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Modeling epidemics - solving differential equation

  1. Nov 19, 2016 #1
    • Moved from a technical math section, so no template was used
    I am given a modified SIR model in which the rate of decrease of susceptibles S is proportional to the number of susceptibles and the square-root of the number if infectives, I. If the number R of those who have been removed or recovered increases in proportion to the infectives, we have the following equations:
    dS/dt = -α*S*√(I)
    dR/dt = β*I where α and β are positive constants. If the total population, N = S + R + I does not change over time, shade the region of the S-I plane in which I increases.

    To find the region, as far as I understand, I need to first find the differential equation for dS/dt which is
    α*S*√(I) - β*I. Then, I need to consider the following differential equation:
    dI/dS = -1 + {β*√(I)}/α*S}
    which I am not sure how to solve since I cannot separate the variables and the method with integrating factor is not applicable either.
    I also tried considering dS/dI but it looked even worse.

    Thank you in advance!
     
  2. jcsd
  3. Nov 19, 2016 #2

    haruspex

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    Express that algebraically.
    Which tells you what in terms of rate of change of I?
     
  4. Nov 20, 2016 #3
    Hi, thank you for your reply, I am not quite sure what you mean. If I express I algebraically, then I'll obtain I = N - S - R
    But then I am not sure how this is will solve the differential equation since I am considering dI/dS and the expression I = N - S - R also introduces R.
     
  5. Nov 20, 2016 #4

    haruspex

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    No, I mean express algebraically the statement that I increases.
    Why? The question asks in which region of the I-S space I increases with time.
     
  6. Nov 20, 2016 #5
    Hmmmm... I am sorry but I still do not get what you are trying to say. As far as I understand, since the question asks to shade the region of the I-S space in which I increases with time, I need to obtain the function I(S) or S(I) which I can get by solving the differential equation. However, I will increase with time when dI/dt >0, i.e. α*S*√(I) - β*I>0. So, when -dS/dt - dR/dt>0 which implies that -dS/dt>dR/dt but I am not sure how this can help to find the region.
     
  7. Nov 20, 2016 #6

    haruspex

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    I do not follow the logic of that.
    You are almost there! Just simplify.
     
  8. Nov 20, 2016 #7
    Ohhh, I see! So, I get that I < {(α)^2 * (S)^2}/{β^2} which is a parabola starting at the origin. Therefore, the shaded region should be under S-axis (if I plot I as y-axis and S as x-axis).
     
  9. Nov 20, 2016 #8

    haruspex

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    Right.
     
  10. Nov 20, 2016 #9
    Thank you for your help!
     
  11. Nov 21, 2016 #10
    Actually, I have just realised that the shaded area should be the area not under negative x-axis but under the parabola. So, it will be the entire area under x-axis and some of the area above it but below the parabola.
     
  12. Nov 21, 2016 #11

    haruspex

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    Ah, yes, I hadn't read your previous reply carefully enough, sorry. Got to the mention of parabola and stopped.
     
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