| |
Summer 1996
A Model Describing the Response of the Immune System to Mycobacterum Tuberculosis
Christian Herrera, University of California-Los Angeles
Sharon K. Lima, Loyola Marymount University
Roberto Muñoz, Universidad de Puerto Rico-Humacao
Gloria Ramos, University of California-Santa Cruz
Ariel Rodríguez, Universidad de Puerto Rico-Rio Piedras
Claudia Salzberg, Brown University
BU-1364-M
Abstract:
We present a mathematical model to describe the dynamics of the immune system in the presence of the causative agent of tuberculosis, Mycobacterium tuberculosis. We take into consideration the relations between the bacteria, T lamphocytes, and macrophages. We compute the basic reproductive number to determine under which conditions we get certain disease states: no infection, latency, and infection. The behavior depicted by our model, under certain parameters, demonstrates the dynamics of these three conditions of the disease. We consider a treatment and analyze its effect on the dynamics of the system.
Mathematical Models to Study the Outbreaks of Ebola
Jaime Astacio, Universidad de Puerto Rico-Humacao
DelMar Briere, Blackfeet Community College
Milton Guillén, University of California-Santa Cruz
Josué Martínez, Univerity of Texas-Austin
Francisco Rodríguez, California State University-Bakersfield
Noé Valenzuela-Campos, University of California-Davis
BU-1365-M
Abstract: Using S-I-R and S-E-I-R models, it was possible to simulate two Ebola outbreaks: the 1976 outbreak in Yambuku, Zaire and the 1995 outbreak in Kikwit, Zaire. The dynamics of these models are determined by the per-capita death rate of infected individuals and the per-capita effective contact rate of an individual contracting the disease. The basic reproductive number, R0 , determines the infectiousness of the disease. For Ebola, 1.72 ≤ R0 ≤ 8.60, and this implies that Ebola is not as infectious as previously postulated. The results of these outbreak simulations will equip scientists in future outbreaks with information that may enable them to minimize potential deaths.
Stochastic Simulations of a Saptial SIR Model
Judit Camacho, University of California-Santa Cruz
Fernando Carreón, University of Texas-El Paso
Derik Castillo-Guajardo, Universidad Autónoma Metropolitana Unidad Xochimilco
Hugo Jiménez-Perez, Universidad Nacional Autónoma de México
Leticia Montoya-Gallardo, Universidad Nacional Aut&oacte;noma de México
Ricardo Alberto Sáenz, University of Texas-El Paso
BU-1366-M
Abstract: In this paper we consider a stochastic spatial SIR (Susceptible-Infectious-Recovered) model. We assume that the population is distributed in separate cells. The disease is transmitted within the cell by direct contact, and from cell to cell through an external object (vector or vehicle) capable of carrying the disease. We simulate this model in a 10 x 10 grid of cells, and investigate the effects of the relative rates of transmission within and between cells on the predictability and progression of the disease. Results of simulation indicate that as the rate of intercellular transmission increases relative to intracellular transmission, the mean number becoming effected within each cell increases but so does the spatial variability. We also found that the time for the epidemic to run its course reaches a maximum average value at intermediate relative as does the spatial variability.
HIV-1 Replication Rate
Michele J. Arias, University of California-Riverside
Erika T. Camacho, Wellesley College
Rafael B. Castillo, State University of New York-Stony Brook
Delmy Iñiguez, University of California-Riverside
Eliel Melón, Universidad de Puerto Rico-Humacao
Luz E. Parra, Northern Arizona University
BU-1367-M
Abstract: Antiviral drugs have been known to prolong the lives of HIV infected patients. However, it is still uncertain if antiviral drugs affect the rate of virion clearance, the loss of target cells, or both. In this study, we used mathematical models to measure the effects of a protease inhibitor on three infected patients. After analysis of this data, we were able to calculate the lifespans of free virions and infected cells. We found that the antiviral drug decreased not only the number of free virions in the plasma but also the loss of infected target cells. Although this antiviral drug is not a cure for this disease, it can provide insight into creating an effective treatment program for HIV patients.
The Effects of Vaccination in a Core Group
Marina Bobadilla, University of California-Santa Cruz
Sharon A. Lozano, University of Texas-Austin
Jessica M. Maia, Massachusetts Institute of Technology
Julio Villarreal, University of San Diego
Novaline Wilson, University of New Mexico
Roberta Winston, New Mexico State University
BU-1368-M
Abstract: This study examines a mathematical model in which a vaccine without complete effectiveness is applied to a core group. The prevalence of the disease within the core group determines the recruitment rate into the core group. The recruitment function in this model is set up for the case dependent on the proportion of infectious individuals. In particular, we study the possible oscillations of the disease over time caused by the vaccination rate and vaccine efficiency.
A Mathematical Model of the Dynamics of Rickettsia rickettsii in Tick-Host Interactions
Mary Alderete, Arizona State University
Carlos Castillo-Garsow, Ithaca High School
Guarionex Salivia, Universidad de Puerto Rico-Rio Piedras
Carlos Lara-Moreno, Universidad Nacional Autónoma de México
Gina Ramírez, California State University-Dominguez Hills
Monica Yichoy, Cornell University
BU-1369-M
Abstract: This paper studies the dynamics of the tick population affected by Rickettsia rickettsii in order to understand how this disease affects other species. This project modifies the Busenberg-Cooke (BC) model to better account for biological aspects. Mathematical analysis assesses the effect of parameters on the dynamics of the model. One main result is obtained: the populations behavior is found to be chaotic in a region of parameter space that differs from that observed in the BC model. More importantly, the nature of the attractors seems qualitatively different.