RAMS Conference 2022
The Richmond Area Mathematical Sciences (RAMS) Conference at VCU was held virtually and synchronously on Saturday, April 30, 2022.
Plenary Talks
Student Talks
The Diffusive Lotka-Volterra Competition Model in Fragmented Patches I: Coexistence
Ananta Acharya
Institution: UNC Greensboro
Co-authors: S. Bandyopadhyay, J.T. Cronin, J. Goddard, A. Muthunayake & R. Shivaji
Faculty mentor: R. Shivaji
Abstract
We consider competitive model having competition of two species $u$ and $v$ inside the domain with strength of competition $b_1>0$ and $b_2>0$. We consider the linear boundary condition such that a parameter $\lambda$ influences both the differential equation as well as the boundary. On the boundary, we have $\gamma_1, \gamma_2$ involved which are related to the hostility of the exterior domain. We analyze the positive solutions of the model as the parameters $b_1, b_2$ and $\gamma_1, \gamma_2$ vary.
Population Dynamics of the Mariana Eight-Spot Butterfly and Parasitoid Wasps: A Compartment Model Approach
Cabrini Aguon, Andrew Lu
Institution: University of Guam
Co-authors: J. Tae
Faculty mentors: G. Curt Fiedler, Hyunju Oh, Leslie Aquino, and JaeYong Choi
Abstract
This research theoretically modeled the population dynamics between Guam's endangered Mariana Eight-Spot butterfly Hypolimnas octocula marianensis with two species of parasitoid wasps Telenomus remus and Echthromorpha intricatoria. We constructed a compartment model to analyze the influence of the number of host plants available and the number of all other parasitoid targets and computed the basic reproduction number R_0 to determine the proliferation of the respective wasp species in the system. We found control measures directly targeting the infection compartment rates eventually became ideal in controlling R_0 as host plants increased, while control measures targeting non-infection parameters eventually became ideal as other targets increased. In R_0 for T. remus, the most sensitive non-infection parameters were the host plant count and the number of other targets, whereas in R_0 for E. intricatoria, the rate at which larvae matured just slightly surpassed other pupa targets and host plants in sensitivity.
Period Doubling Cascades from Models and from Data
Alexander Berliner
Institution: College of William and Mary
Faculty mentor: Sarah Day
Abstract
Orbit diagrams of period doubling cascades represent systems going from periodicity to chaos. Here, we investigate whether a Gaussian process can be used to reconstruct a system from data through asymptotic dynamics in the orbit diagrams for period doubling cascades. To compare the orbits of a system to the Gaussian process reconstruction, we compute the Wasserstein metric between the point clouds of their obits for varying bifurcation parameter values. Visually comparing the period doubling cascades, we note that the exact bifurcation values may shift, which is confirmed in the plots of the Wasserstein distance. This has implications for studying dynamics from time series data since a reconstruction of a system’s period doubling cascade may lead to unpredictable model behavior in a neighborhood of the true bifurcation parameter value.
Some new expressions and study of the odd integral values of the Riemann Zeta Function
Maitreyo Bhattacharjee
Institution: IACS, Kolkata
Faculty mentor: Bibekananda Maji
Abstract
The Riemann Zeta Function, introduced by Leonhard Euler back in the early 18th century and made popular by the German Mathematician Bernhard Riemann in his landmark 1859 paper titled “On the number of primes less than a given quantity”, has received significant attention from a large number of mathematicians over the centuries. It appears very frequently in numerous areas of mathematics, like Number Theory (especially in the study of distribution of prime numbers), Complex Analysis and even in Physics. Though we have a closed form expression for the even positive integral values of the function, much less is known about the properties of the odd integral values. A major breakthrough was achieved when Apery proved the irrational nature in 1978. On the other hand, there are numerous alternate representations for the function (apart from the standard infinite sum representation) involving various Special Functions. In this talk, we would focus mainly on some new alternate expression and behavior of the function at odd values. We would also discuss a new expression of a related interesting result due to Ramanujan. Overall, the talk would involve topics from both Analytic Number Theory and Analysis.
