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Summer 2001
The Role of Time Delay in the Fitzhugh-Nagumo Equations: The Impact of Alcohol on Neuron Firing
Romel S. Franca, University of Florida
Ivy E. Prendergast, Queens College, CUNY
Eva-Shirley Sanchez, Rutgers University
Marco A. Sánchez, California State Polytechnic University, Pomona
Project supervisors: Fiana Berezovsky, Georgia Institute of Technology
Stephen Wirkus, California State Polytechnic University, Pomona
BU-1577-M
Abstract:
Repeated and frequent alcohol use can have serious repercussions on the nervous system, particularly the brain. Here, we focus on the nerve cell (neuron), the fundamental component of the brain. In 1952, Hodgkin and Huxley received the Nobel Prize in Physiology and Medicine for their research on neuron dynamics (firing) with the help of a mathematical model. Our research begins with the assumption that alcohol impairs neuron dynamics. Hence, we begin with the Hodgkin and Huxley model but its complexity moved us to the Fitzhugh-Nagumo equations, a caricature of the Hodgkin and Huxley model. The Fitzhugh-Nagumo equations include only two variables, the membrane potential and the restoring force. In our research, we assumed that alcohol delays the effect of the restoring force and changes the normal state of the system. We analyzed the dynamics of the Fitzhugh-Nagumo equations with and without delay using computer simulations, the qualitative theory of dynamical systems, and bifurcation theory. Alternative hypotheses are discussed in the conclusions.
Am I Too Fat?: Bulimia as an Epidemic
Beverly Gonzalez, University Of Illinois at Urbana-Champaign
Emilia Huerta-Sanchez, Mills College-Oakland
Angela Ortiz-Nieves, Universidad Metropolitana-Cupey
Terannie Vazquez-Alvarez, Universidad Metropolitana-Cupey
Project supervisor:
Christopher Kribs-Zaleta, University of Texas-Arlington
BU-1578-M
Poster sessions award recipient at the National 2002 AMS convention in San Diego, CA
Abstract:
For at least the past ten years, eating disorders have had a major impact in the physical and mental health of women, particularly young women. Anorexia and Bulimia nervosa are closely linked eating disorders. Anorexia often precedes bulimia. However, there are about 2 million women in college that have been exclusively bulimic. In this article, we focus on the role of college-peer pressure on the dynamics of anorexia-free bulimia. The model looks at bulimia as a progressive disease and explores the impact of intervention (treatment) at two stages of disease progression. The impact of relapse (a common occurrence among bulimics) is taken into account. Extensions and connections to anorexia are discussed.
A Model of β-cell Mass, Insulin, Glucose, and Receptor Dynamics with Applications to Diabetes
Ryan D. Hernandez, Pitzer College
Danielle J. Lyles, University of Texas at San Antonio
Daniel B. Rubin, Stanford University
Thomas B. Voden, University of California, Riverside
Project supervisors:
Stephen Wirkus, California State Polytechnic University, Pomona
Ricardo Oliva, Cornell University
BU-1579-M
Abstract: Approximately 15.7 million people in the United States suffer from diabetes mellitus, of which about 90% are classified as type II. Most cases of type II diabetes mellitus are characterized by high blood glucose levels resulting from chronic insulin resistance, which then leads to significant beta-cell mass reduction from "beta-cell exhaustion" and/or "glucose toxicity". Existing mathematical models of beta-cell mass, insulin, and glucose kinetics contribute to the study of the disease by qualitatively and quantitatively describing different pathways to diabetes. Successful models of a complex system are often malleable, in that they can be extended to include further components, and consequently be a more complete representation of the system. Insulin receptor dynamics have not been previously considered in modeling the gluco-regulatory system, yet are important in the pathogenesis of the disease as chronic insulin resistance is associated with the down-regulation of these receptors at the surface of muscle cells. We incorporate the dynamics of insulin receptors into an existing mathematical model, resulting in a four dimensional system of nonlinear ordinary differential equations. Through analytical calculations and numerical simulations we conclude that coupling receptor dynamics is valuable in that our system extends previous models to include a fourth significant factor in diabetes, gives improved quantitative results in describing beta-cell mass, and provides a theoretical justification for experimentally observed receptor behavior.
Different Strokes for Different Folks: The Evolution of the Distribution of Traits within Heterogeneous Populations
Kara Cunningham, California State University, Bakersfield
Paul Hurtado, University of Southern Colorado
Heather Watson, Claremont McKenna College
Project supervisor:
Georgy Karev, Georgia Technical Institute
BU-1580-M
Abstract: Most population models assume that individuals within a given population are identical, that is, they ignore the fundamental role of variation. By understanding the dynamics within heterogeneous populations, we can more accurately predict population growth and composition. Using computational and analytical techniques, we consider Malthus, logistic and Allee growth models with several different initial parameter distributions. Our results highlight the importance of local dynamics on genetic diversity and, consequently, the role of natural selection on the evolution of traits.
