Chapter D1.3, Problem 44E

Applications

44.    Logistic equation for spread of rumors Sociologists model the spread of rumors using logistic equations. The key assumption is that at any given time, a fraction y of the population, where 0 ≤ y ≤ 1. knows the rumor, while the remaining fraction 1 – y does not. Furthermore, the rumor spreads by interactions between those who know the rumor and those who do not. The number of such interactions is proportional to y(1 – y). Therefore, the equation that describes the spread of the rumor is y′(t) = ky(1 – y), where k is a positive real number. The number of people who initially know the rumor is y(0) = y₀, where 0 ≤ y₀ = 0.1

1. a. Solve this initial value problem and give the solution in terms of k and y₀.
2. b. Assume k = 0.3 weeks^–1 and graph the solution for y₀ = 0.1 and y₀ = 0.7.
3. c. Describe and interpret the long-term behavior of the rumor function, for any 0 ≤ y₀ ≤ 1.