Dynamical analysis of a stochastic $SEI_{u}I_{r}RS$ epidemic model with vaccination and log-normal Ornstein-Uhlenbeck process

In this study, we develop and investigate a stochastic $SEI_{u}I_{r}RS$ (Susceptible-Exposed-Undetected infected-Detected infected-Recovered-Susceptible) epidemic model with vaccination of newborns and the disease transmission rate driven by a log-normal Ornstein-Uhlenbeck process. By establishing a series of Lyapunov functions, we derive sufficient conditions for persistence in the mean of the disease in the long term under the condition $\mathcal{R}_{0}>1$ and these criteria are applied to guarantee the existence of an invariant probability measure of the stochastic system. Subsequently, we also derive sufficient conditions for eradication of the disease when the parameters $\mathcal{R}_{0}< 1$ and $\mathcal{R}_{0}^{E}< 1$. Finally, two numerical examples are given to confirm the theoretical results. This work can help us implement interventions to regulate the disease dynamics.