A stochastic SIHR epidemic model with general population-size dependent contact rate and Ornstein–Uhlenbeck process: dynamics analysis

The purpose of this work is to investigate a novel stochastic SIHR epidemic model, which includes a general incidence rate and mean-reversion Ornstein–Uhlenbeck process. Firstly, the existence of global positivity of the solution is testified by Lyapunov function. Secondly, this disease will be eradicated if the reproduction number \(\mathcal {R}_{0}^{s}<1\) ${\mathcal{R}}_0^s<1$ . Otherwise, if the reproduction number \(\mathcal {R}_{0}^{*}>1\) ${\mathcal{R}}_0^{\ast }>1$ , then the system has a stationary distribution, which means that the pandemic will persist. In addition, an explicit expression of the probability density function for a linear system near quasi-endemic equilibrium is obtained under certain conditions. Finally, a series of numerical simulations is carried out to validate the theoretical conclusions.