# Maths and the Spread of Infectious Diseases Session 2: Lucky Dip?

### Possible approach

Video clip 2.2

Video clip 2.4

Video clip 2.5

Video clip 2.6

motivate graphs

After watching the first video clip, encourage your learners to note down the assumptions it made and compare their answers (Charlie and Claire - perhaps we could add 'Hints' for possible assumptions to support some learners); these may include always passing the virus along to 2 other people, if infected you recover and rejoin the group, the members of the group always stay the same - no-one leaves or joins, and the members do not move. Explain that developing a good mathematical modelling is a balance between including sufficient factors to be realistic but not too many to overcomplicate it. What sort of changes would they wish to make to the existing model - collect answers and consider their pros/cons. Focus discussion on the lack of movement in The Standing Game, in real-life we tend to move around and meet different people. We could of course try this in the classroom but we're reaching a point where acting out the model is becoming more difficult in the classroom so we can turn to IT to help us do so more safely and efficiently many more times than we could do in the same time in real-life.

Ask learners if they've come across a Lucky Dip - perhaps at a school fair - and discuss how it works. Then introduce the Lucky Dip interactivity ID 15073 which works in a similar way except the prize is something that the winner might not want..a virus! The default setting for Lucky Dip has the same settings as The Standing Game ie the disease is spread to two more people each time. However, this time they can re-enter the draw, so they could be chosen twice but they will have had the virus so they will be immune the 2nd time around...what might the model do then?

Start off the Lucky Dip interactivity by clicking on the 'Run One Generation' tab noting that the interactivity, just like The Standing Game, assumes that one person is infected at the beginning and that they can infect two people (show this on screen). Explain that there are 26 people in this version of the model and encourage the learners to predict the number of infections each week over the next few months. Then repeatedly press 'Run One Generation' tab to reveal the number of infections each week. Compare the outcome from this model to their own predictions. Do they match? If not, what has happened during the run? Look at the accompanying graph, noting its shape. Re-run the model several times with the same settings. Note that the resulting graph often resembles a hill or bell, indicative of the Normal distribution we often find with infectious diseases. Encourage learners to suggest reasons behind the shape of the curve, which differs from the previous session's geometric sequence which kept increasing along a 'J' curve whereas this normal curves increases and then decreases. Ask if everyone in the Lucky Dip got it each time before the run ended, probably not? Why not? Draw out that since there were so many immune people later on, they had fewer people to pass it too so the virus died out even though there were still some who never caught it. Explain this is the concept of 'herd immunity' - not everyone needs to catch the virus because if enough catch it and recover, there are much fewer to infect later on.

Note issues with this model, especially that its set-up features a consistent group of people, no-one leaves or enters the group whereas in real-life we do this all the time as we move to different classes or travel on the bus home, or perhaps join up with friends later in the day for out-of-school activities. There is an opportunity here to discuss how the government managed this issue during Covid by introducing bubbles to help keep people in smalller groups and bring the infection to a close along the lines of the normal curve.

Now encourage suggestions for changing the model to represent the infectiousness of different diseases, eg changing r from 2, and the size of the group. Note the shape of the graph each time.

#### Video clip of Julia wondering whether our models and their bell-shaped curves reflect real-life epidemic fairly represent real-life by comparing to visual of 1978 flu epidemic in boys school. Revisit idea that  we can add more and more factors to a model, but they do not necessarily make it any more useful. We need to know which factors are the most important to include to help us use mathematics to better understand an epidemic - and exploring those will be the focus of our 3rd session

If time allows, explore other real-life examples where the data tends to follow the normal curve eg shoe sizes across the class or their heights (noting this is important data for businesses, to avoid over-ordering and risk having unsold stock of hoes or clothes later in the year, they need to stock larger amounts of the more popular sizes). Learners may enjoy watching Matt Parker's short video clip of a Galton Board in action (2'24") where the pattern of the fallen ball bearings tends to resemble a bell curve.

Closing video from Julia reflecting on today's learning and noting that learners should now be ready to explore the third session in this series which moves things further forward by exploring  the spread of infectious diseases in different settings and introduces another interactivity for them to investigate.