Kitty developing rough patches under his fur :(

My 3.5 months old Persian kitten started developing rough patches under his fur. I took him to a vet and he diagnosed it as eczema. He has given him 3 shots of antibiotic so far and a cream to apply on the patches. It's the 5th day now and two new patches are developed. Moreover my extremely playful kitten doesn't play at all now instead just sleeps ALL day. His feed however his perfectly normal, infact he is eating more than usual.
What should I do about those eczema patches? When will they completely cure?

My understanding is that eczema can be recurring, so this is something he may always have, however, diet can play an important part in keeping it under control.  Did your Vet discuss that part with you? 

BTW, I'm not familiar with giving an antibiotics for eczema, but does he have open sores from it?  That could be the reason.    Does he scratch at them?  How did your Vet determine this was eczema?  What is the cream you are to apply?