# SIR Model Specifics

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.