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Stability and Hopf Bifurcation for a Delayed SIR Epidemic Model with Logistic Growth

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  • Yakui Xue
  • Tiantian Li

Abstract

We study a delayed SIR epidemic model and get the threshold value which determines the global dynamics and outcome of the disease. First of all, for any τ, we show that the disease‐free equilibrium is globally asymptotically stable; when R0 1, the disease will persist. However, for any τ ≠ 0, the existence conditions for Hopf bifurcations at the endemic equilibrium are obtained. Besides, we compare the delayed SIR epidemic model with nonlinear incidence rate to the one with bilinear incidence rate. At last, numerical simulations are performed to illustrate and verify the conclusions.

Suggested Citation

  • Yakui Xue & Tiantian Li, 2013. "Stability and Hopf Bifurcation for a Delayed SIR Epidemic Model with Logistic Growth," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    • Handle: RePEc:wly:jnlaaa:v:2013:y:2013:i:1:n:916130
    DOI: 10.1155/2013/916130
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    References listed on IDEAS

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    1. Yakui Xue & Junfeng Wang, 2012. "Backward Bifurcation of an Epidemic Model with Infectious Force in Infected and Immune Period and Treatment," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-14, July.
    2. Yakui Xue & Junfeng Wang, 2012. "Backward Bifurcation of an Epidemic Model with Infectious Force in Infected and Immune Period and Treatment," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    3. Xu, Rui & Ma, Zhien, 2009. "Stability of a delayed SIRS epidemic model with a nonlinear incidence rate," Chaos, Solitons & Fractals, Elsevier, vol. 41(5), pages 2319-2325.
    4. 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. Feng, Xiaozhou & Lei, Yang & Xie, Fei & Li, Changtong & Wang, Yuzhen & Wang, Xiaojia & Li, Tingting, 2026. "Stability analysis and Hopf bifurcation in a time-delayed fractional epidemic model," Applied Mathematics and Computation, Elsevier, vol. 515(C).

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