Studying the effects of viruses and toxins on honey bees is important in order to understand the danger these important pollinators are exposed to. Hives exist in various environments, and different colonies are exposed to varying environmental conditions and dangers. To properly study the changes and effects of seasonality and pesticides on the population dynamics of honey bees, the presence of each of these threats must be considered. This study aims to analyze how infected colonies grapple more deeply with changing, seasonal environments, and how toxins in pesticides affect population dynamics. Thus, it addresses the following questions: How do viruses within a colony affect honey bee population dynamics when the environment is seasonal? How can the effects of pesticides be modeled to better understand the spread of toxins? This project is a continuation of my own undergraduate work in a previous class, MAT 350: Techniques and Applications of Applied Mathematics, with Dr. Yun Kang, and also utilizes previous research conducted by graduate students. Original research focused on the population dynamics of honey bee disease interactions (without considering seasonality), and a mathematical modeling approach to analyze the effects of pesticides on honey bees. In order to pursue answers to the main research questions, the model for honey bee virus interaction was adapted to account for seasonality. The adaptation of this model allowed the new model to account for the effects of seasonality on infected colony population dynamics. After adapting the model, simulations with arbitrary data were run using RStudio in order to gain insight into the specific ways in which seasonality affected the interaction between a honey bee colony and viruses. The second portion of this project examines a system of ordinary differential equations that represent the effect of pesticides on honey bee population dynamics, and explores the process of this model’s formulation. Both systems of equations used as the basis for each model’s research question are from previous research reports. This project aims to further that research, and explore the applications of applied mathematics to biological issues.

In this project we focus on COVID-19 in a university setting. Arizona State University has a very large population on the Tempe Campus. With the emergence of diseases such as COVID-19, it is very important to track how such a disease spreads within that type of community. This is vital for containment measures and the safety of everyone involved. We found in the literature several epidemiology models that utilize differential equations for tracking a spread of a disease. However, our goal is to provide a granular look at how disease may spread through contact in a classroom. This thesis models a single ASU classroom and tracks the spread of a disease. It is important to note that our variables and declarations are not aligned with COVID-19 or any other specific disease but are chosen to exemplify the impact of some key parameters on the epidemic size. We found that a smaller transmissibility alongside a more spread-out classroom of agents resulted in fewer infections overall. There are many extensions to this model that are needed in order to take what we have demonstrated and align those ideas with COVID-19 and it’s spread at ASU. However, this model successfully demonstrates a spread of disease through single-classroom interaction, which is the key component for any university campus disease transmission model.