In this talk, we consider a Stackelberg mean field game model between a principal and a mean field of agents evolving on a finite state space, motivated by models of epidemic control in large populations. 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 to optimize its own objective. Later, we 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 use an innovative numerical approach based on Monte Carlo simulations and machine learning tools for stochastic optimization. Finally, we briefly discuss another game formulation for a continuum of non-identical players evolving on a finite state space where their interactions are represented by a graphon.