GLOBAL DYNAMICS OF DETERMINISTIC AND STOCHASTIC SIRS EPIDEMIC MODELS

In this paper, we analyze the dynamic behavior of Heesterbeek et al. [ 12 ] obtained saturating contact rate applied to SIRS epidemic model. We define two threshold values, the deterministic basic reproduction number $ R_0 $ and the stochastic basic reproduction number $ R_0^s $, by comparing the value with one to determine the persistence and extinction of the disease. For deterministic model, if $ R_0 1 $, the system admits a unique endemic equilibrium which is locally asymptotically stable. For stochastic model, we also establish the threshold value $ R_0^s $ for disease persistence and extinction. Finally, some numerical simulations are presented to illustrate our theoretical results. Our results prove that large stochastic perturbation will lead to the extinction of diseases with probability one, revealing the significant influence of stochastic perturbation on diseases and the importance of incorporating stochastic perturbation into deterministic model.