Calculus: Project 1

Before working one of the the specific epidemic projects, you should review the basic assumptions that we made in building with our S-I-R model in Chapter 2 of the text. Write a derivation of the model in your own words as a first section of your project.

1.2 Basic Assumptions We make the following assumptions about the disease in formulating our model:

1. S-I-R: Individuals all fit into one of the following categories:

Susceptible: those who can catch the disease

Infectious: those who can spread the disease

Removed: those who are immune and cannot spread the disease

2. The population is large, but fixed in size and confined to a well-defined region.

3. The population is well mixed; ideally, everyone comes in contact with the same fraction of people in each category every day.
1.3 Derivation of the Equations of Change
In order to describe the way individuals move between the susceptible, infected and removed compartments, we used two parameters, b and c, and the product in our S-I-R model. The values are specific to a particular S-I-R disease.
PARAMETERS FOR THE S-I-R MODEL

The value of b is observed clinically, since b=1/the number of days an individual is infectious.
The parameter c is called the "contact number" because it gives the average number of close contacts each infective has over the course of the disease. The value of c is measured indirectly through the invariant equation given in Chapter 2 of the main text. Describe this measurement briefly in your project if you use measured values of c. Observed values of c are given in the herd immunity project below.
We measure the contact number and then compute the daily contact rate .
THE CONTINUOUS S-I-R VARIABLES
In order to develop a model that applies to different populations, we used fractional variables.

where n is the (fixed) size of the total population. Because of our assumptions,

Figure 1.6: S-I-R Disease Compartments
THE S-I-R DIFFERENTIAL EQUATIONS

These equations describe the rates at which people are getting sick and recovering. Each project in this chapter should begin with a derivation in your own words that explains how the terms asi and bi describe these rates of change. See Chapter 2 for help. The S-I-S project will derive slightly different equations.

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