# Richmond Area Mathematical Sciences Conference at VCU

## The 2020 conference has been canceled in response to COVID-19 virus concerns and will be rescheduled in spring 2021.

# Talks accepted for RAMS 2020

## Had the conference not been canceled, the following talks would have been given.

**Faten Alamri**^{(PhD)}: Nonparametric monotonic spline**Tmader Alballa**^{(PhD)}: Bayesian technique for relating genetic polymorphisms to diffusion tensor images of cocaine users brains**Mohammed Alshehri**^{(PhD)}: A dynamical sampling scheme for solving a generalized heat equation**George Andrews, Savannah Crawford, Mathew Hasty, Matthew Kearney**^{(UG)}: Statistics in deformations of large knots**Yemeen Ayub**^{(PhD)}: Applications of diffusion maps to dimensionality reduction**Sri Vibhaav Bankuru, William Hou, Samuel Kossol, Parsa Mahmoudi**^{(UG)}: A game-theoretic model of Monkeypox to assess vaccination strategies**Jay Bisen, Faheem Farooq, Manaeil Hasan, Akhil Patel**^{(UG)}: Owner-Intruder contests with information asymmetry**Emily Cheng, Neeha Gambirrhao, Rohani Patel, Aufia Zhowandai**^{(UG)}: A game-theoretical analysis of Poliomyelitis vaccination**Matthew Elpers**^{(UG)}: Stability of delayed differential equations**Jeff Evans**^{(MA)}: Modeling breathing mechanism in a computer model of respiratory mechanics in preterm infants**Amelia Guthrie, Jessica Nguyen, Monie Thach, Antenyse Wade**^{(UG)}: Consumer-resource behavior in plant-pollinator interactions**Benjamin Ingram**^{(UG)}: Low sensitivity to group members inhibits cooperation on evolving multiplayer networks**Lubna Kadhim**^{(MA)}: Analysis of a couple of dynamical systems associated with cancer treatment**Aidan Kierans**^{(UG)}: Bootstrap percolation via automated conjecturing**Jackman Liu, Calvin Reedy, Kartikey Sharma, Scarlett Sun**^{(UG)}: Codes from difference sets**Kevin McCall**^{(PhD)}: 2-bootstrap percolation in Kneser graphs**Fatemeh Norouzi**^{(PhD)}: A new study of fractional-order financial system via homotopy analysis**Pravalika Putalapattu**^{(HS)}: Region specific modeling: A novel approach to modeling heterogeneity in biological models**Vooha Putalapattu**^{(UG)}: Game-theoretical model of retroactive hepatitis B vaccination in China**Kelly Reagan**^{(PhD)}: Saving lives, limbs and healthcare costs: A mathematical modeling approach towards reducing hospital-acquired infections**Hui Sui**^{(UG)}: K-Means clustering for longitudinal chemical mixtures analysis in LIFECODES dataset**Grant Wagner**^{(UG)}: A modified Lotka-Volterra model with nonlinear terms**Steve Wheatley**^{(PhD)}: Characterizations of 2 - homeomorphic spaces**Carter Williams**^{(UG)}: Special self-dual codes**Emily Wu**^{(UG)}: Modeling a genetic switch with different phases

### Nonparametric monotonic spline

**Faten Alamri**^{(PhD)}, Virginia Commonwealth University

Coauthor: David Edwards

Faculty mentor: Dr. Edward Boone

**Abstract:** Toxicologists are concerned about the chemicals we are exposed to in our daily life. One big concern is determining the largest dosage of a chemical that we could be exposed to without having an adverse side effect? These studies typically are conducted using rodents in a laboratory setting where the experiment involves varying the dosage of the chemical and recording measurements associated with adverse side effects. This research develops a new monotonic nonparametric model using a Bayesian approach that allows tolerable dosages to be found without the need to specify a "correct" model. Two simulated and two real world datasets are used to illustrate the method and gain insights.

