The outbreak of COVID-19 in 2020 has led to a surge in interest in the research of the mathematical modeling of epidemics. Many of the introduced models are so-called compartmental models, in which the total quantities characterizing a certain system may be decomposed into two (or more) species that are distributed into two (or more) homogeneous units called compartments. We propose herein a formulation of compartmental models based on partial differential equations (PDEs) based on concepts familiar to continuum mechanics, interpreting such models in terms of fundamental equations of balance and compatibility, joined by a constitutive relation. We believe that such an interpretation may be useful to aid understanding and interdisciplinary collaboration. We then proceed to focus on a compartmental PDE model of COVID-19 within the newly-introduced framework, beginning with a detailed derivation and explanation. We then analyze the model mathematically, presenting several results concerning its stability and sensitivity to different parameters. We conclude with a series of numerical simulations to support our findings.

中文翻译：

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在连续体力学框架中制定的扩散反应隔室模型：在 COVID-19、数学分析和数值研究中的应用

2020 年 COVID-19 的爆发导致对流行病数学模型研究的兴趣激增。许多引入的模型是所谓的隔室模型，其中表征某个系统的总量可以分解为两个（或更多）物种，这些物种分布到两个（或更多）称为隔室的同质单元中。我们在此提出一种基于偏微分方程 (PDE) 的隔室模型公式，该方程基于连续介质力学熟悉的概念，根据平衡和相容性的基本方程解释这些模型，并结合本构关系。我们相信这样的解释可能有助于理解和跨学科合作。然后，我们继续关注新引入的框架内的 COVID-19 分区 PDE 模型，从详细的推导和解释开始。然后，我们对模型进行数学分析，提出了几个关于其稳定性和对不同参数的敏感性的结果。我们以一系列数值模拟结束，以支持我们的发现。