Modeling Epidemic Spread with Strategic Vaccination and Socialization: a Mean Field Game Analysis

We study a game-theoretic model of epidemic control in a large population with finitely many groups and non-cooperative individuals. In the model, individuals jointly choose their socialization levels and vaccination rates, and vaccination is subject to a linear individual cost structure. We derive a forward-backward ordinary differential equations (FBODE) system that characterizes the mean field Nash equilibrium, show that the equilibrium vaccination rate exhibits an at-most one-jump bang-bang structure, and establish the existence of a Carath\'eodory solution to the FBODE. This establishes a realistic interpretation of the vaccination decisions, meaning individuals decide to vaccinate until a time point which is determined by model parameters and then stop after. We further consider a population-awareness extension in which individuals incorporate population infection information into their objective functions, and we prove a similar at-most one-jump bang-bang property under suitable conditions. Finally, we propose a numerical algorithm for solving the FBODE and conduct simulations to validate the theoretical findings. The experiments highlight two main insights: the trade-off between socialization and vaccination, and the greater importance of quarantining infected individuals instead of restricting susceptible individuals.