Part 2: Lucky dip
Julia introduces the 'Lucky Dip model', and the accompanying interactivity, which take into account the impact of herd immunity.
Problem
This problem belongs to the Contagious Maths: Understanding the Spread of Infectious Diseases collection. It follows on from Build your first model.
What can we do to improve our model so that it more accurately represents real life?
What do you consider to be the main limitations of the model we have been using so far?
Introducing the "Lucky Dip" model
In this video clip Julia Gog and Rachel Thomas introduce the Lucky Dip model, simulating what might happen to an imaginary population of 26 people. The infection starts off in the same way as before, but then things change...
A new rule for our model
Watch this video clip to see what Julia decides to do after pulling out some empty capsules.
Can you predict what will happen if they keep going?
What might the graph look like after a few more generations?
The impact of immunity
This final clip completes the simulation and introduces the concept of herd immunity.
What do you think will happen if Julia and Rachel ran the simulation again?
The Lucky Dip interactivity
We can speed up the simulation by using an interactivity, and then compare the graphs of several outbreaks.
The interactivity works like a tombola, where each person is represented by a capsule which is drawn at random. The default setting is R=2.
Begin by clicking 'Run one generation'. You can watch the jumbler/tombola working, or click "Next" to skip the animation.
The tombola will release a single capsule with a red token, which represents a single infected person. The screen will now indicate that there will be '2 infected' in the next generation.
When you click 'Next', you will have a choice:
- Click 'Run one generation' again, to see what happens in the next generation
- Click 'Run to the end', to see a chart showing the number of infections in each generation
The resulting graph may appear similar to the one in the video in several ways, but there may be differences too.
What do you notice when you run the interactivity several times?
What's the same about each outbreak? What's different?
Using the Settings menu (the purple cog) you can change the population size, the R number, and the amount of data shown. After changing the setting click one of the two buttons at the bottom of the settings page (either "In the current window" or "In a resizeable pop up window") to return to the interactivity with the updated settings.
Randomness
If you have used the Lucky Dip interactivity to run simulations of epidemics a few times, you might have started to see differences and similarities between different runs.
In this video, Julia and Rachel ran the simulation four more times:
You might find it interesting to compare the shapes of the graphs you are getting from your simulations of epidemics, with data for real epidemics. What features do they have in common, and how are they different?
The SIR (Susceptible, Infected and Recovered) model
In this video clip Julia explains that the adapted model is an example of an SIR (Susceptible, Infected and Recovered) model, which is the basic model for many infectious diseases.
However, the model could be improved further...
What else do you think should be taken into account in order to improve the model?
This concludes Part 2. These resources continue with Part 3: Everybody is different, where we explore how variability between people affects epidemics. Alternatively, if you are finishing here, you make like to continue to Wrap up and Meet the Researchers.
How schools can use these resources
In the "Teachers' Resources" section you will find suggestions as to how this material might be used in the classroom.
This is the second of four parts, designed to be used in a sequence of lessons - here is a lesson by lesson breakdown.
These Contagious Maths resources were developed and written by Julia Gog and the MMP team, including both NRICH and Plus, and funded by the Royal Society’s Rosalind Franklin Award 2020. We have tailored these resources for ages 11-14 on NRICH, and for older students and wider audiences on Plus.
Teachers' Resources
This is the second of four parts, designed to be used in a sequence of lessons - here is a lesson by lesson breakdown.
Why do this problem?
Lucky Dip aims to extend student understanding of mathematical modelling, following on from Build Your First Model. Students are now encouraged to reflect on the limitations of the earlier doubling model, and consider how it might be adapted and improved. They are then invited to explore the similarities and differences that occur when running several simulations of the updated disease model. In the process, they have the opportunity to develop statistical skills, describing simple relationships between two variables in experimental contexts, and making connections between number relationships and graphical representations.
The teaching materials include video clips featuring Professor Julia Gog introducing the concept of herd immunity, the SIR (Susceptible, Immune and Recovered) model, and explaining how the underlying mathematics supports her work as a disease modeller and researcher.
Possible approach
Remind students that since there is a limited population, the graph must change at some point.
"In today's session you'll be investigating what it might look like when we improve our earlier model."
Introduce the "Lucky Dip" model, which simulates what might happen to an imaginary population of 26 people. Either:
- Show a series of three short video clips, with time in between for discussion:
"We're going to watch a video clip where the infection starts off in the same way as The Standing Disease, but then things change. See if you can figure out what's happened."
Watch the start of the video clip (3min 11sec) where Julia Gog and Rachel Thomas introduce Lucky Dip (an updated version of The Standing Disease model).
Agree that an empty capsule means that the person has already been infected.
Students might suggest that having been infected already, they could now be immune, so they will not infect anyone else, meaning that Julia would not pull out any more capsules for each of the empty capsules.
"Let's watch the continuation of the video clip (2min) to see what Julia Gog decides to do after pulling out an empty capsule"
"What do you predict will happen next? What might the graph look like for the next few generations?"
