** Activity**
Adapted from: Hughes-Hawlett, D. et.al;
__Single Variable Calculus (Third Edition)__; John Wiley & Sons;
New York; 2002 pg. 523-526)
(1) Go over yesterday’s
homework. Make sure to spend sufficient time on problem
(7) to ensure that the students
understand how the rate equations would have to change and
why.
(2) Tell the students that today they are going to construct
a SIR model for a flu epidemic that struck a boarding school
in England. But once they have made the model, they are going
to work with the equations of the model themselves and get
mathematical answers to certain situations. They will then
see how to use these answers to affect the spread of the
epidemic.
(3) Tell the students the following story:
In January 1978, 763 students returned to an all-male boarding
school after their winter vacation. A week later, one boy
developed the flu, followed by two others the next day. By
the end of the month, nearly half the boys were sick. Most
of the school had been affected by the time the epidemic
was over in mid-February.
(Source: Hughes-Hawlett,
D. et.al; __Single Variable Calculus (Third Edition)__; John
Wiley & Sons; New York; 2002 pg.
523-526, which cites the original data from: Communicable
Disease Surveillance Centre (UK); reported in “Influenza
in a Boarding School,” *British Medical Journal*; March
4, 1978)
(4)
Have the students write the basic equations for and
in
a model.
Tell them that there is no immunity loss involved in the
situation.
(5) Have the students go back to the flu at the boarding
school problem and ask them how we can get approximate values
for* a* and *b*.
(6) Have
the students now write the equations for andusing
the values of *a* and *b*.
(7) Tell
the students that we are now going to work with the equation
for ,
the rate of change of the number of infected people in
the population. Ask the students to discuss what
happens to the number of people in the infected population
when (a) ,
(b) , and (c) .
(8) Point
out that is
written in terms of and .
Ask the students to set in the
second equation and calculate the
values of the variables and that
are solutions to the equation.
(9) Ask the students to give a physical interpretation for
each of these equations. Ask the students why 192 susceptible
people could be called a** threshold value** for the epidemic.
(10) Ask the students why knowing the threshold value of
susceptible people may have an affect of the spread of the
infection. If doctors have a vaccine for the disease, how
many students would need to be vaccinated to intervene with
the spread of the disease?
**Homework**
1) Let’s look again at the measles-like disease that
strikes a small college that has a population of 1,800. From
last night’s homework, the equations for and
are:
(a) What is the threshold value for susceptible people to
insure there is not an epidemic?
(b) How many people would need to be vaccinated?
2) Do
you want a high threshold value of or a low threshold value
of ? Explain.
(a)
Write the basic equations for and .
(b)
Using the equation for, write
a formula for finding the threshold value of in
terms of *a* and *b*.
3) Two
epidemics have the same recovery coefficient *b*.
However Disease 1 has a higher transmission coefficient *a* than
Disease
2.
(a) Which disease has a higher threshold value of ?
(b) Which disease needs to have more people vaccinated in
order to reach its threshold value of ?
**Closure**
As students use the rate of change of the infected people
in the population to calculate the threshold value of , they
realize that they can use the mathematical model, even though
it has flaws, to make predictions about the epidemic. |