Mathematical Models in the Life and Social Sciences

MA 432, Spring 2005

Course Syllabus

• Prerequisites

  MA 301 or 341 (intro. level differential equations) and MA 305 or 405 (intro. level linear algebra). Proficiency in a programming language.

  Co-requisite MA 421 or ST 371 (intro. level probability).

  If you have any concerns over the pre- or co-requisites, please come and see me.

• Textbook

  Modeling and Simulation in Medicine and the Life Sciences, 2nd edition. F. C. Hoppensteadt and C. S. Peskin. 2002 (Springer Verlag).

  We shall focus on chapters 7, 8 and 9 of this book. We may cover additional topics that are not in this book.

• Objectives

  The course provides an introduction to mathematical modeling in the life sciences (which will be our main focus) and the social sciences. A primary aim of this course is to provide examples of the deployment of mathematics (and the skills developed in other MA courses) in real-world settings. A secondary aim is to develop additional mathematical skills, as and when they are needed, including computer simulation skills.

  We shall develop an appreciation for:

  • the usefulness of modeling approaches
  • how models are built and developed
  • simulation and analysis techniques
  • fitting models to real-world data
  • interpretation of model behavior
  • model criticism and development

  Our modeling will often revolve around models that can be formulated in terms of matrices or differential equations. Techniques learned in the pre-requisite courses MA301/341 and MA305/405 will be reinforced. Beyond these, we shall examine discrete time models (e.g. difference equations), stochastic models (simple Markov Chain models) and possibly integral equation models.

  A major component of the course will be a final project, which will be carried out in groups (three to four students per group). The projects will involve both a written and an oral presentation.

• Course Topics

  (and rough schedule of the semester: we will be somewhat flexible over timing.)

  • Introduction to Modeling

    (week one)

    • Motivation
    • State variables and parameters
    • Use of numerical approaches
    • Mention of sensitivity analysis and model robustness (ideas will be developed in more detail later)

  • Deterministic Population Dynamics

    (weeks two to six, mainly based on chapter 7 of course text)

    • Bacterial growth: geometric growth in discrete and continuous time
    • Fitting population models (least squares)
    • Age structured models (bacterial, human): Leslie matrix models
    • Microbial ecology
    • Nonlinear models (in discrete and continuous time)
    • Linearization and stability analysis, bifurcations
    • Controlling populations (e.g. models for harvesting populations)
    • Models for interacting populations

  • Stochastic Population Models

    (weeks seven and eight)

    • Markov chain models
    • Simple birth/death models

  • Game Theory

    (weeks nine and ten)

    • Payoff matrices and fitness
    • Prisoner's Dilemma and the evolution of co-operation

  • Epidemics

    (weeks eleven and twelve, mainly based on chapter 9 of course text)

    • Reed-Frost model for transmission within families
    • Discrete and continuous time Kermack-McKendrick models: thresholds, epidemics and epidemic sizes
    • Models for recurrent (endemic) infections
    • Control of infection

  • Genetics

    (final two weeks, may be dropped if earlier topics have over-run, mainly based on chapter 8 of course text)

    • Population genetics: Mendelian inheritance, selection, random genetic drift
    • Biotechnology, including plasmids, gene expression models

  MATLAB will be used for model simulation: MATLAB techniques will be developed as they are needed. Example code will be provided: these pieces of code may be modified and developed to help carry out simulations of other models.

  Class time will be devoted to project preparation, including preliminary presentations (discussion of project topics)