SIAM Review, Volume 63, Issue 2, Page 373-373, January 2021.
The SIGEST article in this issue is “A General Framework for Fractional Order Compartment Models,” by Christopher N. Angstmann, Austen M. Erickson, Bruce I. Henry, Anna V. McGann, John M. Murray, and James A. Nichols. The seminal compartmental publication of Kermack and McKendrick ([24] in the reference list) is approaching its 100th anniversary. Compartmental thinking has proved to be powerful and versatile, and a vast range of compartmental ODE models have been developed and applied in many application fields, perhaps most notably in epidemiology. In recent years, there has been growing interest in the idea of introducing fractional derivatives into classical deterministic ODEs and PDEs as a means to capture complex behavior, for example, around creep, diffusion, and hysteresis. It is no surprise, therefore, that many authors have extended compartmental ODE models to a fractional setting. What sets this SIGEST article apart is that the authors do not change from integer to fractional derivatives in an ad hoc manner; instead they start from first principles. To do this, they follow a thought process that has been used to great effect in many applied mathematics contexts: consider a discrete, stochastic, microscale setting and obtain a deterministic macroscale model by taking a mean field, or ensemble limit, approximation. Fractional calculus then emerges naturally if we assume that particles can be “trapped” so that the probability of escaping a compartment depends on the amount of time spent there. For example, in a model for the spread of disease, an individual with chronic infection may have a chance of recovery (moving from the infected compartment to the recovered compartment) that decreases as the infection persists. Section 2 works through the details for a single compartment model, after which it is straightforward to extend to the multiple compartment case. The general framework is then illustrated in four examples: Susceptible-Infected-Recovered (SIR) and Susceptible-Infected-Susceptible (SIS) models of disease spread, HIV infection in vivo, and chromium clearance in mice. Overall, this systematic framework will be of interest to a wide range of applied mathematicians who model complex systems in life sciences and engineering, and also to applied analysts, numerical analysts, and statisticians who wish to study, simulate, or make predictions with realistic models involving fractional derivatives.

中文翻译：

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SIGEST

SIAM 评论，第 63 卷，第 2 期，第 373-373 页，2021 年 1 月。
本期 SIGEST 文章是“分数阶隔间模型的通用框架”，作者为 Christopher N. Angstmann、Austen M. Erickson、Bruce I. Henry、Anna V. McGann、John M. Murray 和 James A. Nichols。Kermack 和 McKendrick 的开创性分区出版物（参考列表中的 [24]）即将迎来 100 周年。房室思维已被证明是强大且通用的，并且已经开发了大量房室 ODE 模型并将其应用于许多应用领域，也许最显着的是在流行病学中。近年来，人们对将分数导数引入经典确定性 ODE 和 PDE 作为捕获复杂行为（例如，围绕蠕变、扩散和滞后）的方法的想法越来越感兴趣。因此，这并不奇怪，许多作者已将分室 ODE 模型扩展到分数设置。这篇 SIGEST 文章的与众不同之处在于，作者并没有以一种特别的方式从整数变为分数导数。相反，他们从第一原则开始。为此，他们遵循一个在许多应用数学环境中已被广泛使用的思维过程：考虑离散的、随机的、微观的设置，并通过采用平均场或集合极限近似来获得确定性的宏观模型。如果我们假设粒子可以被“捕获”，从而逃离隔间的概率取决于在那里度过的时间，那么分数微积分就会自然而然地出现。例如，在疾病传播模型中，患有慢性感染的个体可能有康复的机会（从受感染的隔室移动到恢复的隔室），随着感染的持续，该机会会降低。第 2 节详细介绍了单隔间模型，然后可以直接扩展到多隔间情况。然后在四个示例中说明了一般框架：疾病传播、体内 HIV 感染和小鼠铬清除的易感-感染-恢复 (SIR) 和易感-感染-易感 (SIS) 模型。总的来说，这个系统的框架将引起广泛的应用数学家的兴趣，他们对生命科学和工程中的复杂系统进行建模，以及希望研究、模拟、