A Stochastic Differential Equation SIRS Epidemic Model With Ratio–Dependent Incidence Rate

In this paper we study global dynamics of a general SIRS epidemic model with a ratiodependent incidence rate and its corresponding stochastic differential equation version. For the deterministic model, we show that the disease reproduction number R0 determines whether there is an endemic outbreak or not: the disease free dynamics occurs if R0 ≤ 1 while the unique endemic steady state is globally stable if R0 > 1. For the stochastic model, we show that its related reproduction number R 0 can determine whether there is a unique disease-free stationary distribution or a unique endemic stationary distribution. In addition, we provide analytic results regarding the stochastic boundedness, permanence/extinction, asymptotic stability as well as its related ergodic properties. One of the most interesting findings is that random fluctuations introduced in our stochastic model can suppress disease outbreak, which can provide us some useful control strategies to regulate disease dynamics. Combining our analytical and numerical results, we conclude that different stochastic modeling approaches can have different dynamical outcomes. This provides us a caution when we derive the stochastic models from a given deterministic model. Email addresses: [email protected] (Yongli Cai), [email protected] (Yun Kang), [email protected] (Weiming Wang∗ ), [email protected] (Min Zhao) ∗ Author to whom any correspondence should be addressed. Preprint submitted to Journal of Mathematical Biology August 12, 2013