%0 Journal Article
%T Survival and Stationary Distribution in a Stochastic SIS Model
%A Yanli Zhou
%A Weiguo Zhang
%A Sanling Yuan
%J Discrete Dynamics in Nature and Society
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/592821
%X The dynamics of a stochastic SIS epidemic model is investigated. First, we show that the system admits a unique positive global solution starting from the positive initial value. Then, the long-term asymptotic behavior of the model is studied: when , we show how the solution spirals around the disease-free equilibrium of deterministic system under some conditions; when , we show that the stochastic model has a stationary distribution under certain parametric restrictions. In particular, we show that random effects may lead the disease to extinction in scenarios where the deterministic model predicts persistence. Finally, numerical simulations are carried out to illustrate the theoretical results. 1. Introduction Mathematical epidemiology describing the population dynamics of infectious diseases has been made a significant progress in better understanding of disease transmissions and behavior of epidemics. Many epidemic models have been described by ordinary differential equations [1¨C11]. These important and useful deterministic investigations offer a great insight into the effects of infectious disease, but in the real world, epidemic dynamics is inevitably affected by the environmental noise, which is an important component in the epidemic systems. As a matter of fact, the epidemic models are often subject to environmental noise; that is, due to environmental fluctuations, parameters involved in epidemic models are not absolute constants, and they may fluctuate around some average values. So, inclusion of random perturbations in such models makes them more realistic in comparison to their deterministic counterparts. In recent years, epidemic models under environmental noise described by stochastic different equations have been studied by many researchers. They introduce stochastic noises into deterministic models to reveal the effect of environmental variability on the epidemic dynamics in mathematical ecology [12¨C20]. For example, Chen and Li [12] discussed the stability of the endemic equilibrium of a stochastic SIR model. Tornatore et al. [13] studied the stability of the disease-free equilibrium of a stochastic SIR model with and without distributed time delay. Ji et al. [14] discussed a multigroup SIR model with perturbation, where they showed that if the basic reproduction number , then the solution of the model oscillates around the disease-free steady state, whereas, if , there is a stationary distribution. However, this field is still in its infancy. The classical epidemic models proposed by Kermack and McKendrick in 1927 [21] have been
%U http://www.hindawi.com/journals/ddns/2013/592821/