The outbreak of COVID-19 in 2020 has led to a surge in the interest in the mathematical modeling of infectious diseases. Disease transmission may be modeled as compartmental models, in which the population under study is divided into compartments and has assumptions about the nature and time rate of transfer from one compartment to another. Usually, they are composed of a system of ordinary differential equations in time. A class of such models considers the Susceptible, Exposed, Infected, Recovered, and Deceased populations, the SEIRD model. However, these models do not always account for the movement of individuals from one region to another. In this work, we extend the formulation of SEIRD compartmental models to diffusion–reaction systems of partial differential equations to capture the continuous spatio-temporal dynamics of COVID-19. Since the virus spread is not only through diffusion, we introduce a source term to the equation system, representing exposed people who return from travel. We also add the possibility of anisotropic non-homogeneous diffusion. We implement the whole model in libMesh, an open finite element library that provides a framework for multiphysics, considering adaptive mesh refinement and coarsening. Therefore, the model can represent several spatial scales, adapting the resolution to the disease dynamics. We verify our model with standard SEIRD models and show several examples highlighting the present model’s new capabilities.

2020 年 COVID-19 的爆发导致人们对传染病数学建模的兴趣激增。疾病传播可以建模为隔间模型，其中研究中的人群被分成隔间，并假设从一个隔间转移到另一个隔间的性质和时间速率。通常，它们由时间上的常微分方程组组成。一类这样的模型考虑了易感人群、暴露人群、感染人群、康复人群和死亡人群，即 SEIRD 模型。然而，这些模型并不总是考虑个人从一个地区到另一个地区的流动。在这项工作中，我们将 SEIRD 隔室模型的公式扩展到偏微分方程的扩散反应系统，以捕捉 COVID-19 的连续时空动态。由于病毒传播不仅仅是通过扩散，我们在方程系统中引入了一个源项，代表从旅行中返回的暴露人群。我们还增加了各向异性非均匀扩散的可能性。我们在libMesh，一个开放的有限元库，为多物理场提供了一个框架，考虑了自适应网格细化和粗化。因此，该模型可以表示多个空间尺度，使分辨率适应疾病动态。我们使用标准 SEIRD 模型验证我们的模型，并展示几个突出当前模型新功能的示例。