### Bayesian technique for relating genetic polymorphisms to diffusion tensor images of cocaine users brains

**Tmader Alballa**^{(PhD)}, Virginia Commonwealth University

Coauthors: Dr. Liangsuo Ma, Dr. Gerard Moeller

Faculty mentor: Dr. Edward Boone

**Abstract:** Past investigations utilizing Diffusion Tensor Imaging (DTI) have demonstrated that cocaine use disorder is related with corrupted white matter. By applying Bayesian models averaging using multiple linear regression models in DTI, we have demonstrated that there is a relationship between the impaired white matter in cocaine use disorder subjects and genetic variables. This work explored the two-way and three-way interactions between GAD1a (SNP: rs1978340) and GAD1b (SNP: rs769390) polymorphisms and Years of cocaine use. The study suggests that the two-way interaction of GAD1a and Years of cocaine use has more negative impact on diffusion in the white matter of the brain comparison to the other interactions in the study.

### A dynamical sampling scheme for solving a generalized heat equation

**Mohammed Alshehri**^{(PhD)}, George Mason University

Faculty mentor: ?????

**Abstract:** We give a general formulation of the initial value problem for a generalized version of the heat equation, and aim at producing a sequence of approximate solutions via a time-space trade off, which converges to the exact solution under appropriate assumptions. We study the correlation between the number of measurements that are needed to recover the initial problem to a prescribed accuracy, give precise estimates for the time instances when these measurements need to occur, and provide an optimal reconstruction algorithm under the assumption that the initial problem is in a Sobolev class.

### Statistics in deformations of large knots

**George Andrews, Savannah Crawford, Mathew Hasty, Matthew Kearney**^{(UG)}, George Mason University

Faculty mentor: Dr. Sean Lawton

**Abstract:** The goal of this project is to discover trends in knots based on a measure of "complexity." Here complexity is measured by a specific knot invariant: the Krull dimension of the character variety of the knot group. For our project, we need only look at 'large" knots as all "small" knots have dimension 1. Given a knot K, we look at its complement in 3-space, $K^C$. Then, we take the fundamental group of this space $\pi_1(K^C)$ and calculate the character variety, $\chi(\pi_1(K^C))$. Finally, we calculate the Krull dimension of the character variety. This dimension is a knot invariant; that is, if two knots yield different dimensions, they are fundamentally different.

### Applications of diffusion maps to dimensionality reduction

**Yemeen Ayub**^{(PhD)}, George Mason University

Coauthors: Orton Babb, Aneesh Malhotra, Ryan Vaughn

Faculty mentor: Dr. Tyrus Berry

**Abstract:** The geometric prior assumes that data lies on or near a Riemannian manifold and has lead to the active field of manifold learning. Using diffusion maps, a result from manifold learning, we approximate the Laplace-Beltrami operator of the underlying manifold and attempt to reduce the dimensionality of the data while preserving certain properties of the original manifold such as the intrinsic curvature and volume. As there are many ways to do this, we look for the minimal embedding that reduces dimension while minimizing the extrinsic curvature. Doing so gives an effective and efficient way to interpret high dimensional data sets with minimal loss in information.

### A game-theoretic model of Monkeypox to assess vaccination strategies

**Sri Vibhaav Bankuru, William Hou, Samuel Kossol, Parsa Mahmoudi**^{(UG)}, Virginia Commonwealth University

Faculty mentors: Dr. Jan Rychtar, Dr. Dewey Taylor

**Abstract:** Monkeypox (MPX) is a zoonotic disease similar to smallpox. Its fatality rate is about 11% and it is endemic to the Democratic Republic of the Congo. In this paper, we analyze a compartmental model of MPX dynamics. We show that there are three equilibria - disease free, fully endemic and previously neglected semi-endemic (with disease existing only among humans). The existence of semi-endemic equilibrium has severe implications should the MPX virus mutate to increased viral fitness in humans. We find that MPX is controllable and can be eradicated in a semi-endemic equilibrium by vaccination. However, in a fully endemic equilibrium, MPX cannot be eradicated by vaccination alone.

