Any moms who have kids that are allergic to cats or dogs, what were your kids symptoms? And what age…

Any moms who have kids that are allergic to cats or dogs, what were your kids symptoms? And what age did you find out about it? My son is a month we have a cat and I stay away from the cat as much as possible but if I go into a room where the cats been he gets a real bad rash all over his face and body. I am assuming it’s that but I want some opinions. I am going to talk to his doctor about it. Tia 😊