  We present an approach to studying and predicting the spatio-temporal
progression of infectious diseases. We treat the problem by adopting a partial
differential equation (PDE) version of the Susceptible, Infected, Recovered,
Deceased (SIRD) compartmental model of epidemiology, which is achieved by
replacing compartmental populations by their densities. Building on our recent
work (Computational Mechanics, 66, 1177, 2020), we replace our earlier use of
global polynomial basis functions with those having local support, as
epitomized in the finite element method, for the spatial representation of the
SIRD parameters. The time dependence is treated by inferring constant
parameters over time intervals that coincide with the time step in
semi-discrete numerical implementations. In combination, this amounts to a
scheme of field inversion of the SIRD parameters over each time step. Applied
to data over ten months of 2020 for the pandemic in the US state of Michigan
and to all of Mexico, our system inference via field inversion infers
spatio-temporally varying PDE SIRD parameters that replicate the progression of
the pandemic with high accuracy. It also produces accurate predictions, when
compared against data, for a three week period into 2021. Of note is the
insight that is suggested on the spatio-temporal variation of infection,
recovery and death rates, as well as patterns of the population's mobility
revealed by diffusivities of the compartments.
