During these past months, I've heard a lot about some pandemic modelling techniques, specially the so-called SIR model. Before I begin, I'd like to stress that my interest and question are just a matter of curiosity and I don't work with anything even remotely related to this topic. So, I became curious with it and decided to understand it at least superficially. Here are the three differential equations of the model: $$\frac{dS}{dt} = -\beta I S $$ $$\frac{dI}{dt} = \beta IS - \gamma I$$ $$\frac{dR}{dt} = \gamma I$$
Here, $S, I$ and $R$ stand for the number of susceptible, infected and recovered individuals, respectivelly and $\beta, \gamma$ are two parameters of the model. The point that got me curious was the following. I've read that a constraint to this model is $S+ I+R = N$, where $N$ is the total number of people of, say, a given community. Well, this formula surely makes sense to me, but I don't understand why isn't $N$ also a function of time, $N= N(t)$. It seems reasonable, since the pandemic causes a certain number of deaths. It seems that, when you set $N$ to be fixed, you are modelling a pandemic that causes no deaths. Is that a limitation of the model or is it just me misunderstanding something?