The stability of the positive solution for a fractional SIR model

Abstract

In order to deal with non-differentiable functions, a modification of the Riemann–Liouville definition is recently proposed which appears to provide a framework for a fractional calculus which is quite parallel with classical calculus. Based on these new results, we study on the fractional SIR model in this paper. First, we give the general solution of the fractional differential equation. And then a unique global positive solution of the SIR model is obtained. Furthermore, we get the Lyapunov stability theory. By using this stability theory, the asymptotic stability of the positive solution is analyzed.

Stability of a fractional HIV/AIDS model

“…Theorem 3.2. Let α ∈ (0, 1) and (6) be such that R 0 > 1. Then the unique endemic equilibrium Σ * (7) of the fractional order system (3) is uniformly asymptotically stable in the interior of Ω (4).…”

“…In [5], a fractional order SEIR model with vertical transmission within a nonconstant population is considered, and the asymptotic stability of the disease free and endemic equilibria are analyzed. The stability of the endemic equilibrium of a fractional order SIR model is studied in [6]. A fractional order model of HIV infection of CD4+ T-cells is analyzed in [7].…”

Epidemic Dynamics of a Fractional-Order SIS Infectious Network Model

“…Based on the fractional Lyapunov stability theory, some scholars have analyzed the equilibrium point stability and transmission dynamics of the fractionalorder neural networks and the fractional HIV/AIDS epidemic systems [14][15][16]. Guo [17] studied a fractional-order SIR model in detail, and some theoretical results about the existence and stability of the unique positive solution were obtained. Furthermore, Rostamy et al discussed the existence of multiple equilibrium points in the SIR model, indicating that selecting appropriate fractional-order parameters can expand the stability region of the equilibrium point [18].…”

A fractional order SIR epidemic model with nonlinear incidence rate

“…In epidemiology, many works involving fractional order derivative have been done, and most of them are mainly concerned with SIR-type models with linear incidence rate [9][10][11][12]. In [13], Saeedian et al studied the memory effect of an SIR epidemic model using the Caputo fractional derivative.…”