d1(d2+γ)(δ+d3)Iβg(I)+d1bA(δ+d3)βg(I)=0. Denote (46)φ(I)=-[d3(d2+γ)+δd2]I+A[δ+(a+b)d3],ψ(I)=d1(d2+γ)(δ+d3)Iβg(I)-d1bA(δ+d3)βg(I).

Now, we consider equation φ(I)=ψ(I), which is equivalent to (45). By hypothesis (H), φ(I) is monotonically decreasing on (0,+) and ψ(I) is monotonically increasing on (0,+). Let d=min{d1,d2,d3}; then we have 0<bA/(d2+γ)<A/d. By calculating, we obtain (47)φ(bAd2+γ)=-[d3(d2+γ)+δd2]bAd2+γ+A[δ+(a+b)d3]=A[δ(1-b)d2+δγ+ad3(d2+γ)]d2+γ>0,φ(Ad)=-[d3(d2+γ)+δd2]Ad+A[δ+(a+b)d3]=-Ad[d3(d2+γ)+δd2-d(δ+(1-c)d3)]=-Ad[d3(γ+dc)+(δ+d3)(d2-d)]<0,ψ(bAd2+γ)=0,ψ(Ad)>0. Hence, there exists a unique I*(bA/(d2+γ),A/d) such that φ(I*)=ψ(I*). Furthermore, we have S*=((d2+γ)I*-bA)/βg(I*)>0 and R*=(cA+γI*)/(δ+d3)>0. This implies that model (1) has a unique endemic equilibrium E*=(S*,I*,R*).

Now, we study the global stability of endemic equilibrium E*; we have the following result.

Theorem 12.

Endemic equilibrium E* of model (1) is always globally asymptotically stable.

Proof.

Model (1) becomes into the following form: (48)Sn+1-Sn=-(βg(In+1)+d1)(Sn+1-S*)+δ(Rn+1-R*)0000000000-βS*(g(In+1)-g(I*)),In+1-In=βg(In+1)(Sn+1-S*)-(d2+γ)(In+1-I*)000000000+βS*(g(In+1)-g(I*)),Rn+1-Rn=γ(In+1-I*)-(d3+δ)(Rn+1-R*). We also have (49)In+1-In=bA+βSn+1g(In+1)-(d2+γ)In+1=In+1[bAIn+1+βSn+1g(In+1)In+1-(d2+γ)]=In+1[bAIn+1+βSn+1g(In+1)In+1-bA+βS*g(I*)I*]=In+1[bAIn+1-bAI*+βSn+1g(In+1)In+1-βS*g(I*)I*iiiiiiiiii+βSn+1g(I*)I*-βSn+1g(I*)I*]=In+1[+βg(I*)I*(Sn+1-S*)bA(1In+1-1I*)iiiiiiiiii+βSn+1(g(In+1)In+1-g(I*)I*)iiiiiiiiii+βg(I*)I*(Sn+1-S*)].

