A new aftertreatment technique for the Mittag-Leffler function of a fractional nonlinear SIR-epidemic model

Laila F. Seddek; Abdelhalim Ebaid; Essam R. El-Zahar; Mona D. Aljoufi; Laila F. Seddek; Abdelhalim Ebaid; Essam R. El-Zahar; Mona D. Aljoufi

doi:10.3934/math.20251016

AIMS Mathematics

2025, Volume 10, Issue 10: 22869-22882. doi: 10.3934/math.20251016

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A new aftertreatment technique for the Mittag-Leffler function of a fractional nonlinear SIR-epidemic model

1.
Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, P.O. Box 83, Al-Kharj 11942, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia

Received: 09 August 2025 Revised: 12 September 2025 Accepted: 18 September 2025 Published: 09 October 2025
MSC : 34A08, 34A34, 92D30

This paper introduced a new aftertreatment technique for solving a fractional nonlinear Susceptible-Infectious-Recovered (SIR)-epidemic model. The proposed approach reformulated the power series solution via incorporating the Laplace transform and its inverse. Basically, it applied the Laplace transform to convert the series solution into different Pad$ \acute{e} $-approximants involving Laplace's parameter with arbitrary order. This procedure facilitated the method of deriving the inverse Laplace transform explicitly, as a final step, using basic special functions in fractional calculus. Our analysis was capable of obtaining a sequence of closed-form approximations in terms of the Mittag-Leffler functions. The current results may be provided for the first time regarding the Mittag-Leffler solution of the fractional SIR-model. Additionally, the outcome of this analysis revealed that our approach was not only effective but also applicable to a wide range of fractional differential equations and systems.
- fractional ordinary differential equation,
- epidemic,
- Mittag-Leffler,
- initial value problem
Citation: Laila F. Seddek, Abdelhalim Ebaid, Essam R. El-Zahar, Mona D. Aljoufi. A new aftertreatment technique for the Mittag-Leffler function of a fractional nonlinear SIR-epidemic model[J]. AIMS Mathematics, 2025, 10(10): 22869-22882. doi: 10.3934/math.20251016

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Abstract

This paper introduced a new aftertreatment technique for solving a fractional nonlinear Susceptible-Infectious-Recovered (SIR)-epidemic model. The proposed approach reformulated the power series solution via incorporating the Laplace transform and its inverse. Basically, it applied the Laplace transform to convert the series solution into different Pad$ \acute{e} $-approximants involving Laplace's parameter with arbitrary order. This procedure facilitated the method of deriving the inverse Laplace transform explicitly, as a final step, using basic special functions in fractional calculus. Our analysis was capable of obtaining a sequence of closed-form approximations in terms of the Mittag-Leffler functions. The current results may be provided for the first time regarding the Mittag-Leffler solution of the fractional SIR-model. Additionally, the outcome of this analysis revealed that our approach was not only effective but also applicable to a wide range of fractional differential equations and systems.

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