The Richmond Area Mathematical Sciences (RAMS) Conference at VCU will be held **virtually and asynchronously** on Saturday, May 1, 2021.

The panels below contain the abstracts of the accepted talks.

**Presenters:** Emily Cheng, Neeha Gambhirrao , Rohani Patel, Aufia Zhowandai**Institution:** Virginia Commonwealth University**Faculty Mentor:** Jan Rychtar, 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. Our main objective is to see whether mandatory vaccination policy is needed or if polio could be almost eradicated by a voluntary vaccination. 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. We conclude that the mandatory vaccination policy is therefore needed to achieve a complete

eradication.

**Presenters:** Allison Hodgkins, Essence Pearl, Paul Spears**Institutions:** Regent University, Norfolk State University, Virginia Tech **Co-author:** Cesar Engelhard (University of Virginia)**Faculty mentors:** Jan Rychtar, Dewey Taylor

**Abstract:** Guinea-worm disease (GWD) was thought to be almost eliminated in Chad when it reemerged in 2010. The disease now shows a peculiar pattern of spreading along Chari River and its tributaries, rather than clustering around a particular drinking water source. We create a mathematical model of GWD that includes the population dynamics of the parasite as well as the dynamics of its hosts (copepods, fish, humans, and domestic dogs). We calibrate our model based on data from the literature and validate it on the recent GWD annual incidence data from Chad. The effective reproduction number predicted by our model agrees well with the empirical value of roughly 1.25 derived directly from the data. Our model thus supports the hypothesis that the parasite now uses fish as intermediate transport hosts. We predict that GWD transmission can be most easily interrupted by avoiding eating uncooked fish and by burying the fish entrails to prevent transmission through dogs. Increasing the mortality of copepods and even partially containing infected dogs to limit their access to water sources is another important factor for GWD eradication.

**Presenter:** Lubna Kadhim**Institution:** Morgan State University**Faculty mentor:** Xuming Xie

**Abstract: **In this paper, 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. Cancer is one of the leading causes to death in many parts of the world. Most of the processes are modeled by partial differential equations (PDE), but the models in this paper are given by dynamic systems (ordinary differential equations (ODE)). The evolution of cancer in a tissue can be modeled by a system of PDEs, which describe the convective and reaction-diffusion process for cell densities and nutrient concentrations in the tumor. The interaction among cells and molecules are also represented in the PDEs. The tumor surface moves with the same velocity as cancer cells. In this paper, the first dynamical system model has derived from two PDE models trying to answer this question: Are the T cells able to eradicate the cancer? This question was addressed in some cases where the tumor is spherical. The first drug a checkpoint inhibitor which disables checkpoint receptor on the T cells thus enables the T cells to remain active and kill cancer cells. The second drug is a dose of the oncolytic virus, a virus that is genetically programmed to invade only cancer cells, multiplies within them, and cause their death. By analyzing the fixed points and their linear stability, we study the asymptotic property of the tumor surface and the solution to the dynamical systems and its dependence on the dose levels of the drugs.

**Presenter:** Sohail Maiwand**Institution:** Virginia Commonwealth University**Co-authors:** Ali Chouhan, Matt Ngo, Vooha Putalapattu (VCU)**Faculty Mentors:** Jan Rychtar, 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 potentially 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. Given the uncertainty about the long-term efficacy of the HepB vaccinations, we study several scenarios. When the waning rate is relatively high, we show that this retroactive vaccination should be a necessary component of any HepB eradication effort. When the vaccine offers long-lasting protection, the voluntary retroactive vaccination brings the disease incidence to sufficiently low level. Also, we find that the optimal vaccination rates are almost independent of the vaccination coverage at birth. Moreover, it is in an individual’s self-interest to vaccinate (and potentially re-vaccinate) at a rate just slightly above the vaccine waning rate.

**Presenter:** Josephine Messina **Institution:** James Madison University**Faculty mentor:** Prabhashi Withana Gamage

**Abstract: **In recent years, the United States has had increasing rates of gun violence and more frequently occurring mass shootings. While firearm violence is an issue internationally, the United States has arguably the worst gun violence of all developed countries. Gun control policies and their potential effect on gun violence rates have been debated for years; therefore, it is vital to determine which gun control policies would be effective at reducing gun violence rates in the United States. This analysis focuses on evaluating the relationships between gun violence and gun control policies in the United States. Multiple linear regression models were created for state firearm mortality, suicide, robbery, and assault rates in order to determine if gun control policies had a significant effect on reducing violence. Models were created for both 2005 and 2018 in order to see how gun violence rates and gun control policies have changed in past years. The majority of gun control policies in 2005 seemed to be ineffective at preventing gun violence. However, several gun control policies were shown to have a significant negative relationship with firearm mortality and suicide rates in 2018. Implications for future policy and recommendations for future studies are discussed.

**Presenters:** Mana Nasseri, Yashwant Mirajkar, Kirubel Kentiba**Institution:** Virginia Commonwealth University**Co-author:** Noble Chowdhury (VCU)**Faculty Mentor:** Jan Rychtar, Dewey Taylor

**Abstract:** We consider a game theoretical model of kleptoparasitic interaction between two individuals, the Owner and the Intruder. The Owner is in possession of a resource and must decide whether to defend the resource against the Intruder or flee. If the Owner defends, the Intruder must decide whether to fight with the Owner or flee. The outcome of the fight depends on the hunger of the individuals, the hungrier the individual is, the more likely they are to win the fight. We consider three scenarios:(a) both individuals know their own and their opponent’s hunger, (b) individuals only know their own hunger but not that of their opponent, and (c) individuals do not know their own nor the opponent’s hunger levels. We determine Nash equilibrium strategies in each scenario. We conclude that Owner is generally willing to defend more often than the Intruder is willing to attack. Also, the Intruder’s payoff is largest in the full information case; but the Owner may benefit in the no information or partial information cases when the cost of the fight is neither too large nor too small.

