It's a moot point if there is a legal agreement - I took many a sick child to her dad's for the weekend when we BOTH would have preferred that she stay home with me on that particular weekend. It was her dad's weekend, though, and illness not serious enough to warant a hospitalization was not enough to change that schedule.
Of course a divorced mom would want to keep her sick child with her and think it was "better" for her to stay in her own bed with her own routine. That doesn't matter at all, though, when there is a legal custody arrangment. No judge would alter the agreement on that basis and the non-custodial parent would have every right to insist that he have his regular visitation.
Both my kids spent this weekend in separate bathrooms throwing up. It is just 24 bug (according to the pediatrician) and hardly warrants hospitalization, but believe me, no matter what the legal agreement is, I would not send them anywhere.
Disclaimer: I am not divorced and this is not a personal issue for me.

