In this paper, an investigation and analysis of a nonlinear fractional-order SIR epidemic model with Crowley–Martin type functional response and Holling type-II treatment rate are established along the memory. The existence and stability of the equilibrium points are investigated. The sufficient conditions for the persistence of the disease are provided. First, a threshold value, [Formula: see text], is obtained which determines the stability of equilibria, then model equilibria are determined and their stability analysis is considered by using fractional Routh-Hurwitz stability criterion and fractional La-Salle invariant principle. The fractional derivative is taken in Caputo sense and the numerical solution of the model is obtained by L1 scheme which involves the memory trace that can capture and integrate all past activity. Meanwhile, by using Lyapunov functional approach, the global dynamics of the endemic equilibrium point is discussed. Further, some numerical simulations are performed to illustrate the effectiveness of the theoretical results obtained. The outcome of the study reveals that the applied L1 scheme is computationally very strong and effective to analyze fractional-order differential equations arising in disease dynamics. The results show that order of the fractional derivative has a significant effect on the dynamic process. Also, from the results, it is obvious that the memory effect is zero for [Formula: see text]. When the fractional-order [Formula: see text] is decreased from [Formula: see text] the memory trace nonlinearly increases from [Formula: see text], and its dynamics strongly depends on time. The memory effect points out the difference between the derivatives of the fractional-order and integer order.

中文翻译：

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带记忆的分数阶 SIR 流行病模型的全局动力学

在本文中，沿着记忆建立了具有Crowley-Martin型功能反应和Holling II型治疗率的非线性分数阶SIR流行病模型的调查和分析。研究了平衡点的存在性和稳定性。为疾病持续存在提供了充分条件。首先，得到一个阈值，[公式：见正文]，确定平衡的稳定性，然后确定模型平衡，并利用分数Routh-Hurwitz稳定性准则和分数La-Salle不变量原理对其进行稳定性分析。分数导数采用Caputo意义，模型的数值解通过L1方案获得，该方案涉及可以捕获和整合所有过去活动的记忆轨迹。同时，采用Lyapunov泛函方法，讨论了地方性平衡点的全局动力学。此外，还进行了一些数值模拟来说明所获得的理论结果的有效性。研究结果表明，应用的 L1 方案在计算上非常强大，并且可以有效地分析疾病动力学中出现的分数阶微分方程。结果表明，分数阶导数的阶数对动态过程有显着影响。此外，从结果来看，[公式：见正文]的记忆效应显然为零。当分数阶 [公式：参见文本] 从 [公式：参见文本] 减少时，记忆轨迹从 [公式：参见文本] 非线性增加，并且其动态很大程度上取决于时间。