Raves, Clubs, and Ecstasy: The Impact of Peer Pressure
Melissa Castillo-Garsow, Ithaca High School
Leilani Henson, Howard University
Marcin Mejran, Stuyvesant High School
Karen R. Rios-Soto, University of Puerto Rico-Mayaguez
Project supervisor:
Baojun Song, Cornell University
BU-1581-M
Abstract: MDMA (3,4 Methylenedioxymethamphetamine), commonly known as ecstasy, is a synthetic psychoactive drug, which in recent years has gained popularity among young adults who frequent raves and nightclubs. In fact, the Drug Enforcement Administration reported a 500% increase in the use of ecstasy between 1993 and 1998. In this study, a system of four nonlinear differential equations is used to model the peer-driven dynamics of ecstasy use. It is found that two backwards bifurcations describe situations when sufficient peer pressure can cause an epidemic of ecstasy use despite conditions predicting the opposite trend. Furthermore, factors, which have the greatest influence on ecstasy use, as predicted by the model, are highlighted. The impact of education is also explored, and the results of simulations using parameter values, are shown to illustrate some of the possible outcomes.
To Bt or Not to Bt? Balancing Spatial Genetic Heterogeneity to Control the Evolution of Ostrinia nubilalis
Conrad Miller, Southwestern University
Andres Munoz, City College of New York
Fernando Pena, University of Texas at San Antonio
Rosalyn Rael, Western New Mexico University
Project supervisor:
Abdul-Aziz Yakubu, Howard University and Cornell University
BU-1582-M
Abstract: Genetically modified corn crops have been developed to reduce the impact of potentially devastating agricultural pests such as the European Corn Borer, (Ostrinia nubilalis). Continuous exposure to Bt toxins in genetically modified corn results in the increased prevalence of European Corn Borers that are resistant to these toxins. In this article, we first analyze the evolution of resistance in a uniform environment using a system of nonlinear difference equations. The evolution of resistance is then simulated in spatially explicit environments based on the biology of the insect and using parameters found in the literature. The optimum initial conditions and various stripe patterns on a cornfield that will minimize the evolution of a resistant population are explored numerically.
Compressive Failure of Single Fiber Composites
Jermaine Baldwin, University of Chicago
Sherlyne Paret, SUNY Albany
Carlos A. Torre, Cornell University
Project supervisors:
S. L. Phoenix, Cornell University
Mason Porter, Cornell University
BU-1583-M
Abstract:
We present a model of a single fiber composite subject to a compressive axial load. The purpose is to better understand the origin of fiber failure as strain increases and describe the events subsequent to the initial fracture. Our model, a single fiber in a matrix, is a fourth order differential equation with a parameter in the second derivative dependant on the applied load/strain. This equation produces the oscillating solutions seen in experiments that describe the buckling phenomenon. We develop a model in order to show critical buckling failure effects, critical loads, and shape the fiber takes when fracture occurs and load is applied.
Metapopulation Models with Age Structure
Jermaine Baldwin, University of Chicago
Sherlyne Paret, SUNY Albany
Carlos A. Torre, Cornell University
Project supervisor:
Abdul-Aziz Yakubu, Howard University and Cornell University
BU-1584-M
Abstract: A principal aim of population biologists is to understand the role of intraspecific competition at the metapopulation level (populations of populations). We study the dynamics of a two-patch age-structured metapopulation model where the local (patch) intraspecific competition regimes are of the same type (scramble or contest) or mixed (scramble and contest) types. Metapopulations behave as single patch systems under the same competition regime whenever dispersal is symmetric and all local populations find themselves under contest competition regimes. However, multiple attractors are possible whenever a local patch is under scramble competition regime. The results of this research demonstrate that dispersal between patches, and age-structure provide an evolutionary advantage.
Mouse in the House: Looking at the Spread of Hantavirus in Houses Through the Deer Mouse Population
Brandon J. Brown, University of California, Irvine
Edgar Cabral, University of California, Irvine
Tiffany R. Hegg, Mesa State College
Project supervisors:
Carlos Castillo-Garsow, Cornell University
Baojun Song, Cornell University
BU-1585-M
Abstract: The Sin Nombre Virus is part of the Bunyaviridae family that causes hantavirus pulmonary syndrome. The deer mouse, the primary host of Sin Nombre Virus, supports a prevalence of about 25% in its adult population. Since deer mice are typically found in fields, homes, and barns, we examine the risk of infection Sin Nombre Virus poses on humans by looking at the dynamics of the deer mouse population as it moves through homes and barns in rural areas within western Colorado. Hence, the barn and the house are our epidemiological units and, consequently, it is initially assumed that each unit is in one of three infestation states, that is, at zero, low or high mouse infestation. The threshold that governs the likelihood of an epidemiological outbreak is computed. Explicit spatial simulations of small communities that involve the movement of mice and their seasonally driven reproductive capacities are carried out. The impacts of control measures are tested in the stochastic frameworks.
