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Codimension-Two Bifurcations of a Simplified Discrete-Time SIR Model with Nonlinear Incidence and Recovery Rates

Author

Listed:
  • Dongpo Hu

    (School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China)

  • Xuexue Liu

    (School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China)

  • Kun Li

    (School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China)

  • Ming Liu

    (Institute of Automation, Qufu Normal University, Qufu 273165, China)

  • Xiao Yu

    (School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China
    Shanghai Research Institute for Intelligent Autonomous Systems, Tongji University, Shanghai 201210, China)

Abstract

In this paper, a simplified discrete-time SIR model with nonlinear incidence and recovery rates is discussed. Here, using the integral step size and the intervention level as control parameters, we mainly discuss three types of codimension-two bifurcations (fold-flip bifurcation, 1:3 resonance, and 1:4 resonance) of the simplified discrete-time SIR model in detail by bifurcation theory and numerical continuation techniques. Parameter conditions for the occurrence of codimension-two bifurcations are obtained by constructing the corresponding approximate normal form with translation and transformation of several parameters and variables. To further confirm the accuracy of our theoretical analysis, numerical simulations such as phase portraits, bifurcation diagrams, and maximum Lyapunov exponents diagrams are provided. In particular, the coexistence of bistability states is observed by giving local attraction basins diagrams of different fixed points under different integral step sizes. It is possible to more clearly illustrate the model’s complex dynamic behavior by combining theoretical analysis and numerical simulation.

Suggested Citation

  • Dongpo Hu & Xuexue Liu & Kun Li & Ming Liu & Xiao Yu, 2023. "Codimension-Two Bifurcations of a Simplified Discrete-Time SIR Model with Nonlinear Incidence and Recovery Rates," Mathematics, MDPI, vol. 11(19), pages 1-24, September.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:19:p:4142-:d:1251968
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    References listed on IDEAS

    as
    1. Md. Enamul Hoque, 2020. "An early estimation of the number of affected people in South Asia due to Covid-19 pandemic using susceptible, infected and recover model," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 31(10), pages 1-7, October.
    2. Pájaro, Manuel & Fajar, Noelia M. & Alonso, Antonio A. & Otero-Muras, Irene, 2022. "Stochastic SIR model predicts the evolution of COVID-19 epidemics from public health and wastewater data in small and medium-sized municipalities: A one year study," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    3. Li, Bo & Liang, Houjun & He, Qizhi, 2021. "Multiple and generic bifurcation analysis of a discrete Hindmarsh-Rose model," Chaos, Solitons & Fractals, Elsevier, vol. 146(C).
    Full references (including those not matched with items on IDEAS)

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