In this paper, we investigate the stochastic dynamics of a simple epidemic model incorporating the mean-reverting Ornstein-Uhlenbeck process analytically and numerically. We define two threshold parameters, the stochastic demographic reproduction number R-d(s) and the stochastic basic reproduction number R-0(s), to utilize in identifying the stochastic extinction and persistence of the disease. We find that the stochastic disease dynamics can be determined by the environment fluctuations which measured by the intensity of volatility and the speed of reversion: the larger intensity of volatility or the smaller speed of reversion can suppress the outbreak of the disease, the smaller intensity of volatility or the the higher speed of reversion can enhance the outbreak of the disease. Furthermore, via numerical simulations, we find that the stochastic model has an endemic stationary distribution which leads to the stochastic persistence of the disease. Our results show that mean-reverting process is a well-established way of introducing stochastic environmental noise into biologically realistic population dynamic models. (C) 2018 Elsevier Inc. All rights reserved.

In this paper, we investigate the stochastic extinction and persistence of a parasite–host epidemiological model. We show that the global dynamics of the stochastic model can be governed by the basic reproduction number R0S: if R0S<1, under mild extra conditions, the disease goes to extinction with probability one and the disease-free dynamics occurs; while R0S>1, under mild extra conditions, the disease persists and endemic dynamics occurs almost surely, the solutions of the stochastic model fluctuate around the steady state of the deterministic model, and a unique stationary distribution can be found. Based on realistic parameters of Daphnia-microparasite system, numerical simulations have been performed to verify/extend our analytical results. Epidemiologically, we find that: (1) Large environment fluctuations can suppress the outbreak of disease; (2) The distributions are governed by R0S; (3) The noise perturbations can be beneficial to control the spread of disease on average.