Who Says We R0 Ready for Change?
Nicolas Crisosto, University of California-Berkeley
Project supervisors:
Carlos Castillo-Chávez, Cornell University
Christopher Kribs-Zaleta, University of Texas-Arlington
Stephen Wirkus, California State Polytechnic University, Pomona
BU-1586-M
Abstract: A primary assumption in this article is that individual-based learning tends to use local information to increase an individual's "fitness" while collaborative learning, based on the sharing of information, knowledge and resources, increases group fitness. We frame our discussion about the importance of cooperative learning at the community and individual level with theories of intellectual development based on the views of Vygotsky ("individualistic") and Piaget ("social"), and mediated by concepts and ideas from the fields of epidemiology and evolutionary biology. Our approach is motivated by evolutionary biology metaphors and modeled via epidemiological (contact) processes. Furthermore, using a simple cooperative learning model, we address the belief that sharp community thresholds characterize separate learning cultures such that one must cross a tipping point to move from one culture to the other. Our results allow us to discuss the impact of individual learning on community intellectual development and its resilience to change.
Effects of Education, Vaccination and Treatment on HIV Transmission in Homosexuals with Genetic Heterogeneity
Sara Del Valle, University of Iowa
Arlene Morales Evangelista, Arizona State University
María Cristina Velasco, Universidad del Valle, Columbia
Project supervisors:
Christopher Kribs-Zaleta, University of Texas- Arlington
Carlos Castillo-Chávez, Cornell University
BU-1587-M
Poster sessions award recipient at the National 2001 SACNAS convention in Phoenix, AZ
Abstract:
Genetic studies report the existence of a mutant allele Δ32 of CCR5 chemokine receptor gene at high allele frequencies (~10 %) in Caucasian populations. The presence of this allele is believed to provide partial or full resistance to HIV. In this study, we look at the impact of education, temporarily effective vaccines and therapies on the dynamics of HIV in homosexually active populations. In our model, it is assumed that some individuals possess an allele (like Δ32 of CCR5) that prevents the successful invasion or replication of HIV. Our model therefore differentiates by genetic and epidemiological status and naturally ignores the reproduction process. Furthermore, HIV infected individuals are classified as rapid, normal or slow progressors. In this complex setting, the basic reproductive number R0 is derived in various situations. The separate or combined effect of therapies, education and vaccines are analyzed. Our results support the conclusions of Shu-Fang Hsu Schmitz that some integrated intervention strategies are far superior to those based on a single approach.
Disease Dynamics on Small-World and Other Networks
Gerardo Chowell-Puente, Cornell University
Fabio Sanchez, Cornell University
Project supervisors:
Juan P. Aparicio, Universidad de Belgrano, Argentina
Carlos Castillo-Chávez, Cornell University
BU-1588-M
Abstract: In 1998 Watts and Strogatz introduced the concepts of small-world networks and in the process expanded our views of computer, social, and biological networks. In this project, we built epidemics on small world and other networks. Epidemic outbreaks of communicable and sexually transmitted diseases are modeled on small-world and two-node networks, respectively. The results of simulations are compared to those obtained from homogeneous mixing (mean field) epidemic models. Scaling relationships between transmission rates for epidemics on small-world, random and homogeneous mixing populations are established empirically. The transmission dynamics of gonorrhea in heterosexually active populations with multiple partners is used to illustrate the spread of disease on two-node networks. Strategies for disease control are explored.
Social Mobility and the Evolution of Tuberculosis
Elmer De La Pava - Salgado, Corporacion Autonoma de Occidente, Colombia
Beatriz Salguero - Rivera, Corporacion Autonoma de Occidente, Colombia
Project supervisor:
Juan P. Aparicio, Universidad de Belgrano, Argentina
BU-1589-M
Abstract:
The explicit causes of the historic decline of mortality and morbidity rates of tuberculosis (TB) have not yet been clearly understood. Two different hypotheses have been proposed: a) The influence of public health programs against transmission of tuberculosis. b) The improvement of living standards, which decreased the likelihood of progression to active-TB; and Hypothesis (b) was already tested for the simple case of homogeneous population. Nevertheless, it is known that there exists a strong positive correlation between incidence of active-TB and poverty. In this work we introduce some degree of population heterogeneity. Population is divided in two classes: One is below the poverty level and the other one is above it is assumed that risk of progression to active-TB is greater in the population living below poverty level. United States data on poverty levels (measured by annual household income) is used in order to approximate the time evolution of the size of the population living below poverty levels.