Abstract Fractal-fractional (FF) differential and integral operators having the capability to subsume features of retaining memory and self-similarities are used in the present research analysis to design a mathematical model for the rubella epidemic while taking care of dimensional consistency among the model equations. Infectious diseases have history in their transmission dynamics and thus non-local operators such as FF play a vital role in modeling dynamics of such epidemics. Monthly actual rubella incidence cases in Pakistan for the years 2017 and 2018 have been used to validate the FF rubella model and such a data set also helps for parameter estimation. Using nonlinear least-squares estimation with MATLAB function lsqcurvefit, some parameters for the classical and the FF model are obtained. Upon comparison of error norms for both models (classical and FF), it is found that the FF produces the smaller error. Locally asymptotically stable points (rubella-free and rubella-present) of the model are computed when the basic reproduction number ℛ 0 { {\mathcal R} }_{0} is less and greater than unity and the sensitivity is investigated. Moreover, solution of the FF rubella system is shown to exist. A new iterative method is proposed to carry out numerical simulations which resulted in getting insights for the transmission dynamics of the rubella epidemic.

中文翻译：

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分形-分数算子在优化阶数风疹流行建模中的作用

摘要 本研究分析中采用分形分数（FF）微分和积分算子来设计风疹流行的数学模型，同时注意模型方程之间的维数一致性，具有包含保留记忆和自相似特征的能力。 . 传染病在其传播动力学方面有历史，因此 FF 等非本地算子在此类流行病的动力学建模中发挥着至关重要的作用。巴基斯坦 2017 年和 2018 年的每月实际风疹发病病例已被用于验证 FF 风疹模型，这样的数据集也有助于参数估计。使用MATLAB函数lsqcurvefit进行非线性最小二乘估计，得到经典模型和FF模型的一些参数。通过比较两种模型（经典模型和 FF）的误差范数，发现 FF 产生的误差较小。当基本再生数 ℛ 0 { {\mathcal R} }_{0} 小于和大于 1 时，计算模型的局部渐近稳定点（无风疹和存在风疹），并研究敏感性。此外，FF风疹系统的解决方案显示存在。提出了一种新的迭代方法来进行数值模拟，从而深入了解风疹流行的传播动力学。当基本再生数 ℛ 0 { {\mathcal R} }_{0} 小于和大于 1 时，计算模型的局部渐近稳定点（无风疹和存在风疹），并研究敏感性。此外，FF风疹系统的解决方案显示存在。提出了一种新的迭代方法来进行数值模拟，从而深入了解风疹流行的传播动力学。当基本再生数 ℛ 0 { {\mathcal R} }_{0} 小于和大于 1 时，计算模型的局部渐近稳定点（无风疹和存在风疹），并研究敏感性。此外，FF风疹系统的解决方案显示存在。提出了一种新的迭代方法来进行数值模拟，从而深入了解风疹流行的传播动力学。