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This study develops a stochastic SIRS compartmental model for exploring the transmission dynamics of infectious diseases, integrating the Beddington–DeAngelis incidence rate and vaccination. In the deterministic case, the reproduction number ℛ0 is derived, and the global dynamics is analyzed using the Lyapunov function with respect to ℛ0. The outcomes underscore that ℛ0 completely governs the overall dynamics of the system. In the stochastic case, the primary challenge arises from the two-dimensional boundary system, preventing the Fokker–Planck equation from obtaining the density function of the invariant measure. To address the weak convergence property regarding the invariant measure for both the stochastic system and its corresponding two-dimensional boundary system, the concept of limit measures is introduced. The theoretical results indicate that the persistence and extinction of the infectious disease are entirely determined by the Lyapunov exponent λ, representing the long-term growth rate. Numerical simulations further support these findings. [ABSTRACT FROM AUTHOR]

AIDS is a chronic infectious disease that has been having a major impact on human health and social stability. In this paper, based on the deterministic SIATR model, a stochastic SIATR model is developed to account for the spread of AIDS by introducing the Ornstein–Uhlenbeck process. First, the existence and uniqueness of the global positive solution of the model are investigated. Second, a threshold R0E is set: when R0E < 1, the disease will become extinct; when R0E > 1, the disease will persist. Then, it is shown that there exists a stationary distribution for the model when R0E > 1. On this basis, we derive the exact expression for the probability density function of the model in the neighborhood of the quasi-equilibrium state. Finally, the results of the previous proof are verified by several numerical simulations. [ABSTRACT FROM AUTHOR]

According to the transmission characteristics of COVID-19, this paper proposes a stochastic SAIRS epidemic model with two mean reversion Ornstein–Uhlenbeck processes and saturated incidence rates. We first prove the existence and uniqueness of the global solution in the stochastic model. Using several suitable Lyapunov methods, we then derive the extinction and persistence of COVID-19 under certain conditions. Further, stationary distribution and ergodic properties are obtained. Moreover, we obtain the probability density function of the stochastic model around the equilibrium. Numerical simulations illustrate our theoretical results and the effect of essential parameters. Finally, we apply the model to investigate the latest outbreak of the COVID-19 epidemic in Guangzhou city, China. [ABSTRACT FROM AUTHOR]

In this work, we consider a stochastic epidemic model with vaccination, healing and relapse. We prove the existence and the uniqueness of the positive solution. We establish sufficient conditions for the extinction and the persistence in mean of the stochastic system. Moreover, we also establish sufficient conditions for the existence of ergodic stationary distribution to the model, which reveals that the infectious disease will persist. The graphical illustrations of the approximate solutions of the stochastic epidemic model have been performed. [ABSTRACT FROM AUTHOR]