Faculty Mentors

Marco Aldi and Daniele Grandini

Project Details

The goal of this project is to investigate a dictionary between graph theory and the theory of Lie algebras. The simplest form of this construction, due to Dani and Mainkar, assigns a Lie algebra to each graph. While it is known that these Lie algebras completely characterize the isomorphic class of the associated graph, the graph theoretic information is repackaged in a non-trivial way that is not yet completely understood. The focus of this project is to bridge this gap by systematically studying well-known graph-theoretic notions through the lenses of Lie theory. By doing so we aim at advancing our understanding of both areas of mathematics.

Mentors

Dewey Taylor and Jan Rychtar

Project Details

We will pick a preventable disease of their interest, do a literature search and build a compartmental ODE model (typically a variation of a recently published model) incorporating the disease prevention. The models are often quite complex and incorporate a lot of details about the disease progression, see the papers from previous years. We will solve for disease-free and endemic equilibria and determine conditions for their stability using the effective reproduction number. We will then use these results to get an incentive function which quantifies the net benefits of the preventive actions compared to the net benefits of no action at all when the population vaccination coverage (or the average use of the preventive action) is p. The incentive function is typically decreasing and concave down on the interval [0,pEL] where pEL is the population level of preventive action needed for disease elimination. The solution of the game, the Nash equilibrium, is given as the (unique) root of the incentive function. Also we will typically have to carefully calibrate the ODE model, perform sensitivity and uncertainty analysis, as well as validate the model and its findings on independent sets of data.

Mentors

Dewey Taylor and Jan Rychtar

Project Details

Visceral leishmaniasis (VL) is a deadly neglected tropical disease caused by a parasite Leishmania donovani. It spreadsvia sand flies. We create and analyze a compartmental model of VL transmission and incorporate the use of insecticide treated bed nets (ITNs). We found that in order for VL elimination to occur then ITN usage needs to 96.5% or higher. This is only slightly higher than the proportion reported in  Fortunato et al. 2021, which reported a needed proportion of 96% to achieve VL elimination.

Publications

TBA

Conference Presentations

1. Cameron Davis, Elizabeth Javor, Sonja Rebarber, Jan Rychtar, Dewey Taylor (2022). Mathematical Model of Visceral Leishmaniasis. UOG-VCU REU Symposium

Mentors

Dewey Taylor and Jan Rychtar

Project Details

As infectious diseases continue to threaten communities across the globe, people are faced with a choice to vaccinate, or not.  Many factors influence this decision, such as the cost of the disease, the chance of contracting the disease, the population vaccination coverage, and the efficacy of the vaccine. While the vaccination games in which individuals decide whether to vaccinate or not based on their own interests are gaining in popularity in recent years, the vaccine imperfection has been an overlooked aspect so far. We investigate the effects of an imperfect vaccine  on the outcomes of a vaccination game. We use a simple SIR compartmental model for the underlying model of disease transmission. We model the vaccine imperfection by adding vaccination at birth and maintain  a possibility for the vaccinated individual to become infected. We derive explicit conditions for the existence of different Nash equilibria, the solutions of the vaccination game.  The outcomes of the game depend on the complex interplay between  disease transmission dynamics (the basic reproduction number), the relative cost of the infection, and the vaccine efficacy. We show that for diseases with relatively low basic reproduction numbers, there is a little difference between outcomes for perfect or imperfect vaccines and thus the simpler models assuming perfect vaccines are good enough. However, when the basic reproduction number is above a certain threshold, then, unlike in the case of a perfect vaccine,  there can be multiple equilibria.  Moreover, unless there is a mandatory vaccination policy in place that would push the vaccination coverage above the value of unstable Nash equilibrium, the population could eventually slip to the ``do not vaccinate'' state. Thus, for diseases that have relatively high basic reproduction numbers, the potential for the vaccine not being perfect should be explicitly considered in the models.

Publications

1. I. B. Augsburger, G. K. Galanthay, J.H. Tarosky, J. Rychtar, D. Taylor (2022). Imperfect vaccine can yield multiple Nash equilibria in vaccination games. Mathematical Biosciences 356: 108967. DOI: 10.1016/j.mbs.2023.108967
2. I. B. Augsburger, G. K. Galanthay, J.H. Tarosky, J. Rychtar, D. Taylor (2022). Voluntary vaccination may not stop monkeypox outbreak: A game-theoretic model. PLOS NTD 16(12):e0010970. DOI: 10.1371/journal.pntd.0010970

Conference Presentations

1. Ian Augsburger, Grace Galanthay, Jacob Tarosky, Jan Rychtar, Dewey Taylor (2022). Imperfect Vaccination Yields Multiple Nash Equilibria. UOG-VCU REU Symposium. Virtual/University of Guam.

