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

‎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.