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A fractional-order epidemic model with time-delay and nonlinear incidence rate

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  • Rihan, F.A.
  • Al-Mdallal, Q.M.
  • AlSakaji, H.J.
  • Hashish, A.

Abstract

In this paper, we provide an epidemic SIR model with long-range temporal memory. The model is governed by delay differential equations with fractional-order. We assume that the susceptible is obeying the logistic form in which the incidence term is of saturated form with the susceptible. Several theoretical results related to the existence of steady states and the asymptotic stability of the steady states are discussed. We use a suitable Lyapunov functional to formulate a new set of sufficient conditions that guarantee the global stability of the steady states. The occurrence of Hopf bifurcation is captured when the time-delay τ passes through a critical value τ*. Theoretical results are validated numerically by solving the governing system, using the modified Adams-Bashforth-Moulton predictor-corrector scheme. Our findings show that the combination of fractional-order derivative and time-delay in the model improves the dynamics and increases complexity of the model. In some cases, the phase portrait gets stretched as the order of the derivative is reduced.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:chsofr:v:126:y:2019:i:c:p:97-105
    DOI: 10.1016/j.chaos.2019.05.039
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    References listed on IDEAS

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    1. Abdeljawad, Thabet & Al-Mdallal, Qasem M. & Jarad, Fahd, 2019. "Fractional logistic models in the frame of fractional operators generated by conformable derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 119(C), pages 94-101.
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    6. Qasem Al-Mdallal & Kashif Ali Abro & Ilyas Khan, 2018. "Analytical Solutions of Fractional Walter’s B Fluid with Applications," Complexity, Hindawi, vol. 2018, pages 1-10, February.
    7. Al-Mdallal, Qasem M. & Abu Omer, Ahmed S., 2018. "Fractional-order Legendre-collocation method for solving fractional initial value problems," Applied Mathematics and Computation, Elsevier, vol. 321(C), pages 74-84.
    8. 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.
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    Cited by:

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    2. Postavaru, O. & Anton, S.R. & Toma, A., 2021. "COVID-19 pandemic and chaos theory," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 181(C), pages 138-149.
    3. Lee, Chaeyoung & Li, Yibao & Kim, Junseok, 2020. "The susceptible-unidentified infected-confirmed (SUC) epidemic model for estimating unidentified infected population for COVID-19," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).
    4. Das, Parthasakha & Das, Samhita & Upadhyay, Ranjit Kumar & Das, Pritha, 2020. "Optimal treatment strategies for delayed cancer-immune system with multiple therapeutic approach," Chaos, Solitons & Fractals, Elsevier, vol. 136(C).
    5. Asif, Muhammad & Ali Khan, Zar & Haider, Nadeem & Al-Mdallal, Qasem, 2020. "Numerical simulation for solution of SEIR models by meshless and finite difference methods," Chaos, Solitons & Fractals, Elsevier, vol. 141(C).
    6. 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.
    7. Teresa Faria, 2021. "Permanence for Nonautonomous Differential Systems with Delays in the Linear and Nonlinear Terms," Mathematics, MDPI, vol. 9(3), pages 1-20, January.
    8. Ahmad, Shabir & Ullah, Aman & Al-Mdallal, Qasem M. & Khan, Hasib & Shah, Kamal & Khan, Aziz, 2020. "Fractional order mathematical modeling of COVID-19 transmission," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).
    9. Ruofeng Rao & Zhi Lin & Xiaoquan Ai & Jiarui Wu, 2022. "Synchronization of Epidemic Systems with Neumann Boundary Value under Delayed Impulse," Mathematics, MDPI, vol. 10(12), pages 1-10, June.
    10. Ihtisham Ul Haq & Numan Ullah & Nigar Ali & Kottakkaran Sooppy Nisar, 2022. "A New Mathematical Model of COVID-19 with Quarantine and Vaccination," Mathematics, MDPI, vol. 11(1), pages 1-21, December.
    11. Ghanbari, Behzad & Djilali, Salih, 2020. "Mathematical analysis of a fractional-order predator-prey model with prey social behavior and infection developed in predator population," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).
    12. Tsvetkov, V.P. & Mikheev, S.A. & Tsvetkov, I.V. & Derbov, V.L. & Gusev, A.A. & Vinitsky, S.I., 2022. "Modeling the multifractal dynamics of COVID-19 pandemic," Chaos, Solitons & Fractals, Elsevier, vol. 161(C).
    13. Wang, Ning & Qi, Longxing & Cheng, Guangyi, 2022. "Dynamical analysis for the impact of asymptomatic infective and infection delay on disease transmission," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 200(C), pages 525-556.

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