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I'm trying to understand the meaning of the parameters of the following model:

\begin{align*} \frac{dS}{dt}&=-\alpha I S+b-dS\\[5pt] \frac{dI}{dt}&=\alpha IS-\delta I-\mu I-dI\\[5pt] \frac{dR}{dt}&= \delta I -dR\\[5pt] \frac{dD}{dt}&=\mu I \end{align*}

where $b$ the birth rate, $d$ the natural death rate, $\alpha$ the infection transmission rate, $\delta$ the recovery rate, and $\mu$ the death rate from infection.

I know that to find $\delta$ we can use that $1/\delta$ is the average time of contagiousness so (for example) $\delta=\frac{1}{5}\text{ day}^{-1}$.

I didn't find an explanation to find the other parameters with for example an average time before dying for $\mu$. So my question is do the other parameters have to be expressed in $\text{days}^{-1}$ too?

Intuitively, I would put in $b$ the number of births by day like $83,000$ in France, so $b=83000 \text{ day}^{-1}$?

I'm confused with the consistency of the units and the meaning.

Thank you for your help.

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    $\begingroup$ I took a look at your profile, and I think you probably should read math.stackexchange.com/help/someone-answers, or it's going to be increasingly difficult for you to motivate people to answer your questions. $\endgroup$ Commented Apr 20, 2021 at 11:45

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One fundamental fact about equations (differential or otherwise) that you need to remember is that when you are adding together different terms they all need the same units. So, for instance, if you consider $$\frac{dS}{dt}=-\alpha I S+b-dS$$ then all of $\frac{dS}{dt}, \alpha IS, b$, and $dS$ need to share common units. If $\frac{dS}{dt}$ has units of $\text{individuals}\cdot\text{day}^{-1}$, then every other term, including $b$, needs to also be in units of $\text{individuals}\cdot\text{day}^{-1}$. For the term $\alpha IS$, since $I$ and $S$ both have units of $\text{individuals}$, then to have an overall unit of $\text{individuals}\cdot\text{day}^{-1}$ then $\alpha$ needs to have units $\text{individuals}^{-1}\cdot\text{day}^{-1}.$ On the other hand, since $dS$ only includes one factor with units of $\text{individuals}$, then $d$ must have units $\text{day}^{-1}$. The units in the other terms can be found by following a similar process.

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