In this model, let the number of those not yet ill, but susceptibles, be S(t); let the number currently infected be I(t). The rate at which the susceptibles are infected is given by:
S'(t) = dS/dt = -alpha(SI)
(Where alpha > 0 is a constant). The rate at which the number of those infected can change depends on both the susceptibles getting ill and others recovering; this process is described by:
I'(t) = dI/dt = alpha(SI) - beta(I)
(Where beta > 0 is a constant). Form the differential equation I(S) and, given that I = 1 when S = N, find the solution for I(S).
The part that really throws me is the stuff in blue :S how do I get the function I(S)? I tried dividing I'(t)/S'(t) to give dI/dS then integrating but this seemed totally wrong... Am I missing some sort of elementary trick here? Help very appreciated.