Uhh, don't know where you live, but that's not the way the hospitals work here in FL. I spent three years consulting for a problem hospital in South FL, and I can guarantee you the building has lots and lots of return air ducts in it.
The building is slightly pressurized and that pressure is released through entrance doors and the elevator shafts.
If a large building is cooled using 100% fresh air, it won't be "slightly pressurized," it will be highly pressurized, and there will be pretty much a gale exiting whenever an outside door is opened.
Not saying it wouldn't be a good idea to operate hospitals this way, simply that it's not actually how they work.
In fact, here in FL the state regulatory agency has a requirement that hospitals operate at neutral pressure to the exterior. It's a remarkably stupid requirement, since among other things it's just not possible, but the requirement is there.
In a hot/humid climate, hospitals, like all other buildings, should operate on a slight positive pressure to help prevent intrusion of humid air, and to ensure that building envelope cavities are generally filled with dry "inside air" rather than humid "outside air."
Properly run hospitals do, of course, have specially pressurized rooms for immune-compromised patients (positive pressure) and highly infectious patients (negative pressure).
Not sure what they do when the patient is both immune-compromised and highly infectious. Start making funeral arrangements, I guess.
BTW, for a large building to operate on conditioned 100% fresh air, it would have to be designed to allow for exhaust from all over the building, not just at doors and elevator shafts.
Otherwise, air would stagnate in the center of the building and contaminants would build up.
I worked on these systems both in the Navy and after I got out. Believe it or not aircraft carriers use the same basic system. NO return air. Yes the buildings would pressurize but not it's not that much. Duct size, fan capacity determine and regulate that along with a damper on the louvers at the outside intake itself.
At night when things calmed down I could sit in my shop close to the air handlers and near the elevators I could tell you when someone entered the building by hearing the pressure change.
Now some spaces may have return but it would not likely be the patients area. This would likely happen because a specific piece of equipment had it's own cooling unit. Modern facilities may now have return older ones do not. The pressure is enough that if you have to take up a commode to replace the wax ring air will come up from the piping. The air handlers themselves? It took a bit of strength to open the door into the machinery room due to the pressure. We had one-three hundred ton chiller and three air handlers that cooled a six story 200 bed nursing home and other spaces. But I've also seen hospital systems and even a hotels system No return duct. The would take up too much real-estate within the buildings.