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Logistic equation for an epidemic When an infected person
Chapter 7, Problem 34E(choose chapter or problem)
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
\(\frac{d P}{d t}=k P\left(1-\frac{P}{A}\right), \quad \ P(0)=P_{0}\)
where k is a positive infection rate, A is the number of people in the community, and \(P_{0}\) 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 \(P_{0}\).
b. Graph the solution in the case that k =0.025, A =300, and \(P_{0}=1\).
c. For fixed values of k and A, describe the long-term behavior of the solutions for any \(P_{0}\) with \(0<P_{0}<A\).
Questions & Answers
QUESTION:
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
\(\frac{d P}{d t}=k P\left(1-\frac{P}{A}\right), \quad \ P(0)=P_{0}\)
where k is a positive infection rate, A is the number of people in the community, and \(P_{0}\) 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 \(P_{0}\).
b. Graph the solution in the case that k =0.025, A =300, and \(P_{0}=1\).
c. For fixed values of k and A, describe the long-term behavior of the solutions for any \(P_{0}\) with \(0<P_{0}<A\).
ANSWER:Step 1 of 6
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 P0is the number of infected people at t =0. The model assumes no recovery or intervention.