**Presenter:** Dylan Ryan**Institution:** Virginia Commonwealth University**Faculty Mentor:** Marco Aldi

**Abstract:** Generalized geometry is used in physics to study supersymmetry in the context of string theory. This research involves integrating mathematics and physics in this way by focusing on generalized Cauchy-Riemann F-structures (CRF-structures), specifically on low-dimensional solvable Lie algebras. The goal of this project is to analytically describe the geometry behind extra dimensions (in addition to space and time), especially those that are simple enough to be described mathematically, but complex enough to not be trivial. This research is of interest to both physicists and mathematicians, as it would account for the observed behavior of subatomic particles and contribute to the study of string theory.

**Presenter:** Mark Schwitzerlett**Institution:** Virginia Commonwealth University**Faculty Mentors:** H. Reed Ogrosky, Ihsan Topaloglu

**Abstract:** Viscous liquid film flows in a tube arise in numerous industrial and biological applications, including the transport of mucus in human airways. Previous modeling studies have typically used no-slip boundary conditions, but in some applications the effects of slip at the boundary may not be negligible. We derive a long-wave model based on lubrication theory which allows for slippage along the boundary. Linear stability analysis verifies the impact of slip-length on the speed, growth rate, and wavelength of the most unstable mode. Nonlinear simulations demonstrate the impact of slip-length on plug formation and wave dynamics. These simulations are conducted for flows driven by gravity, core flow, or a combination of the two.

**Presenter:** Essak Seddiq**Institution:** Virginia Commonwealth University**Faculty Mentor:** Glenn Hurlbert

**Abstract:** Graph pebbling is a network optimization model for satisfying vertex demands with vertex supplies (called pebbles), with partial loss of pebbles in transit. The pebbling number of a demand in a graph is the smallest number for which every placement of that many supply pebbles satisfies the demand. The t-Target Conjecture posits that the largest pebbling number of a demand of fixed size occurs when the demand is entirely stacked on one vertex. This truth of this conjecture could be useful for attacking many open problems in graph pebbling, including the famous conjecture of Graham (1989) involving graph products. It has been proven for complete graphs, cycles, cubes, and trees. In this paper we consider 2-paths, split graphs, and Kneser graphs, important classes of graphs in graph structure theory, graph coloring, and algorithms. Using recently developed cost-related methods and induction, we prove the $t$-Target Conjecture for all 2-paths, split graphs of minimum degree 3, and Kneser graphs with $k=2$ and $m\ge 5$, and build tools potentially useful for attacking other graphs as well, such as, we believe, more general classes of chordal graphs.

**Presenter:** Jamie Shive**Institution:** Virginia Commonwealth University**Faculty Mentor:** Richard Hammack

**Abstract:** A graph G on n vertices is palindromic if there is a vertex-labeling bijection f:V(G) --> {1,2, ... ,n} with the property that for any edge vw of G, there is an edge xy of G for which f(x)=n-f(v)+1 and f(y)=n-f(w)+1. This notion was defined and explored in a recent paper by Robert Beeler. The paper gives sufficient conditions on the factors of a Cartesian product of graphs that ensure the product is palindromic but states that it is unknown whether the conditions are necessary. We prove that the conditions are indeed necessary. Further, we prove parallel results for the strong product, lexicographic product, and direct product of graphs.

**Presenter:** David Sykes**Institution:** Texas A&M University**Faculty Mentor:** Igor Zelenko

**Abstract:** The local differential geometry of Levi-nondegenerate CR structures is well understood due in large part to classical results of Cartan, Tanaka, Chern, and Moser, and yet comparatively little is known about other CR structures. This talk will feature some of the key new ideas that have been introduced recently to study Levi-degenerate CR structures, namely constructions that we have recently applied to solve local equivalence problems for 2-nondegenerate CR structures. The main result that we arrive to in this talk is a sharp upper bound for the dimension of certain CR manifolds’ symmetry groups. This talk is on joint work with Igor Zelenko.

**Presenters:** Joy Watson, Casey Glasser, Anna Fortunato**Institutions:** Virginia State University, Virginia Tech, University of Richmond**Faculty mentors:** Yongjin Lu, Jan Rychtar, Dewey Taylor

**Abstract:** Visceral leishmaniasis (VL) is a deadly neglected tropical disease caused by a parasite Leishmania donovani and spread by female sand flies Phlebotomus argentipes. There is conflicting evidence regarding the role of insecticide treated nets (ITNs) on the prevention of VL. Numerous studies demonstrated the effectiveness of ITNs. However, KalaNet, a large

trial in Nepal and India did not support those findings. The purpose of this paper is to gain insight into the situation by mathematical modelling. We expand a mathematical model of VL transmission based on the KalaNet trial and incorporate the use of ITNs explicitly into the model. One of the major contributions of this work is that we calibrate the model based on the available epidemiological data, generally independent of the KalaNet trial. We validate the model on data collected during the KalaNet trial. We conclude that in order to eliminate VL, the ITN usage would have to stay above 96%. This is higher than the 91% ITNs use at the end of the trial which may explain why the trial did not show a positive effect from ITNs. At the same time, our model indicates that asymptomatic individuals play a crucial role in VL transmission.

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- Last Updated: 05/10/2021
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