Citation: Baba, I.A.; Humphries, U.W.; Rihan, F.A.; Valdés,J.E.N. Fractional–Order Modeling and Control of COVID-19 with Shedding Effect. Axioms 2023, 12, 321. https:// doi.org/10.3390/axioms12040321 Academic Editors: Darjan Karabaševi´ c and Martin Bohner Received: 19 December 2022 Revised: 30 January 2023 Accepted: 1 February 2023 Published: 24 March 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). axioms Article Fractional–Order Modeling and Control of COVID-19 with Shedding Effect Isa A. Baba 1,2 , Usa W. Humphries 2, *, Fathalla A. Rihan 3,4 and J. E. N. Valdés 5 1 Department of Mathematics, Bayero University, Kano 700241, Nigeria 2 Department of Mathematics, Faculty of Science, King Mongkuts University of Science and Technology Thonburi (KMUTT), Bangkok 10140, Thailand 3 Department of Mathematical Sciences, College of Science, UAE University, Al Ain 15551, United Arab Emirates 4 Department of Mathematics, Faculty of Science, Helwan University, Cairo 11795, Egypt 5 Facultad de Ciencias Exactas y Naturales y Agrimensura, Universidad Nacional del Nordeste, Corrientes Capital 3400, Argentina * Correspondence: usa.wan@kmutt.ac.th Abstract: A fractional order COVID-19 model consisting of six compartments in Caputo sense is constructed. The indirect transmission of the virus through susceptible populations by the shedding effect is studied. Equilibrium solutions are calculated, and basic reproduction ratio (that depends both on direct and indirect mode of transmission), existence and uniqueness, as well as stability analysis of the solution of the model, are studied. The paper studies the effect of optimal control policy applied to shedding effect. The control is the observation of standard hygiene practices and chemical disinfectants in public spaces. Numerical simulations are carried out to support the analytic result and to show the significance of the fractional order from the biological viewpoint. Keywords: mathematical model; fractional order; Caputo; optimal control; shedding effect; COVID-19 1. Introduction COVID-19 surfaced in the world at the end of the year 2019. It undermined many sectors such astransport, the economy, education systems, sports, entertainment, etc. The pandemic killed and infected many. The nature and mode of the spread of COVID-19 outbreak are still not completely understood. Researchers are geared towards finding vaccines to curtail the spread of the virus. The idea is to limit the number of new infections and subsequent deaths due to the pandemic. Due to the scarcity of vaccines, many countries in the world adopt non-pharmaceutical measures such as lockdown, airport closures, use of sanitizers and social distancing. There is a great deal of research in the literature with regard to the pandemic, both from a theoretical and practical point of view [1–7]. It is estimated that 75% of infected individuals recover without showing serious symp- toms and many achieve e natural recovery [8]. Throat infection, chest pain, runny nose or nasal congestion, losing smell and taste, vomiting, diarrhea and nausea are some of the symptoms of COVID-19. In most cases, these symptoms appear slowly. It is also believed that elderly people can observe serious complications compared to their younger counter- parts. On average, infected individuals spend 7–14 days before showing symptoms [9]. In many cases, it takes 14 days before mild cases recover [10]. The transmission of COVID-19 occurs mostly via either a direct (through contaminated air by tiny droplets and airborne particles containing the virus) or an indirect (through contaminated surfaces) method. The virus is released from the mouth of infected individuals through either sneezing or coughing and is shed into the environment in the form of micro-particles in the air. This shedding effect is of paramount significance in studying COVID-19 transmission. Although diagnostic tests and vaccine treatments are now available to curb the spread of the disease, Axioms 2023, 12, 321. https://doi.org/10.3390/axioms12040321 https://www.mdpi.com/journal/axioms