To make the model we need to make some assumptions about the disease we are going to model. First, we assume that the disease is self-limiting or curable, so that everyone who gets it recovers in a matter of weeks. Secondly, we assume that the disease confers permanent immunity, so that once someone recovers he or she cannot get the disease again. These are reasonable assumptions which are actually true of many real diseases, for example measles.
The progress of a disease in a population consists of 2 processes, people getting sick, and people recovering. To consider these processes mathematically, that is in terms of numbers, one begins by dividing the population into groups. Thus, for example, the mathematical description of a person getting well would be the decrease of the group of sick people by one, and the increase by one of the group of recovered people. So the first step in making a model of the disease is to define the groups. If one considers such a disease in some population, the population can be divided into three groups, at any given time.
Susceptible
The most obvious division is into the sick and the not sick. But the people who are not sick can be further divided into those who have already had the disease, and those who have not. Those who have not yet had the disease are those who are susceptible to it.
Recovered
The well people who have already had the illness and are immune are the recovered part of the population.
Infected
Those people who currently have the disease and are capable of transmitting it to others are infected.
Let us use S, I, and R to denote the number of people in each of the susceptible, infected, and recovered groups.
As time passes people will move from one of these groups to the other. If someone gets sick she moves from the S group to the I group. If someone recovers she moves from the I group to the R group. Since each one of the groups will be changing we will have a rate of change for each group. These rate of change will be denoted, as in the Rabbits example, by the ' symbol. So the 3 rates are S', I' and R'. There will be some relationship between the 3 rates of change since we are assuming that the total number of people in the population is constant.