# Maths and the Spread of Infectious Diseases Session 1: the Standing Disease

### Possible approach

Video clip 1

video clip 2

video clip 3

growth calculator

video clip 4

video clip 6

Begin by watching Video Clip 1 which introduces the session to your learners (length of clip =1 min 3 sec)

After watching Video Clip 1, introduce the 'Odd One Out' activity (needs further development!) which revises their prior learning of recognising and extending arithmetic sequences. The task is intended to encourage meaningful discussion by featuring three cards with different sequences (two arithmetic, one geometric). We could present this as a Concept Cartoon, enabling students to click on ideas from students eg 'Mo thinks Sequence B is the Odd One Out because...' or as a simple slide.

Sequence A: 101, 109, 117, 125, 133, ...

Sequence B: 240, 210, 180, 150, 120, ..

Sequence C: 2, 4, 8, 16, 32, ...

Suggestions for the 'Odd One Out' might include A (only sequence with odd numbers), B (the only decreasing sequence) or perhaps C (only doubling sequence). Challenge the learners to suggest the next two terms in each sequence and explain how they worked them out.  Recap the definition of an arithmetic sequence, noting that Sequences A and B are both arithmetic sequences.

Now watch Video Clip 2 where Julia introduces The Standing Game for the learners to explore themselves afterwards (length of clip = 50 seconds)

Try The Standing Game activity introduced in Video Clip 2:

This first volunteer should then stand up and "infect" two learners by pointing to them.
These two learners then stand up, having been infected.

These two learners then each infect two others in the classroom, and so on, until the whole class has stood up.

Ask the learners how many steps it has taken to infect their class.

Reflect together on the effect making a small change has on the spread of the infection. Explain that this geometric sequence is an example of exponential growth.

Video Clip 3  - Lego Video Clip 3 (clip length = 1 in 34 seconds)

Compare APs and GPs using same step size - note what's the same and what's different between them including shape of their respective graphs. Move on to explore other step sizes for smallpox (3), polio (5) and measles (18). Use this  growth calculator to explore ore ideas.

Show Video Clip 4 which makes the connections between the mathematics of geometric sequences and the modelling infectious diseasses (2 minutes 17 seconds)

Continue the discussion abut the effect of a small change in r making a large difference to the next terms in a sequence by exploring the 'Rice and Chessboard' story. There are many versions of this well-known tale but they mostly start by placing a single grain of rice on the first square of a chessboard, then doubling the number of grains on subsequent squares. This can be acted out in the classroom until the numbers become unmanageable, at which point you might like to share this video clip (2'01") which continues the action in a student-friendly manner.

An alternative activity which reinforces this key concept is the 'Christmas Present Dilemma'. Would the students rather receive £10 one Christmas, £20 the next, £30 after that and so on OR receive £100 the first Christmas, £90 the next and so on OR receive £1 the first Christmas, £2 the second, £4 and so on? Teachers report that classroom discussions and justifications for decisions often lead to debates about life expectancy and actuaries!

Remind the students that they discovered earlier in the session that a geometrical sequence for r=2 had a characterisitic 'J'-shaped curve. Encourage them to investigate whether that's always the case by using this Desmos application, and noting the effect on the shape of changing r (note: a is the initial number infected).

If time allows, you could also explore r numbers below 1. For example, by considering a bouncing ball which rebounds to half the height after each bounce; if it is dropped from a height of 128cm, how many bounces will it take to bounce less than 1cm? Again, learners could visualise this effect using graphs and bar models.

Taking things further, they could also investigate folding a sheet of paper in half and in half again. How many folds do they think they could make? How high might their folded paper reach? They might enjoy viewing this short TED Talk video clip (3'48") Exponential Growth: How Folding Paper Can get You To The Moon.

Watch Video Clip 5 (1 inute 35 seconds) where Julia reflects on the limitations of geometric sequences which overlook real-life factors which can affect transmission (such as immunity, and the difference between a geometric sequence which continues to infinity and a limitations of the world population, perhaps introducing the example of researchers reporting a possible 77 trillion cases of smallpox a few years ago before handing back to teachers to discuss why that was an issue).

The final class discussion for this first session could begin by acknowledging that 77 trillion cases far out-numbers the population of the globe (perhaps the modellers of this potential outbreak appears to have got carried away with their simulation). This reminds us that we need to question our models when calculating how big an outbreak might become. Such an example should act as a reminder to students to check their answers to ensure they are reasonable answers to give.

### Key questions

Estimate how many steps it would take to infect their school, town, country or the world etc if r=2.
It takes approximately 33 steps to infect globe (as there are 2n new cases in generation n of the outbreak).

If they repeat the outbreak, with a different person starting, will they get the same result?

What would happen if each person pointed to 3 or 4 people instead of 2?