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Summer 1997
A Competition Model for Advertised Companies
Victor A. Chacón, Occidental College
Patricia Fuentes, Loyola Marymount University
George F. Gonzalez, Rice University
Mario A. Mendieta, Universidad Autónoma Metropolitana-Iztapalapa
BU-1420-M
Abstract:
Television advertising plays a major role in influencing people to purchase a company's product. In this project, we focus on the competition between two of the top beer companies, Budweiser and Coors Light. Both companies compete for beer drinkers as well as for television advertisements. To explain this competition, we develop an S-I-S (Susceptible-Infected-Susceptible) model utilizing five differential equations in which a susceptible person is an average beer drinker who watches television, and an infected person is one who purchases one of the two major brands. Two versions of this basic model are developed. The first version of the model takes into account only competition between Budweiser and Coors Light for beer drinkers. The second version takes into account both the competition for consumers as well as for television air time. One finds that under certain conditions, one of four states occur. In one state, neither Budweiser nor Coors Light attract any beer drinkers through television advertising. In two complementary cases one beer company influences many beer drinkers to purchase its particular brand. Finally, both beer companies influence enough beer drinkers to purchase their particular brand so that they both coexist in the television advertising market.
Mean Time to Extinction of Source-Sink Metapopulation for Different Spatial Considerations
Agustín Izquierdo-Sabido, Universidad Nacional Autónoma de México
Jessica Lasky, Cornell University
Mark Muktoyuk, Oregon State University
Selim A. Sabillón, Palo Alto College
BU-1421-M
Abstract: In a stochastic spatial source-sink process, we model the spread of a single species through a patchy landscape (i e. one that arises as a result of increasing urban sprawl). This landscape contains discrete patches, known as sink and source patches, that are associated with low and high habitat quality respectively. First, we generate different spatial relations of sink and source patches and place an initial population randomly throughout the grid of patches. We then simulate several discrete generations in which the species is exposed to different probabilities of reproduction and death, and we observe the mean time to extinction of the species with different combinations of patch distributions. Finally, we show that the size of source patch groupings has an effect on the mean time to extinction.
Can We Get a Head Start on Head Lice?
Veronica Ayala-Prado, University of California-Irvine
Claudia A. Catalán, Loyola Marymount University
Nohora Milena Londoño-Alzante, Universidad de Quindío-Armenia
Joaqúin Rivera-Cruz, Universidad de Puerto Rico-Cayey
BM-1422-M
Abstract: Head lice, Pediculus humanus capitis, a common parasite found world-wide, affect children most often. School staffs frequently lack the necessary knowledge and expertise to control lice epidemics. The observed properties of the distribution of lice in humans result from a "macroparasitic" model. In this article, we provide a simple model to better understand the mechanisms behind the distribution of head lice among humans. We also provide an epidemiological model that models the transmission dynamics of lice in humans. The basic reproductive number is computed and involves two terms: the transmission of lice to of susceptible individuals by those with few and many lice, respectively. We establish conditions for the eradication and persistence of lice in humans.
Dynamics of Rubella Virus in Populations with Different Vaccination Policies
Jaqueline Flores, Pomona College
Cristina Garcia, Pomona College
Carlos W. Melgar, University of California-Riverside
Catalina Saenz, Wellesley College
BU-1424-M
Abstract: Rubella is a contagious disease that affects individuals around the world. This mild disease becomes critical when a susceptible pregnant woman is infected. The fetus has a high risk of developing congenital rubella syndrome that leads to malformations and stillbirths. Interaction between countries with different vaccination policies can lead to an increase of rubella cases in either country. In this study we use mathematical models of differential equations to analyze how the interaction between Mexico and the United States affects the dynamics of Rubella. We show that the United States and Mexico must develop a dual policy to succeed in eradicating the rubella virus.
