1. ## logistic model

using $y'=k(M-y)y$

I know how to solve this problem, my issue is getting the k.

M=1000 (total population)
at time t=0 there are 80 cases and 4 hours later half the total population (500) are infected.

I need to find the time when 90% of the population is infected.

I need to find my k to solve it though. initially I though $\frac{500-80}{4-0}= 105$ but that's incorrect

2. ## Re: logistic model

what is your solution in terms of t and k for the initial value problem?

3. ## Re: logistic model

$y(t) = \frac{1000}{\frac{23}{2} e^{-1000kt} +1}$

but I can't figure out the proper value for k

4. ## Re: logistic model

this is correct. Just solve $y(4)=500$

I get

Spoiler:
$k=\dfrac{\log\left(\dfrac{23}{2}\right)}{4000}$