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Summer 2005
Disease Spread as a Function of Socioeconomic Status in Manhattan
Julie Blackwood, Rochester Institute of
Technology
Carlos Chiquete, University of Arizona
Russell Latterman, Arizona State University
Stephen Small, Norfolk State University
MTBI-02-01M
Abstract: When studying the spread of disease, it is impressive to consider variable factors such as population distributions, the interactions between differing populations, and socioeconomic factors. We use a network model of interacting nodes with contact rates dependent on population size and socioeconomic status to explore the disease spread across the twelve districts of Manhattan, New York City. Influenza was chosen as an example due to its short infection period and negligible disease-related deaths in comparison to prevalence levels. Since transmission occurs primarily through casual contact, proportionate mixing is incorporated in the model. Numerical simulation of the model and sensitivity analysis of its parameters are then used to identify critical factors, dependent on socioeconomic status, responsible for the severity of the epidemic. Vaccination strategies are also implemented to explore what methods will have the greatest effects on the dynamics of the model.
An Epidemiological Approach to the Spread of Political Third Parties
Karl Calderon, University of Arizona
Clara Orbe, Brown University
Azra Panjwani, University of California at Berkeley
Daniel M.Romero, Arizona State University
Christopher Kribs-Zaleta, University of Texas at Arlington
Karen Ríos-Soto, Cornell Univeristy
MTBI-02-02M
Abstract: Third political parties are influential in shaping American politics. In this work we study the spread of third parties ideologies in a voting population where we assume that party members are more influential in recruiting new third party voters than non-member third party voters. The study is conducted using an epidemiological model with nonlinear ordinary differential equations as applied to a case study, the Green Party. Through the analysis of our system we obtain the party-free and member-free equilibria as well as two endemic equilibria. We identify two threshold parameters in our model that describe the different possible scenarios for the political parties and their spread. Our system produces a backward bifurcation that helps identify conditions under which a third party can thrive. We perform a sensitivity analysis to the threshold conditions in order to isolate those parameters to which our model is most sensitive. We explore all results through deterministic simulations and refer to data from the Green Party in the state of Pennsylvania as a case study-dependent vital rates. These rates are important as the rates change with the accumulation of "errors".
Epidemiology as Related to the Phylogenetic Analysis of the Evolution of the Influenza Virus
Amanda Criner, University of Maine
Brandon Hale, Murray State University
Lorena Morales-Paredes, University of Alabama at Huntsville
David Roy Estrella Segura, Arizona State University
Ariel Cintron-Arias, Cornell University
MTBI-02-03M
Abstract: The evolution of the influenza virus is characterized by continual changes to its surface structures due to antigenic drift and antigenic shift. The host immune system must alter antibodies in response to the ever-changing virus allowing for the persistence of influenza in a host population. The spread of related strains through a susceptible population with regard to their phylogenetic distance from a parental strain during a season is examined, as well as the within host dynamics. Very little work has been done to integrate phylogenetic analysis of evolution with the epidemiological spread of the influenza virus. In this study, an attempt is made to couple these two scale by using infection rates that are defined as functions of phylogenetic distances between strains. Competition between strains is focused on and strain prevalence for outbreaks during several seasons (2000-2004, inclusive) is examined at various levels: global, regional, and for New York City. Coexistence is found to only be possible between very similar strains, otherwise competitive exclusion or extinction of all strains occurs. Stochastic simulations at the cellular level indicate that the immune system is most effective when the virus has little variability, so the rapid mutation of influenza is an effective strategy in evading the immune system. Similar simulations for the population level show that a strain's prevalence depends largely on the effect on the antigenic structure as a result of the locations of amino acid mutations.
An Epidemic Model of HSV-1 with Vaccination
Asela Acosta, Texas A&M International
University
Efrat Bar-Zohar, Arizona State University
Sa'ul Blanco, Cornell University
Dori Luli, North Central College
Linda Q. Gao, North Central College
MTBI-02-04M
Abstract: The most common type of herpes is Herpes Simplex Virus type 1 (HSV-1), and it is commonly known to cause oral herpes - cold sores and fever blisters. Recent studies show that an HSV-1 vaccine was successful in the lab for animals such as guinea pigs and mice. It is estimated that the vaccine will be available for human use in the next couple of years. Encouraged by those studies, we have formulated a simple SVID model studying the disease transmission dynamics with treatment and vaccination. In this project we find the vaccination-treatment reproductive number, the equilibrium solutions, and their stability. We conduct sensitivity and uncertainty analysis for the reproductive number. We estimate the parameters based on the previous works and perform numerical simulations. Finally, we compare different types of treatment and vaccination strategies to find an optimal combination of them and its relative cost in reducing the prevalence of HSV-1 infection in the population.
