Asymptotic behavior of a stochastic hybrid SIQRS model with vertical transmission and nonlinear incidence

We studied a class of a stochastic hybrid SIQRS model with nonlinear incidence and vertical transmission and obtained a threshold $\mathrm{\Delta }$ to distinguish behaviors of the model. Concretely, the disease was extinct exponentially when $\mathrm{\Delta }<0$. If $\mathrm{\Delta }>0$, the model we discussed admitted an invariant measure. A new class of the Lyapunov function was constructed in proving the latter conclusion. Some remarks were presented to shed light on the major results. Finally, several numerical simulations were provided to test the reached results.