Optimal Incentives to Mitigate Epidemics: A Stackelberg Mean Field Game Approach

Abstract

Motivated by the models of epidemic control in large populations, we consider a Stackelberg mean field game model between a principal and a mean field of agents whose states evolve in a finite state space. The agents play a noncooperative game in which they control their rates of transition between states to minimize an individual cost. The principal influences the nature of the resulting Nash equilibrium through incentives 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 the intensities of their interactions, and the principal is a regulator acting with nonpharmaceutical 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 illustrating the impact of the agents' and the regulator's optimal decisions in two specific models: a basic SIR model with semiexplicit solutions and a more complex model with a larger state space.