How long can it take a rash due to allergic reactions to cats to heal?

My kid's doctor refuses to run allergy tests. He broke out in a blistery rash that looks like poison ivy, and it lasted a month. We suspected he might be allergic to cats, and was exposed about 12 hours before breaking out, and was around them for 5 days and then once we came home it took a month for him to totally heal despite being on prednisone. Can it take up to a month for a child with eczema to heal from an allergic skin reaction to animals?