An epidemic model with Beddington–DeAngelis functional response and environmental fluctuations

Abstract

An epidemic model with the saturated incidence rate and the environmental fluctuations is investigated in this paper. We study the extinction and the persistence in the mean, and stationary distribution of the solution as well. By contradiction, we firstly show that stochastic epidemic model admits a unique global positive solution for any given positive initial value. Further, when R˜0>1 is valid, we derive the persistence in the mean, and also prove the existence of an ergodic stationary distribution by constructing moderate functions. By comparison theorem of stochastic differential equations and properties of inequalities, the extinction of the solution is finally derived when R0<1 and ν<0 hold, where ν indicates the exponential rate for decline. Meanwhile, the distribution for the density of the susceptible is estimated. As a consequence, numerical simulations and illustrative examples are separately carried out to support the main results of this paper.