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

• Published in: Advances in difference equations Vol. 2021; no. 1; pp. 2 - 16
• Main Author: Alqahtani, Rubayyi T.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.01.2021; Springer Nature B.V; SpringerOpen
• Subjects: Analysis; Backward bifurcation; Difference and Functional Equations; Differential equations; Fractional model; Functional Analysis; Hospital bed; Mathematical models; Mathematical Models of Infectious Diseases; Mathematics; Mathematics and Statistics; Nonlinear recovery rate; Numerical analysis; Operators (mathematics); Ordinary Differential Equations; Partial Differential Equations; SIR model; Stability; Stability analysis; Steady state; Nonlinear recovery rate; Stability; Fractional model; SIR model; Backward bifurcation; Hospital bed
• ISSN: 1687-1847; 1687-1839; 1687-1847
• DOI: 10.1186/s13662-020-03192-w

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 is less than unity and unstable when R 0 > 1 . The analysis shows that the phenomenon of backward bifurcation occurs when 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.