# Student Research

In addition to teaching, one of my favorite things to do is to work with students on research projects. My goal is to help a student see that there are questions in mathematics that don't always have nice pretty textbook answers, and that the process of discovering a perhaps-less-than-perfect answer is a solution in itself. My focus tends to be on biomathematics problems, but I am willing to work with students with whatever projects they bring to me, be in fractals or educational spending. I enjoy being "the student" and letting my students teach me what they have learned.

## Independent Student Research (Academic Year)

**Jessica McElwain** (Mathematics with a Concentration in Mathematical Modeling), 2019-2020.

**Jessica McElwain**(Mathematics with a Concentration in Mathematical Modeling), 2019-2020.

*Exploring Global Sensitivity Analysis with a Physiologically-based Pharmacokinetic (PBPK) Model of Bromochloromethane (BCM)**Abstract:*Bromochloromethane (BCM) is an unregulated chemical used in water chlorination that can harm the liver and kidneys when inhaled or ingested. This project takes information given by a previous paper, Cuello et al., (2012), and performs global sensitivity analysis on the model. There are certain parameters, V_{max}and K_{gsh}, where metabolites may cause cancer. Performing global sensitivity analysis can point to the most sensitive parameters. It is important to study these sensitive parameters to understand the behavior of the chemical. The next steps are to explore different distributions to see if that affects the order of sensitivity with the parameters in the model.

**Contributed to work published in: **

Sawyer, M.E., **McElwain, J,. **and **Kenney, J.W**. (2021). Applications of global sensitivity analysis to the optimization of a dermal PBPK model of bromochloromethane.* **Missouri Journal of Mathematical Sciences.* 33 (2), 137-150. DOI: 10.35834/2021/3302137

More details on this work can be found in the SNHU Publications subpage.

**-- -- -- -- -- **

**Cuello, W.S., Janes, T.A.T, Jessee, J.M., Venecek, M.A., **Sawyer, M.E., Eklund, C.R., Evans, M.V., (2012). Physiologically based pharmacokinetic (PBPK) modeling of metabolic pathways of bromochloromethane in rats. * Journal of Toxicology,* vol. 2012, Article ID 629781.

**John Kenney** (Mathematical Modeling with Physics), 2019-2020.

**John Kenney**(Mathematical Modeling with Physics), 2019-2020.

*Optimization of a Physiologically-based Pharmacokinetic (PBPK) Model of Bromochloromethane (BCM)**Abstract:*Building on a model developed in Cuello et al., (2012), this project explores the optimization of a physiologically-based pharmacokinetic model of bromochloromethane with a novel adaption of a dermal exposure compartment; this will allow comparison of fits of metabolic parameters to dermal and inhalation exposure routes.

**Contributed to work published in: **

Sawyer, M.E., **McElwain, J,.** and** Kenney, J.W**. (2021). Applications of global sensitivity analysis to the optimization of a dermal PBPK model of bromochloromethane.* Missouri Journal of Mathematical Sciences.* 33 (2), 137-150. DOI: 10.35834/2021/3302137

More details on this work can be found in the SNHU Publications subpage.

**-- -- -- -- -- **

**Cuello, W.S., Janes, T.A.T, Jessee, J.M., Venecek, M.A., **Sawyer, M.E., Eklund, C.R., Evans, M.V., (2012) Physiologically based pharmacokinetic (PBPK) modeling of metabolic pathways of bromochloromethane in rats. J*ournal of Toxicology*, vol. 2012, Article ID 629781.

**Marisa Jellison **(Mathematics and English Language & Literature), 2018–2019.

**Marisa Jellison**(Mathematics and English Language & Literature), 2018–2019.

*Validating a Physiologically-based Pharmacokinetic (PBPK) Model of Bromochloromethane (BCM)**Abstract:*Bromochloromethane (BCM) is a byproduct of water disinfection and is potentially toxic to humans. Several studies have been done to determine metabolic parameters. This project is centered around validating a physiologically-based pharmacokinetic (PBPK) BCM model in rats, presented in Cuello et al., (2012). Confidence intervals around parameters will be discussed as well as a general assessment of the model.

**-- -- -- -- -- **

**Cuello, W.S., Janes, T.A.T, Jessee, J.M., Venecek, M.A., **Sawyer, M.E., Eklund, C.R., Evans, M.V., (2012). Physiologically based pharmacokinetic (PBPK) modeling of metabolic pathways of bromochloromethane in rats. *Journal of Toxicology*, vol. 2012, Article ID 629781.

