The final rate equation we need--the one for I'--reflects what is already clear from the compartment diagram: every loss in I is due to a gain in R, while every gain in I is due to a loss in S.
If you add up these three rates you should get the overall rate of change of the whole population. The sum is zero. Do you see why?
You should not draw the conclusion that the only use of rate equations is to model an epidemic. Rate equations have a long history, and they have been put to many uses. Isaac Newton (1642-1727) introduced them to model the motion of a planet around the sun. He showed that the same rate equations could model the motion of the moon around the earth and the motion of an object falling to the ground. Newton created calculus as a tool to analyze these equations. He did the work while he was still an undergraduate--on an extended vacation, curiously enough, because a plague epidemic was raging at Cambridge!
Today we use Newton's rate equations to control the motion of earth satellites and the spacecraft that have visited the moon and the planets. We use other rate equations to model radioactive decay, chemical reactions, the growth and decline of populations, the flow of electricity in a circuit, the change in air pressure with altitude--just to give a few examples. You will have an opportunity in the following chapters to see how they arise in many different contexts, and how they can be analyzed using the tools of calculus.
The following diagram summarizes, in a schematic way, the relation between our model and the reality it seeks to portray.
The diagram calls attention to several facts. First, the model is a part of mathematics. It is distinct from the reality being modelled. Second, the model is based on a simplified interpretation of the epidemic. As such, it will not match the reality exactly; it will be only an approximation. Thus, we cannot expect the values of S, I, and R that we calculate from the rate equations to give us the exact sizes of the susceptible, infected, and recovered populations. Third, the connection between reality and mathematics is a two-way street. We have already travelled one way by constructing a mathematical object that reflects some aspects of the epidemic. This is model-building. Presently we will travel the other way. First we need to get mathematical answers to mathematical questions; then we will see what those answers tell us about the epidemic. This is interpretation of the model. Before we begin the interpretation, we must do some mathematics.