  The SIR model is one of the most prototypical compartmental models in
epidemiology. Generalizing this ordinary differential equation (ODE) framework
into a spatially distributed partial differential equation (PDE) model is a
considerable challenge. In the present work, we extend a recently proposed
model based on nearest-neighbor spatial interactions by one of the authors
in~\cite{vaziry2022modelling} towards a nonlocal, nonlinear PDE variant of the
SIR prototype. We then seek to develop a set of tools that provide insights for
this PDE framework. Stationary states and their stability analysis offer a
perspective on the early spatial growth of the infection. Evolutionary
computational dynamics enable visualization of the spatio-temporal progression
of infection and recovery, allowing for an appreciation of the effect of
varying parameters of the nonlocal kernel, such as, e.g., its width parameter.
These features are explored in both one- and two-dimensional settings. At a
model-reduction level, we develop a sequence of interpretable moment-based
diagnostics to observe how these reflect the total number of infections, the
epidemic's epicenter, and its spread. Finally, we propose a data-driven
methodology based on the sparse identification of nonlinear dynamics (SINDy) to
identify approximate closed-form dynamical equations for such quantities. These
approaches may pave the way for further spatio-temporal studies, enabling the
quantification of epidemics.
