Working Mathematically with Viruses
Years 8 - 12

This investigation has several levels of challenge. You don't have to do it all. Like a video game, you can stop at any level then return later for the next level. Keep good journal notes so you know the level you reached.

 

Background

Viruses have been part of human life since ... forever.
  • We already have successful ways of living with them or treating most of them.
  • Example: Many people deal with the annual influenza virus ('flu) by getting vaccinated.
However, new viruses are always developing.
  • Usually we have no protection against new viruses, so they spread around the world.
  • Example: Spanish 'Flu spread around the world between 1918 and 1920.
  • It is estimated that between 17 million and 50 million people died because of Spanish 'Flu.
  • When a virus spreads around the world the spread is called a Pandemic.
Working Mathematically with Viruses is about new viruses.
  • It is written at the time of the COVID-19 virus, which arrived at the end of 2019.
  • COVID stands for CO-rona VI-rus D-isease.
  • Future new viruses will behave in a similar way.

Quick Links

Acknowledgements

This activity developed from a successful lesson sequence explored by Aaron Peeters with his Year 9 class at Manurewa High School, New Zealand. Aaron wrote:
The overall thread ... was to help students understand (a) the current situation we are in and (b) the role that maths plays in helping us understand that current situation, with (c) the technical maths skills required to understand those two aims fitting in where necessary.

Microsoft Excel software is by Dr. Ian Lowe, Melbourne.

Preparation

  • A collection of small objects to use as pretend people.
    The objects need to be different on each side.
    You could use plastic screw caps, playing cards or...
    print this People Page and cut out the squares.
  • Print this square line paper.
  • You need a computing device with Microsoft Excel compatible software.
  • One calculator (there's one on your phone)
  • Write the title of this challenge and today's date on a fresh page in your maths journal.

Exploration 1: Virus Basics

  • Click the image on the right to launch the first video.
  • Watch it through then try the exploration below and record in your journal.

Have fun exploring Working Mathematically with Viruses.

Click this image to begin the video.
 

The Reproduction Number (R) is the number of people to whom the virus is transmitted by 1 infected person.
    1. Use your people to make a model of infection for a virus with R = 2.
      (If you are using the paper tiles start with blank side up.)
      Start with one person then show the next 3 or 4 time periods.

    2. Sketch your model in your journal.
      (Using arrows can be useful in a sketch.)

    3. Put numbers on the sketch to show new cases and the total active cases at each step.
  1.        A table can also be used to model the problem.
           It helps you to organise data so you can look for patterns.

    1. Assume the time period for each step is 1 week.
      Make a table like the one shown and fill in the gaps.

    2. Use the patterns in the table to predict the values for Week 10.

      If you just use numbers for your model, which you are doing in the table,
      you have to remember that the numbers are representing people.

  2.        Repeat this investigation for a virus with R = 3.

  3.        A graph is a sketch or drawing of the data in a table.
           Sometimes a graph makes it easier to see the patterns in a table.

    1. Print this square line paper.
    2. Mark Weeks 0 to 4 along the bottom.
    3. Draw three graphs of total Active Cases on this piece of paper.
      One for R = 1, one for R = 2 and one for R = 3.
      Use a different colour for each graph.
    4. Stick it in your journal.
    5. Comment about what the graphs would look like for R greater than 3 (R > 3).
 
If you aren't sure about how to sketch your model, look for ideas in the Answers & Discussion section below.
 

 

 

R = 2

Week New Cases Active Cases
0 1 1
1 2 3
2 4 7
3 8 15
4 31
5
6
7

Using Real Data

In Melbourne, Victoria, filming of The Masked Singer episodes was stopped because of an outbreak of COVID-19 virus. The programme had been operating under strict health regulations. Cast and crew of over 200 people had to enter self-isolation. The first case was reported on Friday August 21st, 2020. By August 25th the total was 16 cases.
Using the information you have learnt so far:

  • Which R value is most likely to apply to this population?
  • What is the most likely time period involved in transmission of the virus in this situation?
Reference: Sydney Morning Herald

 

When a new virus appears in the world, we don't know its R value and we don't know how to stop it.
What happens if it just keeps infecting with nothing to stop it?

