SIAM Review, Volume 63, Issue 2, Page 375-392, January 2021.
Compartment models are a widely used class of models that are useful when considering the flow of objects, people, or energy between different labeled states, referred to as compartments. Classic examples include SIR models in epidemiology and many pharmacokinetic models used in pharmacology. These models are formulated as sets of coupled ordinary differential equations, but in recent years there has been increasing interest in generalizations involving fractional differential equations. The majority of such generalizations have been performed in an ad hoc manner by replacing integer order derivatives with fractional derivatives. Such an approach does allow for the incorporation of history effects into the models, but may be problematic in a number of ways, such as breaking conservation of matter. To overcome these problems we have developed a systematic approach for the inclusion of fractional derivatives into compartment models by deriving the deterministic governing equations from an underlying physical stochastic process. This derivation also reveals the connection between these fractional order models and age-structured models. Unlike the ad hoc addition of fractional derivatives, our approach ensures that the model remains physically reasonable at all times and provides for an easy interpretation of all the parameters in the model. Illustrative examples, drawn from epidemiology, pharmacokinetics, and in-host virus dynamics, are provided.

中文翻译：

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分数阶隔室模型的通用框架

SIAM 评论，第 63 卷，第 2 期，第 375-392 页，2021 年 1 月。
隔间模型是一类广泛使用的模型，在考虑不同标记状态（称为隔间）之间的物体、人员或能量流动时非常有用。经典例子包括流行病学中的 SIR 模型和药理学中使用的许多药代动力学模型。这些模型被表述为一组耦合的常微分方程，但近年来，人们对涉及分数阶微分方程的推广越来越感兴趣。大多数这样的概括是通过用分数导数替换整数阶导数以特别的方式进行的。这种方法确实允许将历史影响纳入模型，但在许多方面可能存在问题，例如打破物质守恒。为了克服这些问题，我们开发了一种系统方法，通过从基础物理随机过程导出确定性控制方程，将分数导数包含在隔间模型中。这种推导还揭示了这些分数阶模型和年龄结构模型之间的联系。与分数导数的临时添加不同，我们的方法确保模型始终保持物理合理性，并提供对模型中所有参数的简单解释。提供了从流行病学、药代动力学和宿主内病毒动力学中提取的说明性示例。这种推导还揭示了这些分数阶模型和年龄结构模型之间的联系。与分数导数的临时添加不同，我们的方法确保模型始终保持物理合理性，并提供对模型中所有参数的简单解释。提供了从流行病学、药代动力学和宿主内病毒动力学中提取的说明性示例。这种推导还揭示了这些分数阶模型和年龄结构模型之间的联系。与分数导数的临时添加不同，我们的方法确保模型始终保持物理合理性，并提供对模型中所有参数的简单解释。提供了从流行病学、药代动力学和宿主内病毒动力学中提取的说明性示例。