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What Does Math Have to Do with Getting Sick?

By: Brink Harrison


Time: 1 days
Preparation Time: none
Materials: Teacher Cheat Sheet: What Does Math Have to Do with Getting Sick?
Time line for SARS outbreak

Abstract
Students build a SIR (Susceptible-Infected-Recovered) model for an epidemic moving through a population. Students develop rate equations for the rate of change in the number of susceptible people with respect to time, the rate of change in the number of infected people with respect to time, and the rate of change in the number of recovered people with respect to time. They will use these rate equations to calculate the number of susceptible people, infected people, and recovered people are in a population at a given day of an epidemic.

Objectives

Students will be able to:-

  • Explain the meanings of all of the terms and variables used to develop the rate of transmission of a disease
  • Explain how the rate of transmission of a disease is used in determining the rate of change in the number of susceptible people in the population.
  • Explain how the rate of recovery is calculated
  • Explain how the rate of transmission of a disease and the rate of recovery tare used to determine the rate of change in the number of infected people in the population.
  • Calculate the number of susceptible people, infected people, and recovered people are in a population at a given day of an epidemic

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
Almost since the beginning of recorded history epidemics have developed suddenly in populations and then ended just as suddenly, leaving part of the population untouched. The influenza epidemic of 1918-19 killed at least 20 million people overall, more than half a million in the United States. More recently, the Ebola virus and the SARS epidemic of 2002-3 have caused worldwide concern.

It is rarely possible to compare possible control strategies during an actual epidemic. However, mathematical modeling of epidemics is a promising tool for comparison of possible strategies as the examination of models may allow tentative conclusions even before the nature of a new disease is understood. For this reason, disease transmission modeling is being studied and promises to improve our understanding of how epidemics spread and how better to control an epidemic.

An early triumph of mathematical epidemiology was the formulation in 1927 of a simple SIR model by a public health physician, W. O. Kermack, and a biochemist, A. G. McKendrick. Kermack and McKendrick divided the population being studied into three classes. The first class, S, denotes the number of individuals susceptible to the disease, that is, not (yet) infected. The second class, I, represents the number of infected individuals, assumed infectious and able to spread the disease by contact with susceptible individuals. The third class, R, is the number of individuals infected and then removed from the possibility of being re-infected or of spreading infection. Removal is carried out through isolation from the rest of the population, immunization against infection, recovery from the disease with immunity against re-infection, or through death caused by the disease. (These characterizations of removed members are quite different from an epidemiological perspective and of course also from a human point of view, but are equivalent from a modeling point of view that takes into account only the state of an individual with respect to the disease.)

The Kermack-McKendrick epidemic model makes very simple assumptions about the rates of disease transmission and removal. One of the assumptions is that the disease is transmitted from one individual of a population to another by direct contact. (Thus it is not applicable to diseases which are transmitted by a vector , that is, diseases which are transmitted back and forth between two populations such as mosquitoes and birds as in West Nile virus.)

The model contains only two parameters (the values of which are to be determined from observed data) and could be applied to many diseases transmitted by direct contact. In addition, other aspects of a real epidemic are neglected to keep a model sufficiently simple to be amenable to mathematical analysis. For example, a crucial hypothesis in the Kermack-McKendrick model is that the mixing of the population is homogeneous. In real epidemics, it usually turns out that most infected people do not transmit the infection to others evenly, but a few “superspreaders” may pass on the infection to many others.

When these assumptions are translated into mathematical statements of the transition rates between classes the result is a pair of equations, called differential equations, which represent the rates of change of the numbers of susceptible and infective members of the population. These equations can be analyzed by relatively simple mathematical methods, and the analysis predicts that an epidemic will pass through the population without infecting the entire population.

Of course, it is important to remember that a model is only a model and will not be an accurate description of all aspects of an epidemic. A real epidemic differs considerably from an idealized model. The initial stages of an epidemic are not at all as pictured in a model of Kermack-McKendrick type. While a more detailed model might be a better description of a specific disease, it would require more parameters, which, in turn, requires more data be collected. Since data are often incomplete and inaccurate because of under-reporting and incorrect diagnosis at the beginning of an epidemic, a simple model may give better predictions.
Adapted from: http://www.mitacs.math.ca/main.php?mid=10000199&pid=158&ciy=2005&cim=3&aid=3

Related and Resource Websites
http://www.math.smith.edu/Local/cicchap1/node2.html
http://plus.maths.org/issue14/features/diseases/
http://www.recipeland.com/encyclopaedia/index.php/Endemic_(epidemiology)
http://www.mitacs.math.ca/main.php?mid=10000199&pid=158&ciy=2005&cim=3&aid=3
http://www.pims.math.ca/pi/current/page05-09.pdf
http://www.pims.math.ca/pi/current/page10-13.pdf
http://www.bondy.ird.fr/~bacaer/oldsars/node20.html
http://www.biomedcentral.com/1471-2334/3/19

 

 

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.

Embedded Assessment
The discussions among the students as they develop the equations for S', I', and R' will allow the teacher to informally assess how well the students understand the flow into and out of each portion of the population as well as how these rates are related.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


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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