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Stability and Hopf bifurcation for an epidemic disease model with delay

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  • Sun, Chengjun
  • Lin, Yiping
  • Han, Maoan

Abstract

A predator–prey system with disease in the prey is considered. The stability of the positive equilibrium and the existence of Hopf bifurcation with delay τ in the term of degree 2 is investigated, where τ is regarded as a parameter. It is found that there are stability switches, and Hopf bifurcations occur when the delay τ passes through a sequence of critical values. Using the normal form theory and center manifold argument, the explicit formulae which determine the stability, direction and other properties of bifurcating periodic solutions is derived.

Suggested Citation

  • Sun, Chengjun & Lin, Yiping & Han, Maoan, 2006. "Stability and Hopf bifurcation for an epidemic disease model with delay," Chaos, Solitons & Fractals, Elsevier, vol. 30(1), pages 204-216.
    • Handle: RePEc:eee:chsofr:v:30:y:2006:i:1:p:204-216
    DOI: 10.1016/j.chaos.2005.08.167
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    References listed on IDEAS

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    1. Yang, Hong-Yong & Tian, Yu-Ping, 2005. "Hopf bifurcation in REM algorithm with communication delay," Chaos, Solitons & Fractals, Elsevier, vol. 25(5), pages 1093-1105.
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    Cited by:

    1. Hu, Guang-Ping & Li, Xiao-Ling, 2012. "Stability and Hopf bifurcation for a delayed predator–prey model with disease in the prey," Chaos, Solitons & Fractals, Elsevier, vol. 45(3), pages 229-237.
    2. Zhao, Huitao & Lin, Yiping, 2009. "Hopf bifurcation in a partial dependent predator–prey system with delay," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 896-900.
    3. Gan, Qintao & Xu, Rui & Yang, Pinghua, 2009. "Bifurcation and chaos in a ratio-dependent predator–prey system with time delay," Chaos, Solitons & Fractals, Elsevier, vol. 39(4), pages 1883-1895.
    4. Jiang, Zhichao & Wei, Junjie, 2008. "Stability and bifurcation analysis in a delayed SIR model," Chaos, Solitons & Fractals, Elsevier, vol. 35(3), pages 609-619.
    5. Li, Xue-Zhi & Li, Wen-Sheng & Ghosh, Mini, 2009. "Stability and bifurcation of an SIS epidemic model with treatment," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 2822-2832.
    6. Nasir, Hanis, 2022. "On the dynamics of a diabetic population model with two delays and a general recovery rate of complications," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 200(C), pages 571-602.
    7. Liu, Junli & Zhang, Tailei, 2009. "Bifurcation analysis of an SIS epidemic model with nonlinear birth rate," Chaos, Solitons & Fractals, Elsevier, vol. 40(3), pages 1091-1099.
    8. Jingyi Zhao & Chunhai Gao & Tao Tang, 2022. "A Review of Sustainable Maintenance Strategies for Single Component and Multicomponent Equipment," Sustainability, MDPI, vol. 14(5), pages 1-22, March.
    9. Zhang, Xue & Zhang, Qing-ling & Zhang, Yue, 2009. "Bifurcations of a class of singular biological economic models," Chaos, Solitons & Fractals, Elsevier, vol. 40(3), pages 1309-1318.
    10. 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.
    11. Poria, Swarup & Poria, Anindita Tarai & Chatterjee, Prasanta, 2009. "Synchronization threshold of a coupled n-dimensional time-delay system," Chaos, Solitons & Fractals, Elsevier, vol. 41(3), pages 1123-1124.

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