Looking at Community Issues Through the Lens of Mathematical Modeling

Mathematical modeling is an important tool for analyzing and understanding the world around us. Throughout history, it has been instrumental in advancements made in the sciences. Recently, mathematical modeling has been used to gain insight in the social sciences, specifically social issues. Thus, the overall goal for this dissertation is to build up the mathematical modeling toolkit for addressing social issues, especially by applying and developing techniques in network theory. We accomplish this by looking at three social issues facing our communities.

In Chapter 2, we use flow networks to develop an alternative method for calculating and analyzing the basic reproductive number for infectious disease outbreaks, which is the expected number of secondary infections produced by a typical infectious person in a susceptible population. We developed this method to be more accessible to non-mathematical audiences, as the basic reproductive number is important for public health officials and the general population in understanding infection risk and responding to outbreaks. We show our method is equivalent to traditional methods and provide instructions on its implementation.

Chapter 3 applies immuno-epidemiological modeling techniques to study violence spread through exposure. Recently, there has been a push to understand violence as a public health issue. We expand this analogy between violence and infectious diseases to formulate a susceptible-exposed-infectious model. We then provide stability and equilibrium analysis and run example numerical simulations to show that the insights gained from a mathematical model might help identify effective interventions.

The fourth chapter uses network modeling to explore how public transportation can help connect food desert residents to grocery stores. According to the USDA, food deserts are census tracts that experience high poverty rates and limited grocery store access. We select five cities and formulate network models where the food deserts and grocery stores are the nodes and transit lines are the edges. We analyze these networks through centrality measures and provide policy suggestions to improve public transit use in increasing food access.

In total, this work presents examples of how novel mathematical models (especially using network theory) can help address societal problems and effect meaningful change.