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The stationary distribution in a class of stochastic SIRS epidemic models with non-monotonic incidence and degenerate diffusion

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  • Tuerxun, Nafeisha
  • Wen, Buyu
  • Teng, Zhidong

Abstract

A class of stochastic SIRS epidemic models with non-monotonic incidence and degenerate diffusion is investigated. By using the Lyapunov function method, the existence of global positive solutions and the ultimate boundedness with probability one are obtained. By using the Markov semigroups theory, Fokker–Planck equation and Khasminskiǐ functions, the existence of unique stationary distribution for the model is established. That is, when the stochastic basic reproduction number R0S>1 and some extra conditions are satisfied then the distribution density of any positive solutions of the model converges to a unique invariant density as t→+∞. Finally, the main conclusions and open problems are illustrated and verified by the numerical simulations.

Suggested Citation

  • Tuerxun, Nafeisha & Wen, Buyu & Teng, Zhidong, 2021. "The stationary distribution in a class of stochastic SIRS epidemic models with non-monotonic incidence and degenerate diffusion," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 182(C), pages 888-912.
    • Handle: RePEc:eee:matcom:v:182:y:2021:i:c:p:888-912
    DOI: 10.1016/j.matcom.2020.03.008
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    References listed on IDEAS

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    1. Cai, Yongli & Jiao, Jianjun & Gui, Zhanji & Liu, Yuting & Wang, Weiming, 2018. "Environmental variability in a stochastic epidemic model," Applied Mathematics and Computation, Elsevier, vol. 329(C), pages 210-226.
    2. Rudnicki, Ryszard, 2003. "Long-time behaviour of a stochastic prey-predator model," Stochastic Processes and their Applications, Elsevier, vol. 108(1), pages 93-107, November.
    3. Guo, Wenjuan & Cai, Yongli & Zhang, Qimin & Wang, Weiming, 2018. "Stochastic persistence and stationary distribution in an SIS epidemic model with media coverage," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 492(C), pages 2220-2236.
    4. Wang, Weiming & Cai, Yongli & Ding, Zuqin & Gui, Zhanji, 2018. "A stochastic differential equation SIS epidemic model incorporating Ornstein–Uhlenbeck process," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 509(C), pages 921-936.
    5. Lin, Yuguo & Jin, Manli, 2019. "Ergodicity of a regime-switching epidemic model with degenerate diffusion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 526(C).
    6. Wen, Buyu & Rifhat, Ramziya & Teng, Zhidong, 2019. "The stationary distribution in a stochastic SIS epidemic model with general nonlinear incidence," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 524(C), pages 258-271.
    7. Cai, Yongli & Kang, Yun & Wang, Weiming, 2017. "A stochastic SIRS epidemic model with nonlinear incidence rate," Applied Mathematics and Computation, Elsevier, vol. 305(C), pages 221-240.
    8. Lahrouz, Aadil & Omari, Lahcen, 2013. "Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence," Statistics & Probability Letters, Elsevier, vol. 83(4), pages 960-968.
    9. Liu, Qun & Jiang, Daqing & He, Xiuli & Hayat, Tasawar & Alsaedi, Ahmed, 2019. "Stationary distribution of a stochastic predator–prey model with distributed delay and general functional response," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 513(C), pages 273-287.
    10. Lin, Yuguo & Wang, Libo & Dong, Xiaowan, 2019. "Long-time behavior of a regime-switching SIRS epidemic model with degenerate diffusion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 529(C).
    11. Carletti, M. & Burrage, K. & Burrage, P.M., 2004. "Numerical simulation of stochastic ordinary differential equations in biomathematical modelling," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 64(2), pages 271-277.
    12. Xiaoting Fan & Yi Song & Wencai Zhao, 2018. "Modeling Cell-to-Cell Spread of HIV-1 with Nonlocal Infections," Complexity, Hindawi, vol. 2018, pages 1-10, August.
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