Suppose that a given population can be divided into two parts: those who have a given disease and can infect others, and those who do not have it but are susceptible. Let x be the proportion of susceptible individuals and y the proportion of infectious individuals; then x + y = 1. Assume that the disease spreads by contact between sick and well members of the populations, and that the rate of spread dy/dt is proportional to the number of such contacts. Further, assume that members of both groups move about freely among each other, so that the number of contacts is proportional to the product of x and y. Since x = 1 y, we obtain the initial value problem

dy/dt = y(1 y), y(0) = y0,

where is a positive proportionality factor, and y0 is the initial proportion of infectious individuals.

(a) Find the equilibrium points for the differential equation, and determine whether each is asymptotically stable (heads towards the equilibrium point), unstable (heads away from the equilibrium point), or semistable (stable on one side, unstable on the other).

(b) Solve the initial value problem and verify that the conclusions you reached in part (a) are correct. Show that y(t) 1 as t , which means that ultimately the disease spreads through the entire population.