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Global stability and Hopf bifurcations of an SEIR epidemiological model with logistic growth and time delay

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  • Xu, Rui
  • Wang, Zhili
  • Zhang, Fengqin

Abstract

In this paper, an SEIR epidemiological model with saturation incidence and a time delay describing the latent period of the disease is investigated, where it is assumed that the susceptible population is subject to logistic growth in the absence of the disease. By analyzing the corresponding characteristic equations, the local stability of a disease-free equilibrium and an endemic equilibrium is discussed. The existence of Hopf bifurcations at the endemic equilibrium is established. By means of Lyapunov functionals and LaSalle’s invariance principle, it is proved that if the basic reproduction number is less than unity, the disease-free equilibrium is globally asymptotically stable and the disease dies out; if the basic reproduction number is greater than unity, sufficient conditions are obtained for the global stability of the endemic equilibrium. Numerical simulations are carried out to illustrate some theoretical results.

Suggested Citation

  • Xu, Rui & Wang, Zhili & Zhang, Fengqin, 2015. "Global stability and Hopf bifurcations of an SEIR epidemiological model with logistic growth and time delay," Applied Mathematics and Computation, Elsevier, vol. 269(C), pages 332-342.
    • Handle: RePEc:eee:apmaco:v:269:y:2015:i:c:p:332-342
    DOI: 10.1016/j.amc.2015.07.084
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    References listed on IDEAS

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    1. Jansen, H. & Twizell, E.H., 2002. "An unconditionally convergent discretization of the SEIR model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 58(2), pages 147-158.
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    3. Tipsri, S. & Chinviriyasit, W., 2015. "The effect of time delay on the dynamics of an SEIR model with nonlinear incidence," Chaos, Solitons & Fractals, Elsevier, vol. 75(C), pages 153-172.
    4. Li, Xue-Zhi & Zhou, Lin-Lin, 2009. "Global stability of an SEIR epidemic model with vertical transmission and saturating contact rate," Chaos, Solitons & Fractals, Elsevier, vol. 40(2), pages 874-884.
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    Cited by:

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    2. Wanduku, Divine, 2017. "Complete global analysis of a two-scale network SIRS epidemic dynamic model with distributed delay and random perturbations," Applied Mathematics and Computation, Elsevier, vol. 294(C), pages 49-76.
    3. Mbabazi, Fulgensia Kamugisha & Mugisha, J.Y.T. & Kimathi, M., 2018. "Modeling the within-host co-infection of influenza A virus and pneumococcus," Applied Mathematics and Computation, Elsevier, vol. 339(C), pages 488-506.
    4. Zhu, Linhe & Liu, Mengxue & Li, Yimin, 2019. "The dynamics analysis of a rumor propagation model in online social networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 520(C), pages 118-137.
    5. Wang, Qi & Xiang, Kainan & Zhu, Chunhui & Zou, Lang, 2023. "Stochastic SEIR epidemic models with virus mutation and logistic growth of susceptible populations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 212(C), pages 289-309.
    6. Zhao, Danling & Sun, Jianbin & Tan, Yuejin & Wu, Jianhong & Dou, Yajie, 2018. "An extended SEIR model considering homepage effect for the information propagation of online social networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 512(C), pages 1019-1031.

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