Doctoral Dissertations

Date of Award

5-2021

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mathematics

Major Professor

Suzanne Lenhart

Committee Members

Judy Day, Steven Wise, Louis Gross

Abstract

Mathematical modeling is a useful technique to describe dynamics happening within events and allows one to address questions and test hypotheses that may be not be feasible to study in reality. This work uses mathematical models to describe two such phenomena, one relating to immunology and the other to the spread of infectious diseases.

Celiac disease is a hereditary autoimmune disease that affects approximately 1 in 133 Americans. It is caused by a reaction to the protein gluten found in wheat, rye, and barley. After ingesting gluten, a patient with celiac disease may experience a range of unpleasant symptoms while small intestinal villi, essential to nutrient absorption, are destroyed in an immune-mediated process. The only known treatment for this disease is a lifelong gluten-free diet and there is currently no drug treatment. The first study provides a mathematical framework to better understand the effects of immune activation on gut health. This mathematical model uses a system of ordinary differential equations to track changes in small intestinal cell densities and relates them to the Q-MARSH score, a criterion used in the diagnosis of celiac disease. The model can be used to investigate and analyze the immune response and various theories behind the progression of this disease by focusing on understanding the dynamics of the small intestine in situations mirroring healthy function and celiac disease. By doing so, we can assist in further quantifying and augmenting diagnostic measures and investigate potential therapies to mitigate the negative effects of celiac disease.

Clostridioides difficile (C. difficile) is the leading cause of infectious diarrhea and one of the most frequently identified healthcare-acquired infections in United States hospitals. C. difficile is typically contracted after antibiotic use, when healthy gut microbiota that prevent colonization is compromised. Colonized patients, both symptomatic and asymptomatic, shed C. difficile endospores that can survive for long periods on surfaces outside the host and are resistant to many commonly-used disinfectants. Transmission pathways can include contact with endospores on fomites, objects likely to carry infection. The second modeling study investigates the relative contribution of two environmental pathways to C. difficile transmission in healthcare settings. Due to the small hospital ward size, patient and pathogen populations are simulated stochastically and compared with the average behavior described by a system of ordinary differential equations. The results can be utilized to examine the role surfaces with varying touch frequencies contribute to patient colonization with C. difficile in healthcare settings.

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