Using a Mathematician's Tool

Hands-on models, sketches, drawings, tables and graphs are all tools of a mathematician.
Here is another one. Open this Excel spreadsheet.

Note: This spreadsheet will not work in the Mac program called Numbers.

It should open to 1 Virus Basics.
  • If it doesn't choose Tab 1 at the bottom of your screen.

  • Then choose the View menu and Full Screen on the ribbon so it looks like the picture below.

  • Use Esc to return to the page layout view.

  • Each small yellow rectangle is one uninfected person. A rectangle changes to purple when it becomes 1 infected person.

  • In this sheet you enter the R value and follow the red instructions to make the virus infect the population.

  • The default population size is 200 people.

  • The maximum population size available is 1000 people.

  • Enter R = 2 and 'go'. Don't tap F9 yet.

  • Look down the table to Week 10. Check your predictions for New Cases and Active Cases.

  • Delete 'go' and start again using R = 3. Check your Week 10 predictions.
Population Size = 200

  1. Enter R = 1. Enter 'go'. Tap F9 until the Yellow cells stop changing to Purple.

    Note: On some keyboards you will need to hold the Fn button whenever you tap F9.

    1. Check the table to see which week it stopped.
    2. Count the Purple Squares to know the active cases.
    3. The F9 action stopped because the pattern is clear already.
      Predict the number of weeks to reach 200 active cases.
      (Hint: Week numbers start at zero. Week 0 <> 1 Active Case.)

  2. Choose three (or more) R values. For each one find the week in which the Active Cases reaches 200 people. That means 200 Active Cases.
    For each choice, before you experiment, predict the number of weeks.
    Record your predictions and explorations.

Population Size = Your choice

  1. Change the population size to your own choice. The maximum is 1000.
    Check that the correct number of boxes have turned yellow.

  2. Predict the week in which the Active Cases will reach (or be greater than) your population size. Record the week.

  3. Enter R = 1. Enter 'go'.
    Tap F9 and watch the Active Cases box until the week it reaches, or passes, your population size. Record the week.

  4. Choose three (or more) R values. For each one find the week in which the Active Cases reach (or are greater than) your population.
    For each choice, before you experiment, predict the number of weeks.
    Record your predictions and explorations.

Population of the Earth

The human population of the earth is 78 billion.
That's 7,800,000,000.
Ten digits.

R = 1 means one infected person passes the virus to one new person, who passes it to one new person, who passes it to one ... and so on.
So, if R = 1, the week in which Active Cases reaches the world's population size is 78 billion - 1.
That's about 150 million years.

But what happens if R is bigger than 1?

Option 1, Virus Basics, can help us find out.

  1. Suppose R = 15. Predict and record the week in which the earth's population will be reached.
    Do you think the spreadsheet could go that far?
    Enter R = 15, try it out.

    Keep watching the Active Cases box above the grid so you know when 7,800,000,000 is getting close.
    (Something special happens when you reach or go greater than the Earth's Population.)

    Record the result and comment.

Even if all the Yellow cells change to Purple, the spreadsheet will continue to calculate when you tap F9.
  1. Predict for R = 2, 3, 4, 5, 6, then explore to find the week in which the Active Cases reach the population of the earth.
    Record your predictions and explorations in a table.

  2. Imagine a new virus has appeared.
    Remember, when a new virus appears, we don't know its R value.
    Use your data to prepare a brief report to a politician to explain what might happen if nothing is done to battle the virus.

Your report is called a 'worst case scenario'. For each R value in your report you are showing the worst that could happen.
Discuss with someone at home, or class mates, things about the world that would make it harder for the virus to spread to everywhere.

Exploration 2: Recovery

Note:
In this model, Recovered means all who have already recovered plus the new cases from two weeks before.
This is what is called a 'cumulative statistic'.
As long as there are new cases it keeps on growing (accumulating).

