Preparation for students with few previous math courses

A Biologists Guide to Mathematical Modeling in Ecology and Evolution by Sarah P. Otto and Troy provides an excellent introduction to the development and analysis of mathematical models. This textbook can be downloaded as an ebook from many university libraries.

For a student with few math courses, in preparing for the summer school, I would recommend:

  • Chapter 2. How to construct a model. This is an excellent introduction to learning to make your own models. You should be able to solve Problems 2.1-2.6 and 2.8. Also, on this topic, you can read Chapter 1. of Keeling and Rohani, the text book for the Mathematical Epidemiology course at the summer school.

  • Chapter 3. Deriving Classic Models in Ecology and Evolutionary Biology. Building your own models requires a working knowledge of the foundational models. You can omit 3.3. Problems 3.12-3.13.

  • Primer 1. Functions and approximation. I wouldn’t read this section, but you should know that it is good reference material for understanding the shape of functions.

  • Chapter 5. Equilibria and stability of one variable models. This is an important chapter, because ultimately you should have a mastery of Chapter 8, and Chapter 5 is foundational in that progression. The summer school models will mostly be continuous time, \(dn(t)/dt\) (also referred to as differential equations) rather than discrete time, \(n(t+1)\), (also referred to as recursion or difference equations), so if you are time-limited focus on understanding equilibrium and stability for continuous time models. Problems: 6.4a-c, 6.6, 6.8b-d, 6.9.

  • Primer 2 Linear Algebra Again, I wouldn’t read this, but as we consider multivariable models, if you find yourself struggling, it could be necessary to study this chapter on linear algebra.

  • Chapter 7. Equilibria and stablity analysis - linear models with multiple variables. Epidemiological models, generally, will not be linear ordinary differential equations, but nonetheless, understanding linear differential equation models is foundational to understanding non-linear models (the type that appears in the mathematical epidemiology literature). Furthermore, most non-linear models are approximated well by linear models given some restrictive assumptions. While no one desires to unnecessarily make restrictive assumptions, linear models have general solutions (details are given in Chapters 6 and 9), and that aspect of linear models is sometimes very powerful, such that linear models can be useful even if your results hold only under restrictive assumptions. Again, you want to focus your reading on continous time models. You should work towards being able to solve Problem 7.3.

  • Chapter 8 Equilibria and Stability Analysis - nonlinear models with multiple variables. 8.1, 8.2 and 8.5 only. Problems: 8.1, 8.4a-c, 8.12.

Learning mathematics requires giving yourself the time to read and think, but it is also very necessary to practice solving problems to make sure you fully understand what is written.