Environmental variability in a stochastic epidemic model

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 Rds and the stochastic basic reproduction number R0s, 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.