The threshold of a stochastic SIVS model with saturated incidence based on logarithmic Ornstein-Uhlenbeck process

Abstract

In this study, considering the spread of infectious diseases satisfies the logarithmic Ornstein-Uhlenbeck process, we construct a SIVS model with saturated incidence. By constructing a number of suitable Lyapunov functions, a sufficient condition for the disease to persist for a long time is obtained when $R_0^s>1$ which it is an essential condition to prove the existence of stationary distribution for stochastic system. At the same time, we also obtain that the disease will be died out when $R_0^s<1$. Furthermore, by solving the corresponding matrix equation, the exact expression of the probability density function for the stochastic model around the quasi-endemic equilibrium point is obtained. In the end, numerical simulations are conducted to support the theory we obtain.