Stationary distribution of a stochastic generalized SIRI epidemic model with reinfection and relapse

In this paper, we propose and investigate the stochastic SIRI epidemic model with two generalized incidence rate functions. We firstly study the existence and uniqueness of the globally positive solution to the stochastic SIRI model with positive initial value. Then we obtain sufficient conditions for the extinction of the disease in the stochastic epidemic model, and find that the large noise can make the disease die out exponentially. Meanwhile, we obtain that the solution to the stochastic model has a unique stationary distribution when $\widetilde{R_0^s}$ is greater than one. Our results show that the intensity of white noise can affect the dynamical behaviors of the model. Finally, we use numerical simulation to illustrate theoretical results, and apply both the stochastic and deterministic models to analyze the outbreak of COVID-19 epidemic in Serbia.