The current coronavirus disease 2019 (COVID-19) pandemic has highlighted the crucial role of mathematical models in predicting, assessing, and controlling potential outbreaks. Numerous modeling studies using statistics or differential equations have been proposed to analyze the COVID-19 dynamics, with network analysis and cluster analysis also being adapted to understand disease transmission from multiple perspectives. This dissertation explores the use of network science and mathematical models to improve the understanding of infectious diseases. Chapter 1 provides an introduction to infectious disease modeling, its history, importance, and challenges. It also introduces network science as a powerful tool for understanding the complex interactions between individuals that can facilitate disease spread. Chapter 2 develops a statistical model that describes HIV infection and disease progression in a men who have sex with men cohort in Japan receiving a Pre-Exposure Prophylaxis (PrEP) program. The cost-effectiveness of the PrEP programwas evaluated by comparing the incremental cost-effectiveness ratio over a 30-year period against the willingness to pay threshold. Chapter 3 presents an ordinary differential equations model to describe disease transmission and the effects of vaccination and mobility restrictions. Chapter 4 extends the ODE model to include spatial heterogeneity and presents partial differential equations models. These models describe the combined effects of local transmission, transboundary transmission, and human intervention on COVID-19 dynamics. Finally, Chapter 5 concludes the dissertation by emphasizing the importance of developing relevant disease models to understand and predict the spread of infectious diseases by combining network science and mathematical tools.

Social insect groups, such as bees, termites, and ants, epitomize the emergence of group-level behaviors from the aggregated actions and interactions of individuals. Ants have the unique advantage that whole colonies can be observed in artificial, laboratory nests, and each individual's behavior can be continuously tracked using imaging software. In this dissertation, I study two group behaviors: (1) the spread of alarm signals from three agitated ants to a group of 61 quiescent nestmates, and (2) the reduction in per-capita energy use as colonies scale in size from tens of ants to thousands. For my first experiment, I track the motion of Pogonomyrmex californicus ants using an overhead camera, and I analyze how propagation of an initial alarm stimulus affects their walking speeds. I then build an agent-based model that simulates two-dimensional ant motion and the spread of the alarmed state. I find that implementing a simple set of rules for motion and alarm signal transmission reproduces the empirically observed speed dynamics. For the second experiment, I simulate social insect colony workers that collectively complete a set of tasks. By assuming that task switching is energetically costly, my model recovers a metabolic rate scaling pattern, known as hypometric metabolic scaling. This relationship, which predicts an organism's metabolic rate from its mass, is observed across a diverse set of social insect groups and animal species. The results suggest an explicit link between the degree of workers' task specialization and whole-colony energy use.

Need-based transfers (NBTs) are a form of risk-pooling in which binary welfare exchanges

occur to preserve the viable participation of individuals in an economy, e.g. reciprocal gifting

of cattle among East African herders or food sharing among vampire bats. With the

broad goal of better understanding the mathematics of such binary welfare and risk pooling,

agent-based simulations are conducted to explore socially optimal transfer policies

and sharing network structures, kinetic exchange models that utilize tools from the kinetic

theory of gas dynamics are utilized to characterize the wealth distribution of an NBT economy,

and a variant of repeated prisoner’s dilemma is analyzed to determine whether and

why individuals would participate in such a system of reciprocal altruism.

From agent-based simulation and kinetic exchange models, it is found that regressive

NBT wealth redistribution acts as a cutting stock optimization heuristic that most efficiently

matches deficits to surpluses to improve short-term survival; however, progressive

redistribution leads to a wealth distribution that is more stable in volatile environments and

therefore is optimal for long-term survival. Homogeneous sharing networks with low variance

in degree are found to be ideal for maintaining community viability as the burden and

benefit of NBTs is equally shared. Also, phrasing NBTs as a survivor’s dilemma reveals

parameter regions where the repeated game becomes equivalent to a stag hunt or harmony

game, and thus where cooperation is evolutionarily stable.

This dissertation discusses the Cournot competition and competitions in the exploitation of common pool resources and its extension to the tragedy of the commons. I address these models by using potential games and inquire how these models reflect the real competitions for provisions of environmental resources. The Cournot models are dependent upon how many firms there are so that the resultant Cournot-Nash equilibrium is dependent upon the number of firms in oligopoly. But many studies do not take into account how the resultant Cournot-Nash equilibrium is sensitive to the change of the number of firms. Potential games can find out the outcome when the number of firms changes in addition to providing the "traditional" Cournot-Nash equilibrium when the number of firms is fixed. Hence, I use potential games to fill the gaps that exist in the studies of competitions in oligopoly and common pool resources and extend our knowledge in these topics. In specific, one of the rational conclusions from the Cournot model is that a firm's best policy is to split into separate firms. In real life, we usually witness the other way around; i.e., several firms attempt to merge and enjoy the monopoly profit by restricting the amount of output and raising the price. I aim to solve this conundrum by using potential games. I also clarify, within the Cournot competition model, how regulatory intervention in the management of environmental pollution externalities affects the equilibrium number of polluters. In addition, the tragedy of the commons is the term widely used to describe the overexploitation of open-access common-pool resources. Open-access encourages potential resource users to continue to enter the resource up to the point where rents are exhausted. The resulting level of resource use is higher than is socially optimal, and in extreme cases can lead to the collapse of the resource and the communities that may depend on it. In this paper I use the concept of potential games to evaluate the relation between the cost of resource use and the equilibrium number of resource users in open access regimes. I find that costs of access and costs of production are sufficient to determine the equilibrium number of resource users, and that there is in fact a continuum between Cournot competition and the tragedy of the commons. I note that the various common pool resource management regimes identified in the empirical literature are associated with particular cost structures, and hence that this may be the mechanism that determines the number of resource users accessing the resource.