A Stochastic SIS Epidemic Model with Ornstein-Uhlenbeck Process and Brown Motion

Abstract

This paper studies a stochastic differential equation SIS epidemic model, disturbed randomly by the mean-reverting Ornstein-Uhlenbeck process and Brownian motion. We prove the existence and uniqueness of the positive global solutions of the model and obtain the controlling conditions for the extinction and persistence of the disease. The results show that when the basic reproduction number Rs0 < 1, the disease will extinct, on the contrary, when the basic reproduction number Rs0 > 1, the disease will persist. Furthermore, we can inhibit the outbreak of the disease by increasing the intensity of volatility or decreasing the speed of reversion ϑ, respectively. Finally, we give some numerical examples to verify these results.

Dynamical analysis and numerical simulation of a stochastic influenza transmission model with human mobility and Ornstein-Uhlenbeck process

“…This causes a unreasonable result that the variance σ 2 T → +∞ as T → 0, meaning that the modified parameter β(t) will drastically change in a short period which conflicts with continuous environment. To deal with this shortcoming, various stochastic models adapt the assumption that the modified parameter satisfies the mean-reversing process [24,25,26,27] instead of the linear functions of white noise. Inspired by these brilliant ideas, in this paper, we suppose that the transmission rate β is the log-normal Ornstein-Uhlenbeck process, that is to say…”