**Mary Shakshober** (Mathematics and Graphic Design). 2017–2018.

**Mary Shakshober**(Mathematics and Graphic Design). 2017–2018.

*Fractals as a Mathematically Aided Art Form*. https://youtu.be/zVCY8-PpedQ**Co-Advisor:**Tracy Dow

**Kevin Tasley **(Economics and Finance). 2016–2017.

**Kevin Tasley**(Economics and Finance). 2016–2017.

Tasley, K. (2016). Modeling Public-Education Spending vs. Allocation as Independent Factors of Educational Outcomes.

*Undergraduate Economic Review*. 13, 1. http://digitalcommons.iwu.edu/uer/vol13/iss1/10**Co-Advisor**: Adam Gilbert*Abstract:*This paper explores and expands upon the work of Hanushek and Wößmann (2007) whose accumulated findings propose increased educational spending provides only marginal returns in terms of student’s cognitive outcomes. This study constructs an OLS regression model to explore the significance of U.S. state education spending and financial allocations as independent factors of state-level average ACT scores over a 10-year time series. The model additionally accounts for self-selection and socio-economic status. The results of this study support Hanushek and Wößmann’s conclusions while also demonstrating evidence that shifts in allocations towards instructional spending, as opposed to increasing total expenditures, could have a more substantial impact on returns to educational quality.

## Course-Based Research (Semester-Long)

## BIG Problems in Mathematics

*BIG Problems in Mathematics *was developed to help students explore Business, Industry, and Government problems using mathematical techniques. Depending on the semester, students are offered the opportunity to take their work to the MAA Mathfest, a national conference held every summer. This course is now a staple for our Concentration in Mathematical Modeling and runs every Spring in conjunction with one of our more mathematical modeling courses to provide the "reality" to the "mathematics".

This course is sponsored in part by the Mathematical Association of America (MAA) and Society of Industrial and Applied Mathematics (SIAM) program (PIC-Math) is provided by the National Science Foundation (NSF grant DMS-1722275) and the National Security Agency (NSA).

### Spring 2019 *BIG Problems in Mathematics *

*BIG Problems in Mathematics*

In the Spring 2019 *BIG Problems in Mathematics *course, students worked with the US Environmental Protection Agency (US EPA) to investigate a "real-world" relevant problem on water disinfection byproducts (specifically bromochloromethane) using physiologically-based pharmacokinetic modeling and conducted sensitivity analysis (Morris' Method) to validate the results. Two students from the course further explored their results, and successfully contributed to a **peer-review publication** in the Missouri Journal of Mathematical Sciences (see above for Jessica McElwain and John Kenney).

### Spring 2022 *BIG Problems in Mathematics*

*BIG Problems in Mathematics*

The Spring 2022 *BIG Problems in Mathematics* had students exploring the effects of disruption of natural gas supply on European nations. Using linear programming and linear combinations of weighted cost constraints, groups were able to develop two distinct models that agreed on the ratio of the risk of several countries in Europe.

Photo credit: Zachary Davidson, 2022

*BIG Problems in Mathematics* student quotes:

*BIG Problems in Mathematics*student quotes:

"I think this class has really tested everything that I have learned so far. It puts your skills to the test and helps you learn how to do a big project in a group without a professor holding your hand. "

"Before this class, I didn't do undergraduate research ever. This class gives me a brief perception of how to work on research by yourself without a supervisor directly giving you instruction on how to do it step by step. This helps me understand how to work on it by myself and become more self-motivated. As I am 100% [going to] do my masters in the future and I still hesitate on doing a doctorate in the future, the project helped me realize how to research and find some resources in a clear way."

"I think this class gave me a great experience that I will bring to my career. I learned how to communicate with a group, to work on a long-term project and stay on task, to try and figure things out on my own, and I learned new skills that I can add to my resume."

Students were not the only individuals excited about the *BIG Problems in Mathematics* course. A member of the SNHU faculty, Dave Humphreys, filmed a brief documentary of the course experience as a way of showcasing the different ways that the School of Arts and Sciences at SNHU is engaging students outside of the lecture experience.

Video Credit: David Humphreys

## Research Experience for Undergraduates (Summer)

I worked with several REU (Research Experience for Undergraduates) groups in 2011-2013 as a graduate assistant at North Carolina State University (NCSU). A brief description of each project can be found in the subpage for the corresponding year.

In addition, I returned to the NCSU REU in 2014 and 2015 to introduce students to the typesetting language LaTeX and the mathematical computing language MATLAB (specifically in the context of solving differential equations and optimization with basic tools). Resources for these introduction courses can be found here.