Why life expectancy calculate as 1/μ in SIR model?

Question

In the SIR model, when the death rate is μ, life expectancy is calculated as 1/μ. Can anyone explain it intuitively?

Answer 1

Here is an intuitive explanation:

Let $L$ be life expectancy (in time units used in the model). Consider a population of size $N$ with ages uniformly distributed from $0$ to $L$. Then, as one unit of time passes, the fraction of individuals of age $>L$ will be exactly $\frac{1}{L}N$.

Answer 2

When you have a death term that looks like: $S' = -\mu S$, you can solve for $S(t) $ explicitly. This gives: $$S(t) = S_0e^{-\mu t}.$$ The underlying assumption is that the death process is exponentially distributed. For the moment, let $S_0 = 1$. Then integrate $S(t)$ from $t=0$ to $\infty$, you get: $$\int_0^\infty S(s)ds = \int_0^\infty e^{-\mu s}ds = \frac{1}{\mu},$$ which gives you the expected duration in the $S$ compartment assuming an exponential removal rate of $\mu$.