  The design of coherent and efficient policies to address infectious diseases
and their consequences requires to model not only epidemics dynamics, but also
individual behaviors, as the latter has a strong influence on the former. In
our work, we provide a theoretical model for this problem, taking into account
the social structure of a population. This model is based on a Mean Field Game
version of a SIR compartmental model, in which individuals are grouped by their
age class and interact together in different settings. This social
heterogeneity allows to reproduce realistic situations while remaining usable
in practice. In our game theoretical approach, individuals can choose to limit
their contacts by making a trade-off between the risks incurred by infection
and the cost of being confined. The aggregation of all these individual choices
and optimizations forms a Nash equilibrium through a system of coupled
equations that we derive and solve numerically. The global cost born by the
population within this scenario is then compared to its societal optimum
counterpart (i.e. the optimal cost from the society viewpoint), and we
investigate how the gap between these two costs can be partially bridged within
a constrained Nash equilibrium for which a governmental institution would
impose lockdowns. Finally we consider the consequences of the finiteness of the
population size $N$, or of a time $T$ at which an external event would end the
epidemic, and show that the variation of these parameters could lead to first
order phase transitions in the choice of optimal strategies. In this paper, all
the strategies considered to mitigate epidemics correspond to
non-pharmaceutical interventions (NPI), and we provide here a theoretical
framework within which guidelines for public policies depending on the
characteristics of an epidemic and on the cost of restrictions on the society
could be assessed.
