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Analysis of the deterministic and stochastic SIRS epidemic models with nonlinear incidence

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  • Liu, Qun
  • Chen, Qingmei

Abstract

In this paper, the deterministic and stochastic SIRS epidemic models with nonlinear incidence are introduced and investigated. For deterministic system, the basic reproductive number R0 is obtained. Furthermore, if R0≤1, then the disease-free equilibrium is globally asymptotically stable and if R0>1, then there is a unique endemic equilibrium which is globally asymptotically stable. For stochastic system, to begin with, we verify that there is a unique global positive solution starting from the positive initial value. Then when R0>1, we prove that stochastic perturbations may lead the disease to extinction in scenarios where the deterministic system is persistent. When R0≤1, a result on fluctuation of the solution around the disease-free equilibrium of deterministic model is obtained under appropriate conditions. At last, if the intensity of the white noise is sufficiently small and R0>1, then there is a unique stationary distribution to stochastic system.

Suggested Citation

  • Liu, Qun & Chen, Qingmei, 2015. "Analysis of the deterministic and stochastic SIRS epidemic models with nonlinear incidence," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 428(C), pages 140-153.
    • Handle: RePEc:eee:phsmap:v:428:y:2015:i:c:p:140-153
    DOI: 10.1016/j.physa.2015.01.075
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    References listed on IDEAS

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    1. Wang, Fengyan & Wang, Xiaoyi & Zhang, Shuwen & Ding, Changming, 2014. "On pulse vaccine strategy in a periodic stochastic SIR epidemic model," Chaos, Solitons & Fractals, Elsevier, vol. 66(C), pages 127-135.
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    3. 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.
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    Cited by:

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    6. Jiang, Daqing & Liu, Qun & Shi, Ningzhong & Hayat, Tasawar & Alsaedi, Ahmed & Xia, Peiyan, 2017. "Dynamics of a stochastic HIV-1 infection model with logistic growth," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 469(C), pages 706-717.
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    21. Wen, Buyu & Teng, Zhidong & Li, Zhiming, 2018. "The threshold of a periodic stochastic SIVS epidemic model with nonlinear incidence," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 508(C), pages 532-549.
    22. Settati, A. & Lahrouz, A. & Zahri, M. & Tridane, A. & El Fatini, M. & El Mahjour, H. & Seaid, M., 2021. "A stochastic threshold to predict extinction and persistence of an epidemic SIRS system with a general incidence rate," Chaos, Solitons & Fractals, Elsevier, vol. 144(C).

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