Acetylcholinesterase (AChE) inhibition by chemical toxicants such as organophosphates, nerve agents, and carbamates can lead to a series of adverse health outcomes including seizures, coma, and death. An adverse outcome pathway (AOP) is a framework that describes a series of biologically measurable key events (KEs) leading from some molecular initiating event (MIE) to an adverse outcome (AO) of regulatory significance, all developed and hosted in the AOP Wiki. A quantitative AOP (qAOP) is a mathematical model that predicts how perturbations in the MIE affect KEs based on the key event relationships (KERs) that define the AOP. The purpose of this thesis was to expand upon the KERs that define the AOP for AChE inhibition leading to neurodegeneration in order to better understand the effects of AChE inhibitors and the risks they pose to ecosystems, wildlife, and human health. In order to reduce the resources and time spent for chemical toxicity testing, a qAOP was developed based on the available quantitative data and models that supported the AOP. A literature review for the collection of qualitative evidence and quantitative data in support of the AOP was performed resulting in further expansion of the relationships between key events (KERs) through construction of additional KER description pages. A model evaluation was performed by comparing the qAOP model predictions with experimental data, with a subsequent sensitivity analysis of unknown parameters. The qAOP model simulates the MIE through its fifth KE (KE 5) and KE 7. Model predictions compared to experimentally measured data either under- or overpredicting multiple KEs warranting additional refinement such as a formal parameter optimization. Overall, more data amenable to qAOP model development are needed. To aid qAOP model development, the presentation of data in the AOPWiki may be improved by presenting the quantitative data in the AOP Wiki in a tabular format and allowing for the hosting of mathematical models or raw data. With these recommendations in mind, and through continued AOP construction in the AOP Wiki, new qAOP models will be developed, ultimately supporting chemical risk assessment and the mitigation of effects upon exposed individuals and wildlife populations.

A leading crisis in the United States is the opioid use disorder (OUD) epidemic. Opioid overdose deaths have been increasing, with over 100,000 deaths due to overdose from April 2020 to April 2021. This dissertation presents two mathematical models to address illicit OUD (IOUD), treatment, and recovery within an epidemiological framework. In the first model, individuals remain in the recovery class unless they relapse. Due to the limited availability of specialty treatment facilities for individuals with OUD, a saturation treat- ment function was incorporated. The second model is an extension of the first, where a casual user class and its corresponding specialty treatment class were added. Using U.S. population data, the data was scaled to a population of 200,000 to find parameter estimates. While the first model used the heroin-only dataset, the second model used both the heroin and all-illicit opioids datasets. Backward bifurcation was found in the first IOUD model for realistic parameter values. Additionally, bistability was observed in the second IOUD model with the heroin-only dataset. This result implies that it would be beneficial to increase the availability of treatment. An alarming effect was discovered about the high overdose death rate: by 2038, the disease-free equilibrium would be the only stable equilibrium. This consequence is concerning because although the goal is for the epidemic to end, it would be preferable to end it through treatment rather than overdose. The IOUD model with a casual user class, its sensitivity results, and the comparison of parameters for both datasets, showed the importance of not overlooking the influence that casual users have in driving the all-illicit opioid epidemic. Casual users stay in the casual user class longer and are not going to treatment as quickly as the users of the heroin epidemic. Another result was that the users of the all-illicit opioids were going to the recovered class by means other than specialty treatment. However, the relapse rates for those individuals were much more significant than in the heroin-only epidemic. The results above from analyzing these models may inform health and policy officials, leading to more effective treatment options and prevention efforts.

Recent experimental and mathematical work has shown the interdependence of the rod and cone photoreceptors with the retinal pigment epithelium in maintaining sight. Accelerated intake of glucose into the cones via the theoredoxin-like rod-derived cone viability factor (RdCVF) is needed as aerobic glycolysis is the primary source of energy production. Reactive oxidative species (ROS) result from the rod and cone metabolism and recent experimental work has shown that the long form of RdCVF (RdCVFL) helps mitigate the negative effects of ROS. In this work I investigate the role of RdCVFL in maintaining the health of the photoreceptors. The results of this mathematical model show the necessity of RdCVFL and also demonstrate additional stable modes that are present in this system. The sensitivity analysis shows the importance of glucose uptake, nutrient levels, and ROS mitigation in maintaining rod and cone health in light-damaged mouse models. Together, these suggest areas on which to focus treatment in order to prolong the photoreceptors, especially in situations where ROS is a contributing factor to their death such as retinitis pigmentosa (RP). A potential treatment with RdCVFL and its effects has never been studied in mathematical models. In this work, I examine an optimal control with the treatment of RdCVFL and mathematically illustrate the potential that this treatment might have for treating degenerative retinal diseases such as RP. Further, I examine optimal controls with the treatment of both RdCVF and RdCVFL in order to mathematically understand the potential that a dual treatment might have for treating degenerative retinal diseases such as RP. The RdCVFL control terms are nonlinear for biological accuracy but this results in the standard general theorems for existence of optimal controls failing to apply. I then linearize these models to have proof of existence of an optimal control. Both nonlinear and linearized control results are compared and reveal similarly substantial savings rates for rods and cones.

