Mathematical & Theoretical Biology Institute

 

Summer Undergraduate Research Program Colloquium 2008

 

Presentations are in the PSH-153

Thursday July 31, 2008

 

 

 

 

 

8:00 AM - 8:25 AM

A Mechanism for Stabilizing Oscillations in Certain Nonlinear Systems Possessing Different time Scales

Raquel M. Lopez, Department of Mathematics and Statistics, Arizona State University

Erika T. Camacho, Department of Mathematics and Statistics, Arizona State University-West

Sergei K. Suslov, Department of Mathematics and Statistics, Arizona State University

Abstract

There are many natural, physical, and biological systems that exhibit multiple time scales.  For example, the dynamics of a population of ticks can be described in continuous time during their individual life cycle yet discrete time is used to describe the generation of offspring.  These characteristics cause the population levels to be reset periodically.  A similar phenomenon can be observed in a sociological college drinking model in which the population is reset by the incoming class each year, as described in the 2006 work of Camacho et al.  With the latter as our motivation we analytically and numerically investigate the mechanism by which solutions in certain systems with this resetting characteristics stabilize.  We analyze certain one-dimensional and two-dimensional nonlinear systems, and try to generalize our results to higher dimensions.

8:30 AM - 8:55 AM

The Effects of Clustered Unvaccinated Individuals on Epidemic Outbreak

Hamilton Hoxie Ackerman, Department of Mathematics and Statistics, Boston University

Isaac J. Michaud, Department of Mathematics and Statistics, University of Maine

Shannon Reed, Department of Mathematics, University of Missouri - Columbia

Andre Robinson, Department of Mathematics, Medgar Evers College

Graduate Mentors:

Mat Gluck, Department of Mathematics, University of Florida,

Naala Brewer, Department of Mathematics and Statistics, Arizona State University

Griselle Torres-Garcia, Department of Mathematics and Statistics, Arizona State University

Faculty Advisor:

David E. Hiebeler, Department of Mathematics and Statistics, University of Maine

Abstract

An SIR epidemiological household model is constructed and studied to understand the effects that clusters of unvaccinated individuals have on the disease dynamics of a population. The model contains two levels of mixing where individuals make more intra-household than inter-household contacts. Stochastic simulations, numerical solutions are utilized to explore the model. A new extension of the basic reproductive number that incorporates more spatial information by predicting the average number of tertiary infections caused by a single infected individual is introduced to describe the threshold behavior and severity of an epidemic. Using these methods we show that clustering of unvaccinated individuals always leads to more severe epidemics.

 

9:00 AM - 9:25 AM

The Effects of Maternal Age on the Prevalence of Autism

Melissa A. Bilbao, Department of Mathematics and Statistics, California State Polytechnic University

Alexander D. Castro, Department of Mathematics, University of Illinois at Urbana-Champaign

Tyler A. Rigazio, Department of Mathematics and Statistics, University of Maine

Graduate Mentors:

Naala Brewer, Department of Mathematics and Statistics, Arizona State University

Kimberly Rude, Department of Mathematics, Montclair State University

Faculty Advisors:

Stephen Tennenbaum, Department of Mathematics and Statistics, Arizona State University

Baojun Song, Department of Mathematics, Montclair State University

Xiaohong Wang, Department of Mathematics and Statistics, Arizona State University

Abstract

Autism's cause is unknown, but suggested causes are often attributed to genetic or environmental factors. This research examines whether advancing maternal age contributes to the increasing prevalence of autism. The model used to achieve this objective generates values that represent the proportion of offspring expected to be diagnosed with autism, provided their mother belongs to a specific age class. The age class consisting of ages 40-44 was most affected. From these values, projections were made about the prevalence of autism in future populations, specifically, for the United States and California. These projections predict a continued increase in the prevalence of autism.

9:30 AM - 9:55 AM

Analyzing the Treatment of Hepatitis C Infection with Respect to Resulting Hemolytic Anemia.

