Application of piecewise fractional differential equation to COVID-19 infection dynamics

Xiao-Ping Li; Haifaa F Alrihieli; Ebrahem A Algehyne; Muhammad Altaf Khan; Mohammad Y Alshahrani; Yasser Alraey; Muhammad Bilal Riaz

doi:10.1016/j.rinp.2022.105685

Application of piecewise fractional differential equation to COVID-19 infection dynamics

Results Phys. 2022 Aug:39:105685. doi: 10.1016/j.rinp.2022.105685. Epub 2022 Jun 4.

Authors

Xiao-Ping Li¹, Haifaa F Alrihieli², Ebrahem A Algehyne², Muhammad Altaf Khan³, Mohammad Y Alshahrani⁴, Yasser Alraey⁴, Muhammad Bilal Riaz^{5

6

3}

Affiliations

¹ School of Mathematics and Information Science, Xiangnan University, Chenzhou, 423000, Hunan, PR China.
² Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia.
³ Institute for Ground Water Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, South Africa.
⁴ Department of Clinical Laboratory Sciences, College of Applied Medical Sciences, King Khalid University, P.O. Box 61413, Abha, 9088, Saudi Arabia.
⁵ Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, 1/15 Stefanowskiego St., 90-924 Lodz, Poland.
⁶ Department of Mathematics, University of Management and Technology, 54770, Lahore, Pakistan.

Abstract

We proposed a new mathematical model to study the COVID-19 infection in piecewise fractional differential equations. The model was initially designed using the classical differential equations and later we extend it to the fractional case. We consider the infected cases generated at health care and formulate the model first in integer order. We extend the model into Caputo fractional differential equation and study its background mathematical results. We show that the fractional model is locally asymptotically stable when $R_{0} < 1$ at the disease-free case. For $R_{0} \leq 1$ , we show the global asymptotical stability of the model. We consider the infected cases in Saudi Arabia and determine the parameters of the model. We show that for the real cases, the basic reproduction is $R_{0} \approx 1.7372$ . We further extend the Caputo model into piecewise stochastic fractional differential equations and discuss the procedure for its numerical simulation. Numerical simulations for the Caputo case and piecewise models are shown in detail.

Keywords: Mathematical model; Numerical results; Piecewise differential equations; Saudi Arabia cases.