Part 1: Build your first model
With just some simple arithmetic, Julia introduces us to a basic mathematical model of the spread of infectious diseases, and then introduces us to R, one of the most important numbers in disease modelling.
Problem
This problem belongs to the Contagious Maths: Understanding the Spread of Infectious Diseases collection.
How do we use maths to understand how an infectious disease might spread?
If each person goes on to infect two others, how many generations would it take to infect...
Your school?
Your local community?
The nearest city to you?
Your country?
The whole world?
R: the reproduction ratio
If you have an infected person, how many people do they go on to infect on average? As Julia explains, this is a very important question.
You can explore the effect of different values of R using this Desmos application.
Adapting mathematical models
In this video, Julia encourages us to consider the limitations of the model we have been using so far.
What assumptions did you need to make to work out your answers so far?
How realistic do you think they were?
This concludes Part 1. These resources continue with Part 2: Lucky Dip, the follow-up activity which introduces a more advanced model, and the concept of herd immunity.
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 first 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 first of four parts, designed to be used in a sequence of lessons - here is a lesson by lesson breakdown.
Why do this problem?
Many students struggle to appreciate the difference between geometric and arithmetic progressions. This activity encourages students to investigate geometric progressions through engaging contexts, in readiness for exploring a mathematical disease model which uses doubling.
The teaching materials include video clips featuring Professor Julia Gog explaining how the underlying mathematics supports her work as a disease modeller and researcher.
Possible approach
Starter Option 1: Paper Folding
Each student will need a sheet of newspaper (or similar large sheet).
- choosing thinner paper
- folding it in one direction only
- using much larger paper.
"One of the students in the class managed to fold her paper a record-breaking 12 times!" Click on this link and ask "Looking at the photo, how do you think she managed to do that?"
Using the photo evidence, the class can work out that teenage student Britney Gallivan used toilet roll (for its length) and folded it in one direction only.
"How many layers of paper would there be after Britney folded her paper 12 times?" Allow time for the class to extend their table to display the data for 12 folds. Note the approximate height of Britney's folded pile of toilet paper.
Extra Resources for Paper Folding
Here is a video (3min 17sec) showing how St. Mark's students set a new paper folding record of 13 folds using just over 10 miles (53,000 feet) of toilet paper and the 3rd floor of MIT's famous Infinite Corridor. Why did they need approximately 16km (10 miles) of paper?
If it was possible to carry on folding, what would be the height of each folded pile of paper beyond 13 folds?
Challenge students to suggest a well known landmark of an equivalent height to the folded pile of paper after 20, 30, 40... folds, and then show them this TED Ed animation (3min 49sec) How folding paper can get you to the moon.
Starter Option 2: Mathematician's Footsteps
Share the video clip (1min 40sec) below:
Ask students to explain the difference between how Ada moves and how Florence moves. If instead, Ada moved 1m each step, but Florence moved 10cm, then 20cm, then 40cm, and so on, how long would it take Florence to catch up with Ada? How many steps would it take Ada and Florence to walk 100m? How many steps would it take to complete the Coast-to-coast walk (313km)? Or substitute local distances between towns or similar.
A spreadsheet can be used to highlight the difference between Arithmetical and Geometric growth, or students might like to use this growth calculator.
Starter option 3: The Chessboard and Rice Legend
Share the famous legend about the origin of chess. When the inventor of the game showed it to the Emperor of India, the Emperor was so impressed by the new game that he asked the inventor to name his reward. The inventor asked that the Emperor give him one grain of rice for the first square on the chessboard, two grains of rice for the second square, four grains of rice for the third square, and so on for all the 64 squares of the chess board.
Ask students how many bags of rice they think the Emperor thought he might have to give to the inventor (it might help to have a bag of rice as a prompt).
This video clip (2min 01sec) shows how the amounts of rice increase as you move up the chessboard, and this video clip (1min 25sec) shows the amounts of rice on the same board, with scaling used to show the larger amounts. You can also find instructions on how to make your own board.
