Abstract

Mathematical and theoretical biology or, biomathematics, is a branch of biology which employs theoretical analysis, mathematical models and abstractions of the living organisms to investigate the principles that govern the structure, development and behavior of the systems, as opposed to experimental biology which deals with the conduction of experiments to prove and validate the scientific theories. The field is sometimes called mathematical biology or biomathematics to stress the mathematical side, or theoretical biology to stress the biological side. Theoretical biology focuses more on the development of theoretical principles for biology while mathematical biology focuses on the use of mathematical tools to study biological systems, even though the two terms are sometimes interchanged. 

Mathematical biology aims at the mathematical representation and modeling of biological processes, using techniques and tools of applied mathematics. It can be useful in both theoretical and practical research. Describing systems in a quantitative manner means their behavior can be better simulated, and hence properties can be predicted that might not be evident to the experimenter. This requires precise mathematical models. 

Methods of mathematics and computer science have become important tools in analyzing the spread and control of infectious diseases. Partnerships among computer scientists, mathematicians, epidemiologists, public health experts, and biologists are increasingly important in the defense against disease. 

Projects

Dynamics of Eastern Equine Encephalitis Infection Rates: A Mathematical Approach

The Eastern Equine Encephalitis Virus (EEEV) is an erratic and deadly neurological disease that spans across the northeastern coast of the United States. To determine the rate at which the virus is spread between the Black-Tailed Mosquito (Culiseta melanura) and select avian species we began by analyzing the migration patterns of both the mosquito and the avian species. It was found that certain species of avians shared similar, or even identical, flight patterns with the Black-Tailed Mosquito. Through this research, we develop and analyze a system of Ordinary Differential Equations (ODEs) to gain insight into how and when transmission of the virus to avians is at its highest. We incorporate a host stage-structured model where the avian host group is split into two categories, adults and young-of-the-year birds (YOY). Using this we explored the extent to which fluctuations occurred in transmission rates according to host/vector abundances, mosquito biting rate, and type of host. We evaluate the hypothesis that YOY avians are more readily exposed to the mosquito vector as they lack a defense mechanism, unlike their adult counterpart using the compartmental model.

Collaborators: Aurod Ounsinegad

Chemical espionage: modeling the relationship between the Pieris brassicae butterfly and Trichogramma wasps

Our project, Chemical Espionage, describes the situation of the butterfly species Pieris brassicae, the large cabbage white butterfly, which uses chemical signals to attract mates. This, however, presents a new problem for the females as the chemical aphrodisiac can attract too many males and make it difficult for her to lay her eggs in a secure locale. To combat this, males of the species emit an anti-aphrodisiac which deters other males from approaching the mated female. However, the use of this anti-aphrodisiac has an unexpected consequence for the butterflies as it attracts two species of the Trichogramma wasp, parasitic wasps which can detect the anti-aphrodisiac and will use it to locate mated females so they can prey on the eggs she lays. Once a Trichogramma wasp locates a mated female butterfly, they land on the female and ride on it to find where the female lays her eggs. Once located, the wasps will lay their eggs and the larva of the wasps will consume the eggs of the butterfly. The optimal solution for these butterflies is one in which a steady population of eggs are laid to replenish the butterfly population while reducing the amount of eggs lost to wasps. The best solution for the wasps, however, is one in which a balance is struck between the amount of anti-aphrodisiac used and the growing population of the butterflies. In order to find the equilibrium between these two scenarios where both can coexist without driving the other to extinction we created a system on differential equation to model the situation. We started by modeling only the butterfly populations and then introduce the wasps. After parameterizing the model, we search for equilibrium and stability. Simulations are done to show the different scenarios between the two species.

Collaborators: Dashon Mitchell

Parameter Estimation and Simulation of Bacteriophage Infection Model

Bacteriophages are a class of viruses that infect and destroy bacteria. For this reason, they are an emerging focus in research due to their potential use in treating antibiotic resistant bacterial infections. Tarleton State University, as part of the international HHMI SEAPHAGES bacteriophage discovery program, is working on an interdepartmental project between the biology and mathematics departments to create a mathematical model for the complexity of bacteriophage infection of host cells. A system of 4n+1 differential equations were derived to model these interactions. This system of equations was then used to simulate results for the infection outcomes of the different populations. These results were then evaluated in search of equilibria for populations in a given parameter space. It is our hope that these simulated equilibria can be used to identify infection outcomes for bacteriophages used in clinical settings to speed up testing before usage and effectively improve patient outcomes.

