Which sociological perspective argues that sick people fill a societal role, and are stigmatized by society if they spend longer in that role than is appropriate?

I believe the answer is: functionalism

In functionalism perspectve, we see society as one complex system, and each citizens have their own function in order to maintain the complex system.

When a group of citizen fail to do their function, the productivity of other group of citizens would also be disturbed. Because of this, the people who fail to fulfill their 'function' is looked down upon.

answer; i believe that the correct answer is culture based resources;

if interdependence implies a two-way dependence, and if each partner is equally dependent on the other for success, then there is an equal commitment to the making the partnership successful. the advantage is sustainability of the relationship, the disadvantage is that decisions are "democratic" and therefore may take longer to reach and more likely to be a compromise (which is not always the best solution). as the interdependence moves away from equality, the decisions become more influenced by the more independent partner. in the extreme the relationship becomes more dictatorial and more like "master-slave". generally the relationship has less commitment in both directions and is more likely to be temporary.

if one generalizes (and one shouldn't), equity partnerships, which is probably what is implied by interdependence, can be slow to progress or change but durable. the extreme alternative, independent-dependent partnerships tend to be efficient but more fragile.

hope this but you will have to site your own evidence. : )