### Owner-Intruder contests with information asymmetry

**Jay Bisen, Faheem Farooq, Manaeil Hasan, Akhil Patel**^{(UG)}, Virginia Commonwealth University

Faculty mentors: Dr. Jan Rychtar, Dr. Dewey Taylor

**Abstract:** We consider kleptoparasitic interactions between two individuals - Owner and Intruder - and model the situation as a sequential game in an extensive form. Owner is in a possession of a valuable resource when it spots Intruder. Owner has to decide whether to defend the resource; if the Owner defends, the Intruder has to decide whether to fight with the Owner. The individuals may value the resource differently and we distinguish three information cases: (a) both individuals know resource values to both of them, (b) individuals know only their own valuation, (c) individuals do not know the value at all. We solve the game in all three cases. We find that it is typically beneficial for the individuals to know as much information as possible. However, we identify several scenarios where knowing less seems better. We also show that an individual may or may not benefit from their opponent knowing less. Finally, we consider the same kind of interactions but with the reversed order of decisions. We find that typically the individual initiating the interaction has an advantage. However, when individuals know only their own valuation and not the valuations to their opponents, it is sometimes better when the opponent initiates.

### A game-theoretical analysis of Poliomyelitis vaccination

**Emily Cheng, Neeha Gambirrhao, Rohani Patel, Aufia Zhowandai**^{(UG)}, Virginia Commonwealth University

Faculty mentors: Dr. Jan Rychtar, Dr. Dewey Taylor

**Abstract:** Poliomyelitis is a worldwide disease that has nearly been eradicated thanks to the Global Polio Eradication Initiative. Nevertheless, the disease is currently still endemic in three countries. In this paper, we incorporate the vaccination in a two age-class model of polio dynamics. We perform game theoretical analysis and compare the herd immunity vaccination levels with the Nash equilibrium vaccination levels. We show that the gap between two vaccination levels is too large and the mandatory vaccination policy is therefore needed to achieve a complete eradication.

### Stability of delayed differential equations

**Matthew Elpers**^{(UG)}, Virginia Commonwealth University

Faculty mentor: Dr. Oleksandr Misiats

**Abstract:** Second order linear differential equations model many physical systems in science and engineering. Some physical systems have have components which add a delay into the system like micro electronic mechanical systems (MEMS). By finding the characteristic equation of the delay equation we can find stable solutions of a second order delay equation using Eulers method.

### Modeling breathing mechanism in a computer model of respiratory mechanics in preterm infants

**Jeff Evans**^{(MA)}, Virginia Commonwealth University

Faculty mentor: Dr. Laura Ellwein Fix

**Abstract:** A circuit-based compartmental model of differential equations was developed by Dr. Ellwein Fix to model respiratory mechanics in preterm infants. The model represents the dynamics behind atelectasis as a result of chest-wall compliance due to under mineralization. One feature not represented by the model is occasional asynchronous breathing patterns that are more prominent in the preterm infant population. We explored two model modifications to address this: 1. Application of several different non-sinusoidal piecewise analytic forcing functions representing diaphragm pressure; and 2. Addition of model compartments to differentiate between rib cage and abdominal movements. Simulated model output generated with variations of the modified model was compared against pleural pressure, airflow, and tidal volume data from Abbasi and Bhutani, 1990. In particular, flow-volume loops showed discrepancies in lung volume during breathing that, while may be appropriate for the adult lung, did not match the data for preterm infants. This can lead to further development of the model via alterations to the forcing functions.

### Consumer-resource behavior in plant-pollinator interactions

**Amelia Guthrie, Jessica Nguyen, Monie Thach, Antenyse Wade**^{(UG)}, Virginia Commonwealth University

Faculty mentor: Dr. Jan Rychtar

**Abstract:** Pollinator decline due to loss of biodiversity and pesticides has become a biologically significant issue with the additional pressures of climate change. Loss of pollinator density influences the distribution of plant mating systems and stable plant evolution strategies by affecting the cost-benefit relationship between plants which must invest resources into pollinator attraction strategies in outcrossing or self-pollination in selfing. We modified an adaptive dynamics model that incorporates reproductive, cost-dependent, and density-dependent factors to examine delayed selfing (i.e. pollination not mediated by pollinators) and extended the model by incorporating the decline of pollinator density resulting from external effects. Emergence of mutant plant strategies was incorporated into the model at stable equilibria for pollinator and plant populations and examined using pairwise invasibility plots.

