Activity
(adapted
from http://www.math.smith.edu/Local/cicchap1/node2.html )
1. Give
the students the following situation:
There is an epidemic going through a population of 50,000 people. Suppose that
today there are 2100 people infected and 2500 people have already recovered,
which makes them immune. We assume that nobody is going to die from the epidemic
and nobody moves into the population. How many people will be infected tomorrow
or in two days? How many people will have recovered?
2. Ask
the students, “How
can we use math to help predict how many people will get sick on a given day
during an epidemic?” Give the students time
to think and discuss this. Hopefully somebody will come up with the idea of
making a mathematical model of the basic aspects of an epidemic.
3. Point
out that almost all mathematical models of a disease start
out from
basic premise that the population can be subdivided into a set of distinct
groups, depending upon each group’s experience with respect to the disease.
One of the simplest yet most commonly used models for an epidemic is the SIR
model,
which divides the total population into susceptible individuals, infected individuals,
and recovered/removed individuals. Specifically:
S
= the number of susceptible people, the people in the population
who are not sick, but who could become sick.
I = the number of infected people, the people who are currently sick and can
infect
the susceptible people.
R = the number of recovered or removed people, the people who have been sick
and
can no longer infect others or be re-infected themselves.
A good
illustration of this situation, where all three groups
are mixed together to make the total population can be
found at http://www.math.smith.edu/Local/cicchap1/node3.html
4. Ask
the students to use what they know of the spread of diseases
to sketch three graphs as an epidemic spreads
through a totally susceptible population. The first is
a graph of the number of susceptible people (S )
versus time.
The second is a graph of the number of infected people
(I ) versus time. The third is
a graph of the number of recovered/removed
people (R ) versus time
5. Ask
the students how they will go about answering the following
questions,” Suppose
we know the values of S, I, and R today, how can we figure
out what they will be
tomorrow, or the next day, or a week or a month from
now?”
6. Stress
that if we want to know the values of S, I,
and R in the future,
we should pay attention to the
rates
at which these
quantities change. To make it easier to refer to
them, let's denote the three rates by S', I',
and R' where
S' =
Rate of change of number of susceptible people with
respect to time
I' = Rate of change of number
of infected people with respect to time
R' = Rate of change of number
of recovered/removed people with respect to time.
If
this is a calculus class, the students should recognize
that
S', I',
and R' are derivatives and know they
are rates of
change of these variables with respect to
time.
7.
Tell the students that we’ll begin with finding an
equation for S'. Ask them what
they know about the equation for S' from
the graph they made of S' versus
time.
8. Tell
them that S' = rate at which
susceptible people get sick = the
rate of transmission of the disease from an infected
person to a susceptible person. In other words, the rate
equation for S' will depend on both S and
I because transmission of the
disease involves contact between susceptible and
infected persons. Ask them to think about how to write
an expression
for the rate of transmission of the disease to a susceptible
person from an infected person.
9. Tell
the students that we will look at the rate of change in
the number of recovery/removed
people, R'. Ask the students
to think about which group of the population R' is
related to and why.
10. Now
we can deal with the rate of change in the number of infected
people, I'. Ask the students to
think about
how the number of infected people changes. Tell them
that these
changes will lead them directly to an equation for
I'
11. Have
the students write all three equations one under another
and add them together to get the overall
rate
of change of the population. Ask the students
to discuss what this means about our model for the epidemic.
12. Now
we can see the predictive power of the SIR model.
Have the students
go back to the measles epidemic mentioned at the beginning of the lesson.
Tell
the students
that the value of a=0.00003 and b=0.07.
Have the students enter these values into the rate equations.
13. Remind
the students that right now 2100 people are currently infected
and 2500 have already recovered out
of a total population of 50,000.
Have the students
calculate the following:
a) The
number of susceptible people right now
b) The value of S' using the
current numbers
c) The value of I' using the
current numbers
d) The value of R' using the
current numbers
e) The number of susceptible people tomorrow
f) The number of infected people tomorrow
g) The number of recovered people tomorrow
14. Have
the students use tomorrow’s values S1, I1,
and R1 of and
to calculate and , the values of and for two days from
now.
15. Now
would be a good time to remind the students that we use
the SIR model for predictions only. Their
calculated values of S, I,
and R are not the exact size
of the susceptible, infected, and recovered groups in the population because
the
model is based on a very simple interpretation of the epidemic. The numbers
are only approximations, and not very realistic approximations at that.
For
example: in our rate equation R'=0.07I,
we are claiming that 7% of the infected population recovers
everyday. This doesn’t
make any sense in reality. In the first days of the epidemic,
virtually nobody has had the disease long enough to recover,
which means the actual recovery rate will be much less that 0.07I. Towards
the end
of the epidemic there will be very few people infected and many people
who have recovered, which means the recovery rate will
be much higher than 0.07I. Clearly the
model does not mirror reality very well.
However,
our model which, while imperfect, still captures some of
the aspects of the epidemic.
The purpose of the model is to gain insight
into the workings
of an epidemic, and to think of ways of intervention to reduce its
effects. The over-simplifications in the model will be
justified if it leads to
inferences which help us understand how an epidemic works and how we
can deal with it.
Later,
if we wish, the model could be refined, by replacing the simple expressions
with others that better mirror the reality of the actual spread of
the epidemic through
the population. Homework
A
measles-like epidemic strikes
a small college that has a population
of 1,800. As of today, there
are 80 infected people and 31
recovered/removed people. It
is also known that an average
susceptible person only comes
in contact with about 0.3% of
the infected population each
day. Doctors have also found
out that the infection is transmitted
in only one contact out of six
contacts. The daily recovery/removal
rate is approximately 25% of
the infected population.
1.
Find the value of a, the rate
of
transmission of the disease
between susceptible people
and infected people and the value
of b, the rate of recovery/removal
of infected people.
2. Use the
calculated values of a and b to
make a SIR model for the disease. In other
words, find the equations for S', I',
and R'.
3. Calculate
the value of S, the number of susceptible
people
in the population as of
today.
4. Using the
values of a, b, S, I,
and R , calculate the
values
of and
S', I',
and R'.
5. Find the
values of S1, I1,
and R1, the number of susceptible
people,
the number of infected
people, and the number
of recovered/removed
people in the population
tomorrow (one day from
today)
6. Use
tomorrow’s
values of S1, I1,
and R1 to
calculate
S2, I2,
and R2,
the values of S, I, and R for
two days from now.
Some
relatively mild
illnesses, like
the common cold, return
to infect you again
and
again. For a while,
right after
you recover from
a cold, you are
immune. But that
doesn't last; after
some weeks
or months,
depending on the
illness, you become
susceptible
again. This means
there is now a
flow from
the recovered
population
to the susceptible.
Call this immunity
loss, and
use to denote
the coefficient
of immunity loss. This
means the
basic equations
for S', I',
and R' would
need to change
from:
S'=
-aSI
I'= aSI -bI
R'=
bI
7. What would be
the new equations
for S', I',
and R', taking
into account
immunity loss.
Explain your
reasoning.
Closure
The
students should realize that the SIR model has flaws and
does not give an exact picture of the epidemic as it moves
through a population. However, they should also realize that
they can use the model to make predictions about the epidemic.
Tomorrow they will use the rate of change of the infected
people in the population I' to
calculate the threshold value of S necessary
for the epidemic to spread.
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