Collaborators:

Abigail Ballard

Dr. Keith Emmert

Dr. Dustin Edwards

Predator-Prey Model for Largemouth Bass and Channel Catfish

Largemouth bass and Channel catfish are considered highly common freshwater sport fish. They are highly-populated and can be found in almost any lake in Texas. However, they tend to be overfished and must be restocked in order to attract anglers. The goal of this project was to create a model to analyze the predator-prey relationship between these two types of fish and determine how best to find populations at healthy levels. By finding the conditions for healthy levels of co-existence for each fish, pond and lake owners are able to make better decisions about restocking. The data used has been taken from several sources and uses averages from the studies to fit to the differential equation model. With this data, equilibria are found and analyzed to determine parameter ranges where co-existence is possible. With the information from the basic model, a more advanced model can be made that will account for water level/quality and look at higher volumes of these populations and the issues of overfeeding these fish. 

Collaborators: Trace Patterson

Imposter Syndrome: Classifying Salamanders with Computer Vision and Artificial Neural Networks

Salamanders serve as important tetrapod models for developmental, regeneration and evolutionary studies. In order to tell certain species of salamander apart, genetic sequencing is used which can be difficult as the sequence could be rather lengthy and takes time and equipment many places do not have. There are also many museums with specimens for which DNA sequencing cannot be done, thus the need for another method for identifying salamanders. This project aims to create a deep learning model that is able to take in a picture of a salamander and return its classification.
To do this we built a Convolution Neural Network and supplied it with high resolution images provided by Washington University and the Smithsonian. With these images and five classifications of salamanders, the model is able to predict with high accuracy the correct classification of salamander. With more images and higher-performance computers, the model could produce more accurate predictions.

Collaborators:

Preston Ward

Dr. Kyle O'Connell (Outside Tarleton)

SCUDEM Competition

The student competition takes place over a week-long period that begins at each team’s individual home campus and culminates on Saturday, at a regional host site. At 8:00 AM EST on Monday before the competition, student teams can access three modeling scenarios involving differential equations, posted at our SIMIODE website. These teams will work at their home institution, developing approaches and solutions to their chosen modeling scenario. The scenarios are designed so that every team may experience success in modeling, building their skills and confidence in differential equations. Each team will prepare a draft Executive Summary and 10 minute Presentation to bring to the regional host site on Competition Saturday. There, student teams will work on a small modification of the modeling scenario they have selected (for example, effects of new assumptions, variables or changes in parameters) for inclusion in their final submissions.


Over the past years, teams from Tarleton have competed in this competition and won top honors for their work. Some have continued the projects and turned them into full-fledged research over the following years. Students train with professors for the weeks before the competition to prepare. This is a great way to see what modeling with differential equations is all about. 

Student Presentations

Regional
123rd Annual Meeting of the Texas Academy of Science

Stephen F. Austin University – Nacogdoches, TX February 28-29, 2020
⦁    Parameter Estimation and Simulation of Bacteriophage Infection Model –          Abigail Ballard (Co-mentors: Keith Emmert and Dustin Edwards) 
Texas Oklahoma Regional Undergraduate Symposium
Cameron University – Lawton, Ok February 15, 2020

