  In this paper, we present a game-theoretic model describing voluntary social
distancing during the spread of an epidemic. The payoffs of the agents depend
on the social distancing they practice and on the probability of getting
infected. We consider two types of agents, the non-vulnerable agents who have a
small cost if they get infected, and the vulnerable agents who have a higher
cost. For the modeling of the epidemic outbreak, we consider a variant of the
SIR (Susceptible-Infected-Removed) model involving populations of susceptible,
infected, and removed persons of vulnerable and non-vulnerable types. The Nash
equilibria of this social distancing game are studied. The main contribution of
this work is the analysis of the case where the players, desiring to achieve a
low social inequality, pose a bound on the variance of the payoffs. In this
case, we introduce and characterize a notion of Generalized Nash Equilibrium
(GNE) for games with a continuum of players. Through numerical studies, we show
that inequality constraints result in a slower spread of the epidemic and an
improved cost for the vulnerable players. Furthermore, it is possible that
inequality constraints are beneficial for non-vulnerable players as well.