Adderall remains to be one of the most commonly abused drug among college campuses. Although it is a prescription drug that is primarily used to treat attention deficit hyperactivity disorder (ADHD), it has become one of the many "study drugs" due to its usage among college students during stressful school times, such as exams, where increased concentration and energy levels are thought to improve work efficiency. However, Adderall is notable for having a high potential for abuse and a risk of psychological and physical side effects. In this paper, we conducted a mathematical analysis on an existing epidemiological model of Adderall abuse. We started by verifying the positivity of solutions using techniques from dynamical systems because this is a population model dealing with people. Then, we found and investigated different equilibrium solutions to analyze their stability using both analytical and graphical approaches. Finally, the results were tied back into the Adderall model and conclusions were drawn.

Patients suffering from Retinitis Pigmentosa (RP), the most common type of inherited retinal degeneration, experience irreversible vision loss due to photoreceptor degeneration. The preservation of cone photoreceptors has been deemed medically relevant as a therapy aimed at preventing blindness in patients with RP. Cones rely on aerobic glycolysis to supply the metabolites necessary for outer segment (OS) renewal and maintenance. The rod-derived cone viability factor (RdCVF), a protein secreted by the rod photoreceptors that preserves the cones, accelerates the flow of glucose into the cone cell stimulating aerobic glycolysis. This dissertation presents and analyzes ordinary differential equation (ODE) models of cellular and molecular level photoreceptor interactions in health and disease to examine mechanisms leading to blindness in patients with RP.

First, a mathematical model composed of four ODEs is formulated to investigate the progression of RP, accounting for the new understanding of RdCVF’s role in enhancing cone survival. A mathematical analysis is performed, and stability and bifurcation analyses are used to explore various pathways to blindness. Experimental data are used for parameter estimation and model validation. The numerical results are framed in terms of four stages in the progression of RP. Sensitivity analysis is used to determine mechanisms that have a significant affect on the cones at each stage of RP. Utilizing a non-dimensional form of the RP model, a numerical bifurcation analysis via MATCONT revealed the existence of stable limit cycles at two stages of RP.

Next, a novel eleven dimensional ODE model of molecular and cellular level interactions is described. The subsequent analysis is used to uncover mechanisms that affect cone photoreceptor functionality and vitality. Preliminary simulations show the existence of oscillatory behavior which is anticipated when all processes are functioning properly. Additional simulations are carried out to explore the impact of a reduction in the concentration of RdCVF coupled with disruption in the metabolism associated with cone OS shedding, and confirms cone-on-rod reliance. The simulation results are compared with experimental data. Finally, four cases are considered, and a sensitivity analysis is performed to reveal mechanisms that significantly impact the cones in each case.

College campuses are one of the most common places for substance abuse. Typically, these substances are thought of to be alcohol, marijuana, and cocaine. However, Adderall is the second most commonly abused drug on college campuses. It is used to treat attention deficit hyperactivity disorder (ADHD). Adderall increases attention span and focus, so it is also commonly used as a study drug. Students frequently buy Adderall from a friend with a prescription, and use it to stay up all night cramming for an exam or finishing a project. This is a topic that not much research has been done on since Adderall only became widely used starting in the mid 2000’s. Since it is unethical to run experiments to learn more about Adderall use, and there is a limited amount of data online, a different approach had to be taken to explore this issue further. As a mathematics major, I determined that the best way to do so was to create an SIR mathematical model. In this model we have five different populations, or compartments: the population susceptible to Adderall use, people who use Adderall with an Adderall prescription, people who use Adderall without an Adderall prescription, people with an Adderall prescription stop using Adderall, and people without an Adderall prescription stop using Adderall. We also observed the rates at which people move between each population. Using this model, we created a set of differential equations to analyze and run simulations with. Looking at steady state, equilibrium points, stability, best and worst-case scenarios, and parameter impact, we drew conclusions and came up with possible courses of action. Overall, creating this model taught me not only about drug abuse, but about how useful mathematical modeling can be, especially concerning substance abuse.

