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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?

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  • $\begingroup$ In a basic SIR model, mortalities are included in the recovered population. So if the case fatality rate is 1%, then 1% of the Rs have actually died, and 99% of the R's have recovered and survived. $\endgroup$
    – user44143
    Commented May 20, 2020 at 16:20
  • $\begingroup$ @MattF. thanks for your answer! If we include mortalities in the recovered population, then the formula $S+I+R = N$ totally makes sense but I wonder why not to consider $N = N(t)$ instead. I imagine it is because the equations would be much more difficult to solve, but it is just a guess. Anyway, woudn't we expect the model to be, idk, a bit more accurate if we consider $N$ as a time-dependent parameter? $\endgroup$ Commented May 20, 2020 at 16:36
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    $\begingroup$ Add the three equations and see what happens. $\endgroup$ Commented May 20, 2020 at 16:56
  • $\begingroup$ $\frac{d}{dt}(S+I+R) = 0$. But this is just a consequence of $N$ being a constant, right? $\endgroup$ Commented May 20, 2020 at 17:11
  • $\begingroup$ in the socalled "SIR model with vital dynamics" birth and death processes are added, so $N$ becomes time dependent. $\endgroup$ Commented May 20, 2020 at 19:18

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