Symmetric Dual-Wind Discontinuous Galerkin Methods for a Parabolic Obstacle Problem
Satyajith Bommana Boyana
Institution: UNC Greensboro
Co-authors: Tom Lewis, Aaron Rapp, Yi Zhang
Faculty mentor: Yi Zhang
Abstract
Many applications such as phase transition problems, elasto plastic material behavior, option pricing, etc deal with parabolic obstacle problems. The theoretical and the numerical analysis of such obstacle problems is challenging since the problem is nonlinear due to the presence of the obstacle function. In this research project, we proposed and studied a fully discrete scheme to solve the parabolic variational inequality with a general obstacle function in $\mathbb{R}^2$ that uses a symmetric dual-wind discontinuous Galerkin discretization in space and a backward Euler discretization in time. We established the convergence of numerical solutions in $L^\infty(L^2)$ and $L^2(H^1)$ like energy norms and computed the rates. Numerical results are provided to demonstrate the performance of the proposed methods.
A Model of the Dynamics of CRB-G with a Game Theoretical Analysis of the Effectiveness of Control Measures
Jovic Aaron Caasi, Alex Leon Guerrero, and Kangsan Yoon
Institution: University of Guam
Faculty mentors: Hyunju Oh, Leslie Aquino, and Aubrey Moore
Abstract
The coconut rhinoceros beetle Oryctes rhinoceros, or CRB, is an invasive species in Guam that has greatly affected the island's coconut tree population. Native to South and Southeast Asia, it first arrived in Guam in 2007. Various control measures have been used to combat the spread of CRB, but many have been proved ineffective. Strategies used to control the spread of CRB include the removal of coconut tree breeding grounds. We present a mathematical model to
understand the dynamics between CRB and coconut trees. Also, we construct a gametheoretical analysis of the effectiveness of removing moribund and coconut tree breeding grounds that individuals can choose to minimize the CRB damage to coconut trees. We find the maximum relative cost to remove a moribund tree is significantly higher than the maximum relative cost to remove a breeding ground. For future works, we want to consider a mixed strategy analysis comparing both strategies and apply game theory to other control measures.
Modeling and Analysis of Oryctes rhinoceros Behavior
Michael Cajigal, Gabriel Florencio, Ashley Yang
Institution: University of Guam
Faculty mentors: Hyunju Oh, Leslie Aquino, and Aubrey Moore
Abstract
In Guam, the increase of Oryctes rhinoceros, or rhinoceros beetles, — a species of beetles that prioritizes coconut trees as its source of nourishment — has led to a massive ecological problem since its first appearance in 2007. The rhinoceros beetle problem became exacerbated when supertyphoon Dolphin struck in 2015. Since then, the rhinoceros beetle population has increased dramatically causing many more coconut trees on the island to become damaged. This research aims to focus on the behavioral patterns of rhinoceros beetles as it interacts with coconut trees and to model these patterns. The goal of conducting this is to analyze the results collected from modeling these patterns, from which we can discern the best course of action to take in order to help suppress the population enough to either create an equilibrium between the beetles and the coconut trees, or so that the trees reach a sort of herd immunity, where there are no more coconut trees being affected by rhinoceros beetles. When this has been achieved, the solutions can then be applied in the same fashion, modeled, and the data collected in order to provide evidence of the solutions’ efficacy.
On Magic Square Type Sliding Game
Vigneswaran Madappan Chinnasami
Institution: University of South Carolina Salkehatchie
Faculty mentor: Wei-Kai Lai
Abstract
A sliding game is a puzzle that has pieces of numbers, alphabets, or pictures in a grid with exactly one empty space. For any magic square, we can construct a magic square type sliding game by subtracting 1 from each number, and use the place of 0 as the empty space. To solve this game, we try to rearrange all numbers to increasing order from the top row to the bottom row by sliding pieces one at a time. It is known that not all sliding games can be rearranged (solved) this way. Our project will be focusing on how the rotation and reflection impact the solvability of a magic square type sliding game.
Pythagorean Triples and Rational Points
Sayan Dutta
Institution: Indian Institute of Science Education and Research (IISER) Kolkata
Abstract
This talk mainly concerns the age old question of classifying all right angled triangles with integer sides. In other words, we wish to find all integer solutions to the equation, $X^2+Y^2=Z^2$. And we will do that using a method that stands at the poetic bridge between two divine deities of mathematics, namely numbers and geometry. From here, we will move on to the first generalization $X^2+Y^2=nZ^2$ and the second generalization $aX^2+bY^2=cZ^2$. We will also talk about Fermat's Last Theorem, and end with a general note on rational points on elliptic curves.
