  We consider the control of the COVID-19 pandemic through a standard SIR
compartmental model. This control is induced by the aggregation of individuals'
decisions to limit their social interactions: when the epidemic is ongoing, an
individual can diminish his/her contact rate in order to avoid getting
infected, but this effort comes at a social cost. If each individual lowers
his/her contact rate, the epidemic vanishes faster, but the effort cost may be
high. A Mean Field Nash equilibrium at the population level is formed,
resulting in a lower effective transmission rate of the virus. We prove
theoretically that equilibrium exists and compute it numerically. However, this
equilibrium selects a sub-optimal solution in comparison to the societal
optimum (a centralized decision respected fully by all individuals), meaning
that the cost of anarchy is strictly positive. We provide numerical examples
and a sensitivity analysis, as well as an extension to a SEIR compartmental
model to account for the relatively long latent phase of the COVID-19 disease.
In all the scenarii considered, the divergence between the individual and
societal strategies happens both before the peak of the epidemic, due to
individuals' fears, and after, when a significant propagation is still
underway.
