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Math Help - Spread of a disease

  1. #1
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    Spread of a disease

    Under certain circumstances, the rate at which a disease spreads through a population of P individuals is proportional to the product of the number I of people infected and the number not infected

    a) Write and equation for dI/dt

    b) Invert to get dt/dI

    c) Find t as a function of I, assuming time is 0 at the moment when half the population is infected

    d) Find I as a function of t

    Hint for part c: What happens when you combine 1/I+1/(P-1) into a single fraction.

    Any help that you can offer is appreciated!
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  2. #2
    Senior Member
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    Quote Originally Posted by Alice96 View Post
    Under certain circumstances, the rate at which a disease spreads through a population of P individuals is proportional to the product of the number I of people infected and the number not infected

    a) Write and equation for dI/dt

    b) Invert to get dt/dI

    c) Find t as a function of I, assuming time is 0 at the moment when half the population is infected

    d) Find I as a function of t

    Hint for part c: What happens when you combine 1/I+1/(P-1) into a single fraction.

    Any help that you can offer is appreciated!
    a) \frac{dI}{dt}=I(P - I)
    b) \frac{dt}{dI}=\frac{1}{I(P - I)}
    c) \frac{dt}{dI}=\frac{1}{I(P - I)}=\frac{1}{PI} + \frac{1}{P(P-I)}, so dt = \frac{1}{PI} ~dI+ \frac{1}{P(P-I)}~dI. solve this differential equation.
    d) solve part (a)
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