Convergence of a reinforced branching random walk on the triangle
Giordano Giambartolomei
Institution: University College London
Faculty mentor: Nadia Sidorova
Abstract
Edge-reinforced random walks are a well studied type of process with reinforcement, but the effect that branching may have has not yet been investigated. In this model particles move on a triangle: at each discrete time, once on a vertex, they branch at a fixed rate and then they traverse any of the incident edges at random, proportionally to the corresponding number of edge-crossings up to that time. We show the convergence of the proportions of edgecrossings to a random variable through dynamical systems techniques. We describe two nonnegligible asymptotic behaviours thanks to moderate deviations estimates: when none of the edge-crossings proportions vanishes and when only one does. We conjecture that none of the proportions dominates, i.e. none tends to one. Lastly, using martingale estimates, we show that the system does not exhibit a monopolistic regime, i.e. each edge is crossed infinitely many times.
Simulations and Modelling Techniques in a Healthcare Context
Amalia Gjerloev
Institution: University College of London
Faculty mentors: Sonya Crowe, Christina Pagel, Yogini Jani, Luca Grieco
Abstract
The pandemic has put stress on all aspects of life, and unfortunately this has been particularly true for the NHS here in England. The past 2 years has seen a huge influx of patients, a strain on hospital resources, and a demand to operate efficiently in order to save patient lives and distribute vaccinations. One method for addressing these problems is to use Operational Research (OR), which is a branch of mathematics that focuses on using analytical methods and mathematical modelling techniques to problem solve and improve operational decision making. Queuing theory is a commonly used OR technique used to simulate patient flow along a disease pathway. By modelling a patient pathway as a sequence of queues, queueing theory can be used to calculate waiting times, queue lengths, and steady state solutions. However, in more complex queuing networks that consider reneging customers and retrials, the simpler Jackson Network theory becomes inappropriate and fluid and diffusion approximations must be used. For these complex networks, discrete event simulation (DES) can be used and often serves as an ideal visual tool when talking with healthcare professionals. In this talk I will review how queuing theory and DES can be utilized to inform clinicians and healthcare staff, and I will discuss the implications of conducting effective OR research.
Cost and accuracy analysis of group and individual testing strategies: Implications for COVID-19
Ismat Binta Islam
Institution: McMaster University
Faculty mentor: Stephen Walter
Abstract
We compared several group and individual testing strategies in terms of cost and accuracy and then showed which one is more accurate while costing as little as possible for a specified prevalence rate. We designed and compared four testing protocols (First two are group testing (GT) protocols with 2 stages with or without a dilution effect (DE) and last two are
individual testing (IT) protocols with 2 stages and 1 stage respectively). We minimized the expected cost of GT protocols with the optimal group size and estimated the total expected cost of all IT protocols for several prevalence (p) rates. GT strategies are also compared with the IT strategies. We estimated the expected number of false-negative (FN) errors for each protocol, either assuming perfect sensitivity or allowing sensitivity depending on the group size for GT or assuming imperfect sensitivity for IT. Finally, the relationship between the cost and the number of FN cases is examined for each protocol based on lower and higher disease prevalence. We have illustrated these ideas using testing costs associated with screening for COVID-19. Our investigations can assist policymakers in selecting an appropriate protocol in terms of cost or accuracy or both. However, our methods are general, and so they can potentially be applied to other disease screening situations.
An ODE model of yaws elimination in Lihir Island, Papua New Guinea
Presley Kimball, Jacob Levenson, Amy Moore
Institutions: Creighton University, Washington and Lee University, Elon University
Faculty mentors: Jan Rychtar and Dr. Dewey Taylor
Abstract
Yaws is a chronic infection that affects mainly the skin, bone and cartilage and spreads mostly between children. The new approval of a medication as treatment in 2012 has revived eradication efforts and now only few known localized foci of infection remain. The World Health Organization strategy mandates an initial round of total community treatment (TCT) with single-dose azithromycin followed either by further TCT or by total targeted treatment (TTT), an active case-finding and treatment of cases and their contacts. We develop a compartmental ODE model of yaws transmission and treatment for these scenarios. We solve for disease-free and endemic equilibria and also perform the stability analysis. We calibrate the model and validate its predictions on the data from Lihir Island in Papua New Guinea. We demonstrate that TTT strategy is efficient in preventing outbreaks but, due to the presence of asymptomatic latent cases, TTT will not eliminate yaws within a reasonable time frame. To achieve the 2030 eradication target, TCT should be applied instead.