Gloriell M. Cardona-Melendez, University of Puerto Rico-Cayey,

Swati DebRoy, Department Of Mathematics, University of Florida

MinJun Kang, Kyungpook National University, Daegu, Korea

Liana Medina-Rios, Mount Holyoke College

Graduate Mentors:

Edgar Diaz, Department of Mathematics and Statistics, Arizona State University

Anuj Mubayi, Department of Mathematics and Statistics, Arizona State University

Faculty Advisors:

Christopher Kribs, Department of Mathematics, University of Texas-Arlington

Baojun Song, Department of Mathematics, Montclair State University

Abstract

The combination therapy of antiviral peg-IFN and ribavirin (RBV) has evolved as one of the better treatments for Hepatitis-C (HCV). In spite of its success in controlling HCV infection, it has also been associated with treatment related adverse side-effects. The most common among them is hemolytic anemia caused by premature breakdown of hemoglobin in the red blood cell (RBC). More than 67% of treated chronic HCV patients show signs of acute anemia leading to dose reduction or even therapy cessation. This paper extends the basic mathematical model of Dahari et. al. and study the effect of combination therapy in light of anemia. In order to achieve this we introduce RBC and drug concentration in the model. Analysis of this model provides a quantification of the drug concentration that is tolerable by the body without succumbing to hemolytic anemia. This will provide an estimate of the increment in RBC production necessary to keep from hemolytic anemia and settle on a balanced HCV treatment. 

 

10:00 AM - 10:25 AM .

Summer School Opportunities: When there is a will, which is the way?

Ricardo Cordero,  Department of Mathematics & Statistics, Arizona State University

Cristi Guevara,  Department of Mathematics & Statistics, Arizona State University

Erwin Suazo,  Department of Mathematics & Statistics, Arizona State University

Abstract

The path to a Ph. D. in mathematics, and in any field requires hard work and sacrifices. Moreover, meeting the right people and advisors is essential no only for academic guidance, but also to our research and finding the right opportunities. Every summer there are many conferences and summer research programs going on around the world, and chances of finding one in your research area. The application and admission process, and the search for funding, are not easy tasks.We will share our personal experiences attending the 8th International on Harmonic Analysis and PDE conference at Escorial-Spain and CMI summer school on Evolution Equations in Zurich, Switzerland, and how our advisors played a key role in accomplishing this.

 

10:30 AM - 10:55 AM       Coffee and Tea

 

11:00 AM - 11:25AM

A Tale of Two Regions: A Mathematical Model for Chagas'Disease.

Alhaji Cherif, Cornell University

Viviana Garcia Horton, Instituto Tecnologico Autonomo de Mexico

Glorimar Melendez Rosario, Universidad de Puerto Rico-Cayey

William Feliciano, Universidad de Puerto Rico-Cayey

Graduate Mentors:

Jose Vega, Department of Mathematics and Statistics, Arizona State University

Britnee Crawford, Department of Mathematics, University of Texas-Arlington

Faculty Advisors:

Christopher Kribs, Department of Mathematics, University of Texas-Arlington

Fabio Sanchez, OPEN Risk Management team, American Express Inc.

Abstract

In this paper, we model the epidemiological interactions between two populations: one with a virulent strain of Trypanosoma cruzi, which causes Chagas' disease, the other with a non-virulent strain that provides cross immunity against the disease. The chagasic strains of T. cruzi are predominantly found in Latin America. Recent field work has shown that an infective strain is migrating (asymmetrically, via vectors) to the southeastern United States, where the non-virulent strains are native. One explanation of such observation is due to changes in the climate and other environmental conditions. As a result, the southern U.S. is becoming habitable for the virulent strains. The model presented herein describes the effects of the migrating chagasic strains on the prevalence and/or possibility of endemicity of the Chagas' disease in the United States. We use an epidemiological modeling paradigm and an analytical framework of nonlinear dynamics to describe the behavior of the two populations and their symmetric interactions (through a migratory function). Possible ecological and preventive strategies are suggested.

