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Dynamical Behaviors of a Stochastic Susceptible-Infected-Treated-Recovered-Susceptible Cholera Model with Ornstein-Uhlenbeck Process

Author

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  • Shenxing Li

    (School of Mathematics and Statistics, Northeast Petroleum University, Daqing 163318, China)

  • Wenhe Li

    (School of Mathematics and Statistics, Northeast Petroleum University, Daqing 163318, China)

Abstract

In this study, a cholera infection model with a bilinear infection rate is developed by considering the perturbation of the infection rate by the mean-reverting process. First of all, we give the existence of a globally unique positive solution for a stochastic system at an arbitrary initial value. On this basis, the sufficient condition for the model to have an ergodic stationary distribution is given by constructing proper Lyapunov functions and tight sets. This indicates in a biological sense the long-term persistence of cholera infection. Furthermore, after transforming the stochastic model to a relevant linearized system, an accurate expression for the probability density function of the stochastic model around a quasi-endemic equilibrium is derived. Subsequently, the sufficient condition to make the disease extinct is also derived. Eventually, the theoretical findings are shown by numerical simulations. Numerical simulations show the impact of regression speed and fluctuation intensity on stochastic systems.

Suggested Citation

  • Shenxing Li & Wenhe Li, 2024. "Dynamical Behaviors of a Stochastic Susceptible-Infected-Treated-Recovered-Susceptible Cholera Model with Ornstein-Uhlenbeck Process," Mathematics, MDPI, vol. 12(14), pages 1-20, July.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:14:p:2163-:d:1432388
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    References listed on IDEAS

    as
    1. Zhou, Xueyong & Shi, Xiangyun & Wei, Ming, 2022. "Dynamical behavior and optimal control of a stochastic mathematical model for cholera," Chaos, Solitons & Fractals, Elsevier, vol. 156(C).
    2. Fehaid Salem Alshammari & Fahir Talay Akyildiz, 2023. "Epidemic Waves in a Stochastic SIRVI Epidemic Model Incorporating the Ornstein–Uhlenbeck Process," Mathematics, MDPI, vol. 11(18), pages 1-15, September.
    3. Feng, Tao & Qiu, Zhipeng & Meng, Xinzhu & Rong, Libin, 2019. "Analysis of a stochastic HIV-1 infection model with degenerate diffusion," Applied Mathematics and Computation, Elsevier, vol. 348(C), pages 437-455.
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