There are three types of mange affecting cats and dogs; Notoedric Mange (from the Notoedres mite), Demodectic Mange (from the Demodex mite) and Sarcoptic Mange (from the Sarcoptic mite). It is possible for all of these types of mite to be transferred from dogs to cats. However, the demodex mite only causes disease in immunosupressed or young animals. You should try to isolate your affected animal whilst it is receiving treatment for mange.
