  In this paper, we develop a method to estimate the infection-rate of a
disease, over a region, as a field that varies in space and time. To do so, we
use time-series of case-counts of symptomatic patients as observed in the areal
units that comprise the region. We also extend an epidemiological model,
initially developed to represent the temporal dynamics in a single areal unit,
to encompass multiple areal units. This is done using a (parameterized)
Gaussian random field, whose structure is modeled using the dynamics in the
case-counts, and which serves as a spatial prior, in the estimation process.
The estimation is performed using an adaptive Markov chain Monte Carlo method,
using COVID-19 case-count data collected from three adjacent counties in New
Mexico, USA. We find that we can estimate both the temporal and spatial
variation of the infection with sufficient accuracy to be useful in
forecasting. Further, the ability to "borrow" information from neighboring
areal units allows us to regularize the estimation in areal units with high
variance ("poor quality") data. The ability to forecast allows us to check
whether the estimated infection-rate can be used to detect a change in the
epidemiological dynamics e.g., the arrival of a new wave of infection, such as
the fall wave of 2020 which arrived in New Mexico in mid-September 2020. We
fashion a simple anomaly detector, conditioned on the estimated infection-rate
and find that it performs better than a conventional surveillance algorithm
that uses case-counts (and not the infection-rate) to detect the arrival of the
same wave.
