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Model for spread of contagious disease.

  1. Dec 3, 2013 #1
    ImageUploadedByPhysics Forums1386085106.849207.jpg

    Hello guys, I am quite unsure in how to start the modeling for population. If I define y as the number of infected persons and y' as the rate of spread is it correct? I see the number of contacts between infected and non infected persons to be equal to the number of infected persons. Otherwise I have no idea how to form the variables.

    Thank you.
  2. jcsd
  3. Dec 3, 2013 #2
    If [itex]p\in(0,1)[/itex] is the infected proportion, it may be convenient to model things in terms of [itex]x=\log\dfrac{p}{1-p}[/itex], as the latter (which is in one-to-one correspondence with the former) can take any real value.

    It sounds like what is being described is that [itex]\dot p[/itex] is proportional to [itex]e^x[/itex], so say [itex]\dot p = \alpha e^x[/itex] for some [itex]\alpha>0[/itex].

    Then we can compute [itex]\dot x e^x = \dfrac{d}{dt} e^x = \dfrac{(1-p)\dot p - p(-\dot p)}{p^2} = \dfrac{\dot p}{p^2}= \dfrac{\alpha e^x}{p^2}[/itex], and so [itex]\dot x = \dfrac{\alpha}{p^2}[/itex]. One can further verify that [itex]p = \dfrac{1}{1+e^{-x}}[/itex], and so [itex]\dot x = \alpha(1+e^{-x})^2[/itex].

    From the above equation, it's easy to see that:
    - [itex]x[/itex] is strictly increasing over time if [itex]-\infty<x<\infty[/itex], and therefore [itex]p[/itex] is strictly increasing if [itex]0<p<1[/itex].
    - Therefore, [itex]x[/itex] must converge to something finite or infinite. It's easy to see that it can't converge to anything finite over time. Therefore [itex]x\to\infty[/itex]. Thus [itex]p\to 1[/itex] as long as [itex]0<p\leq 1[/itex].

    The only remaining case of [itex]p[/itex] starting at [itex]0[/itex] is easy to consider on its own.
  4. Dec 3, 2013 #3
    Oh whoops, it's simpler than that. Disregard.

    Number of contacts between infected and uninfected should be proportional to [itex]p(1-p)[/itex] maybe?
  5. Dec 3, 2013 #4


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  6. Dec 5, 2013 #5
    Sorry for the late reply. Thank you guys. But I think the simpler version should suffice according to the logistic equation.

    And.. I don't rawly understand what you wrote economicsnerd haha.. My knowledge isn't that advanced yet..
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