  • Click the image on the left to launch the second video.
  • Watch it through then try the exploration below and record in your journal.
Assume, as on the video, that two weeks pass between
when a person is infected and when they recover (or die).
This is probably the situation with the COVID-19 virus.
That's why the isolation period is 14 days for COVID-19.

    1. Use your people to make a recovery model of infection for a virus with R = 2.
      Start with one person then show the next 3 or 4 time periods.

    2. Sketch the model in your journal with numbers on the sketch to show
      New Cases, Recovered Cases and Active Cases at each step.
      OR
      Make a table like the previous one, but with an extra column for Recovered Cases.
  1.        Use your data to predict the values for Week 10.

  2.        Repeat this investigation for a virus with R = 3.

Open this Excel spreadsheet again and choose View / Full Screen.
This time select Tab 2 at the bottom to show the sheet 2 Recovery.

  • Yellow and purple cells mean the same as before, but when an infected person recovers, they are shown as green.

  • The default population size is 200 people.

  • The maximum population size available is 1000 people.

Explore the spreadsheet like you did before:

  • Population: 200
  • Population: Your choice
  • Population of the world
Use at least three different R values.

In each situation you are finding the week in which the Active Cases reaches the chosen population size.

  • For each one make a table that compares your previous numbers (no recovery) to these numbers (recovery).
  • Comment on the effect of including recovery in the model.

Exploration 3: Changing R

Exploration 2 shows that any human immunity which might result from recovery is not going to do anything to halt the progress of a new virus. The virus has already moved on by the time recovery has occurred.

  • Immunity through recovery happens behind the virus.
  • It can't work to stop a new virus.
It is clear from explorations so far that lower values of R mean lower numbers of people are infected. That means:
  • Fewer people spread the virus.
  • Fewer people get sick.
  • Fewer people develop long term lung damage
    or other problems related to fighting the virus.
  • Fewer people die.
In this section we explore what happens if a virus begins to infect a population and we take action to try to get in front of it.
  • In your journal list all the actions a population can take to stop a virus spreading.
    Give your list the heading Public Health Actions.
    You must include hand washing.

Open this Excel spreadsheet again and choose View / Full Screen.
This time select Tab 3 at the bottom to show the sheet 3 Changing R.

  • Click anywhere on the graph.
  • A boundary box will highlight.
  • Drag the boundaries to make the graph almost fit your screen.
    You must be able to see the yellow boxes.
Using Real Data

At the time of preparing this activity, there is no evidence of any long-lasting antibody immunity to COVID-19. In other words the concept of 'get Covid and you can't get it again' is totally unproven and probably proven to be untrue.

Reference: Royal Australian College of General Practitioners 13th July 2020

Interpreting the Graphs

The spreadsheet graphs are built from the table data in Option 2.
Print this data sheet for 11 weeks of data for R=1 through to R=5.

  1. Enter R=1 and 'go'. Tap F9 once.
    Answer these questions in your journal.
    1. How many new cases are there in Week 1?
    2. Why are there two active cases in Week 1?
    3. Explain the the small green line at the bottom of the graph.
    4. Write an ordered pair of numbers for each Week 1 dot, e.g. Purple is ( __, __).
    5. Click anywhere on the graph to highlight the border again.
    6. Hover your mouse over the end of each colour line to check your ordered pairs.
    7. Why is 1 the first number in each pair?
    8. Predict what will happen to each colour line if you tap F9 one more time.
    9. Tap F9 once to check your prediction.
    10. Why has the green line jumped up to (2, 1)?
    11. Explain the red line. Where was it in Week 1?
    12. Predict what will happen if you tap F9 once more to reach Week 3.
    13. Tap F9 once to check your prediction.

    Keep watch on the vertical scale. Sometimes it changes.

    1. Predict what will happen if you tap F9 once more to reach Week 4.
    2. Tap F9 once to check your prediction.

    Did you notice that vertical line changed again?

