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Summer 1999
Competitive Exclusion in a Discrete-Time Epidemic Model with Two Competing Strains
Judith Pérez Velázquez, Fac. Ciencias, UNAM. México
BU-1514-M
Abstract:
In this paper, we study a Susceptible-Infected-Susceptible (S-I-S) discrete-time model with two competing strains and distinct demographic dynamics. We use two different recruitment functions to model the population demography. Discrete models are capable of generating complex dynamics. Analytic and numerical methods are used in this study to separate the impact of demography on the epidemic process. The key issues addressed in this work are those of coexistence and/or competitive exclusion of competing strains in a population with complex dynamics. The principle of competitive exclusion is the most prevalent under the assumptions of this model.
Scramble Versus Contest Competition in a Two Patch System
Beatriz Salguero-Rivera, Corporación Universitaria Autónoma de Occidente- Cali, Colombia
BU-1515-M
Abstract: The population dynamics in two patch discrete-time systems are different if they are modeled by the Logistic and Verhulst equations. First, we analyze one - patch systems modeled with each of the equations. Then we study two patch systems, where the local dynamics between the two patches are coupled with Logistic equations regulated by the scramble competition. We also compare this with the Verhulst equation where the behavior is regulated by contest competition. It is known that fractal basin boundaries of attractors could occur in two-patch systems regulated by the scramble competition. We show that such complex dynamics do not occur in contest competition with dispersion between patches.
Numerical Solution of a Model of Influenza with Two Strains
Elmer de la Pava-Salgado, Corporación Universitaria Autónoma de Occidente- Cali, Colombia
BU-1516-M
Abstract: The epidemiological models with age structure, proportionate mixing, and cross-immunity which have been studied by Castillo Chávez et al. (1989). The removal of age-structure from the two strain models led to damped oscillations. Numerical simulations of a discrete version gave rise to sustained oscillators provided that age structure, two co-circulating viral strains, and cross-immunity was included. The hypothesis that the interaction between cross-immunity and age dependent survivorship may be enough to drive sustained oscillations. In this paper we simulate the continuous time version of this model using an algorithm based on the finite difference method used by Milner et al. (1993). The numerical scheme is proved to converge.
Dynamics of the Spruce Budworm Population under the Action of Predation and Insecticides
Patricia Arriola, California State University of Los Angeles
Irene Mijares-Bernal, Universidad Aut&oactue;noma de Cd. Juárez
Juan Ariel Ortiz-Navarro, University of Puerto Rico-Humacao
Roberto A. Sáenz, Universidad Autónoma de Cd. Juárez
BU-1517-M
Abstract: In this paper we study the dynamics of the spruce budworm system under either bird predation pressure, insecticide budworm disease mortality, or both assuming that the biomass remains constant. A model with two equations describing the dynamics between birds and budworms is proposed. It is shown that under certain conditions this two dimensional system has limit cycles, that is, it predicts the existence of periodic budworm population outbreaks. Numerical simulations illustrate the action of insecticides on the budworm population size. The possibility of controlling the budworm population via insecticides is explored.
Dynamics of a Two-Dimensional Discrete-Time SIS Model
Jaime H. Barrera, Cornell University
Ariel Cintrón-Arias, Cornell University
Nicolas Davidenko, Harvard University
Lisa R. Denogean, Cornell University
Saúl Ramón Franco-Gonzalez, University of California-Irvine
BU-1581-M
Abstract: We analyze a two-dimensional discrete-time SIS model with a non-constant total population. Our goal is to determine the interaction between the total population, the susceptible class and the infective class, and the implications this may have for the disease dynamics. Utilizing a constant recruitment rate in the susceptible class, it is possible to assume the existence of an asymptotic limiting equation which enables us to reduce the system of two-equations into a single, dynamically equivalent equation. In this case, we are able to demonstrate the global stability of the disease-free and the endemic equilibria when the basic reproductive number (R0) is less than one and greater than one, respectively. When we consider a non-constant recruitment rate, the total population bifurcates as we vary the birth rate and the death rate. Using computer simulations, we observe different behavior among the infective class and the total population, and possibly, the occurrence of a strange attractor.
The Ebola Virus: Factors Affecting the Dynamics of the Disease
Rogelio Arreola, University of California-Irvine
Damon Dwayne McDuffy, University of Wisconsin, Stevens Point
Miriam Berenice Mejía, California State University, Long Beach
Anike Imani Oliver, Howard University, Washington D.C.
BU-1519-M
Abstract: We analyze an Ebola epidemic using Susceptible-Infected-Recovered (S-I-R), deterministic and stochastic models. The models include two stages for the infectious class, a recovery class, and a quarantined class. The stochastic model was analyzed using simulations. We report on factors such as epidemic size, mean time to extinction, and the number of fatalities under a wide set of parameters. An increase in the quarantine rate will reduce the size of an epidemic. However, it will not reduce the mean time to extinction of the epidemic. As you quarantine symptomatic infectives at a high rate, the individuals in the asymptomatic stage of infection will mostly spread the disease. As there is a greater difference between per capita infectivity rates, the epidemic size also increases.
