When do I divide by N in SIR and SEIR models?

Question

My question is kind of a duplicate of this

I understand every separate term in the SIR model differential equations:

$ \frac{dS}{dt}=-\beta S I$

$\frac{dI}{dt} = \beta S I - \gamma I$

$\frac{dR}{dt}=\gamma I $

And I have come across and not understood these:

$\frac{dS}{dt}=-\frac{\beta}{N}{S}{I}{N}$

$\frac{dI}{dt} = \frac{\beta}{N}{S}I - \gamma I$

$\frac{dR}{dt}=\gamma I$

I couldn't understand the answer the the question I linked and decided that it wasn't very important. While researching, I noticed that many pages, even wikipedia use the two interchangeably. When I tried to use the model to graph the 2014 EVD epidemic, I also ended up using the two version interchangeably, but it turned out that they require very different $\beta$ and $R_0$.

What does the N mean and what do it do?

How does the N affect $\beta$ and $R_0$?

Thanks in advance.

Answer 1

The difference is in Density dependent and Frequency dependent models. I found this article helpful