Dynamical behavior and density function of a stochasticSIR model with saturated incidence rate under Ornstein-Uhlenbeck process

In this paper, we discuss a stochastic SIR model with saturated incidence rate, which perturbed by log-normal Ornstein-Uhlenbeck (OU) process. The stability of the disease-free equilibriumand endemic equilibrium of deterministic system is initially established. Subsequently, the existenceof unique positive solutions of stochastic systems can be proved through the construction of suitableLyapunov functions. The extinction of stochastic SIR model are demonstrated by introducing thethreshold $R_0^e$. Furthermore, we study the persistence of stochastic SIR model as well as derive itsstationary distribution, and the probability density function around the quasi-endemic equilibrium iscomputed. Finally, theoretical findings are validated through numerical simulations.