A new unconditionally stable implicit numerical scheme for fractional diffusive epidemic model

Yasir Nawaz; Muhammad Shoaib Arif; Wasfi Shatanawi; Muhammad Usman Ashraf; Yasir Nawaz; Muhammad Shoaib Arif; Wasfi Shatanawi; Muhammad Usman Ashraf

doi:10.3934/math.2022788

AIMS Mathematics

2022, Volume 7, Issue 8: 14299-14322. doi: 10.3934/math.2022788

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A new unconditionally stable implicit numerical scheme for fractional diffusive epidemic model

1.
Department of Mathematics, Air University, PAF Complex E-9, Islamabad, 44000, Pakistan
2.
Department of Mathematics and Sciences, College of Humanities and Sciences, Prince Sultan University, Riyadh, 11586, Saudi Arabia
3.
Stochastic Analysis and Optimization Research Group, Department of Mathematics, Air University, PAF Complex E-9, Islamabad, 44000, Pakistan
4.
Department of Mathematics, Faculty of Science, The Hashemite University, P.O. Box 330127, Zarqa, 13133, Jordan
5.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan
6.
Department of Sciences and Humanities, National University of Computer and Emerging Sciences, Islamabad, Pakistan

Received: 04 April 2022 Revised: 20 May 2022 Accepted: 25 May 2022 Published: 01 June 2022
MSC : 34A08, 65F10

This contribution proposes a numerical scheme for solving fractional parabolic partial differential equations (PDEs). One of the advantages of using the proposed scheme is its applicability for fractional and integer order derivatives. The scheme can be useful to get conditions for obtaining a positive solution to epidemic disease models. A COVID-19 mathematical model is constructed, and linear local stability conditions for the model are obtained; afterward, a fractional diffusive epidemic model is constructed. The numerical scheme is constructed by employing the fractional Taylor series approach. The proposed fractional scheme is second-order accurate in space and time and unconditionally stable for parabolic PDEs. In addition to this, convergence conditions are obtained by employing a proposed numerical scheme for the fractional differential equation of susceptible individuals. The scheme is also compared with existing numerical schemes, including the non-standard finite difference method. From theoretical analysis and graphical illustration, it is found that the proposed scheme is more accurate than the so-called existing non-standard finite difference method, which is a method with notably good boundedness and positivity properties.
- fractional numerical scheme,
- conditionally positivity preserving,
- COVID-19 model,
- stability,
- convergence
Citation: Yasir Nawaz, Muhammad Shoaib Arif, Wasfi Shatanawi, Muhammad Usman Ashraf. A new unconditionally stable implicit numerical scheme for fractional diffusive epidemic model[J]. AIMS Mathematics, 2022, 7(8): 14299-14322. doi: 10.3934/math.2022788

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Abstract

This contribution proposes a numerical scheme for solving fractional parabolic partial differential equations (PDEs). One of the advantages of using the proposed scheme is its applicability for fractional and integer order derivatives. The scheme can be useful to get conditions for obtaining a positive solution to epidemic disease models. A COVID-19 mathematical model is constructed, and linear local stability conditions for the model are obtained; afterward, a fractional diffusive epidemic model is constructed. The numerical scheme is constructed by employing the fractional Taylor series approach. The proposed fractional scheme is second-order accurate in space and time and unconditionally stable for parabolic PDEs. In addition to this, convergence conditions are obtained by employing a proposed numerical scheme for the fractional differential equation of susceptible individuals. The scheme is also compared with existing numerical schemes, including the non-standard finite difference method. From theoretical analysis and graphical illustration, it is found that the proposed scheme is more accurate than the so-called existing non-standard finite difference method, which is a method with notably good boundedness and positivity properties.

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