Adverse drug reaction; multiple drugs, possible event(s) per patient. What analysis?

Hello everyone,

I am currently looking for the right method to compare the incidences of a given adverse drug reaction (infection) after the exposure to four different drugs (combinations are also possible) in an observational study. Some patients can function as a case and control (after all, not all medication periods will lead to an infection), but most have no infection at all.
The time of exposure to different drugs varies for every medication period and the infection is assumed to be causal if it occurs during an exposure to the drug or within 1 or 3 months (depending on the drug) since the last exposure.
This many options, and I don't know what type of analysis to perform. I've thought of a Cox-regression with multiple events per patient, a case-crossover design or a mixed-model for logistic regression, but I am lost...

Can you help me pick the right analysis? All the help or additional questions are very much appreciated!