Mentors

Dewey Taylor and Jan Rychtar

Project Details

Yaws is a chronic infection that was thought to affect only humans but was recently found and confirmed in non-human primates (NHP). Unlike in humans, in NHP, the disease is sexually transmitted. We developed the compartmental ODE model for olive baboons. We solved for disease free and endemic equilibria and also performed the stability analysis. We calibrated the model based on the data from Tanzania National Parks. We will use the model to help the parks devise an effective strategy for treatment.

Publications

1. D. Hawkins, R. Kusi, S. Schwab, I. S. Chuma, J. D. Keyyu, S. Knauf, F. Paciencia, D. Zinner, J. Rychtar, D. Taylor (2022). Mathematical modelling of Treponema infection in free-ranging Olive baboons (Papio anubis) in Tanzania, Epidemics 41, 100638. DOI: 10.1016/j.epidem.2022.100638  

Conference Presentations

1. Diamond Hawkins, Solomaya Schwab, Roland Kusi, Jan Rychtar, Dewey Taylor (2022). An epidemiological study of the spread and treatment of treponemal infection in olive baboons. NCUR. 
2. Diamond Hawkins, Solomaya Schwab, Roland Kusi, Jan Rychtar, Dewey Taylor (2021). An epidemiological study of the spread and treatment of treponemal infection in olive baboons. EMaDel MAA meeting. November 13, 2021
3. Diamond Hawkins, Solomaya Schwab, Roland Kusi, Jan Rychtar, Dewey Taylor (2022). An epidemiological study of the spread and treatment of treponemal infection in olive baboons. VCU Undergraduate Research Symposium. Richmond, VA
4. Diamond Hawkins, Solomaya Schwab, Roland Kusi. An epidemiological study on the spread and treatment of treponemal infection in Olive Baboons 2021 SUMS conference at JMU, Virtual via Zoom, December 4, 2021

Mentors

Dewey Taylor and Jan Rychtar

Project Details

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 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 the 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.

Publications

1. P. Kimball, J. Levenson, A. Moore,  J. Rychtar,  D. Taylor. (2022) An ODE model of yaws elimination in Lihir Island, Papua New Guinea, PeerJ 10:e13018. DOI: 10.7717/peerj.13018

Conference Presentations

1. Presley Kimball, Jacob Levenson, and Amy Moore, Jan Rychtar, Dewey Taylor (2022). An ODE model of yaws elimination in Lihir Island, Papua New Guinea. Joint Math Meetings. January 2022
2. Presley Kimball, Jacob Levenson, and Amy Moore, Jan Rychtar, Dewey Taylor (2022). An ODE model of yaws elimination in Lihir Island, Papua New Guinea. NCUR, April 2022. 
3. Presley Kimball. A poster at MAA MathFest "An ODE model of yaws elimination in Lihir Island, Papua New Guinea", August 2022
4. Presley Kimball, Jacob Levenson, and Amy Moore, Jan Rychtar, Dewey Taylor (2022). An ODE model of yaws elimination in Lihir Island, Papua New Guinea. Elon SURF poster symposium. Elon University. 
5. Presley Kimball, Jacob Levenson, Amy Moore, Jan Rychtar, Dewey Taylor (2021). An ODE model of yaws elimination in Lihir Island, Papua New Guinea. Epidemics8 - 8th International Conference on Infectious Disease Dynamics. November 30, 2021
6. Presley Kimball, Jacob Levenson, and Amy Moore, Jan Rychtar, Dewey Taylor (2021). An ODE model of yaws with total community treatment and targeted contact tracing. Southern California Math REU Conference. virtual, Occidental College. 
7. Presley Kimball, Jacob Levenson, and Amy Moore, Jan Rychtar, Dewey Taylor (2021). An ODE model of yaws with total community treatment and targeted contact tracing. Nebraska Conference for Undergraduate Women in Mathematics. 
8. Presley Kimball, Jacob Levenson, and Amy Moore, Jan Rychtar, Dewey Taylor (2021). An ODE model of yaws with total community treatment and targeted contact tracing. 2021 Iowa Section Mathematics Meeting Virtual via Zoom October 8-9, 2021

