  Motivated by models of epidemic control in large populations, we consider a
Stackelberg mean field game model between a principal and a mean field of
agents evolving on a finite state space. The agents play a non-cooperative game
in which they can control their transition rates between states to minimize an
individual cost. The principal can influence the resulting Nash equilibrium
through incentives so as to optimize its own objective. We analyze this game
using a probabilistic approach. We then propose an application to an epidemic
model of SIR type in which the agents control their interaction rate and the
principal is a regulator acting with non pharmaceutical interventions. To
compute the solutions, we propose an innovative numerical approach based on
Monte Carlo simulations and machine learning tools for stochastic optimization.
We conclude with numerical experiments by illustrating the impact of the
agents' and the regulator's optimal decisions in two models: a basic SIR model
with semi-explicit solutions and a more complex model with a larger state
space.
