Solved: Logistic equation for an epidemic When an infected

Chapter 7, Problem 42

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Logistic equation for an epidemic When an infected person is introduced into a closed and otherwise healthy community, the number of people who become infected with the disease (in the absence of any intervention) may be modeled by the logistic equation dP dt = kP a1 - P A b, P102 = P0, where k is a positive infection rate, A is the number of people in the community, and P0 is the number of infected people at t = 0. The model assumes no recovery or intervention. a. Find the solution of the initial value problem in terms of k, A, and P0. b. Graph the solution in the case that k = 0.025, A = 300, and P0 = 1. c. For fixed values of k and A, describe the long-term behavior of the solutions for any P0 with 0 6 P0 6 A.