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Abstract and Applied Analysis
Volume 2014, Article ID 723825, 13 pages
http://dx.doi.org/10.1155/2014/723825
Research Article

Dynamics of a Stochastic SIS Epidemic Model with Saturated Incidence

Department of Applied Mathematics, School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China

Received 6 April 2014; Accepted 19 May 2014; Published 15 June 2014

Academic Editor: Debora Amadori

Copyright © 2014 Can Chen and Yanmei Kang. 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 introduce stochasticity into the SIS model with saturated incidence. The existence and uniqueness of the positive solution are proved by employing the Lyapunov analysis method. Then, we carry out a detailed analysis on both its almost sure exponential stability and its pth moment exponential stability, which indicates that the pth moment exponential stability implies the almost sure exponential stability. Additionally, the results show that the conditions for the disease to become extinct are much weaker than those in the corresponding deterministic model. The conditions for the persistence in the mean and the existence of a stationary distribution are also established. Finally, we derive the expressions for the mean and variance of the stationary distribution. Compared with the corresponding deterministic model, the threshold value for the disease to die out is affected by the half saturation constant. That is, increasing the saturation effect can reduce the disease transmission. Computer simulations are presented to illustrate our theoretical results.