Suppose we know the values of S, I, and R today; can we figure out what they will be tomorrow, or the next day, or a week or a month from now? Basically, this is a problem of predicting the future. One way to deal with it is to get an idea how S, I, and R are changing. To start with a very simple example, suppose the city's Board of Health reports that the measles infection has been spreading at the rate of 470 new cases per day for the last several days. If that rate continues to hold steady and we start with 20,000 susceptible children, then we can expect 470 fewer susceptibles with each passing day. The immediate future would then look like this:
| accumulated | remaining | |||||||||||||
| days after | number of new | number of | ||||||||||||
| today | infections | susceptibles | ||||||||||||
|
|
| ||||||||||||
Of course, these numbers will be correct only if the infection continues to spread at its present rate of 470 persons per day. If we want to follow S, I, and R into the future, our example suggests that 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'. For example, in the illustration above, S is changing at the rate S' = -470 persons per day. We use a minus sign here because S is decreasing over time. If S' stays fixed we can express the value of S after t days by the following formula:
Check that this gives the values of S found in the table when t = 0, 1, 2, or 3. How many susceptibles does it say are left after 10 days?
Our assumption that S' = -470 persons per day amounts to a mathematical characterization of the susceptible population--in other words, a model! Of course it is quite simple, but it led to a formula that told us what value we could expect S to have at any time t.
The model will even take us backwards in time . For example, two days ago the value of t was -2; according to the model, there were
susceptible children then. There is an obvious difference between going backwards in time and going forwards: we already know the past. Therefore, by letting t be negative we can generate values for S that can be checked against health records. If the model gives good agreement with known values of S we become more confident in using it to predict future values.
To predict the value of S using the rate S' we clearly need to have
a starting point--a known value of S from which we can measure
changes. In our case that starting point is S = 20000. This is
called the initial value of S, because it
is given to us at the ``initial time'' t = 0. To construct the
formula 
, we needed to have an initial value as well
as a rate of change for S.
In the following pages we will develop a more complex model for all three population groups that has the same general design as this simple one. Specifically, the model will give us information about the rates S', I', and R', and with that information we will be able to predict the values of S, I, and R at any time t .
