Suppose that a given population can be divided into two

Chapter 2, Problem 22

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Suppose that a given population can be divided into two parts: those who have a givendisease and can infect others, and those who do not have it but are susceptible. Let x be theproportion of susceptible individuals and y the proportion of infectious individuals; thenx + y = 1. Assume that the disease spreads by contact between sick and well members ofthe population and that the rate of spread dy/dt is proportional to the number of suchcontacts. Further, assume that members of both groups move about freely among eachother, so the number of contacts is proportional to the product of x and y. Since x = 1 y,we obtain the initial value problemdy/dt = y(1 y), y(0) = y0, (i)where is a positive proportionality factor, and y0 is the initial proportion of infectiousindividuals.(a) Find the equilibrium points for the differential equation (i) and determine whethereach is asymptotically stable, semistable, or unstable.(b) Solve the initial value problem (i) and verify that the conclusions you reached inpart (a) are correct. Show that y(t) 1 ast , which means that ultimately the diseasespreads through the entire population.