How do we use maths to understand how an infectious disease might spread?
Move the discussion forwards from exploring the abstract concept of doubling, to investigating the modelling of infectious diseases.
"Let's think about how mathematicians use these properties of numbers in their work."
Show the first video clip (1min) in which Julia talks about her work and sets students a challenge!
The video introduces the idea of a model, where one person starts infected. In each generation, everyone infected goes on to infect two others, who become the next generation.
After watching the clip, please explore the consequences of this model. You can do this through thinking about how many generations of infection it takes to infect various populations. If it helps, you can play the Standing Game: choose a student to be the first infected, and they stand up. They select two other students to 'infect' (stand up), who will in turn each 'infect' two other uninfected (seated) students who then also stand up. They will in turn each 'infect' two other uninfected students, and so on. Continue until there are no more students left to 'infect'. Possible questions for the class include:
- How many steps (or 'generations') did it take to infect the class?
- If the Standing Game spreads beyond the classroom, how many generations would it take to infect the school/local community/nearest city/country/world? (choose geographies appropriate for your class!)
Now ask students to compare what would happen if each 'infected' student only infected one other student each time. How long would the disease take to infect the whole class, or whole school in this case?
R: the reproduction ratio
Show the second video clip (2min 27sec) in which Julia uses the geometric progressions with common ratios of 1 and 2, which the students have been investigating so far, to help introduce them to the concept of an R number. After the clip has finished, the students can follow up Julia's challenge of exploring the effect of three people being infected by each infected person (i.e. setting R=3).
Possible questions to ask students include:
- How many generations might it now take to infect their class if R=3?
- How many generations might it now take to infect their school/local community/nearest city/country/the globe?
Classes could also explore different R numbers using this Desmos application.
Adapting mathematical models
Show the third video clip (1min 44sec) in which Julia encourages students to consider the limitations of the doubling model.
Note: this is an important theme in this project, that there isn’t one single model or equation, but that the mathematical representations of real systems can be modified and refined. We will return to this theme more than once.
"What assumptions did you need to make to work out your answers so far?"
"How realistic do you think they were?"
Some of the limitations of the model which students might suggest are:
- assuming that the people who are infected only infect people who have not been infected before
- not allowing for vaccination
- assuming that infected people can't recover from the disease and get infected again
- assuming that anyone can infect anyone else
- assuming that there are always people available to be infected
- assuming everyone infects the same number of people
There are many more possible options of model shortcomings which students might think about, which may depend also on what disease and what context they are thinking of. At this point, it may not be possible to say much about how the model should change to take any of these into account: but this is what we follow up later in the project.
This concludes Part 1 of these resources. You may like to continue to Part 2: Lucky Dip, the follow-up activity which introduces a more advanced model, and the concept of herd immunity.
Key question
What are the limitations of the doubling model?
Possible support
Students could explore this growth calculator which enables them to compare arithmetic and geometric progressions side-by-side, the Pocket Money Problem which uses a familiar setting to explore the different progressions, and this video clip (1min 34sec) which uses Lego figurines to illustrate the differences between the two types of progression.
Possible extension
“Eyebrows were raised when the United States' Public Health Agency Center for Disease Control's model forecast a potential 77 trillion cases of smallpox if it were ever to be accidently released from a laboratory and the epidemic went unchecked.” Ben Cooper (2006)
What do the students notice about this prediction?
Hopefully they will notice that the predicted 77 trillion cases far out-numbers the population of the globe - therefore the modeller 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.
Further resources:
Students might enjoy reading the following articles:
- This article to learn more about Britney's record-breaking paper folding feat
- This article explores exponential growth more generally
- This article explores further the ideas of mathematical modelling in general
Once students have completed these Part 1 activities they might like to explore Part 2: Lucky Dip, the follow-up activity which introduces a more advanced model, and the concept of herd immunity.
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.