To develop the equations, we need to write the mass balance equations for each compartment. These are:
The problem is that 
is not a constant. Let us
try to figure it out. It seems obvious that the rate
must depend on the number of susceptibles,
infecteds and immunes and the probability of
transmitting the disease in a contact. Suppose each
susceptible makes c contacts per unit of time that are
of the disease transmitting type. Then the susceptibles
make cX contacts per unit time. Assume the contacts
are at random with members of the total population,
N=X+Y+Z. Then only the fraction Y/N of the contacts
are with infectious individuals. Let 
be the
probability of transmission in a contact between an
infected and a susceptible. Then the rate susceptibles
become infected must be,
so must be the coefficient of X in that
expression. With that we can write the equations in the
form,
The problem is that equations (4)-(6) form a set of nonlinear compartmental equations and the solution is difficult to obtain. Fortunately, examination of equation (5) tells us most of the important things we need to know about this process. To do that, we rewrite equation (5) in the factored form,
