Analysis of a Kind of Stochastic Dynamics Model with Nonlinear Function

Abstract

In this paper, we establish stochastic differential equations on the basis of a nonlinear deterministic model and study the global dynamics. For the deterministic model, we show that the basic reproduction number $\mathfrak{R}_0$ determines whether there is an endemic outbreak or not: if $\mathfrak{R}_0<1,$ the disease dies out; while if $\mathfrak{R}_0>1,$ the disease persists. For the stochastic model, we provide analytic results regarding the stochastic boundedness, perturbation, permanence and extinction. Finally, some numerical examples are carried out to confirm the analytical results. One of the most interesting findings is that stochastic fluctuations introduced in our stochastic model can suppress disease outbreak, which can provide us some useful control strategies to regulate disease dynamics.