In search of the right methodology to analyze the outcome of repeat hospitalizations

Hello, I’m here to ask for your help.
here’s the method and the context of my study. I’m looking for the most suitable method.
The aim is to see whether the presence of sickle cell disease increases the risk of thrombosis in patients wearing a specific type of venous catheter.
The patients had one or more hospitalizations over time. Each time they are hospitalized, they have had the catheter and we know how long they have worn it. Some may have died without thrombosis, others may have been discharged without thrombosis, others may have had thrombosis. So the dependent variable is binary (thrombosis or no thrombosis). We have the repetitive side of measurements, so intra-patient correlation. Can we evoke the longitudinal side? knowing that nothing happens during the time between hopsitalizations. Patients are not hospitalized and do not wear catheters? Is a multilevel model better suited to this study, with patients as the group factor?
Or should we instead analyze by stay, disregarding patient dependency?
Knowing that there are few events (around thirty), I can’t put many variables in the model. There are a few variables such as gender, age, conditions favoring thrombosis, cancer…
How do I create a good model? Should we calculate a propensity score, or be very selective in our choice of models?
For the descriptive part of the data. How to present statistics. Can we express in terms of incidence the ratio of the number of thromboses per the number of stays?
This is a study based on data already available (i.e. retrospective).

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