Download Ebola Maths Lesson - Modelling the Spread of a Disease

Mathematical Modelling - Diseases
Hello mathematician,
What you are about to read is classified information. As you are aware Ebola has
been spreading rapidly throughout the world. What has not been made public is that
Ebola has been around since 1976 in Western Africa. Since then there has been
three separate epidemics with the current one being the largest. So far there has
been 13,268 cases reported resulting in 4960 deaths.
However a new and potentially dangerous infectious disease has broken out in the
UK. As you are a mathematician the government are urgently seeking your help and
we are asking you to forecast how the disease will spread.
We have been carefully monitoring those people already infected and their friends
and families. Their data show that an infected person typically goes on to infect
either one or two others. We need you to carry out some experiments and report
back to us your findings.
A simulation of the infection using coin tosses.
The number of infected people in each generation of the infection.
Instructions: You decide to simulate the spread of the infection using a coin. You
assume that there was one infected person to start with. You then flip the coin: if
tails comes up, then the infected person goes on to infect one other person,
if heads comes up, they go on to infect two other people. You repeat this
process for every newly infected person. Do this for at least five "generations" of the
infection, keeping track of the new number of infected people at each generation, for
example by plotting the number of infected people against the generation of the
infection.
For example, the first generation consists of the one infected person you started
with. If you get tails in the first round, then two new people are infected, so the
second generation contains a total of three infected people, the initial one and two
new ones. Flip the coin again for each of the new people. If you get one heads and
one tails, then three new people are infected, so the third generation contains a total
of six infected people, and so on.
TASK 1: Carry out the experiment at least five times, drawing a diagram for
each experiment. Work out the total number of infected people at the end
of each experiment. Using the information from one of the experiments
draw a graph (see above) of ‘the number of infected people’ against
‘generation’
How many infected people do you have at the end? Repeat the simulation several
times. In general, do you think that the infection will go on spreading, or will it die
out of its own accord?
STRETCH AND CHALLENGE - What is the average number of sick people at the
end of the experiment? What is the average number of sick people each person
effects (Epidemiologists call the average number an infected person infects the basic
reproduction ratio of the disease)? Which average did you use and why? What do
you think would happen if you continued repeating this process over a longer period
of time? Can you prove this?
Instructions: The government is thinking about introducing measures to reduce the
rate at which infected people go on to infect others, for example, by putting infected
people into quarantine, or giving them anti-viral drugs which reduce the length of
time they're ill.
To find out what effect such measures might have, you decide to see what happens
to your simulation if the number of people an infectious person infects is reduced by
one in each case. So if tails comes up, the person goes on to infect nobody at
all, and if heads comes up they go on to infect only one person. The
experiment starts after two generations when one person has infected two people.
What is the likely course of the infection now?
TASK 2: Carry out the experiment at least five times, drawing a diagram for
each experiment. Work out the total number of infected people at the end
of each experiment. Using the information from one of the experiments
draw a graph (see above) of ‘the number of infected people’ against
‘generation’
How many infected people do you have at the end? Repeat the simulation several
times. In general, do you think that the infection will go on spreading, or will it die
out of its own accord?
STRETCH AND CHALLENGE - What is the average number of sick people at the
end of the experiment? What is the average number of sick people each person
effects (Epidemiologists call the average number an infected person infects the basic
reproduction ratio of the disease)? Which average did you use and why? What do
you think would happen if you continued repeating this process over a longer period
of time? Can you prove this?
SUPER STRETCH AND CHALLENGE 1
Epidemiologists say that an infection will eventually die out if the basic reproduction
ratio is less than 1, and keep on spreading if it is greater than 1. Can you see why?
In the light of this, what should any intervention, like putting people into quarantine,
giving them anti-viral drugs, or vaccinating them, aim to do?
SUPER STRETCH AND CHALLENGE 2
The simulations in Tasks 1 and 2 are simple models of the spread of the infection.
Are they realistic? How could you improve them?
PLENARY
Write a short paragraph reporting you findings back to the government.
How can YOU help the battle against Ebola? So far £98 million has been raised to fight Ebola and
help care for those with the disease. This is only a fraction of the amount needed. If you would like
to donate (with parents/carers permission) then you can go to one of the following websites
http://www.redcross.org.uk/en/Donate-Now
www.oxfam.org.uk/ebola_crisis_appeal
www.unicef.org.uk/Donate-Ebola
General Lesson Plan
Firstly there are many activities/questions following on from the experiments. Please
tailor this to your group to ensure stretch and challenge. Also, some of the teacher
led parts will not be needed so again please tailor this to your group.
Set the scene that they have been approached by the government to carry out an
experiment etc.
Go through an example of how to model an outbreak explaining what is meant by
‘generation’. Explain the task and let them model it.
Show students how to graph one of their experiments.
Let students do this.
Explain the second task – do an example if needed.
Allow time in the lesson to go through the explanations to the additional questions
Plenary – write a summary of their findings to the government. Allow a couple of
students to read out their responses.
End the lesson with how students can donate for Ebola highlighting that they need
parental permission!!!