Infirmary for Modern Era (WW1)
Well for Ancient Era.
Bathhouse for Medieval.
Alchemy Lab for Renaissance. Renaissance was a major overhaul for medicine, so this should be important. Maybe add a "Scientific Method" tech, as it seems most of the innovations for the time where new ideas.
Vaccines for EE.
As for diseases, that could play well into ExCE's attrition. In addition, when exploring far foreign lands, or sending trade ships to far off areas, there's a chance that your unit catches a disease that they've never encountered before (like small pox among the natives). Only way to counter that is to have high enough Health/Science. And once innoculated, affected population/cities gain better immunity.
Of course, Plague outbreaks would be a thing, when a civ has too low health or because it's spread via trade routes, proximity, or biological warfare (if this could be a thing, it would probably have to be passive warfare). Emigration, inner empire trading, and city connections would affect the spread. You could also make it so that your empire can focus on discovering a cure (Set a city's production towards Research also increases chances of finding a cure). Depending on the Era, perhaps there could be a WC proposal to fund for a cure, so that future there may not be a repeat in the future, or as to cure an ongoing disease that has a chance to spread to the rest of the world (ebola?), or perhaps a research agreement between two civs would have the same affect.
For Great Doctors, they could be used to clear a city of disease, or discover a cure to a Plauge. On the subject of Great Doctors, Extra Credits just started doing a series on one of them:
https://www.youtube.com/watch?v=TLpzHHbFrHY
There's a lot that can be done with this. Diseases could also weaken a city's defenses, and certain luxuries can help prevent (Orange + At sea = reduced scurvy)
At this point, it might be easier to mod EUIV into Civ 5 xD