Interactions between Dispersal and Dynamics: Coupled Ricker's Equations
Eratóstenes Flores Torres, Universidad Nacional Autónoma de México
BU-1520-M
Abstract:
We study a one patch model using Ricker's equation, xn-1 = xner-xn, r ≥ 0. We reproduce some results that Hastings (1993) obtained by coupling two discrete time logistic equations. Multiple attractors could occur with dispersion where there is only one attractor without dispersion. The boundary of the basins of attraction of the attractors can be fractal in nature. This makes prediction of the asymptotic behavior of most initial conditions difficult to analyze. Furthermore, we study the same model using Ricker's equation. We show that the qualitative nature of the results for a system of difference equations with dispersion depends on the form of the local dynamics.
Urn, Baby, Urn: A Simpler Approach to Studying Epidemic Models and the Efficacy of Disease-Prevention Policies
Michael Stuart Montgomery Lanham, Centre College
Desiree Mesa, California State Polytechnic University, Pomona
Jesús Francisco Rodríguez, Cornell University
Dianna Soliz Torres, Westminster College
BU-1521-M
Abstract: In this paper, we explore the use of urn models to describe and analyze the long-term effects of various disease-controlling policies on the spread of an infection through a population. These policies include vaccination with all/nothing vaccines or leaky vaccines, ring vaccination with the all/nothing vaccine, and contact tracing. The populations are described via the Susceptible-Infected-Susceptible (SIS) model with demography, the SIS model without demography and the Susceptible-Infected-Recovered (SIR) and Susceptible-Infected-Recovered-Susceptible (SIR) models. We show that these urn models, often simpler than their respective deterministic and stochastic counterparts, yield, in most cases, the same average qualitative dynamics. We also examine the definitions and implications of equilibria and stability in these systems.
Differential Equation Models of Neoadjuvant Chemotherapeutic Treatment Strategies for Stage III Breast Cancer
Edith Aguirre, University of Texas- El Paso
Tametra Smith, Howard University
Jennifer Stancil, University of North Carolina-Chapel Hill
Nicolas Davidenko, Harvard University
BU-1522-M
Abstract: In this study, we investigate different neoadjuvant chemotherapy treatment strategies for patients with Stage III breast cancer. We use two deterministic models to illustrate the effect of chemotherapy on tumor growth and a patient's health. The models consist of a system of three differential equations representing (1) the tumor growth rate, (2) the change in the patient's health (a variable between 0 and 1), and (3) the rate at which the drug combination, cyclophosphamide and adriamycin, dissipates after administration. The intrinsic growth rate of the tumor is exponential in model A, and logistic in model B. In our numerical solutions, success is determined by whether the tumor can be reduced to an operable size (1 x 109 cells) before the patient's health falls to a fatal level (0.1). By varying the frequency and dosage of chemotherapy, we evaluate the effectiveness of various treatment schedules for patients beginning treatment with different sized tumors. We observe that patients diagnosed with smaller tumors show faster success if the chemotherapy dosages are kept constant. On the other hand, patients with larger tumors survive longer when the dosage at each chemotherapy session is proportional to the tumor size and the patient's health.
Probability of False Prostate Cancer Diagnosis Using a Logistic Function Given Age and f/t PSA Ratio
Washington, DC
Aldo Crossa, Wittenberg University
Johnny Guzmán, California State University, Long Beach
Andie Hodge, University of the Virgin Islands
Brisa N. Sánchez, University of Texas, El Paso
BU-1523-M
Poster session award recipient at the National 1999 SACNAS convention in Portland
Poster session award recipient at the National 2000 Joint Math Meeting (AMS, MAA, AWM & NAM) in Washington, DC
Abstract: Prostate specific antigen (PSA) is a protein of free and complex forms found in the bloodstream of patients suffering from prostatic diseases. The free/total PSA ratio is often used in the detection of prostate cancer. We define a logistic function that assigns the probability of having cancer given a specific value of the f/t PSA ratio and the age of the patient. Since the levels of PSA in a serum sample are affected by storage conditions over time, we construct a model to show the effect of improper storage of serum samples on the f/t PSA readings. We then use the logistic function and the model of sample decay over time to determine the probability of a false positive given the storage conditions. In addition we provide an analysis of cut-off values, and a function that predicts the time necessary for the diagnosis of a patient to change from negative to positive given his age, f/t PSA Ratio, and a cut-off value.
An S-I-S Model of Streptococcal Disease with a Class of Beta-Hemolytic Carriers
Kevin Doura, Howard University
Julio D. Meléndez-Morales, University of Puerto Rico, Humacao
Gigi G. Meyer, Virginia Polytechnic Institute and State University
Luis E. Pérez, University of California, Irvine
BU-1524-M
Abstract: We analyzed the dynamics of an epidemic in a population infected with Streptococcal pyogenes (S. pyogenes), the causative agent in strep throat, with a Susceptible--Infected--Susceptible (S-I-S) model that includes an extra class of infectious carriers. Our model represents a three dimensional nonlinear differential equation system, which describes the spread of the disease in a population with three epidemiological classes: susceptible (S), infected (I) and beta-hemolytic carriers (C). We focus on the impact that the classes (I) and (C) have on (S), and the rate at which groups move in and out of the infectious state. Lastly, we study the long term dynamics of the disease in the population.