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Dynamical Analysis of SIR Epidemic Models with Distributed Delay

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  • Wencai Zhao
  • Tongqian Zhang
  • Zhengbo Chang
  • Xinzhu Meng
  • Yulin Liu

Abstract

SIR epidemic models with distributed delay are proposed. Firstly, the dynamical behaviors of the model without vaccination are studied. Using the Jacobian matrix, the stability of the equilibrium points of the system without vaccination is analyzed. The basic reproduction number is got. In order to study the important role of vaccination to prevent diseases, the model with distributed delay under impulsive vaccination is formulated. And the sufficient conditions of globally asymptotic stability of “infection-free†periodic solution and the permanence of the model are obtained by using Floquet’s theorem, small-amplitude perturbation skills, and comparison theorem. Lastly, numerical simulation is presented to illustrate our main conclusions that vaccination has significant effects on the dynamical behaviors of the model. The results can provide effective tactic basis for the practical infectious disease prevention.

Suggested Citation

  • Wencai Zhao & Tongqian Zhang & Zhengbo Chang & Xinzhu Meng & Yulin Liu, 2013. "Dynamical Analysis of SIR Epidemic Models with Distributed Delay," Journal of Applied Mathematics, Hindawi, vol. 2013, pages 1-15, July.
    • Handle: RePEc:hin:jnljam:154387
    DOI: 10.1155/2013/154387
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    Cited by:

    1. Guodong Liu & Xiaohong Wang & Xinzhu Meng & Shujing Gao, 2017. "Extinction and Persistence in Mean of a Novel Delay Impulsive Stochastic Infected Predator-Prey System with Jumps," Complexity, Hindawi, vol. 2017, pages 1-15, June.
    2. Khrennikov, Andrei, 2021. "Ultrametric diffusion equation on energy landscape to model disease spread in hierarchic socially clustered population," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 583(C).

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