  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 (ODE's) 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 \texttt{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.