Mentors

David Chan and Indranil Sahoo

Project Details

In this study, we examine a set of data collected during the early stages of the COVID-19 pandemic that included demographic and support network data. The support network data is comprised of information that quantified the type of support, i.e. emotional or educational, and whether this support was routine or intense. Using this data, models for predicting GPA were created using \textit{Chi-Square Automatic Interaction Detection (CHAID)}, a decision tree algorithm, and \textit{cforest}, a random forest algorithm that uses conditional inference trees. We compare the methods' accuracy and variation in the important variables used by each algorithm. Each algorithm found different variables important for different subsets with some overlap. For white students, different types of educational support were important in predicting GPA excellence, while for nonwhite students, different types of emotional support were important in predicting GPA excellence. The presence of differing types of routine support were important in predicting GPA excellence for cisgender women, while differing types of intense support were important in predicting GPA excellence for cisgender men.

Using demographic and support network data collected during the beginning of the COVID-19 pandemic, linear and nonlinear models are constructed to estimate students' mental health.  We find the most important predictor is PTSD.  We also find differences in important factors for students of different genders and race.  In the linear models the amount of emotional support are important factors for cisgender women, whereas the quality of that support is more important for cisgender men.  In the models focusing on race, the important variables varied for the linear models.  The important variables for White students include more dealing with COVID-19 than the other groups.  For Black and Latinx students, the important variables included emotional support from peers and family.  For Asian students peer emotional support and collaborative learning were important variables.

Publications

1. Kenai Burton-Heckman, Rayna Maleki, Indranil Sahoo, Michael D. Broda, Hollee A. McGinnis, David M. Chan. Predictive Models for Mental Health: An Analysis of College Students and their Social Support Networks, submitted
2. Anthony Frazier, Joethi Silva, Rachel Meilak, Indranil Sahoo, David Chan, and Michael Broda. "Decision Tree-Based Predictive Models for Academic Achievement Using College Students' Support Networks." Journal of Data Science, accepted

Conference Presentations

1. Anthony Frazier, Joethi Silva, Rachel Meilak, Indranil Sahoo, David Chan (2021). Decision Tree-Based Predictive Models for GPA Excellence Using College Students’ Support Networks. Southern California REU Conference. 
2. Anthony Frazier, Joethi Silva, Rachel Meilak, Indranil Sahoo, David Chan (2021). Decision Tree-Based Predictive Models for GPA Excellence Using College Students’ Support Networks. Weber State University Math Club. Weber State University. 

Mentors

Dewey Taylor and Jan Rychtar

Project Description

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.

Publications

1. A. K. Fortunato, C.P. Glasser, J.A. Watson, Y. Lu, J. Rychtar, D. Taylor (2021). Mathematical modelling of the use of insecticide-treated nets for elimination of visceral leishmaniasis in Bihar, India. Royal Society Open Science, 8 (6), 201960. doi:10.1098/rsos.201960

Conference Presentations

1. Dewey Taylor and Jan Rychtar: Mathematical modelling of the use of insecticide treated nets for elimination of visceral leishmaniasis in Bihar, India Epidemics8 Conference, November 30, 2021

Mentors

Dewey Taylor and Jan Rychtar

Project Description

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.

Publications

1. C.A.G. Engelhard, A.P. Hodgkins, E.E. Pearl, P. K. Spears, J. Rychtar, D. Taylor (2021). A mathematical model of guinea worm disease in chad with fish as intermediate transport hosts. Journal of Theoretical Biology, 521, 110683. doi:10.1016/j.jtbi.2021.110683
2. C.A.G. Engelhard, A.P. Hodgkins, E.E. Pearl, P. K. Spears, J. Rychtar, D. Taylor  (2023). Mathematical models of guinea worm disease: A review. UMAP.

Conference Presentations

1. Jan Rychtar. Modeling Guinea worm disease, MAA MD-DC-VA Section, April 23-24, 2021
2. Jan Rychtar and Dewey Taylor. A mathematical model of Guinea worm disease in Chad with fish as intermediate transport hosts (joint presentation with Dewey Taylor), Epidemics8 Conference, November 30, 2021
3. Jan Rychtar Modelling Prevention of Neglected Tropical Diseases, Randolph-Macon College, February 19, 2021
4. Jan Rychtar . talk Modeling Guinea Worm Disease at the MAA Allegheny Mountain Section. April 2, 2022