Solution of the SIR epidemic model of arbitrary orders containing Caputo-Fabrizio, Atangana-Baleanu and Caputo derivatives

Eman A. A. Ziada; Salwa El-Morsy; Osama Moaaz; Sameh S. Askar; Ahmad M. Alshamrani; Monica Botros; Eman A. A. Ziada; Salwa El-Morsy; Osama Moaaz; Sameh S. Askar; Ahmad M. Alshamrani; Monica Botros

doi:10.3934/math.2024894

AIMS Mathematics

2024, Volume 9, Issue 7: 18324-18355. doi: 10.3934/math.2024894

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Solution of the SIR epidemic model of arbitrary orders containing Caputo-Fabrizio, Atangana-Baleanu and Caputo derivatives

1.
Basic Science Department, Nile Higher Institute for Engineering and Technology, Mansoura, Egypt
2.
Section of Mathematics, International Telematic University Uninettuno, CorsoVittorio Emanuele Ⅱ, 39, 00186 Roma, Italy
3.
Department of Mathematics, Faculty of Science, Mansoura University, 35516 Mansoura, Egypt
4.
Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
5.
Basic Science Department, Faculty of Engineering, Delta University for Science and Technology, P.O. Box 11152, Mansoura, Egypt

Received: 26 January 2024 Revised: 21 April 2024 Accepted: 23 April 2024 Published: 31 May 2024
MSC : 34A12, 34A34, 45G05

The main aim of this study was to apply an analytical method to solve a nonlinear system of fractional differential equations (FDEs). This method is the Adomian decomposition method (ADM), and a comparison between its results was made by using a numerical method: Runge-Kutta 4 (RK4). It is proven that there is a unique solution to the system. The convergence of the series solution is given, and the error estimate is also proven. After that, the susceptible-infected-recovered (SIR) model was taken as an real phenomenon with such systems. This system is discussed with three different fractional derivatives (FDs): the Caputo-Fabrizio derivative (CFD), the Atangana-Baleanu derivative (ABD), and the Caputo derivative (CD). A comparison between these three different derivatives is given. We aimed to see which one of the new definitions (ABD and CFD) is close to one of the most important classical definitions (CD).
- Caputo-Fabrizio,
- Caputo and Atangana-Baleanu derivatives,
- SIR epidemic model of arbitrary orders,
- Adomian method,
- unique solution,
- error estimation
Citation: Eman A. A. Ziada, Salwa El-Morsy, Osama Moaaz, Sameh S. Askar, Ahmad M. Alshamrani, Monica Botros. Solution of the SIR epidemic model of arbitrary orders containing Caputo-Fabrizio, Atangana-Baleanu and Caputo derivatives[J]. AIMS Mathematics, 2024, 9(7): 18324-18355. doi: 10.3934/math.2024894

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Abstract

The main aim of this study was to apply an analytical method to solve a nonlinear system of fractional differential equations (FDEs). This method is the Adomian decomposition method (ADM), and a comparison between its results was made by using a numerical method: Runge-Kutta 4 (RK4). It is proven that there is a unique solution to the system. The convergence of the series solution is given, and the error estimate is also proven. After that, the susceptible-infected-recovered (SIR) model was taken as an real phenomenon with such systems. This system is discussed with three different fractional derivatives (FDs): the Caputo-Fabrizio derivative (CFD), the Atangana-Baleanu derivative (ABD), and the Caputo derivative (CD). A comparison between these three different derivatives is given. We aimed to see which one of the new definitions (ABD and CFD) is close to one of the most important classical definitions (CD).

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