How to predict when and why of hospitalization?
Why is potentially hidden in your feature explanations.
Given that your Features give some indication on why, like Patient demographics data, or some other Features that you can include here, you can use them to answer the why.
Like this using shapley and or eli5 you can integrate it with your Standard predictor classes and for the explanations with user friendly API.
BUT i have to prefice that even tough it might be hidden it must not be at the same time, since the predictiveness of time might have Nothing to do real why you are Looking for.