  The outbreak of COVID-19, beginning in 2019 and continuing through the time
of writing, has led to renewed interest in the mathematical modeling of
infectious disease. Recent works have focused on partial differential equation
(PDE) models, particularly reaction-diffusion models, able to describe the
progression of an epidemic in both space and time. These studies have shown
generally promising results in describing and predicting COVID-19 progression.
However, people often travel long distances in short periods of time, leading
to nonlocal transmission of the disease. Such contagion dynamics are not
well-represented by diffusion alone. In contrast, ordinary differential
equation (ODE) models may easily account for this behavior by considering
disparate regions as nodes in a network, with the edges defining nonlocal
transmission. In this work, we attempt to combine these modeling paradigms via
the introduction of a network structure within a reaction-diffusion PDE system.
This is achieved through the definition of a population-transfer operator,
which couples disjoint and potentially distant geographic regions, facilitating
nonlocal population movement between them. We provide analytical results
demonstrating that this operator does not disrupt the physical consistency or
mathematical well-posedness of the system, and verify these results through
numerical experiments. We then use this technique to simulate the COVID-19
epidemic in the Brazilian region of Rio de Janeiro, showcasing its ability to
capture important nonlocal behaviors, while maintaining the advantages of a
reaction-diffusion model for describing local dynamics.
