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.

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 [

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

Until now, the discrete-time SIR and SIRS epidemic models have been extensively studied in many articles; for example, see [

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

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

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 (

In model (

The initial condition for model (

Hypothesis (H) is basic for model (

On the positivity and boundedness of all solutions of model (

For any solution

Model (

When

For any solution

Let

If

For model (

We know that an equilibrium

Secondly, when

When

When

Particularly, for model (

Now, we study the stability of equilibria of model (

Disease-free equilibrium

The necessity is obvious; we only need to prove the sufficiency. Model (

On the global stability of the endemic equilibrium

Endemic equilibrium

The necessity is obvious, we only need to prove the sufficiency. Model (

We consider the following Lyapunov function:

From the above discussion we immediately see that the basic reproduction number

As a consequence of Theorems

For model (

Disease-free equilibrium

Endemic equilibrium

From Corollary

We firstly discuss the existence of equilibria of model (

Model (

From model (

Now, we consider equation

Now, we study the global stability of endemic equilibrium

Endemic equilibrium

Model (

We consider the following Lyapunov function:

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 [

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:

In addition, in this paper, functions

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

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).