Students could sketch what they think the graph might look like. Discuss their ideas before watching the end of the video clip (2min) which completes the simulation and introduces the concept of herd immunity.
"When Julia Gog took the capsules out of the bag, it took quite a long time to carry out the whole simulation! We can speed things up using an interactivity. To begin with, we'll try it together. Then you're going to use it to investigate the patterns between several outbreaks."
- Or carry out a post-it activity with your class:
If you are using the post-it activity then you might like to watch the videos before the lesson, to see what sort of shape graph you might expect to see - in this introduction the students are taking the place of the capsules, and the post-it notes are the tokens (and a student without a post-it note is an "empty capsule").
Give each student in the class a post-it note with a number on it (1 to total number of students in the class). Use a random number generator (for example www.random.org) to select the first student to be infected. Stick their post-it note on the board to show the first generation of infections.
Use the random number generator to select two more students, and (if they still have post-it notes) stick these on the board for the second generation of infections.
At each stage the number of post-it notes shows how many students were infected in the current generation of infections. Each time generate twice as many random numbers as infected students. At some point you will start repeating random numbers indicating that an immune student (one who has already been infected) has come into contact with an infected student - but they will have no post-it note to put on the board and so will not infect anyone else.
Introduce the Lucky Dip interactivity which works like a tombola, where each person is represented by a capsule which is drawn at random. This is a more realistic model than The Standing Disease model.
You'll see that the default setting is R=2, meaning that each infected person can infect two others. Click on 'Run one generation'. The tombola will release a single capsule with a red token, which represents a single infected person. The screen now indicates that there will be '2 infected' in the next generation. Click 'Next'. You now have a choice. To see what happens one generation at a time, click 'Run one generation' again. Otherwise, click 'Run to the end'.
The resulting graph may appear similar to the one in the video in several ways, but there may be differences too.
Allow plenty of time for the students to explore the interactivity for themselves by running it several times.
Ask "What's the same about the graphs for each outbreak?" Possible answers might include:
- rapid acceleration at the beginning of the outbreak
- a peak
- after the rapid acceleration and peak, the shape of the graphs changes direction and drops back to the horizontal axis
- the graphs are asymmetrical
"What are the differences between them?" Possible answers might include:
- the length of the outbreaks
- the height of the peaks
- the total number of people infected
- high variability in the tail of the epidemic (can be long, can be bumpy)
Ask students to explore what happens for different values of R by setting R on the interactivity to take different values between 1 and 4 (you can do this by clicking on the purple cog on the top right). Encourage students to run the interactivity several times and find ways to record their findings.
What do they notice about their results?
Why do you think the number of infected people varies each time you run the interactivity?
Possible discussion points could include the number of generations before the spread of the disease ends, total numbers of the population infected and the shapes of the graphs (in particular how sharp and high the peak is).
In this video clip, Julia discusses how randomness leads to different outcomes.
Julia and Rachel ran the simulation four more times - the numbers of new cases in subsequent generations were:
1, 2, 4, 7, 7, 3, 0
1, 2, 3, 3, 4, 2, 2, 2, 0
1, 2, 4, 6, 5, 5, 0
1, 2, 4, 6, 4, 3, 2, 1, 0
"You might find it interesting to compare the shapes of the graphs you are getting from your simulations of epidemics, with data for real epidemics. What features do they have in common, and how are they different?"
To finish this session, show this video clip (2min 08sec) where Julia Gog explains that the improved model is an example of an SIR (Susceptible, Infected and Recovered) model which is the basic model for many infectious diseases.
The model is still a simplification of reality, but it does seem to capture the essence of how epidemics develop. However, the model could be improved further. Ask students what else they could consider taking into account in order to improve the model. Some suggestions might be:
- that some people move around more than others
- the impact of a vaccination programme
- the impact of social distancing rules
Key questions
"Why do you think the number of infected people varies each time you run the interactivity?"
"What might happen if the R number increases from 2 to 3?"
Possible support
Clicking on the purple cog at the top of the interactivity, and then the 'Extended' statistics tab, will offer students some basic descriptive data relating to the size of the outbreak within the population. This data may help them understand the effect of herd immunity.
Possible extension
Clicking on the purple cog at the top of the interactivity will also allow students to investigate what happens when they change the size of their population or the R number.
Students may be interested in Disease modelling for beginners, a collection of articles which explain how mathematics helps us understand how infectious diseases spread.
Once students have finished these Part 2 activities they may like to explore Part 3: Everybody is different. where we explore how variability between people affects epidemics. Alternatively, if you are finishing here, please continue to Wrap up and Meet the Researchers for a short concluding video and also clips from individual researchers to explore.
These Contagious Maths resources were developed and written by Julia Gog and the MMP team, including both NRICH and Plus, and funded by the Royal Society’s Rosalind Franklin Award 2020. We have tailored these resources for ages 11-14 on NRICH, and for older students and wider audiences on Plus.