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• stochastic SIRSI epidemic model with saturation incidence rate and logistic growth is established. • The existence of a unique global positive solution for the stochastic model are obtained. • We establish the sufficient conditions for the unique ergodic stationary distribution. • We derive the exact expression of probability density function to the stochastic model. Focusing on the results of Rajasekar (2020) and the continuous dynamics of stochastic differential equation (SDE) developed by Mao (1997), a stochastic SIRSI epidemic model with saturation incidence rate and logistic growth is investigated in this paper. First, we propose and prove that the unique solution of stochastic model is globally positive. By constructing some suitable Lyapunov functions, the sufficient condition R 0 h > 1 is obtained for the unique stationary distribution which has ergodicity property. Next, by solving the corresponding Fokker-Planck equation, we derive the approximate probability density function around the quasi-endemic equilibrium of the stochastic system. The above stationary distribution and density function can reveal all statistical properties of the disease persistence. In addition, by comparison with other existing articles, our developed theoretical results and some numerical simulations are introduced at the end of this paper. [ABSTRACT FROM AUTHOR]

In this paper, we present a stochastic hepatitis B virus (HBV) infection model and the dynamic behaviors of the model are investigated. When the fraction of vertical transmission μ ω ν C is not considered to be new infections, the existence and ergodicity of the stationary distribution of the model are obtained by constructing a suitable Lyapunov function, which determines a critical value ρ 0 s corresponding to the basic reproduction number of ODE system. This implies the persistence of the diseases when ρ 0 s > 1. Meanwhile, the sufficient conditions for the extinction of the diseases are derived when ρ 0 T < 0. What is more, we give the specific expression of the probability density function of the stochastic model around the unique endemic quasi-equilibrium by solving the Fokker–Planck equation. Finally, the numerical simulations are illustrated to verify the theoretical results and match the HBV epidemic data in China. • A stochastic HBV infection model is proposed and studied. • Existence and ergodicity of stationary distribution are proved. • Critical value ρ 0 s corresponding to basic reproduction number ρ 0 is obtained. • Density function of stochastic model near quasi-equilibrium is given. • Simulations match the HBV epidemic data in China. [ABSTRACT FROM AUTHOR]