The Cheeger’s constant, also known as the isoperimetric number, is a constant that helps describe the bottleneck present in a graph, if any. Some fields, such as computer networks, have an interest in this constant due to the application of the constant in their field. We examined randomly generated connected graphs and their isoperimetric numbers by developing algorithms to calculate it.

One of the major challenges in education is accurately quantifying a student’s knowledge and skills. Since we cannot directly measure a student’s true intelligence, we rely on test performance, which serves as an imperfect representation of their abilities. This issue arises in many statistical applications where the key problem involves a population in which each individual possesses an underlying ability or trait that cannot be directly observed but can only be inferred through proxy variables. However, these proxies are often contaminated, providing only a noisy or imperfect approximation of the true latent variable.
This project focuses on techniques for recovering latent variables from noisy data. In this context, "recovery" refers to estimating the latent variable using indirect observations. Assuming a linear relationship between the latent trait and the observed proxy variables, we can estimate model parameters and subsequently recover the values of the latent variables.
Specifically, we will examine statistical approaches to latent variable recovery when the test contains items that exhibit differential item functioning (DIF). This means that certain test items do not solely measure the intended knowledge or ability but are also biased toward specific groups. The objective is to develop methods that detect the presence of DIF and adjust for it, allowing for a more accurate estimation of the underlying abilities.
To illustrate these methods, we will use the Holzinger-Swineford dataset, a well-known dataset in psychometrics used to analyze cognitive abilities across multiple domains. This dataset includes 88 observations with scores in five areas: Mechanical Comprehension, Verbal or Visual Comprehension, Algebra Operations, Analytical Operations, and Statistical Reasoning. By applying a linear contamination model, we aim to recover each student's latent ability while accounting for DIF.

Oral Reading Accuracy (ORA) is an important metric for evaluating a student's reading proficiency, measuring how accurately a reader can read words aloud. Traditional ORA evaluations performed by human assessors often require significant time and labor. This study explores the potential of integrating a speech recognition system into ORA assessments to improve efficiency. We analyzed ORA data from 507 elementary school students across ten passages of different lengths and difficulties. Both human evaluators and AI systems recorded the number of words read correctly. The misclassification rates of these scores are divided into two components: True Positive (correct words are identified as correct), and True Negative (incorrect words are identified as correct). This second study expands upon Method of Moments method to estimate these misclassification rates. We apply Generalized Method of Moments which incorporates additional variance information. To compare the two approaches' accuracy, we apply the m-out-of-n Bootstrap method to estimate their standard errors and compare their reductions in estimator variance. Additionally, we introduce a Contaminated Data Solution to address real-world scenarios where true count data is unavailable and only contaminated observed data is observed.

Several viruses can cause cells to fuse into large multinucleated cells called syncytia. Syncytia formation allows the virus to spread without entering the extracellular space, where it might be exposed to immune responses. However, there is evidence that antibodies can also hinder the fusion process. This project uses mathematical analysis to find different possible infection outcomes. A stability analysis of the coinfection model is used to find the fixed points of the model and their stability. This gives us parameter space regions that lead to different possible infection outcomes. Simulations were made to verify the mathematical analysis and see how different syncytia formation properties affect the resulting dynamics. These findings could help develop strategies for controlling viral spread.

Graphene Quantum Dots (GQDs) are nanoscale carbon based graphene sheets that exhibit unique fluorescent properties throughout a wide range of wavelengths. Given their uniquely small size, low toxicity, biocompatibility, and fluorescent capabilities, GQDs have many unique and important roles. To name a few, GQDs are used in drug delivery, fluorescent imaging, and biosensing thanks to their unique ability to fluoresce under different wavelengths of light. Furthermore, there are different types of GQDs with their own unique properties. Knowing this, five amphipathic molecules, called surfactants, were added to two different types of GQDs to test if they would impact the resulting fluorescence. Furthermore, concentrations of these added surfactants were varied to test how different concentrations of a given surfactant might affect the fluorescence for a given GQD. We observed that some of these surfactants provided a beneficial boost to GQDs fluorescence, while others slightly inhibited the fluorescence. Moreover, we saw that the increase in fluorescence varied based on the concentration of surfactant added yielding lower fluorescence for extremely low and high concentrations, while increasing the fluorescence at a more moderate concentration.

With cancer rates increasing at an alarming rate, many traditional methods for cancer treatment begin to feel outdated. This is where engineering nanomaterials, such as Graphene Quantum Dots (GQDs), offer a promising approach to making chemotherapy a more targeted treatment and therefore minimizing the side effects. This study focuses on optimizing drug delivery mechanisms using GQDs, specifically Reduced Graphene Quantum Dots (RGQDs) synthesized via a top-down approach from reduced graphene oxide, and Hyaluronic Acid Graphene Quantum Dots (HAGQDs) synthesized bottom-up from hyaluronic acid. The process is done by loading chemotherapeutics Gemcitabine, Paclitaxel, and Doxorubicin (DOX) HCl onto GQDs through sonication, this is followed by a centrifugal purification which isolates properly drug-loaded GQDs. To evaluate their controlled release, photothermal properties of GQDs are utilized. Samples are excited with an 808 nm laser at 1, 5, and 10 minutes, and they are later compared to a control group. This analysis provides insights into how laser stimulation affects drug release efficiency, paving the way for advancements in GQD based cancer treatments.

