Help With Exponential Function

Question

I'm working on a problem and need help figuring out where I went wrong. The problem states:

"Suppose that $10$ people originally have the virus, and that in the early stages of the virus, the number of people infected is increasing exponentially with $k=1.9$. It is estimated that, approximately $8000$ people become infected."

Based on the information provided by the question, I found a logistic function to model this shown by $P= \frac{8000}{1+499e−^{1.9t}}$ but am told this is incorrect. Can someone explain why?

Additionally another question asks for the vertical coordinate at which the rate of infected people begins to decrease. By solving for $t$ at $8000$, I found $t=3.52$ but again am told this is incorrect. I assume it's incorrect because my equation for $P$ was incorrect?

A wild exponential function has appeared! You use differentiate. Not very effective. — Saketh Malyala, Apr 22 '17 at 23:30
Is it exponential ($f(t)=10e^{kt}$) or logistic ($f(t)=\frac{8000}{1+799e^{-kt}}$)? — AlgorithmsX, Apr 22 '17 at 23:43
@AlgorithmsX I would assume it's logistic because of the implicit limit of 8000 victims. How did you get 799 instead of 499? — JQQ, Apr 23 '17 at 00:28

1123581321 · Answer 1 · 2017-06-16T06:04:13.847

For small values of $t$, $\frac{dP}{dt} \approx kP = 1.9 P$

Since this is a logistic model, $\frac{dP}{dt} = 1.9 P - b P^2$, where $b$ is a constant to be determined.

Use the fact that when $P = 8000, \frac{dP}{dt} = 0$ to solve for $b$.

To find the vertical coordinate at which the rate of infected people, $\frac{dP}{dt}$, begins to decrease find the maximum value of $\frac{dP}{dt}$ by differentiating $ 1.9 P - b P^2$ with respect to $P$, set this to zero, and solving for $P$. This the vertical coordinate, not the corresponding $t$-value.

You should get $b = \frac{1.9}{8000}$ and the vertical coordinate is $P = 4000$.

Help With Exponential Function

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