Concept explainers
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
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.
- a. Find the solution of the initial value problem, for t ≥ 0, 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 a fixed value of k and A, describe the long-term behavior of the solutions, for any P0 with 0 < P0 < A.
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