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Bifurcation Analysis of an SIR Model with Logistic Growth, Nonlinear Incidence, and Saturated Treatment

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  • Ángel G. C. Pérez
  • Eric Avila-Vales
  • Gerardo Emilio García-Almeida

Abstract

There is a wide range of works that have proposed mathematical models to describe the spread of infectious diseases within human populations. Based on such models, researchers can evaluate the effect of applying different strategies for the treatment of diseases. In this article, we generalize previous models by studying an SIR epidemic model with a nonlinear incidence rate, saturated Holling type II treatment rate, and logistic growth. We compute the basic reproduction number and determine conditions for the local stability of equilibria and the existence of backward bifurcation and Hopf bifurcation. We also show that, when the disease transmission rate and treatment parameter are varied, our model undergoes a Bogdanov-Takens bifurcation of codimension 2 or 3. Simulations of the solutions and numerical continuation of equilibria are carried out to generate 2D and 3D bifurcation diagrams, as well as several related phase portraits that illustrate our results. Our work shows that incorporating these factors into epidemic models can lead to very complex dynamics.

Suggested Citation

  • Ángel G. C. Pérez & Eric Avila-Vales & Gerardo Emilio García-Almeida, 2019. "Bifurcation Analysis of an SIR Model with Logistic Growth, Nonlinear Incidence, and Saturated Treatment," Complexity, Hindawi, vol. 2019, pages 1-21, July.
    • Handle: RePEc:hin:complx:9876013
    DOI: 10.1155/2019/9876013
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    1. Li, Jinhui & Teng, Zhidong & Wang, Guangqing & Zhang, Long & Hu, Cheng, 2017. "Stability and bifurcation analysis of an SIR epidemic model with logistic growth and saturated treatment," Chaos, Solitons & Fractals, Elsevier, vol. 99(C), pages 63-71.
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    Cited by:

    1. Kumari, Sangeeta & Upadhyay, Ranjit Kumar, 2021. "Exploring the behavior of malware propagation on mobile wireless sensor networks: Stability and control analysis," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 246-269.

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