Virtual Math Circle: Summer 2025 Research Projects

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About

Math Circle is a virtual summer camp for high school students. You can read more about how the program works.

Cost & Registration

The tuition fee is $1,200 covering all instructional sessions, materials, and resources.

IMPORTANT: Scholarships or financial aid may be available to eligible students.

College Credit

We are excited to announce that Math Circle has now been given approval to offer math credit - one hour of choice college credit* for Math 1999 - to high school students who want to continue their research project after the summer and present at LSU Discover Day in April 2026. Participants have the option to present virtually or in person.

*Choice College Credit: Students may opt to receive regular LSU credit if satisfied with their final grade.

Session 1: Mon, Jun 9, 2025 – Sat, Jun 28, 2025

A Mathematical Model to Mitigate the Spread of Environmentally Transmissible Diseases

Virtual Math Circle Research Proposal

Session
Session 1: Jun 9, 2025 – Jun 28, 2025
Mentor
Hemaho Beaugard Taboe
Ph.D. Candidate in Mathematical Biology
Department of Mathematics
University of Florida

Project Title
A Mathematical Model to Mitigate the Spread of Environmentally Transmissible Diseases
Topic Area
Mathematical Biology, Epidemiology

Helpful Skills and Background
High school calculus and first-order differential equations skills will be useful. These concepts will be taught or revised at the beginning of the project.

Abstract
Mathematical models provide guidance on controlling communicable diseases. In recent years, many scientific papers have focused on epidemic models, particularly due to COVID-19. However, one often overlooked aspect of these studies is the possible transmission of pathogens via surfaces, which we refer to as environmental transmission.

The objective of this study is to develop a basic mathematical model with logistic recruitment into the uninfected (susceptible) population that incorporates transmission from the environment to humans. We will apply this framework to the well-known environmentally transmissible disease, Lassa Fever virus, on which I have published research. Certain model parameters will be sourced from existing literature, while others will be estimated by calibrating the model to weekly confirmed data from my publications. Additionally, we will analyze various control strategies such as surface disinfection, mask-wearing, and treatment of infected persons to determine scenarios that may help mitigate disease spread.

Possible Extension
This project may be extended by incorporating an additional group of individuals who recover from the infection but can contract the disease again after an immunity period. Furthermore, the extended version will investigate the impact of vaccination in the presence of environmental contamination.

Outline/Timeline
Week 1: Introduction to the project, revision of calculus and ordinary differential equations, tutorial on Matlab for solving differential equations, and Overleaf for research documentation.

Week 2: Learning compartmental modeling in epidemiology, refining the proposed model, computing basic reproduction numbers, and estimating parameters from literature and data fitting.

Week 3: Model simulation, answering research questions, assisting students with manuscript writing, and preparing the final research report.

References
  1. Taboe, Hemaho B., Sergei S. Pilyugin, and Calistus N. Ngonghala. "Resolve Lassa Fever Persistence: A Compartmental Model with Environmental Virus-Host-Vector Interaction." (2024). Click here
  2. Nagle, R. Kent, et al. Fundamentals of differential equations and boundary value problems. New York: Addison-Wesley, 1996.
  3. Ngonghala, C. N., Taboe, H. B., Safdar, S., & Gumel, A. B. (2023). "Unraveling the dynamics of the Omicron and Delta variants of the 2019 coronavirus in the presence of vaccination, mask usage, and antiviral treatment." Applied Mathematical Modelling, 114, 447-465. Read more

Combinatorics: Counting Strategies and Applications

LSU Math Circle Research Proposal

Session
Session 1: Jun 9, 2025 – Jun 28, 2025
Mentor
Sabrina Rashid
Graduate Student Instructor
University of South Carolina

Project Title
Combinatorics: Counting Strategies and Applications
Topic Area
Combinatorics
High School Calculus
Not a prerequisite; calculus will not be introduced in this project.

Skills and Background
This research project requires basic algebra and arithmetic as prerequisites. It serves as an excellent introduction to counting methods, sequences, series, combinatorial proofs, and optimization. Students will gain experience with presentation tools such as PowerPoint, ShareLaTeX, and/or TeXShop for their final presentation.

