Question

{Use of Tech} 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
dP/dt=kP(1−P/A),P0=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.
a. Find the solution of the initial value problem, for t≥0, in terms of K, A, and P0.

Accepted Answer

Recognize that the given differential equation is a logistic differential equation of the form $$\frac{dP}{dt} = kP\left(1 - \frac{P}{A}\right)$$, where $$k$$ and $$A$$ are constants, and $$P(t) is $$the function to solve for with initial condition $$P(0) = P_0$. Rewrite the equation to separate variables: $$\frac{dP}{P(1 - \frac{P}{A})} = k \, dt$$. This allows us to integrate both sides with respect to their variables. Simplify the left side by expressing the denominator as P$$\left(1 - \frac{P}{A}\right) = P \left(\frac{A - P}{A}\right) = \frac{P(A - P)}{A}$$. Then, rewrite the integral as $$\int \frac{A}{P(A - P)} \, dP = \int k \, dt$$. Use partial fraction decomposition to break down $$\frac{A}{P(A - P)}$$ into simpler fractions: $$\frac{A}{P(A - P)} = \frac{C}{P} + \frac{D}{A - P}$$. Find constants C$ and D$ by equating numerators. Integrate both sides: $$\int \left(\frac{C}{P} + \frac{D}{A - P}\right) dP = \int k \, dt$$. After integration, solve for P(t$$)$$ explicitly, applying the initial condition P(0$$) = $$P_0 to $$find the constant of integration.

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Key Concepts

Logistic Differential Equation

Initial Value Problem (IVP)

Separation of Variables Method