Fractional-order SIR epidemic model with treatment cure rate

Abstract

In this paper, we will study a fractional-order SIR epidemic model with treatment cure rate. The paper starts with the investigation of some needed preliminaries related to fractional calculus. Next, we investigate the well-posedness of the fractional-order model in terms of positivity and boundedness of solution. Then, we give the basic reproduction number via the next generation matrix method. The model has two steady states, namely, the disease-free equilibrium and the endemic equilibrium. It was proven that when the basic reproduction number is less than unity, then, the disease-free-equilibrium is globally asymptotically stable. While, when the basic reproduction number is greater than unity, then the endemic-equilibrium is globally asymptotically stable. Numerical simulations are carried out in order to confirm the theoretical results. To this end, we will apply a numerical approach based on the fundamental theorem of fractional calculus and a three-step Lagrange polynomial interpolation technique. It is shown that any increase of the fractional order derivative leads to a decrease in the convergence speed towards the studied equilibrium.

Keywords

• SIR epidemic model
• Treatment cure rate
• Global stability
• Numerical simulations