Mathematical model of SIR epidemic system (COVID-19) with fractional derivative: stability and numerical analysis

Abstract

In this paper, we study and analyze the susceptible-infectious-removed (SIR) dynamics considering the effect of health system. We consider a general incidence rate function and the recovery rate as functions of the number of hospital beds. We prove the existence, uniqueness, and boundedness of the model. We investigate all possible steady-state solutions of the model and their stability. The analysis shows that the free steady state is locally stable when the basic reproduction number $R_{0}$ R 0 is less than unity and unstable when $R_{0} > 1$ R 0 > 1 . The analysis shows that the phenomenon of backward bifurcation occurs when $R_{0}<1$ R 0 < 1 . Then we investigate the model using the concept of fractional differential operator. Finally, we perform numerical simulations to illustrate the theoretical analysis and study the effect of the parameters on the model for various fractional orders.

An extended epidemic model with vaccination: Weak-immune SIRVI

“…How the peak time of infection can be forecasted accurately without a full simulation was formulated in [16] , [17] . With the effects of health system, the fractional SIR model was analyzed in [18] . Prediction of the outbreak of COVID-19 in Iran/Isfahan in the long run was shown to be failure with the traditional SIR model in [19] .…”

Asymptotic behavior of a stochastic SIR model with general incidence rate and nonlinear Lévy jumps

“…In [ 15 ], authors assumed that the infection rate of HIV-1 was given by the Beddington–DeAngelis incidence function , obviously, with the different values of a and b , this nonlinear incidence rate can be transformed into Holling-type II or saturation incidence function. Similarly, when Alqahtani performed the stability and numerical analysis of a SIR epidemic system (COVID-19), they also adopted the Beddington–DeAngelis incidence function [ 16 ]. Besides, Ruan et al proposed an epidemic model with nonlinear incidence rate in [ 17 ], where measures the infection force of the disease and measures the inhibition effect from the behavioral change of the susceptible individuals when their number increases or from the crowding effect of the infective individuals.…”

Asymptotic behavior and semi-analytic solution of a novel compartmental biological model

“…To simulate the transmission of disease, the authors [9] looked at the SIR model with a generic incidence rate function and a nonlinear recovery rate. The influence of the health system affects the nonlinear recovery rate.…”