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Dynamics of Fractional-Order Epidemic Models with General Nonlinear Incidence Rate and Time-Delay

Author

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  • Ardak Kashkynbayev

    (Department of Mathematics, Nazarbayev University, Nur-Sultan 010000, Kazakhstan)

  • Fathalla A. Rihan

    (Department of Mathematical Sciences, College of Sciences, UAE University, Al Ain 15551, United Arab Emirates)

Abstract

In this paper, we study the dynamics of a fractional-order epidemic model with general nonlinear incidence rate functionals and time-delay. We investigate the local and global stability of the steady-states. We deduce the basic reproductive threshold parameter, so that if R 0 < 1 , the disease-free steady-state is locally and globally asymptotically stable. However, for R 0 > 1 , there exists a positive (endemic) steady-state which is locally and globally asymptotically stable. A Holling type III response function is considered in the numerical simulations to illustrate the effectiveness of the theoretical results.

Suggested Citation

  • Ardak Kashkynbayev & Fathalla A. Rihan, 2021. "Dynamics of Fractional-Order Epidemic Models with General Nonlinear Incidence Rate and Time-Delay," Mathematics, MDPI, vol. 9(15), pages 1-16, August.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:15:p:1829-:d:607276
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    References listed on IDEAS

    as
    1. Fathalla A. Rihan & Dumitru Baleanu & S. Lakshmanan & R. Rakkiyappan, 2014. "On Fractional SIRC Model with Salmonella Bacterial Infection," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-9, April.
    2. Rihan, F.A. & Al-Mdallal, Q.M. & AlSakaji, H.J. & Hashish, A., 2019. "A fractional-order epidemic model with time-delay and nonlinear incidence rate," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 97-105.
    3. Jing Bai & Guoguang Wen & Ahmed Rahmani & Yongguang Yu, 2015. "Distributed formation control of fractional-order multi-agent systems with absolute damping and communication delay," International Journal of Systems Science, Taylor & Francis Journals, vol. 46(13), pages 2380-2392, October.
    4. Ahmed, E. & Elgazzar, A.S., 2007. "On fractional order differential equations model for nonlocal epidemics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 379(2), pages 607-614.
    5. Ozturk Mizrak, Ozlem & Mizrak, Cihan & Kashkynbayev, Ardak & Kuang, Yang, 2020. "Can fractional differentiation improve stability results and data fitting ability of a prostate cancer model under intermittent androgen suppression therapy?," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
    6. Sene, Ndolane, 2020. "SIR epidemic model with Mittag–Leffler fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 137(C).
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    Cited by:

    1. Vsevolod G. Sorokin & Andrei V. Vyazmin, 2022. "Nonlinear Reaction–Diffusion Equations with Delay: Partial Survey, Exact Solutions, Test Problems, and Numerical Integration," Mathematics, MDPI, vol. 10(11), pages 1-39, May.

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