In recent decades, there has been a growing interest in generalizing differential equations of arbitrary order by replacing integer derivatives with non-integer derivatives. This allows for the construction of new models that can replicate memory effects due to the non-locality of fractional operators. However, this approach can lead to models that suffer from physical misinterpretation, do not maintain mass balance, and alter the original units of model parameters. This paper introduces an approach based on the generalization of exponential by Mittag-Leffler decays, which is associated with waiting time distributions. The model remains physically interpretable and ensures that the dimensions are constructively corrected. We discuss the mean waiting time, the asymptotic behavior, and apply this theory to develop a novel fractional SIRS model. Additionally, we present numerical results.

近几十年来，人们越来越关注通过用非整数导数代替整数导数来推广任意阶微分方程。这允许构建新模型，可以复制由于分数运算符的非局部性而导致的记忆效应。然而，这种方法可能会导致模型遭受物理误解、无法维持质量平衡并改变模型参数的原始单位。本文介绍了一种基于 Mittag-Leffler 衰减指数推广的方法，该方法与等待时间分布相关。该模型保持物理上的可解释性，并确保尺寸得到建设性修正。我们讨论平均等待时间、渐近行为，并应用该理论开发一种新颖的分数 SIRS 模型。此外，我们还提供了数值结果。