I've still got my head in the hospital. And I know just enough about economics to be a total idiot in it. But it strikes me that visiting a patient in a hospital can be examined as an exchange problem (probably not the correct term for what I'm trying to do)--the costs and benefits and how they change over time.
The patient gains from the visit--a break from the boredom, knowledge that someone cares enough to take the trouble, a chance to learn news from the broader world, reaffirmation of family and friendship ties. Note that most of these benefits don't decline in value over a succession of visits. The patient seldom has costs, assuming he is fit enough to receive visitors. (I may be jumping to conclusions--a proud patient can suffer from being seen as incapacitated. That cost may decrease as the patient becomes adjusted to the new role.)
What does the visitor gain from the visit?
The visitor learns the status of the patient, something difficult to assess over the phone. The visitor probably can't learn much news from the patient, except to the extent you can assess what you might face when and if you become a patient. My sense is that the visitor experiences a greater rate of diminishing returns than does the patient. The visitor benefits from showing he's a good person who conforms to social norms. But the sense of self-approbation can decline rapidly.
The visitor has significant costs--a visit is a distraction from the daily routine, which economists seem to assume is a stable balance between costs and benefits.
All of this would suggest that visits should decrease over time.