Abstract
In this paper, we investigate a stochastic susceptible-infective-removed-infective (SIRI) epidemic model with relapse. We show that the densities of the distributions of the solutions can converge in
to an invariant density or can converge weakly to a singular measure under certain condition. We also find the support of the invariant density. Moreover, we establish sharp sufficient criteria for the extinction of the disease in two cases. The results show that the smaller white noise can assure the existence of a stationary distribution which implies the persistence of the disease while the larger white noise can lead to the extinction of the disease.