Chapter D1.3, Problem 34E

Logistic equation for an epidemic When an infected person is introduced into a closed and otherwise healthy community, the number of people who contract the disease (in the absence of any intervention) may be modeled by the logistic equation

     $\frac{dP}{dt}=kp\left(1-\frac{P}{A}\right),P\left(0\right)=P_0,$

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 also assumes no recovery.

1. a. Find the solution of the initial value problem, for t ≥ 0, in terms of k, A, and P₀.
2. b. Graph the solution in the case that k = 0.025, A = 300, and P₀ = 1.
3. c. For a fixed value of k and A, describe the long-term behavior of the solutions, for any P₀ with 0 < P₀ < A.