### Low sensitivity to group members inhibits cooperation on evolving multiplayer networks

**Benjamin Ingram**^{(UG)}<, University of North Carolina Greensboro

Faculty mentor: Dr. Igor Erovenko

**Abstract:** We model a mobile population interacting over an underlying spatial structure using a Markov movement model. Interactions take the form of public goods games, and can feature an arbitrary group size. Individuals choose strategically to remain at their current location or to move to a neighboring location, depending upon their exploration strategy and the current composition of their group. This work builds upon Erovenko et al. (2019), which investigated the effect of network topology on the evolution of cooperation. In this project, we vary the sensitivity to the group composition as part of the exploration strategy of the individuals. We find that low awareness to whom individuals interact with inhibits cooperation independently of the network topology.

### Analysis of a couple of dynamical systems associated with cancer treatment

**Lubna Kadhim**^{(MA)}, Morgan State University

Faculty mentor: Dr. Xuming Xie

**Abstract:** In this work, we consider two dynamical systems associated with cancer treatment. The two dynamical systems are derived from two free boundary problems modeling tumor growth and cancer treatment by combination therapy. By analyzing the fixed points and their linear stability, we study the asymptotic property of the solution and its dependence on the dose levels of the drug.

### Bootstrap percolation via automated conjecturing

**Aidan Kierans**^{(UG)}, Virginia Commonwealth University

Coauthors: Blake Conka, Vinay Gupta, Hudson Lafayette, Craig Larson, Sarah Loeb, Kevin McCall, Andriy Mulyar, Christine Sullivan, Scott Taylor, Evan Wainright, Evan Wilson, Guanyu Wu

Faculty mentor: Dr. Neal Bushaw

**Abstract:** Bootstrap percolation is a simple monotone cellular automaton with a long history in physics, computer science, and discrete mathematics. In k-neighbor bootstrap percolation, a collection of vertices are initially infected. Vertices with at least k infected neighbors subsequently become infected; the process continues until stability is reached. In this paper, we hunt for graphs which can become entirely infected from initial sets which are as small as possible. We used automated conjecture-generating software and a large group lab-based model as fundamental parts of our exploration.

### Codes from difference sets

**Jackman Liu, Calvin Reedy, Kartikey Sharma, Scarlett Sun**^{(UG)}, University of Richmond

Faculty mentor: Dr. James Davis

**Abstract:** We will give a method for constructing error-correcting codes from combinatorial objects called difference sets and describe a construction for an infinite family of codes with maximal error-correcting capability. No prior knowledge of this topic will be required.

### 2-bootstrap percolation in Kneser graphs

**Kevin McCall**^{(PhD)}, Virginia Commonwealth University Bushaw

**Abstract:** A Kneser Graph, denoted K(s, t) is a graph whose vertex set is all t-element subsets of a set with s elements. Two vertices u, v are adjacent if their respective sets have no elements in common. r-neighbor bootstrap percolation, also known as r-bootstrap percolation, is a process in a graph which begins by selecting an initial set of vertices to be infected in the first round. In every subsequent round infected vertices remain infected while uninfected vertices become infected if they have at least r infected neighbors. In 2-Bootstrap Percolation, we begin by selecting two initial infected vertices and in each round infect every vertex with at least two infected neighbors. A graph is 2-Bootstrap Good (2-BG) if it is possible to select two vertices in the graph so that the infection percolates throughout the entire graph. A natural question is to investigate classes of graphs which are 2-BG. We characterize those Kneser graphs which are 2-BG.