A Knowledge Ontology of AI Stock Prediction Technologies and Their Patent Landscape
Bailey Meche
Institution: University of Louisiana at Lafayette
Coauthors: Butsayarin Suwattananuruk, Adeline Chan
Faculty mentors: Charles Trappey, Kumer Das, Amy Trappey
Abstract
Searching for emerging and significant technology in a fast-growing field has become increasingly complex as analytical methodology and related informatics technology advances. This research constructs an initial knowledge ontology of a complex field and employs several methods by which to collect useful technology.  The methods employed and integrated within this patent landscape study could be used in other fields in future studies to efficiently examine patent profiles in any given technical or application domains. The field selected for this study is state-of-art stock market prediction systems. A rising trend of published patents in stock market prediction featuring innovative algorithms demonstrates a demand for artificial intelligence (AI) based methods and their adoptions in the industry. The relevant patents, searched and identified from the Derwent Innovation system, include 606 patents published from 2001 to 2021. Several text- and semantic-based analytical approaches, including patent document clustering, keyword extractions, and Latent Dirichlet Allocation (LDA) for topic identification, are used to refine and validate the AI stock prediction’s knowledge domain in the
ontology schema. To reinforce application, a case study of a top innovator is also included. The research results discover that practical applications of innovation in this field include topics complimentary to stock market prediction, such as investment risk control, portfolio management, and money laundering prevention.
Mathematical modelling Treponema infection in free-ranging Olive baboons (Papio anubis) in Tanzania
Solomaya Schwab
Institution: Cedar Crest College
Coauthors: Diamond Hawkins, Roland Kusi, Idrissa S. Chuma, Julius D. Keyyu, Sascha Knauf, Filipa M.D. Paciencia, Dietmar Zinner, Jan Rychtar, Dewey Taylor
Faculty mentors: Dewey Taylor, Jan Rychtar
Abstract
Yaws is a chronic infection caused by the bacterium Treponema pallidum susp. pertenue (TPE) that was thought to be an exclusive human pathogen but was recently found and confirmed in nonhuman primates. In this paper, we develop the first compartmental ODE model for TPE infection with treatment of wild olive baboons. We solve for disease-free and
endemic equilibria and give conditions on local and global stability of the disease-free equilibrium. We calibrate the model based on the data from Lake Manyara National Park in Tanzania. We use the model to help the park managers devise an effective strategy for treatment. We show that an increasing treatment rate yields a decrease in disease prevalence. This indicates that TPE can be eliminated through intense management in closed population. Specifically, we show that if the whole population is treated at least once every 5-6 years, a disease-free equilibrium can be reached. Furthermore, we demonstrate that to see a substantial decrease of TPE infection to near-elimination levels within 15 years, the whole population needs to be treated every 2-3 years.
Machine-Aided Detection of Atrial Fibrillation through R-R Intervals
Maximilian Wang, Sahil Patel
Institution: Isaac Bear Early College HS
Coauthors: Justin Guo
Faculty mentors: Tracy Chen
Abstract
Atrial Fibrillation (A-FIB) is a heart condition that occurs when the atria fail to beat in coordination with the ventricles, resulting in "irregularly irregular" heartbeats. This can lead to blood clotting and potentially a stroke. Since detecting A-FIB is extremely difficult, a possible solution is an application for devices like Apple Watches to constantly track the heart rate of its user. The program would then use the data collected to predict A-FIB based on the R-peaks and the distance between them, otherwise known as R-R intervals. Various features were used in conjunction with numerous classifiers to create models of prediction. The most prominent among these features was transitions, but all the features combined led to a greater overall accuracy. Out of all the classifiers, Light Gradient Boosting (LGBM) and Extreme Gradient Boosting (XGBoost) had the two highest accuracies, at 97.57% and 97.56%, respectively, as well as the two highest sensitivities and specificities. Ensemble models which combined the outputs of many classifiers were also created, none of which outperformed LGBM and XGBoost. Therefore, it was concluded that Light and Extreme Gradient Boosting, when provided with all features, would be the best algorithms for predicting atrial fibrillation.