A stochastic SIS epidemic model with saturating contact rate

Abstract 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 R 0 s is proved to be a sharp threshold which determines whether there is an endemic outbreak or not. If R 0 s 1 and under mild extra conditions, the disease can be eradicated almost surely whereas if R 0 s > 1 , it has a stationary distribution which leads to the stochastic persistence of the disease. Finally, numerical simulations are presented to illustrate our theoretical results.