Some viruses have the ability to form syncytia. Syncytia are multi-nucleated cells formed via membrane fusion. Syncytia formation allows viruses to spread infection to other cells without entering the extracellular space where it could be exposed to antiviral drugs or immune responses such as antibodies. This project explores how syncytia formation can help viruses avoid antiviral drugs. Drug efficacy parameters are applied to a mathematical model of differential equations to explore the impact of antiviral drugs on cell infection, cell fusion, and viral production to model respiratory syncytial virus. The models show that as syncytia formation increases the drugs become less effective. This information will help physicians treat patients with syncytia forming viruses.

After the COVID-19 pandemic, over 40 million children worldwide are at risk of measles due to delayed vaccination and temporary SARS-CoV-2 viral dominance. The lasting immunosuppression caused by the disease presents a major health threat, and treatment options are urgently needed, especially for low- and middle-income countries. The manuscript by Cox et al. (2024) explores features of canine distemper virus (CDV) in ferrets, using this model as a surrogate for measles to evaluate two possible antiviral treatments, ERDRP-0519 and GHP-88309. Ferrets were infected with a lethal challenge of CDV and treated with either drug or therapeutic vaccination. We aim to characterize both the infection dynamics and efficacy of the two drug treatments using the data from the PBMC (peripheral blood mononuclear cell) associated viremia titers of CDV infected ferrets and the lymphocyte counts measured during the duration of the study. A differential mathematical model was fitted to the experimental data by minimizing the sum of squared residuals (SSR), and errors in the parameter fits were estimated using Monte Carlo Markov Chain (MCMC). We visualized the key parameter distributions for each dataset using histograms, allowing us to directly compare how each treatment influences infection dynamics. The results revealed that ERDRP-0519 reduced viral entry and enhanced clearance while GHP-88309 improved target cell growth and increased the rate of infected cell death. These findings suggest that both drugs are potentially effective measles treatment options, with ERDRP-0519 having a direct antiviral effect and GHP-88309 aiding in immune recovery. Overall, these insights provide a foundation for optimizing treatment strategies and highlight the potential for both drugs to combat measles and related morbillivirus infections, especially in areas with limited resources and vaccines.

Gliomas account for approximately 27% of all primary central nervous system tumors and exhibit highly aggressive growth patterns, making conventional treatments ineffective. Previous research has demonstrated that a replication-competent Sindbis virus (SINV) combined with cytokines (IL-7, IL-12, and GM-CSF) shows promising results in slowing down glioma progression. While prior research demonstrated that SINV combined with cytokines reduces tumor growth, a quantitative understanding of its effects remains limited. This study aims to develop and fit a mathematical model of oncolytic virus infection to data from previous research to quantify key biological processes in glioma treatment. By parameterizing the Sindbis virus-glioma interaction and estimating the effects of cytokine therapy, this model aims to evaluate the efficacy of different SINV variants, with and without cytokine combinations, in controlling tumor growth. We use an Ordinary Differential Equation (ODE) model to describe tumor growth inhibition by the oncolytic SINVs. The model includes variables for uninfected and infected tumor cells, viral load, and cytokine concentration. The data extracted from published tumor growth curves will be used to estimate key parameters, including viral replication rate, tumor growth rate, and cytokine effects. Parameter fitting will be conducted by minimizing the Sum of Squared Residuals (SSR) between model predictions and experimental data. Error in the parameters will be estimated through bootstrapping to find the best fit parameters with 95% confidence intervals. Preliminary analysis suggests that the model effectively captures tumor growth rates observed in the experimental data. Parameter estimation provides insights into the viral infection rate, cytokine-induced tumor suppression, and the timing of viral injections. These findings will help refine our understanding of how the SINVs and cytokine therapy interact in glioma treatment. This study provides a quantitative framework for evaluating the therapeutic effects of an oncolytic SINV combined with cytokines in glioma treatment. By providing parameter estimates for key biological processes, our model can help optimize treatment strategies and guide future experimental research in oncolytic virotherapy.

Mathematical modeling of viral kinetics can be used to gain further insight into the viral replication cycle and virus-host interactions. However, many virus dynamics models do not incorporate the cell-to-cell heterogeneity of virus yield or the time-dependent factor of virus production. A recent study of the kinetics of the vesicular stomatitis virus (VSV) in single BHK cells determined that both the virus production rate and the yield of virus particles vary widely between individual cells of the same cell population. We used the results of this study to determine the distribution that best describes the time course of viral production within single cells. The best distribution was then used to incorporate time-varying production into a standard model of viral kinetics. The best-fit model was determined by fitting potential distributions to cumulative viral production from single cells and comparing the Akaike Information Criterion (AIC). The results show that the best fit for most cells was log-normal. Time-dependent viral production was modeled with an integro-differential equation that incorporated the log-normal probability distribution into a standard constant production model of viral kinetics. This time-dependent model was compared to one of constant production by examining the differences between the viral peak, time of the peak, upslope, downslope, and area under the curve. These findings could have further-reaching implications for helping define the time course and nature of a particular virus infection within the human body as well as improving the dose-timing and efficacy of anti-viral treatments.