The recovery rate is the simplest rate to consider because we are assuming that recovery from an illness is only a matter of time. If we consider some disease like measles, it takes on the average about 14 days to recover. So if we look at the entire infected population today, we can expect to find some who have been infected less than one day, some who have been infected between one and two days, and so on, up to fourteen days. Those in the last group will recover today. In the absence of any definite information about the fourteen groups, let's assume they are the same size. Then 1/14-th of the infected population will recover today: Thus the rate equation for R' says that R' will be equal to 1/14 of the infected population, I.
R' = (1/14) * I
In order to make the model more general, we can define b as the recovery coefficient, which in our specific case we chose to be (1/14). So we get the general formulation
R' = b * I
The rate of transmission, which is the rate of growth of the infected population, I, is not so simple. Since we are considering a contageous disease, in order for a person to become sick she must come into contact with someone who is already sick. If I am a uninfected person, one of the susceptible group, my chances of coming into contact with a sick person will depend on how many sick people there are. If I am a sick person, my chance of coming into contact with a susceptible person will be greater if there are many susceptible people, and smaller if there are few susceptible people.
Here's a way to model the transmission rate. First, let's consider a single susceptible person on a single day and let's say that on that day there are 5000 sick people in the population. Since not every person in the population comes into contact with every other person each day, we should assume that the our healthy subject comes into contact with only a small fraction of the sick population. Suppose there are 5000 infected people, 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
2 =(2/I)*I < p*I
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 susceptible: this is
p*I * S = pS I
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 p S I , we can expect p * q SI new infections per day (i.e., to convert contacts to infections, multiply by q .
This becomes a SI if we define a, for convenience, to be the product q p.
Thus the conclusion is that the rate of new infections is
a * S * I