L.L.
To me, working as a nanny is like working as a stand-in-parent. The parent has chosen this, and YOU have chosen this. As parents, we don't get to take off from our children when they are sick, and expect that we, as parents, will get sick to. I can see keeping a child home from a daycare center where other children will be exposed, but you can't keep a child away from their own home.
That being said, you have a right not to be around a sick child...sort of. You've chosen this as your job...a nanny. You and the family obviously need to revise your contract to address this, however you choose to do so...and that would be between them and you. It seems as though the mother expects you to care for her child, whether her child is sick or not. However, if YOU have an illness that the child hasn't already been exposed to, perhaps she would want you to stay home. Talk to the family!