#### Abstract

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 [1], the population of size at time is divided into three distinct classes: the susceptible class of size , the infectious class of size , and the recovered class at time . 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 of the model and also to obtain the fact that the disease-free equilibrium is globally stable when , as well as the endemic equilibrium exists and is globally stable when .

Until now, the discrete-time SIR and SIRS epidemic models have been extensively studied in many articles; for example, see [2–22] 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 [4], 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 [5], 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 [6], 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 , as well as the global stability of the endemic equilibrium when , are established only for bilinear incidence rate (see [4]). Therefore, motivated by the above works, as an extension of the results given in [4], 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:

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 , and the endemic equilibrium is globally asymptotically stable if and only if .

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 . In the fourth section, we will study the global asymptotic stability of endemic equilibrium of model (1) for case . Lastly, in the fifth section, we will give a conclusion.

#### 2. Preliminaries

In model (1), , , and denote the numbers of susceptible, infected, and recovered individuals at th period, respectively. is the recruitment rate of the total population, parameters , , and are the fraction constants of input to susceptible class , infected class , and recovered class , respectively, and satisfying . , , and represent the death rate of susceptible, infected, and recovered individuals, respectively. Particularly, death rate includes the natural death rate and the disease-related death rate of the infected individuals. is the rate at which recovered individuals lose immunity and return to the susceptible class. is the natural recovery rate of the infective individuals. is the proportionality constant, and the transmission of the infection is governed by a nonlinear incidence rate . In this paper, we always assume that are positive constants.

The initial condition for model (1) is given by Throughout this paper, we always assume that (H) is continuous and monotonically increasing on , , is also monotonically increasing on , and exists with .

*Remark 1. *Hypothesis (H) is basic for model (1). Particularly, with constant ; then assumption naturally holds. Furthermore, if function satisfies that second-order derivative exists and for all , then we easily prove that is monotone increasing on .

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 of model (1) with initial condition (2), it holds that
*

*Proof. *Model (1) is equivalent to the following form:
When , we have
Let and let ; then obviously, . Substituting , into , we obtain that satisfies the following equation:
where . From this, we further obtain that satisfies the following equation:
From hypothesis (H), we obtain that is monotonically increasing on , and, obviously,
If , then we have
and if , then . Hence, there is a unique positive solution such that . Therefore, we have . Further, from (5) we also obtain and .

When , in a similar way, we can obtain , and . By the induction, we finally obtain that , and for all . This completes the proof.

Lemma 3. *For any solution of model (1) with initial condition (2), it holds that
**
where .*

*Proof. *Let ; then from model (1) we have
Hence,
By using iteration method, we obtain
Therefore, it holds that
This completes the proof.

#### 3. Case

If , we have , and , ; then model (1) becomes into the following form: Particularly, when , then and model (1) will become into the following well-known form:

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 Here, is the disease transmission rate, is the average infection period, and implies that 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: On the existence of equilibria of model (15), we have the following result.

Theorem 4. *(1)* If , then model (15) only has a unique disease-free equilibrium , where and .*(2)* If , then model (15) has a unique endemic equilibrium , except for the disease-free equilibrium .

*Proof. *We know that an equilibrium of model (15) satisfies
Firstly, when , we have
from which we obtain the disease-free equilibrium , where and .

Secondly, when , from the second and third equations of (20), we obtain
Substituting into the first equation of (20), we have
Denote
By hypothesis (H), is monotonically decreasing on satisfying
and we also have

When , we have . Then, there is not any such that . Therefore, model (15) only has a unique disease-free equilibrium .

When , we have . Then, there exists a unique such that . Furthermore, we have and . This implies that model (15) has a unique endemic equilibrium . This completes the proof.

*Remark 5. *Particularly, for model (16), the disease-free equilibrium given in Theorem 4 will become into .

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

Theorem 6. *Disease-free equilibrium of model (15) is globally asymptotically stable if and only if .*

*Proof. *The necessity is obvious; we only need to prove the sufficiency. Model (15) can be rewritten as the following form:
We consider the following Lyapunov function:
where
Calculating the difference of along (27), we have
Since , we have . Hence,
Under hypothesis (H), we have for any
Hence,
This implies that
By Lyapunov’s theorems on the global asymptotical stability for difference equations, we directly obtained that the disease-free equilibrium is globally asymptotically stable. This completes the proof.

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

Theorem 7. *Endemic equilibrium of model (15) is globally asymptotically stable if and only if .*

*Proof. *The necessity is obvious, we only need to prove the sufficiency. Model (15) can be rewritten as the following form:
We also have

We consider the following Lyapunov function:
where
Calculating the difference of along (35), we have
Further, from hypothesis (H) and , we have
Hence,
This implies that
By Lyapunov’s theorems on the globally asymptotical stability for difference equations, we directly obtained that the endemic equilibrium is globally asymptotically stable. This completes the proof.

*Remark 8. *From the above discussion we immediately see that the basic reproduction number 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.*(1)*Disease-free equilibrium is globally asymptotically stable if and only if .*(2)*Endemic equilibrium is globally asymptotically stable if and only if .*

*Remark 10. *From Corollary 9, we see that the corresponding results on the global asymptotic stability obtained in [4] 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 [4], 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

We firstly discuss the existence of equilibria of model (1). It is easy to see that if , 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 .*

*Proof. *From model (1) we know that the endemic equilibrium satisfies
By the second and third equations of (43), we can obtain
Substituting (44) into the first equation of (43), we have
Denote

Now, we consider equation , which is equivalent to (45). By hypothesis (H), is monotonically decreasing on and is monotonically increasing on . Let ; then we have . By calculating, we obtain
Hence, there exists a unique such that . Furthermore, we have and . This implies that model (1) has a unique endemic equilibrium .

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

Theorem 12. *Endemic equilibrium of model (1) is always globally asymptotically stable.*

*Proof. *Model (1) becomes into the following form:
We also have

We consider the following Lyapunov function:
where
Calculating the difference of along (48), we have
From hypothesis (H) and , we have
and it is easy to see that
Hence,
This implies that
By Lyapunov’s theorems on the global asymptotical stability of difference equations, we directly obtained that the endemic equilibrium 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 [4] 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 and different death rates , , and . That is, the disease-free equilibrium is globally asymptotically stable if and only if basic reproduction number , and the endemic equilibrium is globally asymptotically stable if and only if .

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: and with distributed delay That is, only under the assumption which functions and are monotonically increasing with respect to , whether we also can obtain that the disease-free equilibrium is globally asymptotically stable if basic reproduction number , and the endemic equilibrium is globally asymptotically stable if .

In addition, in this paper, functions and in model (1) are assumed to be monotonically increasing with respect to . 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 or is not monotonically increasing with respect to .

#### 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).