Has anyone ever had a baby that ended up having bad allergies?

So say you have a cat and a dog that sheds like crazy (Siberian Husky maybe) and you are pregnant. You have other young children too. When it comes time around to have the baby, and you bring him/her home finally you shortly notice that something is wrong. You take them to the doctor and the doctor says it’s pet allergies. What do you do in this sort of situation? Do you get rid of your animals? Do they give the baby medicine and you get to keep the animals or what?

Has this happened to anyone before? If so, please share your story and how you handled it.