In survival analysis, when there are multiple possible events and occurrence of a one doesn't prevent occurence of the other ones, we talk about failure-time data and the Cox model may be used. Instead, when occurrence of one event prevents occurence of the other ones, the Fine and Gray model for competing events is appropriate.
But what happens for intermediate situations? What if we have two events, one of which is conclusive but the other is not? In particular, if one of the events is death, it clearly prevents occurrence of the other event, while if the other event (let's say, hospitalization) occurs, I can still be interested in observing what's happening, until either I observe death, or data get censored. Then, events are not fully competing (hospitalization doesn't prevent observation of death) but at the same time when one specific event occurs (death), the other one cannot be observed anymore.
Which model should be adopted in such case?