Part 3: Everybody is different

Julia refines our model, and the interactivity offers a chance to simulate the outbreak of an infectious disease, recognising that not everyone is the same.

Age

11 to 14

Challenge level

Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving

Being curious Being resourceful Being resilient Being collaborative

Problem

This problem belongs to the Contagious Maths: Understanding the Spread of Infectious Diseases collection. It follows on from Build your first model and Lucky Dip.

How could we take our model further?

In this video clip, Professor Julia Gog introduces the idea of how people can differ in how they are connected to others.

The number of people that infected people will go on to infect will vary.

How will this affect the epidemic?

You can test your ideas by revisiting the Lucky Dip interactivity, and setting R on the interactivity to 'Random 0 to 5' (you can do this by clicking on the purple cog in the top right).

Are the shapes of these graphs any different to the previous ones?

If we're not all the same, what is R?

In this video clip, Julia asks us to consider what R could mean if we are all different.

Once you've had a chance to think about Julia's question, watch the follow-up video in which she answers her question.

In this video clip, Julia explains that R represents an average value across the whole population.

R applies to the whole population. The official definition that researchers use is this: R is the average (ie, the mean) number of people that one infected person goes on to infect.

The Everybody is Different interactivity

The 'Everybody is Different' interactivity below enables you to simulate the outbreak of an infectious disease, recognising that not everyone is the same. Before you get started, it's helpful to know the following information:

Circles represent different people who are susceptible to infection. The colours represent different numbers of nearby people that they could go on to infect (also given by the number displayed on the icons):

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0 contacts (very cautious!)
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1 contact
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2 contacts
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5 contacts (super-spreaders!)

And different shapes represent people who are infected, recovered or immunised:

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are infected and might infect others
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have recovered from infection and are now immune
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are people that you have chosen to immunise (see below), so are also immune

To run your first simulation, click the "Infect a random person" button

You'll see that one person has turned red, it means they have been infected and are liable to spread the disease to people near them.

What do you think will happen next? Can you justify your answer?

You can now investigate whether you were correct, by either watching the epidemic progress one generation at a time, by clicking on the 'Run a generation' tab, or letting it run its course and seeing the outcome, by clicking on 'Run to the end'. Note that when someone turns purple, it means they have recovered from the disease and are immune to it.

Explore the interactivity by infecting random or selected individuals, immunising selected individuals, and varying the value of R.

You can use the R slider to change the average R value across the population.
If you click on a person (or more) you can choose to infect or immunise them.
Or if they are already immune you can decide to make them vulnerable again.

Notice if you change R, the mix of different colours changes to make up the right value for the mean.

Can you anticipate what will happen?

When the model has finished, use the Graph button to view a graph of the whole infection from start to finish. Is it similar to the shape of the graphs in Lucky Dip?

What does the value of R tell us?

In this video clip, Julia explains why R is important to understanding epidemics, particularly whether R is greater than or less than 1.

What else do you think should be taken into account in order to improve the model?

This concludes Part 3. These resources continue with Part 4: Get Moving!, which will introduce models where the population moves about. 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 third 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 third of four parts, designed to be used in a sequence of lessons - here is a lesson by lesson breakdown.

Why do this problem?

Everybody is Different follows on from Build Your First Model and Lucky Dip, and introduces the need for a more realistic model, in which we don't all behave in identical ways. Until now, we've assumed that every infected person exposes the same number of people to the infection. In reality there is variability in the number of people that infected people go on to infect. Students are encouraged to explore the similarities and differences that occur when running several simulations of the updated disease model, and to consider how variability affects the epidemic.

This lesson, and the next, meet the curriculum aims of developing students' Fluency (moving freely between different representations), Reasoning (making connections between number relationships and graphical representations), and Problem Solving skills (begin to model situations mathematically).

The teaching materials include video clips featuring Professor Julia Gog introducing the need for a more realistic model, and then explaining the impact of R being greater than or less than 1.

Possible approach

Everybody is Different is intended as an activity for students who have already explored Build Your First Model and Lucky Dip.

Start by showing this video clip in which Julia introduces the need for a more realistic model, in which we don't all behave in identical ways.

Collect ideas about the closing question 'What do you think that will do to the epidemic?'

Revisiting Lucky Dip, and setting R on the interactivity to 'Random 0 to 5', will allow students the opportunity to explore their ideas. Encourage students to run the interactivity several times and find ways to record their findings. What do they notice about their results? Some differences may become apparent very quickly, e.g that the epidemic can now fizzle out after just one generation, but other differences may be more subtle, such as the length of the epidemic is now more variable.

In this video clip Julia asks 'What do you think R could mean if we can all be different?'

Once students have had a chance to explore what R might mean, and compared suggestions, show this video clip which explains that R represents an average value across the whole population.

Introduce the Everybody is Different interactivity

Begin by displaying the Everybody is Different interactivity and asking the students what they notice. Possible responses include:

the three different colours (these represent different numbers of nearby people that each person could go on to infect - also given by the number displayed in the icons)
the grid shape
the additional controls which include decimals - for the mean R of the population

The meaning of the different colours is explained in the problem page.

Model running the activity by clicking on the 'Infect a random person' tab. The random person will be highlighted in red - encourage the class to predict how the disease might spread from that person to others and justify their responses. Allow time for students to explore the interactivity by infecting random people and varying the value of R.

The Save / Load feature generates a code, which allows you to select and share a layout, which can be used as a common basis for exploration and discussion in the classroom. You can challenge the class to select individuals on the grid who they think would be likely to kick-start an epidemic, and explain their reasoning behind their choices. Allow time for students to test their theories. Then ask students to consider which individuals might be less likely to spread the disease widely, and test out those ideas too.

You can ask how important R is in determining what will happen, and how much it matters who the initial infected person is. What difference does it make if you start by choosing several infected individuals?

Finish by showing this video clip in which Julia explains the impact of R being greater than or less than 1.

Bring the session to a close by noting that the individuals in the models so far, have been static. Why might it be useful to include movement in a model?

Exploring models which incorporate movement is the basis of the final session, Get Moving!.

Key questions

What do you think will happen to the spread of an infectious disease if different people infect different numbers of other people?

Do different layouts affect the spread of a disease?

What else could we consider exploring with our model?

Possible support

Students could take screenshots to help them compare the outcomes of the spread of the disease each time they run the model.

Possible extension

Once students have finished these Part 3 activities they may like to explore Part 4: Get Moving! which looks in more detail at the importance of how people are connected and the effect of travel on 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.

Students may be interested in Disease modelling for beginners, a collection of articles which explain how mathematics helps us understand how infectious diseases spread.

Or search by topic

Number and algebra

Geometry and measure

Probability and statistics

Working mathematically

Advanced mathematics

For younger learners