exponential growth

Question

on our campus of 7700 students, one student returns from thanksgiving break with a contagious virus. after 2 days 40 other students are infected.using the following equation:
P(t)=(KPo)/(Po+(K-Po)e^kt)
K is the max capacity
Po is the initial size
k is a third parameter
t is time

1)identify parameters to be determined
2)use known information to determine parameters(one peice of info is needed for each parameter
3)use the model to answer a question
the question is how many kids will be sick in 17 days?

Answer 1

There are three parameters here, K, P₀ and k, and the problem hands you exactly one piece of information for each. That is what part 2 is hinting at.

K is the maximum capacity, and the virus can reach at most the whole campus, so K = 7700. P₀ is the initial size, and one student came back infected, so P₀ = 1. Those two you just read off the problem. The only one you actually have to solve for is k, and the two-day information is what gives it to you.

After 2 days, 40 other students are infected, so counting the student who brought it back, P(2) = 41. Putting that in:

41 = 7700/(1 + 7699e^2k)

Solving for the exponential gives e^2k ≈ 0.0242609, and then taking the natural log and dividing by 2:

k ≈ −1.8594

Do not let that negative sign worry you. The usual way this model gets written has e^−kt in the denominator, and your version has e^kt, so the same growth simply shows up as a negative k. It is the same curve either way.

Now for the actual question. Putting t = 17 into the model:

P(17) = 7700/(1 + 7699e^{17(−1.8594)}) ≈ 7700

That exponential term works out to roughly 1.9 × 10⁻¹⁴, which is so small that the denominator is essentially just 1, so the model says the entire campus.

If you notice, the interesting part of this problem is not day 17 at all. Running the model day by day, you get 41 on day 2, about 256 on day 3, roughly 1,392 on day 4, about 6,937 on day 6, and 7,680 by day 8. The whole outbreak is essentially finished inside of about nine days, and everything after that is just the curve flattening out against its ceiling. That is what logistic growth does, and it is why a question set 17 days out has such a blunt answer.

So, our final answer is that after 17 days essentially all 7,700 students are sick according to this model.