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Math Help - Determine the constant C such that I(S0) = I0

  1. #1
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    Determine the constant C such that I(S0) = I0

    Course: Self- Study
    Textbook: A Course in Mathematical Biology

    I'm working with the basic SIR model, and I have this system:
    $$\begin{align}
    S'(t) &= -\beta S I \\
    I'(t) &= \beta S I - \alpha I
    \end{align}$$
    We look for solutions in the form $I(S)$ so we divide $I'$ by $S'$ making $$\frac{dI}{dS} = \frac{\alpha}{\beta S} - 1$$
    I have a question that says to determine the constant $C$ such that $I(S_0) = I_0$.
    By solving by separation of variables, we see that $I(S) = \frac{\alpha}{\beta} \ln(S) - S + C$. So
    $$\begin{align}
    I(S_0) &= \frac{\alpha}{\beta} \ln(S_0) - S_0 + C = I_0 \\
    C &= S_0 + I_0 - \frac{\alpha}{\beta} \ln(S_0)
    \end{align}$$
    In my chapter there have not been any initial conditions, so I'm wondering if leaving C in terms of $S_0$ and $I_0$ would be fine, because I'm supposed to create a Matlab function for $I(S)$ with the appropriate constant of integration.
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  2. #2
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    Re: Determine the constant C such that I(S0) = I0

    This is all fine.
    Thanks from MadSoulz
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