We consider the following Lyapunov function: (50)Un=12(Sn-S*+In-I*+Rn-R*)2+k12(Sn-S*)2+k2I*Ing(τ)-g(I*)g(τ)dτ+k32(Rn-R*)2+k4(InI*-1-lnInI*), where (51)k1=d1+d3δ,k2=k1S*,k3=d2+d3γ,k4=(d1+d2)(I*)2βg(I*). Calculating the difference of Un along (48), we have (52)Un+1-Unk1[-(βg(In+1)+d1)(Sn+1-S*)iiiiii+δ(Rn+1-R*)iiiiii-βS*(g(In+1)-g(I*))](Sn+1-S*)+k2[βg(In+1)(Sn+1-S*)iiiiiiii-(d2+γ)(In+1-I*)iiiiiiii+βS*(g(In+1)-g(I*))]×(g(In+1)-g(I*)g(In+1))+k3[γ(In+1-I*)-(d3+δ)(Rn+1-R*)]×(Rn+1-R*)+k4In+1[+βg(I*)I*(Sn+1-S*)bA(1In+1-1I*)iiiiiiiiiiiiii+βSn+1(g(In+1)In+1-g(I*)I*)iiiiiiiiiiiiii+βg(I*)I*(Sn+1-S*)]×(In+1-I*I*In+1)+(-d1Sn+1-d2In+1-d3Rn+1)×(Sn+1-S*+In+1-I*+Rn+1-R*)=-[k1(βg(In+1)+d1)+d1](Sn+1-S*)2-d2(In+1-I*)2-[k3(d3+δ)+d3](Rn+1-R*)2+k2g(In+1)-g(I*)g(In+1)[βS*(g(In+1)-g(I*))iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii-(d2+γ)(In+1-I*)]+k4I*bA(In+1-I*)(1In+1-1I*)+k4I*βSn+1(In+1-I*)(g(In+1)In+1-g(I*)I*). From hypothesis (H) and d2+γ=βS*(g(I*)/I*), we have (53)k2g(In+1)-g(I*)g(In+1)×[βS*(g(In+1)-g(I*))-(d2+γ)(In+1-I*)]0,k4I*βSn+1(In+1-I*)(g(In+1)In+1-g(I*)I*)0, and it is easy to see that (54)k4I*bA(In+1-I*)(1In+1-1I*)0. Hence, (55)Un+1-Un-[k1(βg(In+1)+d1)+d1](Sn+1-S*)2-d2(In+1-I*)2-[k3(d3+δ)+d3](Rn+1-R*)2. This implies that (56)Un+1-Un<0(Sn,In,Rn)(S*,I*,R*). By Lyapunov’s theorems on the global asymptotical stability of difference equations, we directly obtained that the endemic equilibrium E* is globally asymptotically stable. This completes the proof.

5. Conclusion

From the main results obtained in this paper, we see that the results on the global asymptotic stability of the disease-free equilibrium and endemic equilibrium for a discrete-time SIRS epidemic model with bilinear incidence rate obtained in  are directly extended. By constructing new discrete type Lyapunov functions we established the sufficient and necessary conditions on the global asymptotic stability of the disease-free equilibrium and endemic equilibrium for a class of discrete-time SIRS epidemic models with general nonlinear incidence rate βSg(I) and different death rates d1, d2, and d3. That is, the disease-free equilibrium is globally asymptotically stable if and only if basic reproduction number R01, and the endemic equilibrium is globally asymptotically stable if and only if R0>1.

An interesting and important open problem is whether the results obtained in this paper can be extended to the following discrete-time SIRS epidemic models with general nonlinear incidence rate: (57)Sn+1-Sn=aA-βf(Sn+1)g(In+1)-d1Sn+1+δRn+1,In+1-In=bA+βf(Sn+1)g(In+1)-(d2+γ)In+1,Rn+1-Rn=cA+γIn+1-(d3+δ)Rn+1 and with distributed delay (58)Sn+1-Sn=aA-βSn+1j=0mf(j)g(In-j)-d1Sn+1+δRn+1,In+1-In=bA+βSn+1j=0mf(j)g(In-j)-(d2+γ)In+1,Rn+1-Rn=cA+γIn+1-(d3+δ)Rn+1. That is, only under the assumption which functions g(I) and I/g(I) are monotonically increasing with respect to I, whether we also can obtain that the disease-free equilibrium is globally asymptotically stable if basic reproduction number R01, and the endemic equilibrium is globally asymptotically stable if R0>1.

In addition, in this paper, functions g(I) and I/g(I) in model (1) are assumed to be monotonically increasing with respect to I. Obviously, these conditions are rather strong and not easily satisfied in many practical applications. Therefore, an interesting and important open problem is whether the results obtained in this paper can be extended to model (1) with function g(I) or I/g(I) is not monotonically increasing with respect to I.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the referees for their careful reading of the paper and many valuable comments and suggestions that greatly improved the presentation of this paper. This work is supported by the National Natural Science Foundation of China (Grant nos. 11271312 and 11261058), the China Postdoctoral Science Foundation (Grant nos. 20110491750 and 2012T50836), and the Natural Science Foundation of Xinjiang (Grant no. 2011211B08).

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