  Individual behaviors play an essential role in the dynamics of transmission
of infectious diseases, including COVID--19. This paper studies a dynamic game
model that describes the social distancing behaviors during an epidemic,
assuming a continuum of players and individual infection dynamics. The
evolution of the players' infection states follows a variant of the well-known
SIR dynamics. We assume that the players are not sure about their infection
state, and thus they choose their actions based on their individually perceived
probabilities of being susceptible, infected or removed. The cost of each
player depends both on her infection state and on the contact with others. We
prove the existence of a Nash equilibrium and characterize Nash equilibria
using nonlinear complementarity problems. We then exploit some monotonicity
properties of the optimal policies to obtain a reduced-order characterization
for Nash equilibrium and reduce its computation to the solution of a
low-dimensional optimization problem. It turns out that, even in the symmetric
case, where all the players have the same parameters, players may have very
different behaviors. We finally present some numerical studies that illustrate
this interesting phenomenon and investigate the effects of several parameters,
including the players' vulnerability, the time horizon, and the maximum allowed
actions, on the optimal policies and the players' costs.
