Abstract
This paper is devoted to a continuous-time stochastic differential system which is derived by incorporating white noise to a deterministic [Formula: see text] epidemic model with mass action incidence, cure and relapse. We focus on the impact of a relapse on the asymptotic properties of the stochastic system. We show that the relapse encourages the persistence of the disease in the population and we determine the threshold of the relapse rate, above the threshold the disease prevails in the population. Furthermore, we show that there exists a unique density function of solutions which converges in [Formula: see text], under certain conditions of the parameters to an invariant density.
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