The Effects of Migration and Connectivity upon Initial Behavior of Multi-City Endemic Models
Alan Covert, Arizona State University
Laiza Espinoza, Arizona State University West
Matthew Hoffman, New Mexico Institute of Mining and Technology
MTBI-02-05M
Abstract: We consider the behavior of an infectious disease over a system of networked cities with SLIR dynamics. Two models are studies: one with a partition between traveler and resident populations, and another without. R0 is computed for both the partitioned and non-partitioned models. Numerical solutions of the nonlinear dynamical and intercity spread of disease, and system response to dispersal volumes and number of connections between cities is also studied. We consider a theoretical future disease with SARS-like parameters, and run numerical solutions using parameters from past studies on SARS.
Effects of Lifestyle Choices on Atherosclerosis: A Mathematical Approach
Lily I. D. Davidoff, Mount Holyoke
College
Heather Harrington, University of Massachusetts-Amherst
Ludguier D. Montejo, Whitman College
Danielle E. Robbins, University of Maryland Baltimore County
Fabio A. Sánchez, Cornell University
Baojun Song, Montclair University
MTBI-02-06M
Abstract: Cardiovascular diseases (CVDs) cause more than 40% of all deaths in America [5,6]. Genetic predisposition and lifestyles choices such as smoking, poor nutrition, physical inactivity and other negative behavioral actions are common risk factors that increase the probability of developing CVDs. We construct a mathematical model of nonlinear differential equations to describe the dynamics of atherosclerosis, a precursor to other cardiovascular diseases. We divide the population by low and high genetic predisposition to developing CVDs. The population is then divided into healthy and unhealthy classes based on modifiable and non-modifiable lifestyles. We focus on two populations: African Americans and Caucasian Americans because of their different genetic predispositions to CVDs. The basic reproductive number, R 0, is calculated. Local stability of the unhealthy-free equilibrium is established. A sensitivity and uncertainity analysis is performed on the basic reproductive number. We conclude that the number of interactions between healthy an unhealthy individuals play a key role in reducing the progression to atherosclerosis.
Understanding Immigration and Policy Change from a Mathematical Perspective
Wilbert Fern'andez Cuevas, Arizona
State University
Genevieve-Yvonne Toutain, Simon's Rock College of Bard
Darío Alberto Varela, Arizona State University
MTBI-02-07M
Abstract: Every year nearly 300,000 Mexican people cross the border into the United States. This paper is a discussion of the effect of changes in United States policy on Mexican immigration rates. Using three separate compartmental models, we look at Mexican immigration as a whole and conclude immigration is inevitable; a model of the effect of quotas on illegal Mexican immigration; and the effect of the Patriot Act.
The Dynamics of Poverty and Crime
Haiyun Zhao, Stevens Institute of
Technology
Zhilan Feng, Purdue University
Carlos Castillo-Chávez, Arizona State University
MTBI-02-08M
Abstract: Poverty and crime are two maladies that plague metropolitan areas. The economic theory of crime (Becker, 1968) demonstrates a direct correlation between poverty and crime. The model seeks to examine the dynamics of the poverty crime system through stability analysis of a system of ODEs in order to identify cost-effective strategies to combat crime in metropolises.
Ring Vaccination as a Control Strategy for Foot-and-Mouth Disease
Edgar Diaz, Arizona State University
Alicia Urdapilleta, Arizona State University
Gerardo Chowell, Los Alamos National Laboratory
Carlos Castillo-Chávez, Arizona State University
MTBI-02-09M
Abstract: Foot-and-Mouth disease (FMD) is a highly infectious illness of livestock and a serious economic threat. Effort has been placed in modeling various control strategies for eradicating the disease. In this study we will consider a spatial model that incorporates ring vaccination and isolation as a control measure for the dispersal of the epidemic. We found an upper and lower bound of the basic reproductive number for the spatial model in terms of our parameters. Through numerical simulations we were able to show that ring vaccinations is effective in controlling the epidemic. We validate our results by using the dataset based on the 2001 FMD epidemic in Uruguay.
Two Mathematical Models for the Tympanic Membrane
Andrea Bruder, Utah State University
MTBI-02-10M
Abstract: In this project we present two mathematical models for the human tympanic membrane. The eardrum can be viewed as an example of the vibrating drum problem. In the first model, we treat the tympanic membrane as a rectangular region. In the second model the tympanic membrane is considered as a disk. Both models use a wave equations. We study the impact of changes in the membrane tension related to trauma or tumors which cause the frequencies of vibration to either increase or decrease.
The Effect of Immune Response and Combination Drug Treatment on the Progression of Multi-Strain HIV
Mat Gluck, University of California,
Riverside
Maria Osorio, Univeristy of Arizona
Kelly Smith, Clarion Univeristy
Xiaohong Wang, Arizona State University
Zhilan Feng, Purdue University
MTBI-02-11M
Poster sessions award recipient at the National 2006 AMS convention in San Antonio, TX
Abstract: Rapidly mutating HIV strains pose difficulties for effective theraphy. By using a mathematical model, we explore the in-host progression of mutating HIV strains considering both the immune response of the host and a combination of antiviral drugs. The first drug inhibits the entry of the HIV virus into CD4+ cells, while the second is a protease inhibitor. We conduct uncertainty and sensitivity analysis of the parameters in the effective reproductive number, R0. Deterministic simulations are performed to illustrate the random behavior of the independent HIV strains on the progression and severity of the disease.
