  We use daily data from Lombardy, the Italian region most affected by the
COVID-19 outbreak, to calibrate a SIR model individually on each municipality.
These are all covered by the same health system and, in the post-lockdown phase
we focus on, all subject to the same social distancing regulations. We find
that municipalities with a higher number of cases at the beginning of the
period analyzed have a lower rate of diffusion, which cannot be imputed to herd
immunity. In particular, there is a robust and strongly significant negative
correlation between the estimated basic reproduction number ($R_0$) and the
initial outbreak size, in contrast with the role of $R_0$ as a \emph{predictor}
of outbreak size. We explore different possible explanations for this
phenomenon and conclude that a higher number of cases causes changes of
behavior, such as a more strict adoption of social distancing measures among
the population, that reduce the spread. This result calls for a transparent,
real-time distribution of detailed epidemiological data, as such data affects
the behavior of populations in areas affected by the outbreak.

Populations and Evolution;General Economics;Physics and Society;Economics