Efficiency of Symmetric Nash Equilibria in Epidemic Models with Confinements

We consider a non-cooperative game in the SIR model with confinements. Each member of the population is a player whose strategy is her probability of being protected from the epidemic. We assume that for each player, there is a cost of infection per unit time and a cost of being confined, which is linear and decreasing on her confinement strategy. The total cost is defined as the sum of her confinement and infection costs. We present a method for computing a symmetric Nash equilibrium for this game and study its efficiency. We conclude that the Nash equilibrium we obtain leads to fewer confinements than the strategy that minimizes the cost of the entire population.