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We present an approach to studying and predicting the spatio-temporalprogression of infectious diseases. We treat the problem by adopting a partialdifferential equation (PDE) version of the Susceptible, Infected, Recovered,Deceased (SIRD) compartmental model of epidemiology, which is achieved byreplacing compartmental populations by their densities. Building on our recentwork (Computational Mechanics, 66, 1177, 2020), we replace our earlier use ofglobal polynomial basis functions with those having local support, asepitomized in the finite element method, for the spatial representation of theSIRD parameters. The time dependence is treated by inferring constantparameters over time intervals that coincide with the time step insemi-discrete numerical implementations. In combination, this amounts to ascheme of field inversion of the SIRD parameters over each time step. Appliedto data over ten months of 2020 for the pandemic in the US state of Michiganand to all of Mexico, our system inference via field inversion infersspatio-temporally varying PDE SIRD parameters that replicate the progression ofthe pandemic with high accuracy. It also produces accurate predictions, whencompared against data, for a three week period into 2021. Of note is theinsight that is suggested on the spatio-temporal variation of infection,recovery and death rates, as well as patterns of the population's mobilityrevealed by diffusivities of the compartments.

System inference via field inversion for the spatio-temporal progression of infectious diseases: Studies of COVID-19 in Michigan and Mexico