Abstract
Combinatorics is an area of mathematics concerned with counting, organizing, and selecting items. It serves as the foundation for understanding complex concepts such as probability, algorithms (tree), and optimization. This project will help students understand the fundamental ideas of combinatorics, such as the Rule of Sum, Rule of Product, permutations, and combinations.

Students will explore various problems, including scenarios where committees must be selected and organized with distinct or indistinguishable roles. We will examine how introducing constraints, such as specific roles assigned to individuals, affects the total number of arrangements. These exercises will provide a deep understanding of combinatorial techniques and their applications.

Possible Extension
One possible extension is exploring optimization problems such as the Traveling Salesman Problem or resource allocation using graph coloring. Another direction could involve investigating generating functions or recurrence relations for modeling problems in combinatorics.

Outline/Timeline
Week 1: Understanding the Basics and Problem Setup
- Introduction to Combinatorics, Rule of Sum and Product, factorials, permutations, and combinations.
- Computing the number of ways to form committees with and without assigned roles.
- Exploring scenarios with distinct vs. indistinguishable roles.
Week 2: Solving Variations and Adding Constraints
- Introducing constraints such as specific role assignments.
- Using the Inclusion-Exclusion Principle for overlapping constraints.
- Solving problems where roles are indistinguishable but constraints exist.
Week 3: Extensions, Applications, and Presentation
- Generalizing problems to include weighted constraints and overlapping committees.
- Assigning mini-projects where students formulate their own constraints and solve the problem.
- Preparing and delivering presentations summarizing their work.

Final Deliverables
  • A written report detailing solutions to the original problem and variations.
  • A computational model (optional, using Python, Excel, or R).
  • A presentation summarizing the work and key findings.

References
  1. Tucker, A. (2006). Applied Combinatorics (5th ed.). Wiley.
  2. Stanley, R. P. (2011). Enumerative Combinatorics (Vol. 1 and 2). Cambridge University Press.
  3. Lovász, L., Pelikán, J., & Vesztergombi, K. (2003). Discrete Mathematics: Elementary and Beyond. Springer.
  4. Warner and Constenoble. Finite Mathematics (5th ed.). Wiley.
  5. Graham, R. L., Knuth, D. E., & Patashnik, O. (1994). Concrete Mathematics: A Foundation for Computer Science (2nd ed.). Addison-Wesley.

Session 2: Mon, Jul 14, 2025 – Sat, Aug 2, 2025

Modeling COVID-19 with Comorbidities in the USA

Virtual Math Circle Research Proposal

Session
Session 2: Jul 14, 2025 – Aug 2, 2025
Mentor
Hemaho Beaugard Taboe
Ph.D. Student in Mathematical Biology
Department of Mathematics
University of Florida

Project Title
Modeling COVID-19 with Comorbidities in the USA
Topic Area
Mathematical Biology, Epidemiology

Helpful Skills and Background
High school calculus and first-order differential equations skills will be useful. These concepts will be taught or revised at the beginning of the project.

Abstract
The unprecedented COVID-19 pandemic continues to unfold with unimaginable consequences, despite the implementation of stringent control measures since its initial outbreak in China in 2019. The burden of the disease, in terms of its associated death toll, is higher in the United States compared to other parts of the world. One factor cited in the literature to explain this phenomenon is the prevalence of preexisting health conditions among Americans.

The objective of this study is to develop a deterministic basic COVID-19 model that incorporates patients who concurrently have COVID-19 and either Cancer or Diabetes (Comorbidity), as well as those who are solely positive for COVID-19. We will utilize this framework to assess the magnitude of COVID-19-related deaths during the primary wave of the Delta variant in the U.S. Certain parameters of the model will be sourced from existing literature, while others will be estimated by calibrating the model to daily COVID-19 death data. Additionally, various control strategies (such as vaccination and mask-wearing) will be analyzed to determine scenarios in which the burden may be alleviated.

