  The surprisingly mercurial Covid-19 pandemic has highlighted the need to not
only accelerate research on infectious disease, but to also study them using
novel techniques and perspectives. A major contributor to the difficulty of
containing the current pandemic is due to the highly asymptomatic nature of the
disease. In this investigation, we develop a modeling framework to study the
spatio-temporal evolution of diseases with high rates of asymptomatic
transmission, and we apply this framework to a hypothetical country with
mathematically tractable geography; namely, square counties uniformly organized
into a rectangle. We first derive a model for the temporal dynamics of
susceptible, infected, and recovered populations, which is applied at the
county level. Next we use likelihood-based parameter estimation to derive
temporally varying disease transmission parameters on the state-wide level.
While these two methods give us some spatial structure and show the effects of
behavioral and policy changes, they miss the evolution of hot zones that have
caused significant difficulties in resource allocation during the current
pandemic. It is evident that the distribution of cases will not be stagnantly
based on the population density, as with many other diseases, but will
continuously evolve. We model this as a diffusive process where the diffusivity
is spatially varying based on the population distribution, and temporally
varying based on the current number of simulated asymptomatic cases. With this
final addition coupled to the SIR model with temporally varying transmission
parameters, we capture the evolution of "hot zones" in our hypothetical setup.
