  In the wake of the 2020 COVID-19 epidemic, much work has been performed on
the development of mathematical models for the simulation of the epidemic, and
of disease models generally. Most works follow the susceptible-infected-removed
(SIR) compartmental framework, modeling the epidemic with a system of ordinary
differential equations. Alternative formulations using a partial differential
equation (PDE) to incorporate both spatial and temporal resolution have also
been introduced, with their numerical results showing potentially powerful
descriptive and predictive capacity. In the present work, we introduce a new
variation to such models by using delay differential equations (DDEs). The
dynamics of many infectious diseases, including COVID-19, exhibit delays due to
incubation periods and related phenomena. Accordingly, DDE models allow for a
natural representation of the problem dynamics, in addition to offering
advantages in terms of computational time and modeling, as they eliminate the
need for additional, difficult-to-estimate, compartments (such as exposed
individuals) to incorporate time delays. Here, we introduce a DDE epidemic
model in both an ordinary- and partial differential equation framework. We
present a series of mathematical results assessing the stability of the
formulation. We then perform several numerical experiments, validating both the
mathematical results and establishing model's ability to reproduce measured
data on realistic problems.
