  We develop a minimalist compartmental model to study the impact of mobility
restrictions in Italy during the Covid-19 outbreak. We show that an early
lockdown shifts the epidemic in time, while that beyond a critical value of the
lockdown strength, the epidemic tend to restart after lifting the restrictions.
As a consequence, specific mitigation strategies must be introduced. We
characterize the relative importance of different broad strategies by
accounting for two fundamental sources of heterogeneity, i.e. geography and
demography. First, we consider Italian regions as separate administrative
entities, in which social interactions between age classs occur. Due to the
sparsity of the inter-regional mobility matrix, once started the epidemics tend
to develop independently across areas, justifying the adoption of solutions
specific to individual regions or to clusters of regions. Second, we show that
social contacts between age classes play a fundamental role and that measures
which take into account the age structure of the population can provide a
significant contribution to mitigate the rebound effects. Our model is general,
and while it does not analyze specific mitigation strategies, it highlights the
relevance of some key parameters on non-pharmaceutical mitigation mechanisms
for the epidemics.
