Since susceptibles become infected, the compartment diagram above should also have an arrow that goes from S to I and a rate equation for S' to show how S changes as the infection spreads. While R' depends only on I, because recovery involves only waiting for people to leave the infected population, S' will depend on both S and I, because transmission involves contact between susceptible and infected persons.
Here's a way to model the transmission rate. First, consider a single
susceptible person on a single day. On average, this person will
contact only a small fraction, p, of the infected population. For
example, suppose there are 5000 infected children, so I = 5000. We
might expect only a couple of them--let's say 2--will be in the same
classroom with our ``average'' susceptible. So the fraction of contacts
is p = 2/I = 2/5000 = .0004. The 2 contacts themselves can be
expressed as 
contacts per day per susceptible.
To find out how many daily contacts the whole susceptible
population will have, we can just multiply the average number of
contacts per susceptible person by the number of susceptibles: this is
.
Not all contacts lead to new infections; only a certain fraction q do.
The more contagious the disease, the larger q is. Since the number of
daily contacts is pSI, we can expect 
new infections per
day (i.e., to convert contacts to infections, multiply by q). This
becomes aSI if we define a to be the product qp.
Recall, the value of the recovery coefficient b depends only on the illness involved. It is the same for all populations. By contrast, the value of a depends on the general health of a population and the level of social interaction between its members. Thus, when two different populations experience the same illness, the values of a could be different. One strategy for dealing with an epidemic is to alter the value of a. Quarantine does this, for instance; see the exercises.
Since each new infection decreases the number of susceptibles, we have the rate equation for S:
The minus sign here tells us that S is decreasing (since S and I are positive). We call a the transmission coefficient.
Just as people flow from the infected to the recovered compartment when they recover, they flow from the susceptible to the infected when they fall ill. To indicate the second flow let's add another arrow to the compartment diagram. Because this flow is due to the transmission of the illness, we will label the arrow with the transmission coefficient a. The compartment diagram now reflects all aspects of our model.
We haven't talked about the units in which to measure a and b. They must be chosen so that any equation in which a or b appears will balance. Thus, in R' = bI the units on the left are persons/day; since the units for I are persons, the units for b must be 1/(days). The units in S' = -aSI will balance only if a is measured in 1/(person-day).
The reciprocals have more natural interpretations. First of all,
1/b is the number of days a person needs to recover. Next, note that
1/a is measured in person-days (i.e., persons 
days), which
are the natural units in which to measure exposure. Here is why.
Suppose you contact 3 infected persons for each of 4 days. That gives
you the same exposure to the illness that you get from 6 infected
persons in 2 days--both give 12 ``person-days'' of exposure. Thus, we
can interpret 1/a as the level of exposure of a typical susceptible
person.