Delay differential equations for the spatially-resolved simulation of
epidemics with specific application to COVID-19
Abstract
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.
In the present work, 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.