Complex Dynamics of a Stochastic SIR Epidemic Model with Vertical Transmission and Varying Total Population Size

Abstract

In this paper, we propose a stochastic SIR epidemic model with vertical transmission and varying total population size. Firstly, we prove the existence and uniqueness of the global positive solution for the stochastic model. Secondly, we establish three thresholds $\lambda _{1},$ $\lambda _{2}$ and $\lambda _{3}$ of the model. The disease will die out when $\lambda _{1}<0$ and $\lambda _{2}<0,$ or $\lambda _{1}>0$ and $\lambda _{3}<0$, but the disease will persist when $\lambda _{1}<0$ and $\lambda _{2}>0,$ or $\lambda _{1}>0$ and $\lambda _{3}>0$ and the law of the solution converge to a unique invariant measure. Moreover, we find that when $\lambda _{1}<0$ some stochastic perturbations can increase the threshold $\lambda _{2}$, while others can decrease the threshold $\lambda _{2}$. That is, some stochastic perturbations enhance the spread of the disease, but others are just the opposite. On the other hand, when $\lambda _{1}>0$, some stochastic perturbations increase or decrease the threshold $\lambda _{3}$ with different parameter sets. Finally, we give some numerical examples to illustrate obtained results.