As the first step in the modeling process, we identify the independent and dependent variables. The independent variable is time \(t\), measured in days. We consider two related sets of dependent variables.

The first set of dependent variables counts people in each of the groups, each as a function of time:

\(S = S(t)\)	is the number of susceptible individuals,
\(I = I(t)\)	is the number of infected individuals, and
\(R = R(t)\)	is the number of recovered individuals.

The second set of dependent variables represents the fraction of the total population in each of the three categories. So, if \(N\) is the total population \((7,900,000\) in our example), we set

\(s(t) = \frac{S(t)}{N}\),	the susceptible fraction of the population,
\(i(t) = \frac{I(t)}{N}\),	the infected fraction of the population, and
\(r(t) = \frac{R(t)}{N}\),	the recovered fraction of the population.

It may seem more natural to work with population counts, but some of our calculations will be simpler if we use the fractions instead. The two sets of dependent variables are proportional to each other, so either set will give us the same information about the progress of the epidemic.

1. Under the assumptions we have made, how do you think \(s(t)\) should vary with time? How should \(r(t)\) vary with time? How should \(i(t)\) vary with time?
2. Sketch on a piece of paper what you think the graph of each of these functions looks like.
3. Explain why, at each time \(t\), \(s(t) + i(t) + r(t) = 1\).

Next we make some assumptions about the rates of change of our dependent variables:

• No one is added to the susceptible group, since we are ignoring births and immigration. The only way an individual leaves the susceptible group is by becoming infected. We assume that the time-rate of change of \(S(t)\), the number of susceptibles, depends on the number already susceptible, the number already infected, and the rate of contact between susceptibles and infecteds. In particular, suppose that each infected individual has a fixed number \(b\) of contacts per day that are sufficient to spread the disease. Not all these contacts are with susceptible individuals. If we assume a homogeneous mixing of the population, the fraction of these contacts that are with susceptibles is \(s(t)\). Thus, on average, each infected individual generates \(b*s(t)\) new infected individuals per day.

• We also assume that a fixed fraction \(k\) of the infected group will recover during any given day. For example, if the average duration of infection is three days, then, on average, one-third of the currently infected population recovers each day. (Strictly speaking, what we mean by "infected" is really "infectious," that is, capable of spreading the disease to a susceptible person. A "recovered" person can still feel miserable, and might even die later from pneumonia.)