I read on Wikipedia (under "Compartmental models in epidemiology") that the differential equations for the SIR Model was the following,

$$S'(t)=-\frac{\beta}{N}I(t)S(t)$$ $$I'(t)=\frac{\beta}{N}I(t)S(t)-\gamma I(t)$$ $$R'(t)=\gamma I(t)$$

It never states the range of $\gamma$ and $\beta$. However, based on the explanation of $\beta$, which is the transmission effectiveness of the disease and the average number of contacts between people per time multiplied, so theoretically if this average became arbitrarily large, $\beta \to \infty$? So it's range could be $\beta \in [0,\infty)$. Whereas, $\gamma=\frac{1}{D}$, where $D$ is how long someone is infected for which could mean $D \in (0,\infty)$ and as such $\gamma \in (0,\infty)$. However, I'm not entirely sure if these are correct (since I just assumed it based on the explanation), and I haven't found any other sources explaining this.


Theoretically, yes, the ranges of $\beta$ and $D$ are as you stated (although $\beta = 0$ means the disease never spreads, which isn't really helpful). However, SIR is a model, and the purpose of models are to give a good approximation for things like diseases over a limited period of time. This means some parameter values may make the model very accurate, while other values may not even produce a practical model to work with.

Note that $\frac{\beta}{\gamma} = \beta D = R_0$, which is the basic reproduction number of a disease. Values of $R_0$ for common diseases seem to be around 0.5 to 20, depending on the disease. Hopefully this gives you some idea what realistic values of $\beta$ and $D$ are.

Also, this link may also provide more insight on the SIR model.

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