Mathematical Ecology and Epidemiology
Learning guide
How to use this guide
Welcome to Mathematical Ecology and Epidemiology. This guide leads you through the module week by week. It replaces both lecture notes and problem sheets: everything you need is in here, or in the computer notebooks linked from it.
Why mathematical ecology and epidemiology?
Mathematical models are how we predict the ecological effects of climate change, decide how many fish can be caught sustainably and how many people must be vaccinated to prevent an epidemic. During the Covid pandemic the public was told almost daily about the value of \(R_0\). In this module you will learn how such models are built, what they predict, and why. The models are simple enough to be understood with pen and paper, and this deep understanding is what lets you judge more complicated numerical models later. The same techniques apply to models of the economy, the climate or chemical reactions.
How this module works
This module does not follow the usual “lecture first, exercises later” pattern. Instead you discover things first and the lecturer then helps you make sense of what you found. In a typical session you will:
- start with a question or a prediction,
- investigate it, on your own or in a small group, using this guide and a computer notebook, while the lecturer comes round to talk to you,
- discuss what you found with the whole class,
- listen to a short lecture (never more than 15 minutes) that ties the results together.
There are four sessions each week. The last one is in a computer classroom: it starts with a 15-minute assessment quiz on that week’s work and then you start exploring the topic of the following week. So each week you arrive at the first session with observations of your own, which the week then explains.
- Work through the week’s chapter, in class and between classes.
- After each section do the feedback quiz on the VLE. Keep going until you get everything right: you can retry as often as you like.
- Attempt the exam-style questions by hand, without a computer or AI, before looking at the solutions.
- Take the assessment quiz in the last session of the week. It contains questions just like those in that week’s feedback quizzes.
The boxes in this guide
Each section is an investigation made up of the following kinds of activity. The colour of the box tells you which kind it is.
Make a quick guess before calculating or simulating. It does not matter whether your guess is right. What matters is that you commit to one, because then you will notice when you are surprised.
Experiment with a computer notebook. Follow the instructions, change parameter values, and record what you observe. No programming knowledge is needed.
Turn your observations into mathematics with pen and paper, following the guided steps. Where you see a blank you are expected to fill it in.
A task to be done in a group of three or four. Talk, argue, and make sure that everyone in the group can explain the group’s answer.
Go to the feedback quiz of the same number on the VLE. The questions give you immediate feedback and can be retried with new numbers.
These collapsed boxes contain the definitive statement of the results of an investigation. Open them only after you have done the investigation and the class has discussed it. Later they are your reference notes for revision.
A question like those in the final exam. Do it by hand under exam conditions, without a computer, notes or AI.
The solution to an exam-style question. Open it only after a serious attempt.
At the end of each chapter there is a launch investigation for the following week. You do it in the computer classroom at the end of the week and bring your observations to the first session of the next week.
Computer notebooks
The experiments use Python notebooks that run in Google Colab in your web browser. You need a Google account but you do not need to install anything, and you do not need to know how to program. Every notebook is linked from the guide with a button like this: Open in Colab
In a notebook:
- Run a cell by clicking the ▶ button to its left (or press Shift+Enter).
- Always run the first cell, called “Setup”, first. It loads the tools that the rest of the notebook uses.
- Most cells show sliders. Move a slider and the plot is redrawn.
- Some cells ask you to change a number or a formula. Just edit it and run the cell again. If something goes wrong, choose Runtime → Restart session and start again from the setup cell. You cannot break anything.
- To keep your changes, save a copy to your Google Drive (File → Save a copy in Drive).
Using AI
You are welcome to use AI tools during this module. They can explain a piece of code, check a step in a calculation, or suggest another way to think about a concept. Remember, though, that there is no AI and no computer in the assessment quizzes or in the final exam. Use AI to help you understand, not to do the thinking for you. A good test: after a conversation with an AI, can you redo the calculation on a blank sheet of paper?
Prerequisites
We use ordinary differential equations (first-order linear ODEs, separation of variables), Taylor expansion, eigenvalues and eigenvectors of matrices and a little about partial differential equations (separation of variables). The Toolkit appendix collects the methods we use repeatedly.
Feedback on this guide
This guide is new. If you spot a mistake or something that is unclear, please tell the lecturer or email gustav.delius@york.ac.uk.