Global Behaviors of a Class of Discrete SIRS Epidemic Models with Nonlinear Incidence Rate

Zhidong Teng,Long Zhang,Lei Wang

doi:10.1155/2014/249623

Abstract

We study a class of discrete SIRS epidemic models with nonlinear incidence rateF(S)G(I)and disease-induced mortality. By using analytic techniques and constructing discrete Lyapunov functions, the global stability of disease-free equilibrium and endemic equilibrium is obtained. That is, if basic reproduction numberℛ0<1, then the disease-free equilibrium is globally asymptotically stable, and ifℛ0>1, then the model has a unique endemic equilibrium and when some additional conditions hold the endemic equilibrium also is globally asymptotically stable. By using the theory of persistence in dynamical systems, we further obtain that only whenℛ0>1, the disease in the model is permanent. Some special cases ofF(S)G(I)are discussed. Particularly, whenF(S)G(I)=βSI/(1+λI), it is obtained that the endemic equilibrium is globally asymptotically stable if and only ifℛ0>1. Furthermore, the numerical simulations show that for general incidence rateF(S)G(I)the endemic equilibrium may be globally asymptotically stable only asℛ0>1.

Highlights

IntroductionNo matter discrete epidemic models or continuous epidemic models, have been widely studied
During the past decades, no matter discrete epidemic models or continuous epidemic models, have been widely studied
In [13], the author studied a continuous SIRS epidemic model with bilinear incidence rate and obtained that the disease-free equilibrium is globally stable if basic reproduction number R0 ≤ 1 and the endemic equilibrium is globally stable if R0 > 1

Summary

Introduction

No matter discrete epidemic models or continuous epidemic models, have been widely studied. The main research subjects are the computation of the threshold value or basic reproduction number which distinguishes whether the infectious disease will persist or die out, the local and global stability of the disease-free equilibrium and endemic equilibrium, the extinction, persistence, and permanence of the disease, and the bifurcations, chaos, and more complex dynamical behaviors of the models. Enatsu et al in [22] proposed a class of discrete SIR epidemic models with bilinear incidence rate, which are derived from continuous SIR epidemic models with distributed delays by using a variation of the backward Euler method, and obtained that global stability of diseasefree equilibrium and endemic equilibrium. Muroya et al in [23] discussed global stability and permanence of a discrete epidemic model with bilinear incidence rate and for disease with immunity and latency spreading in a heterogeneous host population, which is discretized from the continuous case by using the backward Euler method.

Equilibria and Positivity

Global Stability

Permanence of Disease

Numerical Simulations

Conclusions