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

F Talay Akyildiz,Fehaid Salem Alshammari

doi:10.1186/s13662-021-03470-1

F Talay Akyildiz, Fehaid Salem Alshammari

Open Access

https://doi.org/10.1186/s13662-021-03470-1

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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.

Highlights

The coronavirus is a new viral pneumonia, deadly and rapidly spreading infection that has put great panic around the globe since the break out late 2019
To the best of our knowledge, this is the first time to study a non-local and non-singular derivative operator for the model of the SIR, which has nonlinear incidence and recovery rates, where we examine the existence and uniqueness results with the Atangana– Baleanu derivative by using a fixed-point method
Theorem 3.1 The disease-free equilibrium point E0 is locally asymptotically stable and if R0 < 1 it is unstable if R0 > 1

Summary

Introduction

The coronavirus is a new viral pneumonia, deadly and rapidly spreading infection that has put great panic around the globe since the break out late 2019. In order to overcome this problem, Atangana and Baleanu [13] establish a new model, the so called generalized ABC differential operator involving the Mittag-Leffler (ML) and non-local type kernel, which has an anti-derivative fractional integral operator [14]. This new definition of derivative is shown to be more efficient for the SIR model with generalized incidence rate compared to the other existing fractional models [15] and an exothermic reactions model having a constant heat source in porous media with power problem [16].

Mathematical model and discussion

Simulation

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Conclusion