Stability Analysis and Existence of Solutions for a Modified SIRD Model of COVID-19 with Fractional Derivatives

Bilal Basti,Rabah Djemiat,Imadeddine Berrabah,Noureddine Benhamidouche,Nacereddine Hammami,Farid Nouioua

doi:10.3390/sym13081431

Abstract

This paper discusses and provides some analytical studies for a modified fractional-order SIRD mathematical model of the COVID-19 epidemic in the sense of the Caputo–Katugampola fractional derivative that allows treating of the biological models of infectious diseases and unifies the Hadamard and Caputo fractional derivatives into a single form. By considering the vaccine parameter of the suspected population, we compute and derive several stability results based on some symmetrical parameters that satisfy some conditions that prevent the pandemic. The paper also investigates the problem of the existence and uniqueness of solutions for the modified SIRD model. It does so by applying the properties of Schauder’s and Banach’s fixed point theorems.

Highlights

The coronavirus pandemic was a major worldwide challenge in 2020
Katugampola proposed a generalized derivative in [26,31,32]; he demonstrated the existence of solutions for Caputo–Katugampola fractional differential equations in [26]
We discussed some analytical studies for a modified fractionalorder SIRD mathematical model of the COVID-19 disease, with Caputo–Katugampola’s fractional derivative being used as the differential operator, which unifies the Hadamard and Caputo fractional derivatives into a single form

Summary

Introduction

The coronavirus pandemic was a major worldwide challenge in 2020. The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is the virus responsible for COVID-19 infection. Techniques of decomposition, homotopy, and variation were used to comprehensively analyze the mathematical models [15,16,17,18] Many methods such as the residual power series, symmetry, spectral, Fourier transform, similarity, and collocation methods are used to study and manage differential equations in both fractional and classical orders, along with their systems (for more details see [6,7,8,9,10,11,12,17,18,19,20,21,22,23,24,25,26,27,28,29]). We provide the necessary conditions for obtaining at least one solution or its uniqueness using the fixed point theorems

Preliminary and Necessary Definitions

Analysis for the Modified SIRD Model of the Pandemic

Main Results

Conclusions