Differential Behavior of Vectors Infected with Chagas' disease
Karyn Sutton, Arizona State University
Priscilla Greenwood, Arizona State University
Christopher Kribs-Zaleta, University of Texas at Arlington
Leon Arriola, University of Wisconsin at Whitewater
Carlos Castillo-Chávez, Arizona State University
MTBI-02-12M
Abstract: Chagas' disease, caused by certain strains of the parasite Trypanosoma cruzi, is a vector-borne disease, previously thought to be transmitted solely through the fecal matter of the triatomine vectors after feeding on the mammalian host. However, this mode of transmission is inefficient in the vectors, Triatoma Sanguisuga a subspecies of the reduviid family, prevalent in the Southeastern United States, due to the significant delay between feeding and defecation times. The prevalence in this region, 40-60% thus necessitates an alternative explanation. The hosts in the sylvatic cycle of this region, including opossums, raccoons, and armadillos, to name a few, are known to consume the vectors, although this is a traditionally inefficient way of transmitting the parasite. Recently, vector behavior has been observed to be modified during infection, termed differential behavior, such as feeding more frequently and wandering into broad daylight. The extent to which this affects the disease dynamics warrants investigation and could explain the persistence of T.Cruzi in the sylvatic cycle of this region. To include both modes of transmission, a deterministic model of the disease dynamics has been developed, incorporating both vector-host and predator-prey dynamics. This model is studied to examine how the differential behavior affects the disease dynamics, threshold of infection, and the current endemic equilibrium which is presently the case. Numerical simulations are carried out to verify the theoretical results. We have shown that elevation of consumption of the vector decreases infection levels and could possibly drive the vector population to extinction. Vectors increased vulnerability to predation increases consumption of infected vectors, which decreases prevalence levels but only slightly affects the total population size. Also, increased feeding frequently of the vectors boosts infection levels significantly, and could explain the high prevalence of T.Cruzi in the southeastern United States.
The Role of Transactional Sex in the Spread of HIV/AIDS: A Modeling Perspective
Titus G. Kassem, University of Jos,
Nigeria
Svetlana Roudenko, Arizona State University
Stephen Tennenbaum, Cornell University
Carlos Castillo-Chávez, Arizona State University
MTBI-02-13M
Abstract: The sex industry has been implicated in the spread of HIV across the world. In this article we propose a simple theoretical model consisting of two core groups of interacting heterosexual populations. One of the core group consists of male truck drivers and the other group consists of female sex workers. The truck drivers need for entertainment and female companionship make them use the services of the female sex workers in stop-over towns near major transportation routes. The resulting co-mingling of these sexually active, high-risk populations not only explains high prevalence of HIV in truck drivers and female sex workers and the subsequent spread of the disease in general population, but also points out the magnitude of the problem and the urgency of introducing effective controls. Our model assumes (i) a low level of condom use among the trucking population and female sex workers, (ii) high level of HIV in both truck drivers and female sex workers, and (iii) continuous recruitment in both groups when losses due to AIDS or natural factors occur. We give the complete analysis of the disease free and endemic equilibria. With that we also show the effect of reducing HIV cases in both groups by lowering of HIV transmission rates (e.g. by using condoms.)
An Interspecies Competition Model with Multiscale Interpretations
José Almora, University of North
Carolina at Chapel Hill
Nick Dowdall, Sonoma State University
Benjamin Morin, University of Maine
David Murillo, Arizona State University
MTBI-02-14M
Poster sessions award recipient at the National 2006 AMS convention in San Antonio, TX
Poster sessions award recipient at the Los Alamos National Laboratory 2005 undergraduate research symposium.
Abstract: A method for simulating and analyzing the competition between similar species is presented here by first adapting a Gause type model that allows four fitting parameters [2]. From this deterministic model we derive conditions for the existence and stability of several equilibria, including multiple coexistence equilibria. An agent-based simulation is then created to model the biology of the species on the scale of individual interaction. Changes in the growth scale parameters for the deterministic model are then considered in order to try the replicate the behavior of the agent-based biology. This is done in an attempt to rationalize the growth scale parameters as a tool to capture all possible behaviors. In an effort to observe the dynamics on a different spatial scale, we implement a spatially structured stochastic simulation with parameters chosen to reflect the different outcomes of the agent-based model. This second simulation will track, on a larger scale, the interaction between groups or colonies of the species of interest. The multiscale method presented here allows for a broader interpretation of interspecies competition than is possible through interpretation of interspecies competition than is possible through deterministic analysis. For the purpose of an example, two species of ants will be considered (Solenopsis invicta and S. geminata).