Branch of data science that covers event based time series? Let's say I have discrete events in time, e.g. patients getting sick, and I want to predict whether theses events are indicators of some other underlying event, e.g. a disease outbreak. Usually, one would transform the event based time series into a regular time series by for example aggregating the counts of patients over each week. However, I feel that by this aggregation a lot of information is lost and it is also hard use to multiple features of events, e.g. patients' age, patients' sex, etc. My question is very general: is there a branch of statistics / data science that treats time series as composed of events at arbitrary times rather that aggregated values at evenly distributed intervals?

I tried googling it, but it seemed hard to phrase the question in a way that a search engine understands.
"rather than aggregated values at evenly distributed intervals?" I think that part defines your question very well; i.e that is usually what we generally do in usual time-series problems, merging the events in some discrete time intervals. 

What you can do is that, you can give the time distance between the events as features, without giving evenly distributed intervals. For example, the exact time distances between events of every disease outbreak or every patient getting sick and etc. Moreover, you can do some feature engineering; squared time differences or percentage changes of time difference between the differences between the sequential doubles (x, x-1) and (x-1, x-2) where x-n,.....,x-2, x-1, x are the sequential events, not the time. 

The relevant concept here seems to be "unevenly spaced time-series", the wiki page Wikipedia: Unevenly spaced time series and this researcher's website http://eckner.com/research.html  seems to give enough material to get started with.