Mittag-Leffler stability and bifurcation of a nonlinear fractional model with relapse

Abstract

In this paper, we propose to study a fractional-order SIRI epidemic model with relapse and a general non-linear incidence rate. The existence of solutions, steady states and sufficient conditions to ensure the asymptotic stability are investigated. Meanwhile, a complete analysis of the global stability as well as the stability in the sense of Mittag-Leffler is provided. Indeed, we showed that the disease-free equilibrium is globally asymptotically stable when the basic reproduction number R0 is less or equal to unity. However, it is Mittag-Leffler stable only if R0<1. In addition, the positive equilibrium is shown to be globally asymptotically stable whenever R0>1, but it is not stable in the sense of Mittag-Leffler. An estimate of the rate of convergence is derived for every initial condition. In the second part of this work, we incorporated a time delay into the proposed model to describe the period preceding the relapse. By considering the delay as a bifurcation parameter, it was shown that the system undergoes a Hopf-bifurcation when the delay passes through a critical value τ0 leading to the appearance of limit cycles. Our findings, reveal that the combination of fractional order derivative and time delay enriches the behaviors and increase the complexity of the model. By means of the modified Adams–Bashforth–Moulton predictor–corrector scheme, numerical simulations are implemented to support and illustrate the theoretical results.