Ergodicity and stationary distribution of a stochastic SIRI epidemic model with logistic birth and saturated incidence rate

The present study aims to investigate the dynamic behavior of a stochastic SIRI model with logistic birth and saturated incidence rate. First, we show that the solution of the proposed stochastic SIRI model is global and positive. Then, we established sufficient conditions for the extinction and persistence in mean of the infectious disease. Moreover, by formulating a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the solution of the model. Finally, the theoretical results are verified by some numerical simulations.