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.