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Discrete Dynamics in Nature and Society
Volume 2014, Article ID 472746, 10 pages
http://dx.doi.org/10.1155/2014/472746
Research Article

Lyapunov Functions for a Class of Discrete SIRS Epidemic Models with Nonlinear Incidence Rate and Varying Population Sizes

College of Mathematics and Model Sciences, Xinjiang University, Urumqi 830046, China

Received 21 March 2014; Accepted 21 April 2014; Published 21 July 2014

Academic Editor: Luca Guerrini

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.

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.