⦁    Parameter Estimation and Simulation of Bacteriophage Infection Model –          Abigail Ballard (Co-mentors: Keith Emmert and Dustin Edwards)
⦁    Landing Rovers in Microgravity Environments – Sidney Davis and Jaryd            Domine
⦁    Randomly Walking Away from Gerrymandering – Diana Dinh-Andrus (Co-        mentor: Scott Cook)
⦁    Chemical Espionage – Dashon Mitchell and Aurod Ounsinegad
⦁    A Predator-Prey model for Largemouth Bass and Channel Catfish– Trace            Patterson
SIMIODE A Systemic Initiative for Modeling Investigations & Opportunities with Differential Equations
Texas A&M University-Commerce– Commerce, TX 11/9/2019
⦁    Chemical Espionage – Abigail Ballard, Dashon Mitchell, and Aurod                  Ounsinegad (Meritorious award) (Co-mentor: Bryant Wyatt)
⦁    Landing Rovers in Microgravity Environments – Johnny Seay, Sidney                Davis, and Jaryd Domine (Outstanding award)
15th Annual Texas Undergraduate Mathematics Conference
University of Texas at Tyler – Tyler, TX October 25-26, 2019
⦁    Parameter Estimation and Simulation of Bacteriophage Infection Model –          Abigail Ballard (Co-mentors: Keith Emmert and Dustin Edwards)
⦁    Big Jumps or Little Steps: Fighting Gerrymandering with Random Walks –        Tyra Buchanan and Shawn Brody (Co-mentor: Scott Cook)
100th Annual Meeting of the Texas Section of the MAA 
University of North Texas – Denton, TX March 27-29, 2020
⦁    Parameter Estimation and Simulation of Bacteriophage Infection Model –          Abigail Ballard (Co-mentors: Keith Emmert and Dustin Edwards)
⦁    Chemical Espionage – Dashon Mitchell and Aurod Ounsinegad 
124th Annual Meeting of the Texas Academy of Science
Virtual –February 26-27, 2020
⦁    Increase in exposure rates of the Eastern Equine Encephalitis Virus from            the black-tailed mosquito to avian species: a mathematical approach.                  Aurod Ounsinegad
⦁    Chemical espionage: modeling the relationship between the Pieris                      brassicae butterfly and Trichogramma wasps. Dashon Mitchell
⦁    A road most traveled: utilizing the concept of transit-time to measure                 compactness. Shawn Brody (Graduate Student) (Co-mentor: Scott Cook)
SIMIODE A Systemic Initiative for Modeling Investigations &                      Opportunities with Differential Equations
Virtual–November 14, 2020
⦁    Problem A Oil Decay – Robert Moore, Westen Halcom and Aurod                     Ounsinegad (Outstanding award)
Annual Meeting of the Texas Section of the MAA
Virtual – April 9, 2021
⦁    Increase in exposure rates of the Eastern Equine Encephalitis Virus from            the black-tailed mosquito to avian species: a mathematical approach.                  Aurod Ounsinegad
⦁    Chemical espionage: modeling the relationship between the Pieris                      brassicae butterfly and Trichogramma wasps. Dashon Mitchell
Local
Math Day 2020 (Virtual because of COVID-19)
Tarleton State University – Stephenville, TX April 28 2020
⦁    Parameter Estimation and Simulation of Bacteriophage Infection Model –          Abigail Ballard  (Third Place) (Co-mentors: Keith Emmert and Dustin                Edwards)
⦁    Chemical Espionage –Aurod Ounsinegad 
⦁    A Predator-Prey model for Largemouth Bass and Channel Catfish– Trace            Patterson
Tarleton College of Science and Technology Research Symposium
Tarleton State University – Stephenville, TX April 14, 2020
⦁    Parameter Estimation and Simulation of Bacteriophage Infection Model –          Abigail Ballard (Co-mentors: Keith Emmert and Dustin Edwards)
⦁    Chemical Espionage – Dashon Mitchell and Aurod Ounsinegad
⦁    A Predator-Prey model for Largemouth Bass and Channel Catfish– Trace           Patterson
Tarleton Student Research and Creative Activities Symposium
Tarleton State University – Stephenville, TX March, 19-20, 2020
⦁    Parameter Estimation and Simulation of Bacteriophage Infection Model –          Abigail Ballard (Co-mentors: Keith Emmert and Dustin Edwards)
⦁    Chemical Espionage – Dashon Mitchell and Aurod Ounsinegad
⦁    A Predator-Prey model for Largemouth Bass and Channel Catfish– Trace            Patterson
Math Day 2020 (Virtual because of COVID-19)

Tarleton State University – Stephenville, TX May 5, 2021
⦁    Increase in exposure rates of the Eastern Equine Encephalitis Virus from            the black-tailed mosquito to avian species: a mathematical approach.                  Aurod Ounsinegad
⦁    Chemical espionage: modeling the relationship between the Pieris                      brassicae butterfly and Trichogramma wasps. Dashon Mitchell
⦁    Ebola Outbreak of Guinea in 2014-2015, Ida Bjerregaard Nielsen                        (Graduate Student)
⦁    An Alternative Model for Zika Virus, Justin Sullivan (Graduate Student)
⦁    A Look at MEV-1, Sidney Davis
National:
Society of Mathematical Biology Annual Meeting - Virtual
⦁    Increase in exposure rates of the Eastern Equine Encephalitis Virus from            the black-tailed mosquito to avian species: a mathematical approach.                  Aurod Ounsinegad