Tumor Growth Dynamics: A Deterministic and Stochastic Analysis of the Interaction Between Normal and Abnormal Cells
Brendaliz Acosta, Universidad de Puerto Rico-Cayey
Jaime H. Barrera, Texas A&M University
Ernesto S. Clarke II, Pitzer College
Nicolas Davidenko, Harvard University
Derek Ting, Cornell University
BU-1425-M
Abstract: We study the interactions between normal and abnormal cell populations as they occur in a tumorous growth. The purpose of our research is determine whether the spatial arrangement of abnormal cells in a tissue is a significant factor governing the spread of the tumor. To this end, we model how normal and abnormal cells compete for nutrients using a deterministic model and a spatial stochastic model. We vary nutrient competition rates as well as drug treatment effects for the two cell populations. The deterministic model indicates how the populations interact without consideration of spatial arrangement, while the stochastic model includes this factor. Our results show that different spatial arrangements of cells may cause significant differences in the growth dynamics of the cells even if the initial population sizes are kept constant. We have found that the spatial model reveals some growth dynamics that the deterministic model overlooks. Therefore it is of interest to obtain more realistic spatial models. For this, we need to focus research on the most distinctive factor of the spatial model: how normal and malignant cells on the boundary of a tumor compete for nutrients.
Models for the Transmission Dynamics of Gonorrhea in a Homosexually-Active Population
Sharon K. Lima, Purdue University
Mabel Torres, University of Miami
BU-1426-M
Abstract: The purpose of this project is to study two questions associated with the transmission dynamics of gonorrhea in a homosexually-active population. First, we examine the impact of a partially effective vaccine on gonorrhea dynamics. Second, we analyze the role of an antibiotic-driven mutation with respect to the survival and spread of resistant gonorrhea strains. Third, we study a model that combines the factors -vaccination and multiple strains of gonorrhea-considered in the first two models. We also explore the influence of heterogeneity and age-structure. Finally, we give a partial mathematical analysis of the system, including the computation of its basic reproduction number.
Dynamics of HIV/AIDS in Core Groups in the Presence of a Transient Population
Dámaris Santana-Morant, Universidad de Puerto Rico-Humacao
Moira Zellner, Universidad de Belgrano
BU-1427-M
Abstract:
We investigate the role of a transient population of male individuals who interact with a core group of prostitutes in the dynamics of a sexually transmitted disease. We use a stochastic model to follow the evolution of the individuals of each population, in terms of the disease. The analysis is concentrated in the equilibrium of the process.
We adapt this model to a city on the U.S.-Mexico border, where some of the parameters have already been estimated. In this way, we seek to provide a method of evaluating the effect of U.S. border policy in the transmission of diseases to the U.S.
Three Models for Measles Control
Carlos William Castillo-Garsow, Cornell University
Julio César Villarreal, Cornell University
BU-1428-M
Abstract: We study the dynamics of three models consisting of infant vaccination and booster shots to control the spread of measles. Varying the rates of infant vaccination and administration of booster shots, we alter the characteristics of the basic reproductive number to study the effectiveness of both infant vaccination and booster shot control methods. In two models, we find that total effort on both birth vaccination and boosters lowers cases of recovered and infected individuals to possibly eradicate measles.
Macrophage-Activating and Tissue-Damaging Imune Responses to M.tuberculosis
Ariel E. Rodríguez-Herrera, Stanford University
Guarionex Jordán-Salivia, Universidad de Puerto Rico-Río Piedras
BU-1429-M
Abstract: The two principal immune responses against Mycobacterium tuberculosis are the macrophage-activating response (MAR) and the tissue-damaging response (TDR). In the TDR, T-lymphocytes kill those macrophages that permit uncontrolled bacillary growth in their cytoplasm, this mechanism is believed to be responsible for the caseous necrosis in the host's lung tissue. In the MAR, macrophages are activated by T-lymphocytes in order to digest the bacilli as they are engulfed. With the use of a mathematical model we intend to find a proper interplay of both mechanisms to obtain an effective immune response where tissue damage is minimized and still the disease is eradicated.
Analysis of an Age-structured Epidemic Model with a Chronic State
Ricardo A. Sáenz, University of Texas at El Paso
BU-1430-M
Abstract: In this work we study an age-structured epidemic model with a chronic state for Cytomegalovirus. We divide the population in three groups: susceptible, infectious and chronic, where the chronic state is structured by the amount of time spent in that state. It is assumed that susceptible individuals can be infected by infectious and by chronic. Infectious individuals may recover or become chronic. We study the stability of the model around the disease-free equilibrium and interpret it in terms of the parameters associated with the model's basic reproductive number.