Possible Extension
This project may be extended to countries such as China, France, and Nigeria for comparison purposes. The model will remain the same, but the parameters may vary for each country.

Outline/Timeline
Week 1: Introduction to the project, revision of calculus and ordinary differential equations, tutorial on Matlab for solving differential equations, and Overleaf for research documentation.

Week 2: Learning compartmental modeling in epidemiology, refining the proposed model, computing basic reproduction numbers, and estimating parameters from literature and data fitting.

Week 3: Model simulation, answering research questions, assisting students with manuscript writing, and preparing the final research report.

References
  1. Chatterjee, Sayan, et al. "Association of COVID-19 with comorbidities: an update." ACS Pharmacology & Translational Science 6.3 (2023): 334-354. Click here
  2. Nagle, R. Kent, et al. Fundamentals of differential equations and boundary value problems. New York: Addison-Wesley, 1996.
  3. Eikenberry, Steffen E., et al. "To mask or not to mask: Modeling the potential for face mask use by the general public to curtail the COVID-19 pandemic." Infectious Disease Modelling 5 (2020): 293-308.
  4. Ngonghala, C. N., Taboe, H. B., Safdar, S., & Gumel, A. B. (2023). "Unraveling the dynamics of the Omicron and Delta variants of the 2019 coronavirus in the presence of vaccination, mask usage, and antiviral treatment." Applied Mathematical Modelling, 114, 447-465. Read more

Predicting Future Events Using Markov Chains and Machine Learning

Virtual Math Circle Program: Research Proposal I

Session
Session 2: Jul 14, 2025 – Aug 2, 2025
Mentor
Jacob Kapita
Ph.D. Student
Louisiana State University
Department of Mathematics

Project Title
Predicting Future Events Using Markov Chains and Machine Learning
Topic Area
Stochastic Probability Theory, Machine Learning, Data Science
High School Calculus
Not a prerequisite.

Skills and Background
Basic arithmetic and algebra skills are sufficient for this project. Coding is not required but can be helpful. Necessary skills will be taught as the project progresses. This project serves as an introduction to probability theory, stochastic analysis, data analysis, and machine learning. We will use Python in Google Colab to generate results and LaTeX's Beamer in Overleaf for presentations.

Abstract
Modeling real-life events is vital in various disciplines. Sometimes, we want to predict the likelihood of future random events. Markov Chains provide a good approximation based on the notion that future events depend only on the present state, not past events—making Markov Chains memoryless.

This project will focus on discrete-time Markov Chains, beginning with predicting the next word in a sentence. This will be achieved using machine learning models built on Markov Property principles. Students will explore real-life data modeling and prediction using Python-based Markov Chains.

Possible Extension
The project can be extended to focus on continuous-time Markov Chains in actuarial mathematics (life insurance). We may also explore inference using Markov Chains, determining the probability of being in a particular state after a given number of steps.

Timeline
Week 1: Background
- Basics of probability theory: probability laws, independence, and conditional probability.
- Introduction to Markov Chains and the Markov Property.
- Matrix operations (multiplication), Markov Transition Matrices, and examples.
Week 2: Markov Chains in Machine Learning (Applications)
- Introduction to Python on Google Colab.
- Overview of machine learning concepts and how Markov Chain principles apply.
- Modeling real-life data and predicting future events using Markov Chains in Python.
- Predicting the next word in a sentence using Markov-based models.
- Optionally, students may choose a dataset of interest.
Week 3: Finalizing Results and Presentation
- Introduction to Beamer on Overleaf for research documentation.
- Creating and preparing the final presentation.

References
  1. Fewster, Rachel. "Chapter 8: Markov Chains." Lecture Notes for Stats 325, Stochastic Processes, 2014. Read here
  2. Patel, Vatsal. "Markov Chain Explained." Built In, 2022. Read here
  3. Verma, Yugesh. "A Guide to Markov Chain and its Applications in Machine Learning." Analytics India Magazine, 2021. Read here

Questions?

Contact Isaac Michael <imichael@lsu.edu>.