### A new study of fractional-order financial system via homotopy analysis

**Fatemeh Norouzi**^{(PhD)}, Morgan State University

Faculty mentor: Dr. Gaston N'Guerekata

**Abstract:** In this talk, we are concerned by a dynamic model in Finance governed by a fractional derivative in the sense of Caputo. We prove the existence and uniqueness of the solution, and provide an approximation formula of the solution using the homotopy analysis. Then we examine the stability properties and behaviour of the equilibria of the system. This approach is new and complements recent work which presents only numerical simulations of the model.

### Region specific modeling: A novel approach to modeling heterogeneity in biological models

**Pravalika Putalapattu**^{(HS)}, Thomas Jefferson High School for Science and Technology

**Abstract:** The SIR model for the spread of infectious disease is a standard tool used by epidemiologists to study the spread of an infectious disease in a given population based on parametric input. The model uses a system of differential equations to track population flow, but makes several simplifying assumptions, such as population homogeneity and endemicity; significant deviations from these assumptions are ignored. This severely limits epidemiological understanding of large-scale/mobile epidemics, such as the Coronavirus and Ebola outbreaks. Current methods to account for heterogeneity include Poisson noise-modeling and non-constant parameterization. However, these methods have the following drawbacks: they 1) assume that real data is homoscedastic and 2) require heavy computation. In this project, we developed the region- specific SIR model (rsSIR), which works by sectioning the population into an arbitrary number of dynamic regions and using both the standard SIR differential equations and discrete-time Markov chains to track population flow. Using Python programming and numerical approximation, we compared the fit of generated SIR and rsSIR models for the 1968 Hong Kong epidemic in New York City; the rsSIR displayed over 10x less error. Furthermore, using the rsSIR, we developed a selective sampling method to characterize large populations with less data collection, which vastly cuts down on modeling costs. Lastly, we introduced a computational approach to optimizing the allocation of limited resources, such as treatments or vaccines, between regions in a population. With this approach, epidemiologists can effectively minimize the number of deaths caused by a disease, while considering time and resource constraints.

### Game-theoretical model of retroactive hepatitis B vaccination in China

**Vooha Putalapattu**^{(UG)}, Virginia Commonwealth University

Coauthors: Ali Chouhan, Sohail Maiwand, Matthew Ngo

Faculty mentors: Dr. Jan Rychtar, Dr. Dewey Taylor

**Abstract:** Hepatitis B (HepB) is one of the most common infectious diseases affecting over two billion people worldwide. About one third of all HepB cases are in China. In recent years, China made significant efforts to implement a nationwide HepB vaccination program and reduced the number of unvaccinated infants from 30% to 10%. However, many individuals still remain unprotected, particularly those born before 2003. Consequently, a catch-up retroactive vaccination is an important and especially cost-effective way to reduce HepB prevalence. In this paper, we analyze a game theoretical model of HepB dynamics that incorporates government-provided vaccination at birth coupled with voluntary retroactive vaccinations. We show that this retroactive vaccination should be a necessary component of any HepB eradication effort. Due to the vaccine waning, the optimal vaccination rates are almost independent of the vaccination coverage at birth. Moreover, it is in individual's self-interest to vaccinate (and re-vaccinate) at a rate just slightly below the vaccine waning rate.

### Saving lives, limbs and healthcare costs: A mathematical modeling approach towards reducing hospital-acquired infections

**Kelly Reagan**^{(PhD)}, Virginia Commonwealth University

Coauthors:Dr. Gonzalo Bearman, Ginger Vanhoozer

Faculty mentor: Dr. David Chan

**Abstract:** Hospital-acquired infections (HAIs) impact 1 in 22 hospitalized patients and are one of the leading causes of mortality in the United States. In addition to the direct patient impact, HAIs also cost the US healthcare system about $45 billion a year. Multiple studies have shown that bathing patients with chlorhexidine gluconate (CHG) wipes reduces HAIs. We employed a Markov chain model to assess the impact of CHG bathing on yearly HAIs and associated costs. Although the cost of using CHG wipes over traditional practices is about $4 more per patient, millions of dollars still could be saved when CHG bathing compliance is improved. Furthermore, we examine the effect of active resistors and organizational constipators on the reduced number of potentially prevented HAIs and the increase in associated healthcare costs. These individuals often delay implementation of emerging best practices in infection prevention.

