Question

Multiple Choice The spread of a disease through a community can be modeled with the logistic equation \(\frac{d y}{d t}=\frac{0.9}{1+45 e^{-0.15 t}}\) \(\begin{array}{l}{\text { where } y \text { is the proportion of people infected after } t \text { days. Accord- }} \\ {\text { ing to the model, what percentage of the people in the commu- }} \\ {\text { nity will not become infected? } } \\ {\text { (A) } 2 \%} {\text { (B) } 10 \%} {\text { (C) } 15 \%} {\text { (D) } 45 \%} {\text { (E) } 90 \%}\end{array}\)

Short answer

10% of the people in the community will not become infected (Option B)

Step-by-step solution

Analyze the Differential Equation

The equation \( \frac{dy}{dt} = \frac{0.9}{1+45e^{-0.15t}} \) is a separable differential equation, which means we can move all terms involving \(y\) to one side and all terms involving \(t\) to the other side, then integrate both sides.

Solve the Differential Equation

To solve the differential equation, consider that long-term behavior is dictated by the carrying capacity, which is the upper limiting value as \(t \rightarrow \infty\). As \(t\) becomes large, the exponential \(e^{-0.15t}\) will approach 0, simplifying the equation to \(\frac{dy}{dt} = 0.9\), which implies that \(y\) will approach 0.9 or 90% as \(t \rightarrow \infty\).

Determine the Proportion not Infected

Since \(y\) represents the proportion of people infected and this value is approaching 0.9 or 90%, this implies that 10% of people will not become infected.