Threshold dynamics of a stochastic SIRS epidemic model with transfer from infected individuals to susceptible individuals and log-normal Ornstein-Uhlenbeck process

This paper focused on a stochastic susceptible-infected-recovered-susceptible (SIRS) epidemic model with standard incidence and transfer from infected individuals to susceptible individuals. We assumed that the incidence rate satisfied the log-normal Ornstein-Uhlenbeck process. First, by using stochastic Lyapunov analysis method, the sufficient condition for the existence of stationary distribution was obtained. After that, we established the sufficient criteria for the extinction of the infectious disease. It was worth noting that the dynamical behavior of the considered model was governed by a threshold. In addition, we derived the exact expression of probability density function near the positive equilibrium point of the corresponding deterministic system. Finally, some numerical simulations were carried out to confirm theoretical results.