A stochastic SIS epidemic model with saturating contact rate

In this paper, we formulate and investigate the dynamics of a SIS epidemic model with saturating contact rate and its corresponding stochastic differential equation version. For the deterministic epidemic model, we show that the disease-free equilibrium is global asymptotically stable if the basic reproduction number is less than unity; and if the basic reproduction number is greater than unity, model (1.2) admits a unique endemic equilibrium which is locally asymptotically stable by analyzing the corresponding characteristic equations. For the stochastic epidemic model, the existence and uniqueness of the positive solution are proved by employing the Lyapunov analysis method. The basic reproduction number R0s is proved to be a sharp threshold which determines whether there is an endemic outbreak or not. If R0s1, it has a stationary distribution which leads to the stochastic persistence of the disease. Finally, numerical simulations are presented to illustrate our theoretical results. •A stochastic SIS epidemic model with saturating contact rate is studied.•We establish sufficient conditions for the diseases to die out with probability one.•The basic reproduction number R0s is proved to be a threshold which determines whether there is an endemic outbreak.•Environmental fluctuations plays a positive effect in the control of infectious disease.