### K-Means clustering for longitudinal chemical mixtures analysis in LIFECODES dataset

**Hui Sui**^{(UG)}, UNC Chapel Hill

Faculty mentors: Dr. Rachel Carroll, Dr. Cuixian Chen

**Abstract:** Phthalate and paraben exposure in pregnant women has previously been linked to adverse health outcomes such as preterm birth. Unsupervised clustering methods may serve as useful tools for identifying individuals with shared real-life patterns of chemical exposures. Knowledge of these groupings and their risk of adverse outcomes has the potential to inform targeted public health prevention strategies. This research applies k-means clustering to identify clusters of pregnant women with shared exposure profiles as defined by levels of the urinary phthalate metabolites and parabens using data from a case-control study within the LIFECODES birth cohort. This results in four clusters: low, moderate low, moderate high, and high exposures. These clusters are then examined for demographic composition and association with oxidative stress biomarkers. Finally, various clustering methods are compared with respect to applicability to the data and questions of interest, separation and consistency of clusters, and interpretability. K-means is easy to implement and fairly interpretable. However, in this example, it produces unevenly distributed clusters. In addition, contrast examinations indicate that subjects in different clusters do not have significantly different oxidative stress biomarkers.

### A modified Lotka-Volterra model with nonlinear terms

**Grant Wagner**^{(UG)}, James Madison University

Faculty mentor: Dr. James Liu

**Abstract:** We examine two modifications of the classical Lotka-Volterra equations to account for nonlinear interspecific and intraspecific interactions, by performing a complete analysis of the critical points of both equations. In the first modification, we show that it exhibits similar properties of the classical model; in the second modification, we show that it exhibits new properties not seen in the classical model.

### Characterizations of 2 - homeomorphic spaces

**Steve Wheatley**^{(PhD)}, George Mason University

Faculty mentor: Dr. Ronnie Levy

**Abstract:** In a 2018 paper, Arhangel'skii and Maksyuta give the definition of a 2 - homeomorphism, a topological concept that generalizes the notion of a homeomorphism. In this talk, we give some characterizations of spaces that are 2 - homeomorphic to spaces possessing various topological properties, including compact spaces and discrete spaces. We also show that, although many topological properties are not preserved under the 2 - homeomorphism relation, the property of having finite Cantor-Bendixson height is preserved.

### Special self-dual codes

**Carter Williams**^{(UG)}, Christopher Newport University

Faculty mentor: Dr. Neville Fogarty

**Abstract:** In quantum error correction, CSS codes are a type of stabilizer code constructed from classical codes. Classical error correction uses redundancy and probability to correct noisy errors; however, due to the no-cloning theorem it is impossible to use this method in the quantum state. Instead a stabilizer code is used to append ancilla qubits to qubits to simulate a noiseless qubit channel given a noisy qubit channel. Classical codes can be used to create these stabilizer codes if they satisfy the dual-containing constraint. In this discussion we will talk about the basics of coding theory, Reed-Muller codes, and how we can construct these stabilizer codes from Reed-Muller codes. We explicitly don't assume any expertise in coding theory (classical or quantum), but we hope that the audience has some recollection of basic ideas from linear algebra.

### Modeling a genetic switch with different phases

**Emily Wu**^{(UG)}, University of Richmond

Faculty mentor: Dr. Ovidiu Lipan

**Abstract:** Molecular networks regulate gene expression to help living systems adapt to changes in their environment. We developed a stochastic model of a two-component network found in biological processes such as heat shock response. This network behaves like a "switch" because an input molecular variable causes a change in an output molecular variable. We found that this switch exhibits three distinct phases, which are classified by the mean and variance of the output, as the model parameters change. Our model can be used as a building block for modeling large molecular networks at the core of systems biology.