Hi Stephen,
If you have ArcGIS Pro you can use the Summarize Within tool. If you are using ArcGIS Desktop, you can accomplish this using the Spatial Join tool. Before running the tool, create a new field in the patient address feature class called something like "COUNT". Calculate the value of this to 1 for each row. When executing the Spatial Join you can specify a merge rule to SUM. Here is a very old help document that shows this:
ArcGIS Desktop Help 9.2 - Example of aggregating field values with Spatial Join