Dynamics of a stochastic SEIQR model: stationary distribution and disease extinction with quarantine measures

This paper investigates the dynamics of a stochastic $\mathcal{SEIQR}$ epidemic model, which integrates quarantine measures and a saturated incidence rate to more accurately reflect real-world disease transmission. The model is based on the classical $\mathcal{SEIR}$ framework, with the addition of a quarantined compartment, offering insights into the impact of quarantine on epidemic control. The saturated incidence rate accounts for the diminishing rate of new infections as the susceptible population grows, addressing the limitations of traditional bilinear incidence rates in modeling epidemic spread under high disease prevalence. We first establish the basic reproductive number, $\mathcal{R}_0$, for the deterministic model, which serves as a threshold parameter for disease persistence. Through the stochastic Lyapunov function method, we identify the necessary conditions for the existence of a stationary distribution, focusing on the case where $\mathcal{R}_0^* > 1$, signals the potential long-term persistence of the disease in the population. Furthermore, we derive sufficient conditions for disease extinction, particularly when $\mathcal{R}_S^* < 1$, indicating that the disease will eventually die out despite the inherent randomness in disease transmission. Numerical simulations confirm that environmental noise and quarantine rates shape disease dynamics. Simulations show that more noise or higher quarantine rates speed up disease extinction, offering key policy insights. Our results clarify how quarantine, noise intensity, and disease dynamics interact, aiding epidemic modeling in stochastic settings.