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

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 < P₀ < A.

Transcribed Image Text:

dP kP( 1 dt P(0) = Po, =