Complex mathematical SIR model for spreading of COVID-19 virus with Mittag-Leffler kernel

Abstract

This paper investigates a new model on coronavirus-19 disease (COVID-19), that is complex fractional SIR epidemic model with a nonstandard nonlinear incidence rate and a recovery, where derivative operator with Mittag-Leffler kernel in the Caputo sense (ABC). The model has two equilibrium points when the basic reproduction number $R_0>1$ ; a disease-free equilibrium $E_0$ and a disease endemic equilibrium $E_1$ . The disease-free equilibrium stage is locally and globally asymptotically stable when the basic reproduction number $R_0<1$ , we show that the endemic equilibrium state is locally asymptotically stable if $R_0>1$ . We also prove the existence and uniqueness of the solution for the Atangana-Baleanu SIR model by using a fixed-point method. Since the Atangana-Baleanu fractional derivative gives better precise results to the derivative with exponential kernel because of having fractional order, hence, it is a generalized form of the derivative with exponential kernel. The numerical simulations are explored for various values of the fractional order. Finally, the effect of the ABC fractional-order derivative on suspected and infected individuals carefully is examined and compared with the real data.

Keywords: Atangana–Beleanu–Caputo (ABC) derivatives; Complex fractional SIR model; Coronavirus-19 disease; Fixed-point method; Stability.