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?
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 $P = 4000$.