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SIR epidemic model with Mittag–Leffler fractional derivative

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  • Sene, Ndolane

Abstract

The SIR epidemic model with delay in the context of the fractional derivative with Mittag–Leffler kernel has been considered. The Atangana–Baleanu fractional derivative is a non-singular fractional derivative with Mittag–Leffler kernel. The positivity of the solutions of the SIR model depends strongly on the order of the Atangana–Baleanu–Caputo fractional derivative. We investigate the existence and the uniqueness of our proposed model in terms of the used fractional derivative. The reproduction number related to the SIR epidemic model in our paper is presented. The trivial equilibrium point and the endemic equilibrium point have been proposed. The asymptotic stability for the trivial equilibrium and the endemic equilibrium points have been investigated. The global asymptotic stability of the disease-free equilibrium and the endemic equilibrium have been analyzed in terms of the Lyapunov direct method. The graphical representations of the approximate solutions of the model have been proposed.

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  • Sene, Ndolane, 2020. "SIR epidemic model with Mittag–Leffler fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 137(C).
    • Handle: RePEc:eee:chsofr:v:137:y:2020:i:c:s0960077920302332
    DOI: 10.1016/j.chaos.2020.109833
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    References listed on IDEAS

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    1. Sene, Ndolane, 2020. "Second-grade fluid model with Caputo–Liouville generalized fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 133(C).
    2. Atangana, Abdon, 2018. "Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 505(C), pages 688-706.
    3. Bhatter, Sanjay & Mathur, Amit & Kumar, Devendra & Nisar, Kottakkaran Sooppy & Singh, Jagdev, 2020. "Fractional modified Kawahara equation with Mittag–Leffler law," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
    4. Angstmann, C.N. & Henry, B.I. & McGann, A.V., 2016. "A fractional-order infectivity SIR model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 452(C), pages 86-93.
    5. Sene, Ndolane, 2018. "Stokes’ first problem for heated flat plate with Atangana–Baleanu fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 117(C), pages 68-75.
    6. Atangana, Abdon & Mekkaoui, Toufik, 2019. "Trinition the complex number with two imaginary parts: Fractal, chaos and fractional calculus," Chaos, Solitons & Fractals, Elsevier, vol. 128(C), pages 366-381.
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    Cited by:

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    3. Addai, Emmanuel & Zhang, Lingling & Ackora-Prah, Joseph & Gordon, Joseph Frank & Asamoah, Joshua Kiddy K. & Essel, John Fiifi, 2022. "Fractal-fractional order dynamics and numerical simulations of a Zika epidemic model with insecticide-treated nets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 603(C).
    4. Alzahrani, Faris & Razzaq, Oyoon Abdul & Rehman, Daniyal Ur & Khan, Najeeb Alam & Alshomrani, Ali Saleh & Ullah, Malik Zaka, 2022. "Repercussions of unreported populace on disease dynamics and its optimal control through system of fractional order delay differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 158(C).
    5. Asamoah, Joshua Kiddy K. & Owusu, Mark A. & Jin, Zhen & Oduro, F. T. & Abidemi, Afeez & Gyasi, Esther Opoku, 2020. "Global stability and cost-effectiveness analysis of COVID-19 considering the impact of the environment: using data from Ghana," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
    6. Higazy, M., 2020. "Novel fractional order SIDARTHE mathematical model of COVID-19 pandemic," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).
    7. Masoud Shahmanzari & Fehmi Tanrisever & Enes Eryarsoy & Ahmet Şensoy, 2023. "Managing disease containment measures during a pandemic," Production and Operations Management, Production and Operations Management Society, vol. 32(5), pages 1362-1379, May.
    8. 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.
    9. Nabi, Khondoker Nazmoon & Kumar, Pushpendra & Erturk, Vedat Suat, 2021. "Projections and fractional dynamics of COVID-19 with optimal control strategies," Chaos, Solitons & Fractals, Elsevier, vol. 145(C).
    10. Omame, Andrew & Abbas, Mujahid & Abdel-Aty, Abdel-Haleem, 2022. "Assessing the impact of SARS-CoV-2 infection on the dynamics of dengue and HIV via fractional derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).
    11. Kaviya, R. & Priyanka, M. & Muthukumar, P., 2022. "Mean-square exponential stability of impulsive conformable fractional stochastic differential system with application on epidemic model," Chaos, Solitons & Fractals, Elsevier, vol. 160(C).

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