Suppose that a given population can be divided into two parts:those who have a given disease and can infect others, and those whodo not have it but are susceptible. Let x be the proportion ofsusceptible individuals and y the proportion of infectiousindividuals; then x + y = 1. Assume that the disease spreads bycontact between sick and well members of the population and thatthe rate of spread dy?dt is proportional to the number of suchcontacts. Further, assume that members of both groups move aboutfreely among each other, so the number of contacts is proportionalto the product of x and y. Since x=1?y, we obtain the initial valueproblem
dy?dt = ?y(1 ? y), y(0) = y0, (i) where ? is a positiveproportionality factor, and y0 is the initial proportion ofinfectious individuals.
(a) Find the equilibrium points for the differential equation(i) and determine whether each is asymptotically stable,semistable, or unstable.
(b) Suppose that the equation was instead y? = y(? ? y2). Repeatyour analysis from part (a). Note that your answer will depend onwhether ?<0, ?=0, or ?>0.
