Dynamics and density function analysis of a stochastic SVI epidemic model with half saturated incidence rate

•A SVI epidemic model with half saturated incidence rate is considered.•Local asymptotic stability of the equilibria of deterministic SVI model is obtained.•The threshold for persistence and extinction of disease is derived.•The expression of probability density function around the positive equilibrium of the deterministic model is obtained. In this paper, we study the dynamical behaviors of a SVI epidemic model with half saturated incidence rate. Firstly, the local asymptotic stability of the endemic and disease-free equilibria of the deterministic model are studied. Then for stochastic model, we show that there is a critical value R0s which can determine the extinction and the persistence in the mean of the disease. Furthermore, by constructing a series of suitable Lyapunov functions, we prove that if R0s>1, then there exists an ergodic stationary distribution to the stochastic SVI model. It is worth mentioning that we obtain an exact expression of the probability density function of the stochastic model around the unique endemic equilibrium of the deterministic system by solving the corresponding Fokker-Planck equation, which is guaranteed by a new critical R^0s. Finally, some numerical simulations illustrate the analytical results.