Season 2 Episode 7
In this episode, Francesca describes the mathematics of competing virus strains, with chaotic consequences.
Further Reading
Francesca’s Slides
OOMC Chaotic Viruses.pdf 11 MB
The mathematics here was pretty tricky – I hope that we gave you an impression of some of the ideas that go into a mathematical model, and the sort of results that you can get out.
If you’d like to see more about epidemic modelling, check out Francesca’s previous episode last term on OOMC (season 1 episode 8).
Chaos
We saw something quite strange in Francesca’s talk – there was a bit where a deterministic system had unpredictable almost random output. That’s quite interesting! Here’s an experiment you can try at home to see behaviour like this, just using a calculator (you need one with an “Ans” answer button).
First, type in your favourite number that’s between 0 and 1 (like 0.37) and press equals. Then type $2.5\times \text{Ans}\times\left(1-\text{Ans}\right)$ and press equals, and keep pressing equals again and again!
You should find that the numbers coming out of your calculator settle down to a single number (can you work out what the number is and what it has to do with the expression $2.5x(1-x)$?). This is not chaos, this is a nice stable fixed point.
Next, type in your favourite number that’s between 0 and 1 (like 0.37) and press equals. Then type $3.2\times \text{Ans}\times\left(1-\text{Ans}\right)$ and press equals, and keep pressing equals again and again!
You should find that the numbers coming out of your calculator settle down to a pair of numbers (hard: can you work out what the numbers are and what it has to do with the pair of equations $y=3.2x(1-x)$ and $x=3.2y(1-y)$?). This is also not chaos, this is nice periodic behaviour.
Next, type in your favourite number that’s between 0 and 1 (like 0.37) and press equals. Then type $4\times \text{Ans}\times\left(1-\text{Ans}\right)$ and press equals, and keep pressing equals again and again!
You should find out that the numbers coming out of your calculator… have no recognisable pattern. The numbers go very small, they go close to 1, they go everywhere in between. If you change your starting number just a little bit, then you quite quickly get completely different results. This is chaos!
For more on this, see this Wikipedia page. Chaos theory is a small part of a third-year Oxford course. I don’t often link to lecture notes here, because they’re really complicated, but the Nonlinear Systems lecture notes have some absolutely fantastic images which more people should see; see this page for the lecture notes (14 MB. These notes are from 2020, when the course was lectured by Prof Jon Chapman). See this page for an overview of the Nonlinear Systems course.
Curve sketching
Sketch on the same axes
$$y=e^{-x}\cos (x)+\frac{1}{6}, \qquad y=e^{-x}\sin x+\frac{1}{6}$$
That’s my best attempt at something that looks a bit like one of Francesca’s graphs. Can you make something that looks like the other graph Francesca showed us? If you know about complex numbers, can you simplify those expressions above? If you know about differential equations, can you find a simple differential equation that both satisfy?
If you want to get in touch with us about any of the mathematics in the video or the further reading, feel free to email us on oomc [at] maths.ox.ac.uk.