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Math Help - logistic model

  1. #1
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    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
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  2. #2
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    Re: logistic model

    what is your solution in terms of t and k for the initial value problem?
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  3. #3
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    Re: logistic model

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

    but I can't figure out the proper value for k
    Last edited by Jonroberts74; March 22nd 2014 at 12:39 AM.
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  4. #4
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    Re: logistic model

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

    I get

    Spoiler:
    $k=\dfrac{\log\left(\dfrac{23}{2}\right)}{4000}$
    Thanks from Jonroberts74
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