There are still mathematical predictions in the fight against epidemics. Speedy expansion, ways and procedures for the pandemic control require early understanding when solutions with better computer-based mathematical modeling and prognosis are developed. Despite high uncertainty in each of these models, one of the important tools for public health management system is epidemiology models. The fractional order is shown to be more effective in modeling epidemic diseases, in relation to the memory effects. Notably, recently founded calculus tools, called fractal-fractional calculus, having a fractional order and fractal dimension, enable us to study the behavior of a real-world problem under both fractal and fractional tools. This paper is about the dynamical behavior of a new mathematical model of novel corona disease (COVID-19) under the fractal-fractional Atangana–Baleanu derivative. The considered model has three compartments, namely, susceptible, infected and recovered or removed (SIR). The existence and uniqueness of the model’s solution will be proved via Krasnoselskii’s and Banach’s fixed point theorems, respectively. The stability of the solution in the sense of Hyers–Ulam (HU) will be built up by nonlinear functional analysis. Moreover, the numerical simulations for different values of isolation parameters corresponding to various fractal-fractional orders are analyzed using fractional Adams–Bashforth (AB) method with two-step Lagrange polynomial. Finally, the obtained simulation results are applied to the real data of disease spread from Pakistan. The graphical interpretations demonstrate that increasing the isolation parameters which is caused by strict precautionary measures will reduce the disease infection transmission in society.

中文翻译：

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分形分数导数下巴基斯坦 COVID-19 爆发的 SIR 数学模型的动力学

与流行病的斗争仍然存在数学预测。在开发具有更好的基于计算机的数学建模和预测的解决方案时，需要及早了解大流行控制的快速扩展、方法和程序。尽管这些模型中的每一个都存在很大的不确定性，但公共卫生管理系统的重要工具之一是流行病学模型。与记忆效应相关，分数阶在模拟流行病方面表现得更有效。值得注意的是，最近创建的微积分工具，称为分形-分数微积分，具有分数阶和分形维数，使我们能够研究分形和分数工具下现实世界问题的行为。本文是关于新冠病 (COVID-19) 新数学模型在分形-分数 Atangana-Baleanu 导数下的动力学行为。所考虑的模型具有三个隔间，即易感、感染和恢复或移除 (SIR)。模型解的存在性和唯一性将分别通过 Krasnoselskii 和 Banach 不动点定理证明。Hyers-Ulam (HU) 意义上的解的稳定性将通过非线性泛函分析建立。此外，使用具有两步拉格朗日多项式的分数Adams-Bashforth（AB）方法分析了对应于各种分形分数阶的不同隔离参数值的数值模拟。最后，将获得的模拟结果应用于巴基斯坦疾病传播的真实数据。