  We analyze the spread of COVID-19 by considering the transmission of the
disease among individuals both within and between regions. A set of regions can
be defined as any partition of a population such that travel/social contact
within each region far exceeds that between them. COVID-19 can be eliminated if
the region-to-region reproductive number---i.e. the average number of other
regions to which a single infected region will transmit the virus---is reduced
to less than one. We find that this region-to-region reproductive number is
proportional to the travel rate between regions and exponential in the length
of the time-delay before region-level control measures are imposed. Thus,
reductions in travel and the speed with which regions take action play decisive
roles in whether COVID-19 is eliminated from a collection of regions. If, on
average, infected regions (including those that become re-infected in the
future) impose social distancing measures shortly after active spreading begins
within them, the number of infected regions, and thus the number of regions in
which such measures are required, will exponentially decrease over time.
Elimination will in this case be a stable fixed point even after the social
distancing measures have been lifted from most of the regions.
