Mean-field game (MFG) systems are revolutionary models to describe complex multi-agent dynamic systems, such as competition between asset managers in finance and traffic congestion in population dynamics. They combine mean-field approximation techniques to describe the population with optimal control approaches to characterize a representative player.

This thesis investigates both computation and modeling aspects of the mean-field games. We propose a computational method for nonlocal MFGs. Our approach relies on kernel-based representations of mean-field interactions and feature-space expansions, which yields a dimension reduction. Based on the monotone inclusion formulation, we further generalize the splitting method for nonlocal MFGs to solve a class of non-potential MFGs. In terms of modelings, we integrate the spatial epidemic models with mean-field control (MFC) models to control the propagation of pandemics. We also apply MFCs to study the optimal vaccine distribution strategy. Numerical experiments demonstrate that the proposed model effectively separates infected patients in a spatial domain and transports vaccines efficiently. Finally, we study an inverse MFG problem. We propose a model recovery algorithm to reconstruct the ground metrics and interactions in the running costs with some noisy observations.