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Who makes an Epidemic?

Author: Brink Harrison


Time: 1 day
Preparation Time: None
Materials: Optional Calculus lesson
Teacher Cheat Sheet
Timeline for SARS

Abstract
By analyzing the rate of change in the number of infected people, , in terms of , the number of susceptible people in the population, and , the number of infected people in the population, students will be able to calculate the threshold value for the number of susceptible people needed for the epidemic to occur.


Objectives

Students will be able to:

• Calculate the threshold number of people needed in the susceptible group for the epidemic to occur.


National Math Standards
Problem Solving
• Solve problems that arise in mathematics and in other contexts
• Apply and adapt a variety of appropriate strategies to solve problems
Communication
• Use the language of mathematics to express mathematical ideas precisely
• Communicate their mathematical thinking coherently and clearly to others
Connections
• Recognize and apply mathematics in contexts outside of mathematics
Representation
Use representations to model and interpret physical, social, and mathematical phenomena



Teacher Background
The connection between reality and mathematics is a two-way street. Yesterday the students constructed a mathematical object, a model, which reflected some aspects of an epidemic. That is one direction of travel on this two-way street. Today the students will travel the other way. Working from the model, they will get mathematical answers to mathematical questions that the model raises. They will take the mathematical answers and will see what those answers tell us about the epidemic.


Related and Resource Websites

http://www.math.smith.edu/Local/cicchap1/node2.html
http://www.bondy.ird.fr/~bacaer/oldsars/node20.html
http://www.recipeland.com/facts/Epidemiology

 

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.

Embedded Assessment

The discussions among the students as they use the limited data on the situation to derive the values of a and b will allow the teacher to informally assess how well the students understand the meaning of the transmission rate and recovery rate of an epidemic. Similarly, the discussions on the physical interpretations of the solutions to the equation will allow the teacher to informally assess how well the students understand the concept of a threshold value for the number of susceptible people in the population.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


PULSE is a project of the Community Outreach and Education Program of the Southwest Environmental Health Sciences Center and is funded by:


an
NIH/NCRR award #16260-01A1
The Community Outreach and Education Program is part of the Southwest Environmental Health Sciences Center: an NIEHS Award

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Supported by NIEHS grant # ES06694


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