Search

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]

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]

In this paper, we investigate a two-dimensional avian influenza model with Allee effect and stochasticity. We first show that a unique global positive solution always exists to the stochastic system for any positive initial value. Then, under certain conditions, this solution is proved to be stochastically ultimately bounded. Furthermore, by constructing a suitable Lyapunov function, we obtain sufficient conditions for the existence of stationary distribution with ergodicity. The conditions for the extinction of infected avian population are also analytically studied. These theoretical results are conformed by computational simulations. We numerically show that the environmental noise can bring different dynamical outcomes to the stochastic model. By scanning different noise intensities, we observe that large noise can cause extinction of infected avian population, which suggests the repression of noise on the spread of avian virus. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the stochastic avian influenza model with human-to-human transmission, which is disturbed by both white and telegraph noises. First, we show that the solution of the stochastic system is positive and global. Furthermore, by using stochastic Lyapunov functions, we establish sufficient conditions for the existence of a unique ergodic stationary distribution. Then we obtain the conditions for extinction. Finally, numerical simulations are employed to demonstrate the analytical results. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic epidemic system with both switching noise and white noise is proposed to research the dynamics of the diseases. Nonlinear incidence and vaccination strategies are also considered in the proposed model. By using the method of stochastic analysis, we point out the key parameters that determine the persistence and extinction of the diseases. Specifically, if Rs 0 is greater than 0, the stochastic system has a unique ergodic stationary distribution; while if R* 0 is less than 0, the diseases will be extinct at an exponential rate. [ABSTRACT FROM AUTHOR]