Not sure. We moved into a house which previously had a cat. Lots of hair about, but only carpet was hall and stairs. This has now been replaced and DD excema is rather better than before. Coincidence? Cause/effect? Not cat related just dirt related due to being a minging ancient carpet, and the plaster coming down in the hall? Who knows.
Can strongly recommend a massive clean, including a bottle cleaner down the gaps in the radiators, and pulling out the kitchen cupboards. The fossilized cat turd was an unpleasant discovery, but Id rather have found it than left it...