  This paper introduces a novel approach to spatio-temporal data analysis using
metric geometry to study the propagation of COVID-19 across the United States.
Using a geodesic Wasserstein metric, we analyse discrepancies between the
density functions of new case counts on any given day, incorporating the
geographic spread of cases. First, we apply this to identify the periods during
which the changes in the geographic distribution of COVID-19 were most
profound. The greatest shift occurred between May and June of 2020, when
COVID-19 shifted from mostly dominating the Northeastern states to a wider
distribution across the country. We support our findings with a new measure of
the extent of geodesic variance of a distribution, demonstrating that the
geographic imprint of COVID-19 was most concentrated in May 2020. Next, we
investigate whether the epidemic exhibited meaningful patterns of spatial
reversion, where similar geographic distributions return later. We identify
broad similarity between the spread of COVID-19 across the US between the
second and third waves, and to a lesser extent, the reemergence of the first
wave's Northeastern dominance closer to the present day. This methodology could
provide new insights for analysts to monitor the dynamical spread of epidemics
and enable regional policymakers to protect their localities. More broadly, the
framework we introduce could be applied to a variety of problems evolving over
space and time.
