Mathematical analysis of fractional order co-abuse infection model using power law type kernel

Volume 36, Issue 4, pp 352--370 https://dx.doi.org/10.22436/jmcs.036.04.01

Publication Date: August 10, 2024 Submission Date: December 21, 2023 Revision Date: December 28, 2023 Accteptance Date: July 09, 2024

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Authors

I. U. Haq - Department of Basic Sciences‎, University of Engineering and Technology‎, Khyber‎ ‎Pukhtunkhwa, Anbar, Pakistan. A. Ali - Department of Basic Sciences‎, University of Engineering and Technology‎, Khyber‎ ‎Pukhtunkhwa, Baghdad, Pakistan. A. Ullah - Department of Mathematics, ‎University of Malakand, ‎Haldia‎, ‎721657‎, ‎W.B., ‎Chakdara‎, ‎Dir(L),18000‎, ‎KPK, Pakistan. K. Shah - Department of Mathematics, ‎University of Malakand, ‎P.O‎. ‎Box 66833, ‎Chakdara‎, ‎Dir(L),18000‎, ‎KPK, Pakistan. - Department of Mathematics and Sciences, ‎Prince Sultan University, ‎P.O‎. ‎Box 66833, ‎Riyadh 11586, Saudi Arabia. Th. Abdeljawad - Department of Mathematics and Sciences, ‎Prince Sultan University, ‎Riyadh 11586, Saudi Arabia.

Abstract

In this study‎, ‎we present a mathematical model designed to illustrate the simultaneous occurrence of smoking and heroin co-abuse infections‎. ‎To explore non-negative solutions and identify a stable equilibrium point‎, ‎as well as the fundamental reproductive number‎, ‎we enhance the model by integrating Caputo fractional-order (FO) derivative operators‎. ‎Employing functional analysis concepts‎, ‎we derive several results pertaining to the existence of a unique solution‎. ‎Additionally‎, ‎we utilize the Ulam-Hyres (UH) notion to establish the stability of the model solutions‎. ‎To offer further insights‎, ‎we present numerical results for the fractional-order system using an Euler-type numerical technique‎. ‎These results are visually represented in graphs‎, ‎illustrating the diverse responses of the model under different parameter values.

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Keywords

• Caputo operator
• existence theoory
• numerical techniques
• unique solution‎

MSC

•  26A33
•  34A08
•  35A09

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