Dynamics of a stochastic epidemic model with information intervention and vertical transmission

The dynamic behavior of a stochastic epidemic model with information intervention and vertical transmission was the concern of this paper. The threshold to judge the extinction and persistence of the disease was obtained. Specifically, when $\mathrm{\Delta }<0$ ( $\mathrm{\Delta }$ appears in Section 3), the three classes $I_t$, $M_t$, and $R_t$ appearing in the model go extinct at an exponential rate, and the susceptible class $S_t$ almost surely converges to the solution of the boundary equation exponentially. When $\mathrm{\Delta }>0$, the result that the disease in the model is persistent in the mean and the existence of invariant probability measure are proved by constructing a new form of Lyapunov functions, which results in getting sufficient and nearly necessary conditions for different properties. Moreover, one of the main characteristics of this article was the study of the critical case of $\mathrm{\Delta }=0$ under some conditions. Some examples were listed to confirm the obtained results.