In this thesis, we consider stationary one-dimensional mean-field games (MFGs) with or without congestion. Our aim is to understand the qualitative features of these games through the analysis of explicit solutions. We are particularly interested in MFGs with a nonmonotonic behavior, which corresponds to situations where agents tend to aggregate. First, we derive the MFG equations from control theory. Then, we compute explicit solutions using the current formulation and examine their behavior. Finally, we represent the solutions and analyze the results. This thesis main contributions are the following: First, we develop the current method to solve MFG explicitly. Second, we analyze in detail non-monotonic MFGs and discover new phenomena: non-uniqueness, discontinuous solutions, empty regions and unhappiness traps. Finally, we address several regularization procedures and examine the stability of MFGs.

In this thesis, we will study basic mathematical epidemic models SIR, SIS, and SEIR. Then we will construct a modified model as a combination of SIR and SIS models. First, we will find the explicit solutions for the SIS model. and show no exact solution for the SIR model. Also, find the parametric solution for the SIR model and find a numerical solution by using Euler’s method. Then we find an approximate explicit form of the epidemic curve. Also, we will study parametric influences on the SIR and SIS models. Finally, we will suggest some recommendations to decrease the epidemic’s spread.