A stochastic differential equation SIS epidemic model incorporating Ornstein–Uhlenbeck process

Abstract

In this paper, based on the results of Gray et al. (2011), we propose a new SDE SIS model incorporating mean-reverting Ornstein–Uhlenbeck process, and prove that the stochastic basic reproduction number R0s can be used to identify the stochastic extinction and persistence for the SDE mode: if R0s<1 under mild extra conditions, the disease will be extinct a.s., while if R0s>1, the disease will persist a.s. Epidemiologically, we find that smaller speed of reversion or bigger intensity of volatility can suppress the disease outbreak. Thus, in order to control the spread of the disease, we must decrease the speed of reversion or increase the intensity of volatility.