DIMACS Summer School Tutorial on Dynamic Models of Epidemiological Problems

June 24 - 27, 2002
Rutgers University, Piscataway, NJ

Carlos Castillo-Chavez, Cornell University, Biometrics, cc32@cornell.edu
Herbert Hethcote, University of Iowa, Mathematics, herbert-hethcote@uiowa.edu
Pauline van den Driessche, University of Victoria, Mathematics and Statistics, pvdd@math.uvic.ca
Presented under the auspices of the Special Focus on Computational and Mathematical Epidemiology.

Interest in infectious diseases has increased in recent years for many reasons. New diseases such as Lyme disease, HIV/AIDS, hepatitis C, hantavirus, and West Nile virus have emerged. Antibiotic-resistant strains of tuberculosis, pneumonia, and gonorrhea have evolved. Malaria, dengue, and yellow fever have reemerged and may spread into new regions as climate changes occur. Diseases such as plague, cholera, and hemorrhagic fevers still erupt occasionally. New infectious agents called prions have been discovered to be the cause of mad cow disease (BSE). Human or animal invasions of ecosystems, global warming, environmental degradation, increased international travel, and changes in economic patterns will continue to provide opportunities for new and existing infectious diseases. There is now great concern about the deliberate introduction of diseases such as anthrax, smallpox or plague by bioterrorists.

Mathematical models have become important tools in analyzing the spread and control of infectious diseases. They can lead to a better understanding of infectious disease transmission processes and can be helpful in guiding intervention policy decisions. The model formulation process clarifies assumptions, variables, and parameters; moreover, models provide conceptual results such as thresholds, basic reproduction numbers, contact numbers, and replacement numbers. Mathematical models and computer simulations are useful experimental tools for building and testing theories, assessing quantitative conjectures, answering specific questions, determining sensitivities to changes in parameter values, and estimating key parameters from data. Understanding the transmission characteristics of infectious diseases in communities, regions and countries can lead to better approaches to decreasing the transmission of these diseases. Mathematical models and computer simulations are used in comparing, planning, implementing, evaluating, and optimizing various detection, prevention, therapy and control programs. Epidemiology modeling can contribute to the design and analysis of epidemiological surveys, suggest crucial data that should be collected, identify trends, make general forecasts, and estimate the uncertainty in forecasts.

This tutorial will develop mathematical models for the spread of infectious diseases by starting with the most basic dynamic models and then increasing the complexity to include host-vector situations, multiple groups, variable population size, age-structure, differential-delay equations, and functional differential equations. The models presented will address concepts such as thresholds, basic reproduction numbers, stability of equilibria, global stability, Hopf bifurcation to periodic solutions, multiple endemic equilibria and chaotic behavior. Applications to specific diseases such as tuberculosis, influenza, rubella, chickenpox, whooping cough, and HIV/AIDS will be included. Thus the essentials of model formulation and mathematical analysis will be presented, examples of the use of models to answer questions about specific diseases will be examined, and some of the questions, challenges, and opportunities in theoretical epidemiology will be presented.

On each of the 4 days, there will be 5 presentations followed by short discussions. In addition, at the end of each day there will be a discussion centered around a theme.

The three organizers will give most of the presentations, but there will be some guest speakers on the last 2 days. Titles of the talks are given elsewhere on this web site.

A goal is to go from the most basic models to the research frontiers in mathematical epidemiology. The organizers believe that the material in this tutorial course will be accessible to participants with different backgrounds. Thus this course is suitable for working mathematicians including graduate students, faculty, and postdocs, who know differential equations and want to learn about epidemiology modeling. It is also suitable for epidemiologists, biologists, and biostatisticians with an epidemiology background who want to learn about dynamic modeling of infectious disease transmission and control (some previous exposure to differential equations would be helpful). A mixture of mathematicians and epidemiologists as participants will provide a stimulating atmosphere for the course.

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Document last modified on November 16, 2001.