According to the data above in this thread, there are a number of ways a young person could become infected with HPV and not have been sexually active at all.
That makes it a disease that the state should have a real interest in. However, fair disclosure, I also think the state should be involved in controlling syphilis and gonorhhea, diseases that are not as communicable as is hpv.
Levels of interest could be:
quarantine
forcible innoculation
required innoculation
required innoculation with opt out
innoculation with opt in
public service recommendation.
I would place the plague at the worst level, quarantine. I would place the common cold as a public service recommendation. I’d make hpv a required innoculation with opt out.