Mathematical Modeling of Infectious Diseases

Spring 2006

Classes will be held 13:30-14:45 on Mondays and Wednesdays, in Harrelson Hall, Room 368.

Draft Syllabus

The course will focus on the simplest biological situations, namely directly transmitted infectious diseases. Discussion of more involved settings, such as indirectly transmitted diseases (e.g. malaria and other vector-borne infections) and multi-strain infectious agents (e.g. HIV and influenza) will be given, but with reduced emphasis.

The main emphasis will be on epidemiological dynamics, and the links to ecological (predator/prey) theory. The importance of evolutionary dynamics will be highlighted where appropriate.

Grading

Course Texts

Please check with me before investing in any of the following. No single book is comprehensive for this course, and we shall make use of the primary literature in many cases.

• Anderson and May (1991) 'Infectious Diseases of Humans: Dynamics and Control' (OUP).

  Provides an introduction to both mathematical and biological aspects of the course. We shall follow the broad outline of this book.

• Diekmann and Heesterbeek (2001) 'Mathematical Epidemiology of Infectious Diseases : Model Building, Analysis and Interpretation' (Wiley).

  A more mathematically-oriented book.

• Daley and Gani (2001) 'Epidemic Modelling: An Introduction' (CUP)

  Provides examples involving statistical theory and stochastic modeling.

Course Philosophy

In this course, I am hoping to interest a diverse range of people in the subject matter. As a result, I will try my best to make the course as self-contained as possible. The main thrust of the course will not delve too deeply into details of the mathematics or the minutiae of the biology, but people are free to look more closely at things that interest them.

The assessment of the course will follow a similar philosophy, and so the way in which I assess your performance will depend on your background. Your project can be tailored to suit your strengths and interests, in addition to challenging you regarding new ideas.

Above all, I am more interested in the way in which you are engaged by the subject matter and approach than by whether you can prove some result or give complex details about some specific disease.

I am also open to taking suggestions if there are specific things you want to look at during the semester...

Prerequisites

I aim to make this course as self-contained as possible.

This may mean some repetition for those of you who have taken mathematical biology courses before. Hopefully, we can find a way to work around this.

For those of you whose mathematical skills are rusty: I will try to ease you back into the material, and am more than willing to give 'math clinics' to help you catch up. We will make use of some ideas from calculus, but hopefully not too much. Most of the ideas and concepts won't rely on heavy math, although some details might.

We will hopefully have some time in front of computers, where we can run simulations of the models discussed in lectures. This will help you get a better feel for the behavior of models. I have not yet decided how these labs will work, so I don't know which simulation or mathematical package we will use. There are some nice packages around, which mean you shouldn't need to have programing skills.

Course Topics

• Motivation and Background

  Background: examples of problems and issues

  What is modeling? Types of models, questions that we ask of models, modeling philosophies and limitations of models

• Simple ODE models

  Threshold behavior (invasion/endemnicity) and bifurcations.

  Implications for disease control: herd immunity, vaccination thresholds.

• Heterogeneity in disease transmission

  Data: evidence for the limitations of mass-action model.

  Multi-group epidemiological models.

  Implications for disease control. Optimal vaccination policies must account for heterogeneities

  More complex bifurcation structure (backwards bifurcations)

• Stochastic models

  Invasion and endemnicity conditions reconsidered in the light of stochasticity.

  Stochastic invasion theorems, final epidemic size distributions.

  Importance of population size; limitations of deterministic approaches.

  Bartlett's work on the dynamics of childhood diseases.

• Small Community Models and Statistics

  Stochastic models for disease outbreaks in families and small communities. (e.g. work of Becker and others).

  Parameter estimation.

• Other Topics

  We MAY cover some of the following:

  Evolutionary issues E.g. information gained from phylogenetic trees for diseases such as HIV.

  Spatial issues

  Network models

  Within-host models e.g. HIV dynamics, drug treatment.

  Emergence of drug resistance. Antibiotic resistance, HIV.

• Case Studies

  Real world examples where modeling and statistical approaches have had a major impact. Also discuss when modeling approaches have been less successful, and why.

  Decision making regarding vaccination policies, e.g. differences in rubella vaccination policy between UK and US.

  UK veterinary infections: BSE/CJD, foot and mouth disease. (Examples where modelers' input has informed government policy in real time.)

  HIV drug treatment dynamics.