Many human diseases are contagious: you ``catch'' them from someone who is already infected. Contagious diseases are of many kinds. Smallpox, polio, and plague are severe and even fatal, while the common cold and the childhood illnesses of measles, mumps, and rubella are usually relatively mild. Moreover, you can catch a cold over and over again, but you get measles only once. A disease like measles is said to ``confer immunity'' on someone who recovers from it. Some diseases have the potential to affect large segments of a population; they are called epidemics (from the Greek words epi, upon + demos, the people.) Epidemiology is the scientific study of these diseases.
An epidemic is a complicated matter, but the dangers posed by contagion--and especially by the appearance of new and uncontrollable diseases--compel us to learn as much as we can about the nature of epidemics. Mathematics offers a very special kind of help. First, we can try to draw out of the situation its essential features and describe them mathematically. This is calculus as language. We substitute an ``ideal'' mathematical world for the real one. This mathematical world is called a model. Second, we can use mathematical insights and methods to analyze the model. This is calculus as tool. Any conclusion we reach about the model can then be interpreted to tell us something about the reality.
To give you an idea how this process works, we'll build a model of an epidemic. Its basic purpose is to help us understand the way a contagious disease spreads through a population--to the point where we can even predict what fraction falls ill, and when. Let's suppose the disease we want to model is like measles. In particular,
At any time, that population can be divided into three distinct classes:
Suppose we let S, I, and R denote the number of people in each of these three classes, respectively. Of course, the classes are all mixed together throughout the population: on a given day, we may find persons who are susceptible, infected, and recovered in the same family. For the purpose of organizing our thinking, though, we'll represent the whole population as separated into three ``compartments'' as in the following diagram:
The goal of our model is to determine what happens to the numbers S, I, and R over the course of time. Let's first see what our knowledge and experience of childhood diseases might lead us to expect. When we say there is a ``measles outbreak,'' we mean that there is a relatively sudden increase in the number of cases, and then a gradual decline. After some weeks or months, the illness virtually disappears. In other words, the number I is a variable; its value changes over time. One possible way that I might vary is shown in the following graph.
During the course of the epidemic, susceptibles are constantly falling ill. Thus we would expect the number S to show a steady decline. Unless we know more about the illness, we cannot decide whether everyone eventually catches it. In graphical terms, this means we don't know whether the graph of S levels off at zero or at a value above zero. Finally, we would expect more and more people in the recovered group as time passes. The graph of R should therefore climb from left to right. The graphs of S and R might take the following forms:
While these graphs give us an idea about what might happen, they raise some new questions, too. For example, because there are no scales marked along the axes, the first graph does not tell us how large I becomes when the infection reaches its peak, nor when that peak occurs. Likewise, the second and third graphs do not say how rapidly the population either falls ill or recovers. A good model of the epidemic should give us graphs like these and it should also answer the quantitative questions we have already raised--for example: When does the infection hit its peak? How many susceptibles eventually fall ill?