    1. Explain why the orange and purple lines are now both horizontal.
    2. Explain why the red and green lines keep growing.
    3. For Week 4 predict the first number in each ordered pair?
    4. Hover your mouse over the end of each colour line to check your predictions.
    5. Predict Week 5 then tap F9 to see what happens.
You don't have to do any more of R=1,
but if you want to you can go through to Week 20.

  1. Delete 'go'.
    Enter R=2 and 'go'. Tap F9 once.
    Answer these questions in your journal.
    1. Predict and investigate what each colour will be for each week up to Week 5.
      The table data might help you predict.
      Keep watch on the vertical scale.
    2. Explain in your journal any differences you notice between R=1 and R=2.
      If you want to, you can go through to Week 20.
Combining R=1 and R=2

A virus needs to enter 1 host person before it can spread.
Suppose a new virus enters a host in a population and starts to spread at R=1.

  • Delete 'go'.
  • Enter R=1 and 'go'.
  • Tap F9 until you reach Week 5.
  • If you overshoot, delete 'go' and start again.
Now suppose the conditions change (perhaps a lot of tourists visit for summer) and the virus starts to spread from here at R=2.
  • DON'T delete 'go'.
  • Just Enter R=2.
  • Tap F9 until you reach Week 10.
Your graphs should look like this:

Challenge

Your aim is to bring new cases down to zero (or close) by Week 20.
  • You can change R to any number you want and you can change it more than once.
    (Any number means 0 or above, including decimals.)
  • BUT each time you change it you must choose which item on your Public Health list has been used.
  • Do several experiments until you get as close as you can.
  • Write a short report in your journal.
    (Would screen captures be useful in your report?)
    Remember, your mouse can tell you the ordered pair for any dot.

What happens if...?

The previous challenge assumed that the virus began with R=1 and when conditions changed it became R=2.

  • Some viruses are much more vigorous than that.
  • What happens if the virus arrives with R=2 and at Week 5 it changes to R=3 until Week 10?
Read and record the New Cases value when you reach Week 5.

Read and record New Cases again
after you change to R=3 and reach Week 6.

  • Explain why New Cases jump from 32 to 96 between Week 5 and Week 6.
Read and record New Cases again when you reach Week 10.
  • Is it possible to bring the new cases to zero by Week 20?
  • Experiment with changing R values to see if you can do it.
Remember to say which Public Health Action(s) you would take to make the R value drop.

Your Choice

  • Start the virus with your own R value.
  • Decide the number of weeks that pass before Public Health action is taken.
  • Experiment to find an R value (and the Public Health actions that have to be taken) to bring New Case numbers down to zero by Week 20.
You might have to take extra action to reduce the R value if your first choice doesn't bring it down quickly enough.

Write a story explaining one of your experiments.
Here is an example.

Example

This is my graph. I tried to make it like what happened in my state, Victoria, in 2020 when COVID-19 came.

Algebra On The Side
  • A spreadsheet like this operates with formulas.
  • Formulas can be written in words or symbols.
  • These are our formulas.
New Cases
  • In a pandemic everything starts from New Cases (N).
  • N is the cases that are new this week.
  • If N stays at zero (0) the virus has been beaten.
  • If N has any other value, the virus can spread.
  • The value of N can go up or down.

Recovered Cases

  • Recovered Cases (Rc) are N from 2 weeks before (N2w) plus all new cases before that.
  • From Week 3, Rc can never be 0 because once a person recovers they are always counted in this group.
  • The value of Rc can only go up or stay at the same number.
  • Rc = N2w + [ 1 + ... + N2w-1]

Active Cases

  • Active Cases (A) start in Week 0 with 1 because the first N is immediately active.
  • After that A equals Active Cases from 1 week before (A1w) plus N minus N2w.
  • A = A1w + N - N2w

Total Cases

  • Total Cases (T) is all the people the virus has been in or is still in.
  • It doesn't count people the virus hasn't reached yet.
  • T = Active Cases plus Recovered Cases.
  • T = A + Rc

When it started it was just one overseas traveller and no one was too worried. We didn't know then that it had R=3. Then in Week 4 (only one month) there were 81 New Cases and 108 Active Cases altogether. We had started getting a bit worried and washing our hands more and doing social distancing, but it wasn't enough. We were asked to stay and home and work from home and even not go to school.

