Stochastic dynamics and probability analysis for a generalized epidemic model with environmental noise

In this paper we consider a stochastic SEIQR (Susceptible–Exposed–Infected–Quarantined–Recovered) epidemic model with a generalized incidence function. Using the Lyapunov method, we establish the existence and uniqueness of a global positive solution to the model, ensuring that it remains well-defined over time. We further establish V-geometric ergodicity, which guarantees the exponential convergence of the system’s probability distribution to its stationary measure, providing a quantitative measure of the system’s stability over time. By leveraging Young’s and Chebyshev’s inequalities, we demonstrate the concepts of stochastic ultimate boundedness and stochastic permanence, providing insights into the long-term behavior of the epidemic dynamics under random perturbations. Additionally, we derive conditions for stochastic extinction, which describes scenarios where the epidemic may eventually die out. Finally, we perform numerical simulations to verify our theoretical results and assess the model’s behavior under different parameters.