Mathematical models can be very useful for improving understanding of human behavior and helping forecasting it. In this thesis, we develop and explore three models of human dynamics. One is a model of disease spread coupled to opinion dynamics, another is a model of opinion dynamics, and the third is a model of illegal logging. We build models in the form of dynamical processes on networks for the first two applications and employ optimal control theory to study the third application.
We first introduce a network model of the spread of a disease under the influence of the spread of competing opinions. We describe the network structure in this model as a two-layer multilayer network. On one layer, two opinions --- pro-physical-distancing and anti-physical-distancing --- spread concurrently and compete with each other. On the other layer, the disease evolves and individuals are less likely (respectively, more likely) to become infected if they adopt the pro-physical-distancing (respectively, anti-physical-distancing) opinion. We explore both beneficial and harmful effects of the spread of opinions on disease transmission with mean-field approximations and direct numerical simulations. We also examine how heterogeneous networks with specified interlayer and intralyer degree--degree correlations influence the dynamics.
We then develop an opinion-dynamics model that extends the majority-vote model to two-layer multilayer networks with community structure. We assume that neighbors from different social relationships have different abilities to change an individual's opinions. We find three patterns of steady-state opinion distributions and study phase transitions in the model with a mean-field approximation and direct numerical simulations.
Finally, we study a model of the behavior of uncontrolled loggers in the presence of law-enforcement agencies. We assume that loggers want to maximize a profit function that incorporates the benefit of logging, travel cost, and the the risk of capture by finding optimal travel trajectories and optimal logging duration. We formulate the problem as a static Hamilton--Jacobi equation, which we solve using a fast sweeping method. We use Brazilian rainforest data and demonstrate the importance of geographically targeted patrol strategies using numerical experiments.
