More thoughts on the expected excess LOS

I realised that the expected LOS at a given time $s$, denoted $E[T|s]$, is the same as the life expectancy I'm more used to in life tables. The most common $s$ in this case is life expectancy at birth $E[T|s=0]$, which would be like stay expectancy at admission in the HCAI model.

Of course, life expectancy can be estimated from any age. Your life expectancy changes at you get older by the simple fact that you've already survived up to that point, so your life expectancy at 60 is probably more than your life expectancy at 0.

What we are interested in for the excess LOS measure is average difference LOS over all times in hospital. This is equivalent to the average life expectancy over all ages i.e. we don't know what age some person is so what would our best guess of how long they'll live be. Since a population is not evenly distributed over ages it would make sense to weight the more likely ages more heavily and the less likely ages less so. [I think this is called the overall life expectancy and is closely related to the all-age-all-cause mortality.]

The excess LOS is similar but slightly different. In this case, we're looking at the difference in 2 holding times- the infected "life expectancy" and the uninfected "life expectancy". Equivalently to the life table context,  we want to weight the times at which there are more individuals in the states of interest- not simply the alive state but the infected and non-infected states.

So we prefer times with high probability of being in state 0 and high probability of being in state 1, Put another way, we prefer times with low probability of being in state 2, the death/discharge state.

The probability of being in state 0 is simply the survival probability of not having left it before some time.

The probability of being in state 1 at $s$ is the probability of having jumped to it at some time before $s$ and not having left.

What might be easier is to think in terms of 1-P(being in state 2 at time $s$) since this is a sink state, this is the probability of having entered it at an time up to time $s$ only i.e. a c.d.f.