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 population 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 the number of contacts is proportional to the product of x and y. Since x = 1 − y, we obtain the initial value problem (22) dydt=αy(1−y),y(0)=y0,where α is a positive proportionality factor, and y0 is the initial proportion of infectious individuals. a.Find the equilibrium points for the differential equation (22) and determine whether each is asymptotically stable, semistable, or unstable. b.Solve the initial value problem 22 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.
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 population 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 the number of contacts is proportional to the product of x and y. Since x = 1 − y, we obtain the initial value problem
a.Find the equilibrium points for the differential equation (22) and determine whether each is asymptotically stable, semistable, or unstable.
b.Solve the initial value problem 22 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.
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 4 images