That changed the R value because it made it harder for the virus to find people. I changed my R to 08 and used that to Week 10. Then I only had 22 New Cases and 49 Active Cases. That seemed pretty good so we started to ease restrictions, so I did R=08 for two more weeks to make it three months from the start.

At Week 12, I had only 14 New Cases and 32 Active Cases and we were going well.

But people started to not be so careful and some people did really stupid stuff and things changed for the virus again. Lots of old people got sick. So I put R back to 3. The next week there were 42 Active Cases and I know why. It's because of what R means. Each of the 14 New Cases from Week 12 infected 3 other people and that makes 42.

Two weeks later in Week 15, there were 378 New Cases and 504 Active Cases and the graph was getting really high really quickly. The same public health stuff as last time wasn't going to stop the virus this time. We had to do much more...

The graphs above are not the real Victorian data, although they are a similar shape to the real data.

The next section uses actual Victorian data to show that R really did change.

Using Real Data: Calculating R

One way to calculate Reproductive Factor (R Value) of the COVID-19 virus is to divide the New Cases on any date by the New Cases 5 days before.
Reference: See below in Answers & Discussion

Print this New Cases Data sheet.
It shows new cases in Victoria, Australia, around the peak of both waves of infection in 2020.

  1. For each wave predict then calculate R on the peak day.
    Record the date and the R value.
  2. For each wave choose 3 dates before the peak and 3 dates after.
    Record each date, predict and then calculate R.
  3. Comment on what you notice.
Calculate to 2 decimal places.

 

Exploration 4: Effect of Vaccination

Vaccination is meant to allow the population to live normally again.
  • Click the image on the left to launch the third video.
  • Watch it through then try the exploration below and record in your journal.

Now we know:

  • People recovering from a virus does not prevent the virus spreading.
  • Public Health actions keep people away from the virus and can bring down its Reproduction Factor (R value) and even stop it.
BUT Public Health actions can cost a very, very big amount of money.

AND if there is even one case left in the world it can start all over again.

IF a vaccination is found to fight the virus, the vaccine will be waiting in people's bodies to attack the virus when it arrives.

Challenge

  • Use Options 4 & 5 of the spreadsheet to investigate different percentages of the population being vaccinated.
  • Prepare a report with evidence to advise the government about the vaccination level necessary to make us safe.
  • Write a paragraph explaining why you would or would not want your family to be vaccinated.
Open this Excel spreadsheet again and choose View / Full Screen.
This example will help you plan your investigation.

Open Option 4.

  • Click on the graph and change the border to fit your screen.
  • In Option 4 you can change the R value the same way as Option 3.
  • You can also change the percentage of the population who get vaccinated.
  • Start with R=3 and 0% vaccinated.
  • Spread the virus to Week 4.
  • Record the New Cases and Active Cases.
  • Vaccinate 20% of the population. That means 1 person in every 5 is vaccinated.
  • Spread the virus to Week 20.
  • Record the New Cases and Active Cases now.
  • Delete 'go'.

Open Option 5.

  • Click on the graph and change the border to fit your screen.
  • In Option 5 you can change the R and the percentage the same as Option 4.
  • Start with R=2 and 0% vaccinated.
  • Spread the virus to Week 4.
  • Click on the graph and record the New Cases and Active Cases.
    They should be the same as your Option 4 experiment.
  • Vaccinate 1 person in every 5.
  • STOP and look at the dotted lines.
    They show you what would have happened in Week 5 if no vaccine was available.
  • Spread the virus to Week 20.
  • Click on the graph and record the New Cases and Active Cases now.
    They should be the same as your Option 4 experiment (30,114 and 48,935).
    That is still way, way too many cases, but the dotted lines tell you 20% vaccination has made a difference.
  • Record the values for the dotted lines and explain how much difference 20% vaccination has made to an R=2 virus that spreads for four weeks before vaccination is available. (See below in Answers & Discussion to check your work.)
Now design your own experiments to gather information for the reports you have been asked to prepare.
Note: R=3 was probably the start value of COVID-19.

