Ergodic stationary distribution of a stochastic nonlinear epidemic model with relapse and cure

In this paper, we consider a stochastic general epidemic model with relapse, graded cure and nonlinear incidence rate. We show the existence and uniqueness of a global positive solution for the model under some mild conditions. We also give some sufficient conditions for ensuring the existence of an ergodic stationary distribution for the model. Moreover, we analyse the extinction of the disease described by the model. Finally, we provide some numerical simulations to illustrate our main results.