Dynamics of an epidemic model for COVID-19 with Hattaf fractal-fractional operator and study of existence of solutions by means of fixed point theory

Volume 36, Issue 4, pp 371--385 https://dx.doi.org/10.22436/jmcs.036.04.02

Publication Date: August 15, 2024 Submission Date: January 14, 2024 Revision Date: April 14, 2024 Accteptance Date: July 08, 2024

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Authors

H. EL Mamouni - Laboratory of Analysis‎, ‎Modeling and Simulation (LAMS)‎, ‎Faculty of Sciences Ben M'Scik, ‎Hassan II University of Casablanca, ‎P.O. Box 7955 Sidi Othman, Casablanca, Morocco. Kh. Hattaf - Laboratory of Analysis‎, ‎Modeling and Simulation (LAMS)‎, ‎Faculty of Sciences Ben M'Scik, ‎Hassan II University of Casablanca, ‎P.O. Box 7955 Sidi Othman, Casablanca, Morocco. - Equipe de Recherche en Modelisation et Enseignement des Mathematiques (ERMEM), ‎Centre Regional des Metiers de l'Education et de la Formation (CRMEF), ‎20340 Derb Ghalef, Casablanca, Morocco. N. Yousfi - Laboratory of Analysis‎, ‎Modeling and Simulation (LAMS)‎, ‎Faculty of Sciences Ben M'Scik, ‎Hassan II University of Casablanca, ‎P.O. Box 7955 Sidi Othman, Casablanca, Morocco.

Abstract

‎Coronavirus disease 2019 (COVID-19) is an infectious disease caused by a new virus called severe acute respiratory syndrome coronavirus 2 (SARS-COV-2)‎. ‎To describe the spread of this infectious disease‎, ‎we propose a mathematical model including some important aspects‎, ‎such as the carrier and memory effects as well as the nonlinearity of incidence function‎. ‎The memory effect is described by the Hattaf fractal-fractional derivative‎. ‎Sufficient conditions for the existence and uniqueness of solutions are established by means of Krasnoselskii's fixed point theorem and Banach contraction‎. ‎Furthermore‎, ‎our results show that the proposed fractal-fractional model has one stable disease-free equilibrium when the basic reproduction number satisfies\(\mathcal{R}_{0} \leq 1\) and a unique stable endemic equilibrium when \(\mathcal{R}_{0} > 1\)‎. ‎In addition‎, ‎numerical simulations for different values of fractal and fractional orders are carried out to illustrate the theoretical results‎.

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Keywords

• COVID-19
• SARS-CoV-2
• Krasnoselskii's fixed point theorem
• Hattaf fractal-fractional derivative
• numerical simulations

MSC

•  26A33
•  47H10
•  92D30

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