Just Before You Finish

The main purpose of this activity is to allow you to do the same work as professional mathematicians do when there is a serious virus outbreak.
You have been learning to work like a mathematician.
  • Print this Working Like A Mathematician page.
  • Put a tick beside anything you have done in this activity that is on this list.
  • In your journal, finish this paragraph:
    I was working like a mathematician when I...
  • Also in your journal answer this question:
    What do you know now that you didn't know when you started this investigation?

 

Answers & Discussion

A mathematician's tool you might use for sketching your model in Question 1 is a Tree Diagram.

A tree diagram like this works for 3 or 4
time periods if R=2, but for R=3
the diagram needs much more space.
Imagine the space needed for R=5!

Also a tree diagram, this time for R=3.
Can you explain how it works?
The numbers represent people.
Our model is becoming more mathematical.

Did you remember to use numbers on your sketch to show
total New Cases and total Active Cases?

Calculating R

The activity above with this heading is based on the work of the mathematician who prepared this report published on August 5th 2020.

Option 5 Experiment

For R=2 (4 weeks) then 20% vaccination until Week 20

  • Active Cases (dotted line) would have been;
    one million, five hundred and seventy-two thousand, eight hundred and sixty-four
  • New Cases (dotted line) would have been
    one million, forty-eight thousand, five hundred and seventy-six

 

Send any comments or photos about this activity and we can start a gallery here.

 

Information for Teachers & Parents
Disclaimer
The model developed in this activity is based on mathematics, data and assumptions. It reflects the reality of the COVID-19 pandemic of 2020. It could be applied (with a little variation) to any virus. It is not the model used by authorities for public health purposes. Such models are also based on mathematics, data and assumptions. Our model has been developed for educational purposes as outlined at the head of the page by Aaron Peeters who designed and taught the first draft.

Assumptions
Our key assumptions are:

  • Uniform spread of people in a population being investigated.
  • Two weeks between initial infection and recovery based on the 14 day quarantine required for COVID-19.
  • We chose to combine Recovered Cases and Deaths as one number, because either is a non-active case.
You might want to discuss these with your students.
Mathematics
Our primary purpose in this activity is for learners to experience constructing, applying and interpreting a mathematical model. There are several other maths learning pathways for various year levels which could be developed in this context, including:
  • Power sequences and their symbolic representation
  • Functions developed from the algebra
  • If these functions were differentiated or integrated would they have any meaning in the context?
  • Composite functions, which is what is happening when R is changed in Option 3
  • Index laws, in particular, a context in which a0 = 1 makes sense
  • Rounding and rounding errors (for example, explore Option 1 for R=05 and R=049 and compare)
  • Extending the use of spreadsheets as a learning tool
  • ...
Pedagogy
We have tried to make our teaching craft clear in these notes. We have used:
  • An expectation that time is available
  • Real and relevant context
  • Hands-on exploration
  • Video
  • Multiple visual representations
  • Investigative software
  • Mathematical journalling
  • Scaffolded questioning
  • Unfolding challenges
  • Cross-curriculum learning
  • Student choice
  • Self evaluation
  • ...
With your department colleagues, or your teachers in training:
  • Select one of these and imagine removed for the lesson plan. Discuss.
  • Compare and contrast this activity to one with similar mathematical content which might be presented via a textbook. For example, considering the topic of graphs and their interpretation.
Software
The spreadsheet is locked to prevent any unintended changes while it is in use. If you need an unlocked version contact: doug@blackdouglas.com.au

 

Maths At Home is a division of Mathematics Centre