Long-time behavior and quasi-density function for a stochastic epidemic model with relapse and reinfection

Abstract

A stochastic susceptible–infected–recovered–infected (SIRI) epidemic model with relapse and reinfection is established in this paper. First, we prove that the solution to the epidemic model is unique and globally positive. Next, we determine some sufficient conditions for the extinction of the disease when [Formula: see text] and for the persistence in mean in the case of [Formula: see text]. Furthermore, we prove the existence of at least one ergodic stationary distribution of the stochastic model if [Formula: see text]. Additionally, by solving the corresponding three-dimensional Fokker–Planck equation, it is theoretically shown that the epidemic model has a log-normal probability density function when [Formula: see text], then we obtain the exact expression of density function of the stationary distribution. Finally, we give some numerical simulations to support our theoretical results.