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 ${\mathcal{R}}_0$ is less or equal to unity. However, it is Mittag-Leffler stable only if ${\mathcal{R}}_0<1$. In addition, the positive equilibrium is shown to be globally asymptotically stable whenever ${\mathcal{R}}_0>1,$ 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 ${\tau }_0$ 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.

在本文中，我们建议研究分数阶 $\mathrm{小号}\mathrm{一世}\mathrm{[R}\mathrm{一世}$具有复发和一般非线性发病率的流行病模型。研究了解的存在性，稳态和确保渐近稳定性的充分条件。同时，提供了对全局稳定性以及Mittag-Leffler意义上的稳定性的完整分析。实际上，我们证明了当基本繁殖数达到无病平衡时，它在全局渐近稳定${\mathcal{[R}}_0$小于或等于统一。但是，只有在以下情况下Mittag-Leffler才稳定${\mathcal{[R}}_0<\mathrm{1个}$。此外，无论何时，正平衡都被证明是全局渐近稳定的${\mathcal{[R}}_0>\mathrm{1个}，$但是从Mittag-Leffler的角度来看，它并不稳定。对于每个初始条件，都会得出收敛速度的估计值。在这项工作的第二部分中，我们将时延纳入建议的模型中，以描述复发之前的时期。通过将延迟视为分叉参数，表明当延迟通过临界值时，系统会经历Hopf分支。${\tau }_0$导致出现极限循环。我们的发现表明，分数阶导数和时间延迟的组合丰富了行为并增加了模型的复杂性。通过改进的Adams–Bashforth–Moulton预测器–校正器方案，实现了数值模拟，以支持和说明理论结果。