 

11:30 AM - 11:55 AM

The Effects of Estrogen and Chemotherapy on the Dynamics of Invasive Carcinoma of Breast Cancer Patients

Cindy Coralie Jackson, Department of Mathematics and Statistics, California State Polytechnic University-Pomona

Lindsey K. Lauderdale, Department of Mathematics, University of Illinois at Urbana-Champaign

Nicholas Earl Millett, Department of Mathematics and Statistics, The University of Maine

Samantha Anne Smee, Department of Mathematics, Oregon State University

Adrian Smith, Department of Mathematics, University of Washington

Graduate Mentors:

Kevin Flores, Department of Mathematics and Statistics, Arizona State University

Faculty Advisors:

Faina Berezovskaya, Department of Mathematics, Howard University

Stephen Wirkus, Department of Mathematics, Arizona State University-West

Abstract

In this paper two mathematical models are used to examine two different drug treatment regiments on estrogen receptor-positive breast cancer in women. The first regiment is a combination therapy of estrogen stimulation followed by chemotherapy and the second regiment is a single treatment of chemotherapy alone. The method under consideration is to first use estrogen therapy on a diagnosed advance staged breast cancer patient to move the tumor cells into the proliferating stage, at which time chemotherapy can be applied to kill the proliferating cancer cells. Utilizing both analytical and numerical approaches, a study of the efficacy of combination treatment and the single treatment is completed. Two partially decoupled models are created to study both a healthy cell population and a cancerous cell population in the breast. At first the cancerous cell population (tumor growth) is analyzed separately from the healthy cell population, and then later composed with the healthy cell model to examine the dynamics of cancer and treatment on the body. Both of the populations are divided into quiescent and proliferating stages in order to account for the cell cycle specific treatment that are later applied. The resulting models are given by four dimensional nonlinear systems.

 

12:00 PM - 12:25 PM

A Mathematical Approach to the Nemesis of Consciousness - Alzheimer's: The Cyclic Compartmental Model

Helme Castro, School of Materials, Arizona State University

Sunhwa Choi, Department of Mathematics, Konkuk University, Seoul, Korea

Natalia Rodriguez, Departamento de Matematica-Fisica, Universidad de Puerto Rico-Cayey,

YuJie Shui, Department of Biomedical Engineering, Boston University

Graduate Mentors:

David Tello, Department of Mathematics and Statistics, Arizona State University

Griselle Torres-Garcia, Department of Mathematics and Statistics, Arizona State University

Faculty Advisors:

Joaquin Rivera, Department of Mathematics and Statistics, Arizona State University

Abdessemad Tridane, Department of Mathematics and Statistics, Arizona State University

Abstract

Whether researchers favor the physiological or psychological approach, the agreement remains, Alzheimer's disease (AD) is the enemy of consciousness, especially that of elders. From pharmaceutical products and imaging techniques to mental discussion and memory training, scientists have not discovered a successful method to prevent AD's inevitable progression. However, recent research discovers that rats have acquired certain AD symptoms upon the injection of Amyloid-beta-42 (Aβ₄₂) protein. While Amyloid beta protein has always been a theory to Alzheimer's cause, researchers were unable to distinguish whether this protein is a cause or a symptom until this discovery. With this result, our research suggests that Aβ₄₂ is one of the puzzling reasons for AD. Particularly, the input and output around Aβ₄₂ will be the basis of this mathematical modeling approach. For our purposes, the modeling proposal follows a continuous cycle observed solely through AD patients. Building on the work of Reed et al. (2002), our aim is to describe the fundamental aspects of this common cycle among Alzheimer's patients, and then propose a qualitative initiation of AD. With an emphasis on the dynamics of methionine (Met) and homocysteine (Hcy) in Reed's Methionine Cycle, we monitor the qualitative decrease of Aβ₄₂ level to determine the progression of AD.

 

12:30 PM - 12:55 PM

Introduction to the EM Algorithm

Kenhinde  Salau, Department of Mathematics & Statistics, Arizona State University

Naala Brewer, Department of Mathematics & Statistics, Arizona State University

 

Abstract

The EM algorithm is a well known tool for estimating unknown data points from a set of known, but incomplete data.This intends to be an introduction to this specific form of data estimation. Several types of distributions will be examined and optimized using the EM algorithm, with coding instructions in R, MATLAB, and C++.Comparison between the observed and complete data will be displayed to show the effectiveness of the algorithm.The EM algorithm, as well as other forms of data estimation, have important applications, and ramifications, in the realm of math-biology.

 

1:15 PM       Reception in PSA-206

The MTBI/SUMS Summer Undergraduate Research Program is supported by The National Science Foundation (DMS-0502349), The National Security Agency (DOD-H982300710096), The Sloan Foundation,  and Arizona State University.