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Stability and bifurcation analysis in a delayed SIR model

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  • Jiang, Zhichao
  • Wei, Junjie

Abstract

In this paper, a time-delayed SIR model with a nonlinear incidence rate is considered. The existence of Hopf bifurcations at the endemic equilibrium is established by analyzing the distribution of the characteristic values. A explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by using the normal form and the center manifold theory. Numerical simulations to support the analytical conclusions are carried out.

Suggested Citation

  • Jiang, Zhichao & Wei, Junjie, 2008. "Stability and bifurcation analysis in a delayed SIR model," Chaos, Solitons & Fractals, Elsevier, vol. 35(3), pages 609-619.
    • Handle: RePEc:eee:chsofr:v:35:y:2008:i:3:p:609-619
    DOI: 10.1016/j.chaos.2006.05.045
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    References listed on IDEAS

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    1. Zeng, Guang Zhao & Chen, Lan Sun & Sun, Li Hua, 2005. "Complexity of an SIR epidemic dynamics model with impulsive vaccination control," Chaos, Solitons & Fractals, Elsevier, vol. 26(2), pages 495-505.
    2. Gakkhar, Sunita & Singh, Brahampal, 2006. "Dynamics of modified Leslie–Gower-type prey–predator model with seasonally varying parameters," Chaos, Solitons & Fractals, Elsevier, vol. 27(5), pages 1239-1255.
    3. Gao, Shujing & Chen, Lansun, 2005. "The effect of seasonal harvesting on a single-species discrete population model with stage structure and birth pulses," Chaos, Solitons & Fractals, Elsevier, vol. 24(4), pages 1013-1023.
    4. 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.
    5. Liu, Zhihua & Yuan, Rong, 2006. "Stability and bifurcation in a harvested one-predator–two-prey model with delays," Chaos, Solitons & Fractals, Elsevier, vol. 27(5), pages 1395-1407.
    6. Moghadas, S.M. & Gumel, A.B., 2002. "Global stability of a two-stage epidemic model with generalized non-linear incidence," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 60(1), pages 107-118.
    7. Wang, Fengyan & Zhang, Shuwen & Chen, Lansun & Sun, Lihua, 2006. "Bifurcation and complexity of Monod type predator–prey system in a pulsed chemostat," Chaos, Solitons & Fractals, Elsevier, vol. 27(2), pages 447-458.
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    Cited by:

    1. Raja Sekhara Rao, P. & Naresh Kumar, M., 2015. "A dynamic model for infectious diseases: The role of vaccination and treatment," Chaos, Solitons & Fractals, Elsevier, vol. 75(C), pages 34-49.
    2. Yu, Chunbo & Wei, Junjie, 2009. "Stability and bifurcation analysis in a basic model of the immune response with delays," Chaos, Solitons & Fractals, Elsevier, vol. 41(3), pages 1223-1234.
    3. Cai, Liming & Guo, Shumin & Li, XueZhi & Ghosh, Mini, 2009. "Global dynamics of a dengue epidemic mathematical model," Chaos, Solitons & Fractals, Elsevier, vol. 42(4), pages 2297-2304.
    4. 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.
    5. Dramane Ouedraogo & Ali Traore & Aboudramane Guiro, 2020. "Global Analysis of SIRS Epidemic Model With General Incidence Function and Incomplete Recovery Rates Stochastical Model," Journal of Mathematics Research, Canadian Center of Science and Education, vol. 12(6), pages 100-100, December.
    6. 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.
    7. Li, Kai & Wei, Junjie, 2009. "Stability and Hopf bifurcation analysis of a prey–predator system with two delays," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 2606-2613.

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