SIR epidemic model with Mittag–Leffler fractional derivative

Abstract

Abstract The SIR epidemic model with delay in the context of the fractional derivative with Mittag–Leffler kernel has been considered. The Atangana–Baleanu fractional derivative is a non-singular fractional derivative with Mittag–Leffler kernel. The positivity of the solutions of the SIR model depends strongly on the order of the Atangana–Baleanu–Caputo fractional derivative. We investigate the existence and the uniqueness of our proposed model in terms of the used fractional derivative. The reproduction number related to the SIR epidemic model in our paper is presented. The trivial equilibrium point and the endemic equilibrium point have been proposed. The asymptotic stability for the trivial equilibrium and the endemic equilibrium points have been investigated. The global asymptotic stability of the disease-free equilibrium and the endemic equilibrium have been analyzed in terms of the Lyapunov direct method. The graphical representations of the approximate solutions of the model have been proposed.