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Global Hopf bifurcation and permanence of a delayed SEIRS epidemic model

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  • Jiang, Zhichao
  • Ma, Wanbiao
  • Wei, Junjie

Abstract

In this paper, an SEIRS system with two delays and the general nonlinear incidence rate is considered. The positivity and boundedness of solutions are investigated. The basic reproductive number, R0, is derived. If R0≤1, then the disease-free equilibrium is globally asymptotically stable and the disease dies out. If R0>1, then there exists a unique endemic equilibrium whose locally asymptotical stability and the existence of local Hopf bifurcations are established by analyzing the distribution of the characteristic values. An explicit algorithm for determining the direction of Hopf bifurcations and the stability of the bifurcating periodic solutions is derived by using the center manifold and the normal form theory. Furthermore, there exists at least one positive periodic solution as the delay varies in some regions by using the global Hopf bifurcation result of Wu for functional differential equations. If R0>1, then the sufficient conditions of the permanence of the system are obtained, i.e., the disease eventually persists in the population. Especially, the upper and lower boundaries that each population can coexist are given exactly. Some numerical simulations are performed to confirm the correctness of theoretical analyses.

Suggested Citation

  • Jiang, Zhichao & Ma, Wanbiao & Wei, Junjie, 2016. "Global Hopf bifurcation and permanence of a delayed SEIRS epidemic model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 122(C), pages 35-54.
    • Handle: RePEc:eee:matcom:v:122:y:2016:i:c:p:35-54
    DOI: 10.1016/j.matcom.2015.11.002
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    References listed on IDEAS

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    1. Jiang, Zhichao & Wei, Junjie, 2008. "Stability and bifurcation analysis in a delayed SIR model," Chaos, Solitons & Fractals, Elsevier, vol. 35(3), pages 609-619.
    2. Wen, Luosheng & Yang, Xiaofan, 2008. "Global stability of a delayed SIRS model with temporary immunity," Chaos, Solitons & Fractals, Elsevier, vol. 38(1), pages 221-226.
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    Cited by:

    1. Liu, Qiming & Li, Hua, 2019. "Global dynamics analysis of an SEIR epidemic model with discrete delay on complex network," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 524(C), pages 289-296.
    2. Zhang, Zizhen & Kundu, Soumen & Tripathi, Jai Prakash & Bugalia, Sarita, 2020. "Stability and Hopf bifurcation analysis of an SVEIR epidemic model with vaccination and multiple time delays," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
    3. Mahajan, Shveta & Kumar, Deepak & Verma, Atul Kumar & Sharma, Natasha, 2023. "Dynamic analysis of modified SEIR epidemic model with time delay in geographical networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 629(C).
    4. Duan, Xi-Chao & Yin, Jun-Feng & Li, Xue-Zhi, 2017. "Global Hopf bifurcation of an SIRS epidemic model with age-dependent recovery," Chaos, Solitons & Fractals, Elsevier, vol. 104(C), pages 613-624.
    5. Sharma, Natasha & Gupta, Arvind Kumar, 2017. "Impact of time delay on the dynamics of SEIR epidemic model using cellular automata," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 471(C), pages 114-125.

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