Relating the SIER model, I am trying to understand the intuition of the $\gamma$ paramater. This paramater is the recovery rate. $\gamma$ is fixed and biologically determined. Some authors conider for USA:
$$\gamma = \frac{1}{18}$$
and then asserts that $\frac{1}{\gamma} = 18$ is to match an average illness duration of 18 days.
I can't understand how the inverse of 18 days can represent a recovery rate.
For example, if $\gamma = 5 \hbox{ recovered}/\hbox{day}$. How $1/\gamma = 1 \hbox{ day}/5 \hbox{ recovered}$ can represent a recovery rate?
Can someone explain to me what is the intuition behind this relationship between illness days and recovery rate?
The explanation lies in the exponential decay assumption of the process.
If $\gamma$ is the recovery rate, then the infection population behaves like $I'(t) = -\gamma I(t)$ (e.g. only focus on the removal/recovery rate for now). Solving the differential equation leads to $I(t) = I(0)e^{-\gamma t}$, which has a mean lifetime/removal time of $\frac{1}{\gamma}$, see this Wiki. Hence the reason why the duration is the inverse of the removal rate.