Local and global stability of fractional SAIRS Models

Abstract

In this paper we consider a fractional order SAIRS (Susceptible-Asymptomatic infected-symptomatic Infected-Recovered-Susceptible) model with vaccination. Which represents the interaction of four distinct compartments of people in community with an epidemic. We show that the disease free (resp. endemic) equilibrium is locally asymptotically stable if $\mathcal{R}_0<1$ (resp. $\mathcal{R}_0>1$). Moreover, we prove that the disease free (resp. endemic) equilibrium is globally asymptotically stable if $\mathcal{R}_0$ is less than another threshold $\mathcal{R}_1$ (resp. $\mathcal{R}_0>1$ when $\gamma=0$). To conclude this work we give some remarks with numerical simulations to illustrate our theoretical results.