Mathematics 162                              Laboratory 8                          Week of March 22, 1993
	  
Name: _____________________________   Lab Partner: ___________________________
	  
Consulted with:  ____________________________________________________________

© Lafayette College, 1994.

A Model for an Epidemic

Suppose a contagious disease has appeared in a population.  The disease is not fatal, but people who have recovered are still susceptible to the disease.  We will use sequences to build a mathematical model for the spread of the disease in the population.

Part 0:  The Model

Define the following sequences

In = the proportion of the population that has the disease on day n;
Fn = the proportion of the population that is newly infected on day n;
Cn = the proportion of the population that recovers on day n.

From the definitions, it follows that

In = In – 1 + Fn  –  Cn.

This is called a recursive relationship between these sequences, because the values of I, F, and C on day n depend on the value of I the preceding day.  We will make two more assumptions that can be written mathematically as recursive relationships.  First, we will assume that the new cases of the disease on a day are jointly proportional to the number of people infected and the number who are uninfected the preceding day.  (Jointly proportional means proportional to the product.)  Mathematically,

Fn = a In – 1(1 – In – 1).

(The constant a is positive.)  Our second assumption is that the recoveries on a day are proportional to the number of people infected the previous day.  We will write this

Cn = (1 – b) In – 1.

b is the proportion of infected people who will not be well the next day.  Substituting these two assumptions into our initial equation gives us a recursive definition for I :

In = In – 1 + a In – 1(1 – In – 1) –  (1 – b) In – 1 = (a + b) In – 1–  a In – 12

In class, we showed that if we made the definitions k = a + b and dn = a/k In, then this definition is equivalent to a simpler one for the sequence {dn}:

dn = k dn – 1(1 – dn - 1).

Notice that by definition, 0 < k ≤ a, so 0 < dn ≤ In ≤ 1.

Part 1:  Controlling the Epidemic

Explain in a couple of sentences why the following statement is true:  If the disease is mild, that is, it is not very contagious and it is easy to recover from, then the value of k will be small.

Let's begin by assuming the disease is mild and plot the growth of {dn}.  Suppose that k = 0.9 and d1 = 0.01.  (If you wish, assume that a = 0.9 and b = 0 for  concreteness; then dn = In.)  The following command will define the appropriate sequence (and tell Mathematica to remember the values of terms it has already computed)

k = 0.9;
d[1] = 0.01;
d[n_] := d[n] = k d[n-1](1 - d[n-1])

We can plot the graph of d as we learned in the lab before break, using the command

ListPlot[Table[{n, d[n]}, {n,1,20}]]

What happens to the number of people infected in the first twenty days of the epidemic?  What will happen to the proportion of people infected over the long run?  (You may plot more terms if you wish, to support your answer.)

Now consider the case when the disease is a little less mild; take k = 1.1.  In order to define a new sequence to examine this case, first we must clear the old definition with the command

Clear[d,k]

Now modify the original definition to treat this case (assuming d1 is still 0.01).

Now what happens to the proportion of people infected over the first twenty days? Over the long run?

Part 2:  Analysis of the Model using Limits

Show that if the sequence {dn} has a limit L, then the limit must satisfy the equation:

L = k L (1 – L).

Hint:  Take the limit of the recursive definition as n goes to infinity.

Use the equation you just derived to find the limit of the sequence in each of the two cases considered in Part 1.  You may use Mathematica, but write your answers below.

Public health officials often have some control over the parameters a and b, and hence over k.  For instance, they may be able to decrease b by giving proper treatment and they might be able to decrease a through quarantines, or education.  For what values of k will the disease tend to be eliminated from the population?

Part 3:  Dependence on parameters

We have seen that given two constants:  the value of k, and the initial value of d, we can find a sequence to model an epidemic.  We call these constants parameters. In this part we will examine how the evolution of an epidemic depends on these parameters.

Assuming d1 is still 0.01, investigate the long term effect of an epidemic on a population with each of the following k values: 2.8, 3.84, 4.0.  In each case, write a few sentences describing your conclusion.

Suppose that in the case when k = 4.0, the sequence had a limit.  What would it be?

Finally, consider the effect of changing the initial value d1.  Define a sequence using k = 4.0 and d1 = 0.75, and plot its graph.  Repeat this using k = 4.0 and d1 = 0.7500001.  Compare the long term behavior of these two sequences.

Suppose a public health official said "The disease is not at all mild, the a value is about 3.1 and the b value is 0.9, but we appear to have the epidemic under control. The proportion of the population that is infected has been staying near 58% for a few days now."  What would you reply?  Should the official expect the epidemic to remain under control for long?  What can you tell her concerning the future of this epidemic?  What must be done to eradicate the disease?