Mathematical models are important tools for addressing problems that exceed experimental capabilities. In this work, I present ordinary and partial differential equation (ODE, PDE) models for two problems: Vicodin abuse and impact cratering.

The prescription opioid Vicodin is the nation's most widely prescribed pain reliever. The majority of Vicodin abusers are first introduced via prescription, distinguishing it from other drugs in which the most common path to abuse begins with experimentation. I develop and analyze two mathematical models of Vicodin use and abuse, considering only those patients with an initial Vicodin prescription. Through adjoint sensitivity analysis, I show that focusing efforts on prevention rather than treatment has greater success at reducing the total population of abusers. I prove that solutions to each model exist, are unique, and are non-negative. I also derive conditions for which these solutions are asymptotically stable.

Verification and Validation (V&V) are necessary processes to ensure accuracy of computational methods. Simulations are essential for addressing impact cratering problems, because these problems often exceed experimental capabilities. I show that the Free Lagrange (FLAG) hydrocode, developed and maintained by Los Alamos National Laboratory, can be used for impact cratering simulations by verifying FLAG against two analytical models of aluminum-on-aluminum impacts at different impact velocities and validating FLAG against a glass-into-water laboratory impact experiment. My verification results show good agreement with the theoretical maximum pressures, and my mesh resolution study shows that FLAG converges at resolutions low enough to reduce the required computation time from about 28 hours to about 25 minutes.

Asteroid 16 Psyche is the largest M-type (metallic) asteroid in the Main Asteroid Belt. Radar albedo data indicate Psyche's surface is rich in metallic content, but estimates for Psyche's composition vary widely. Psyche has two large impact structures in its Southern hemisphere, with estimated diameters from 50 km to 70 km and estimated depths up to 6.4 km. I use the FLAG hydrocode to model the formation of the largest of these impact structures. My results indicate an oblique angle of impact rather than a vertical impact. These results also support previous claims that Psyche is metallic and porous.

Pharmacokinetics describes the movement and processing of a drug in a body, while Pharmacodynamics describes the drug's effect on a given subject. Pharmacokinetic/Pharmacodynamic(Pk/Pd) models have become a fundamental tool when predicting bacterial behavior and drug development. In November of 2009, Katsube et al. published their paper detailing their Pk/Pd model for the drug Doripenem and the bacteria P. aeruginosa. In their paper, they determined that there is a dependent relationship between the drug's effectiveness and the dosing strategy of the drug. Therefore, this thesis has applied optimal control in order to optimize the drug's effectiveness, while not burdening the subject with the side effects of the drug. Optimal Control is a mathematical tool used to balance two competing factors. As a result, it has become a useful tool used to make decisions involving complex behavior. By using Optimal Control, the model will maximize the drug's effect on the bacterial population of P. aeruginosa, while minimizing the drug concentration of Doripenem. In doing so, our research will enable doctors and clinicians to maximize a drug's effectiveness on the body, while minimizing side effects.

The retina is the lining in the back of the eye responsible for vision. When light photons hits the retina, the photoreceptors within the retina respond by sending impulses to the optic nerve, which connects to the brain. If there is injury to the eye or heredity retinal problems, this part can become detached. Detachment leads to loss of nutrients, such as oxygen and glucose, to the cells in the eye and causes cell death. Sometimes the retina is able to be surgically reattached. If the photoreceptor cells have not died and the reattachment is successful, then these cells are able to regenerate their outer segments (OS) which are essential for their functionality and vitality. In this work we will explore how the regrowth of the photoreceptor cells in a healthy eye after retinal detachment can lead to a deeper understanding of how eye cells take up nutrients and regenerate. This work uses a mathematical model for a healthy eye in conjunction with data for photoreceptors' regrowth and decay. The parameters for the healthy eye model are estimated from the data and the ranges of these parameter values are centered +/- 10\% away from these values are used for sensitivity analysis. Using parameter estimation and sensitivity analysis we can better understand how certain processes represented by these parameters change within the model as a result of retinal detachment. Having a deeper understanding for any sort of photoreceptor death and growth can be used by the greater scientific community to help with these currently irreversible conditions that lead to blindness, such as retinal detachment. The analysis in this work shows that maximizing the carrying capacity of the trophic pool and the rate of RDCVF, as well as minimizing nutrient withdrawal of the rods and the cones from the trophic pool results in both the most regrowth and least cell death in retinal detachment.