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Math Help - Modelling an Epidemic

  1. #1
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    Modelling an Epidemic

    The number y of persons infected by a highly contagious diesease is modelled by a so-called logistic curve;
    y = \frac {L}{1+ Me^{-kt}},

    where t is the time measured in months from when the outbreak of the disease was discovered. At that time there were 200 infected persons, and after one month there were 1,000 infected. Eventually the number levelled out at 10,000.
    (a) Show that y satisfies the logistic equation
    \frac {dy}{dt} = ky(1-\frac {y}{L}).
    (b) Find the values of hte parameters L,M, and k of the model.
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  2. #2
    Senior Member apcalculus's Avatar
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    Quote Originally Posted by Robb View Post
    The number y of persons infected by a highly contagious diesease is modelled by a so-called logistic curve;
    y = \frac {L}{1+ Me^{-kt}},

    where t is the time measured in months from when the outbreak of the disease was discovered. At that time there were 200 infected persons, and after one month there were 1,000 infected. Eventually the number levelled out at 10,000.
    (a) Show that y satisfies the logistic equation
    \frac {dy}{dt} = ky(1-\frac {y}{L}).
    (b) Find the values of hte parameters L,M, and k of the model.
    For part a), take the differential equation and separate variables, all the y terms and dy on the left, and k dt on the right, then integrate. For the left hand side you will have to use partial fractions but since it is already factored, it should not take a lot of effort.

    After successfully integrating, try to write the relationship in the form as given in the problem.

    For part b) use the two conditions and the 'asymptote' to determine the coefficients. I hope this helps!
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