THE THRESHOLD OF A STOCHASTIC SIR EPIDEMIC MODEL WITH LOGISTIC GROWTH

In this paper, we propose and study a stochastic SIR epidemic model with logistic growth. The threshold dynamics of the stochastic model are governed by a parameter $ R_0^S $. Precisely, the disease goes to extinction at an exponential rate and the distribution of susceptible individuals converges weakly to a unique invariant probability measure provided that $ R_0^S<1 $; whereas if $ R_0^S>1 $, we show that there is a unique ergodic stationary distribution of the positive solutions to the model by constructing a suitable stochastic Lyapunov function. Numerical simulations are introduced to illustrate our theoretical results. The aim of the work is to investigate the impact of environmental noise on the transmission of infectious diseases.