Rabbit mange

Hi
My rabbit seems to have sarcoptic mange. I took him to a vet and he confidently says it is not contagious.
Humans or other animals won't contract the disease and it is transmitted from rabbit to rabbit only.
But some sites on the internet are saying otherwise. I have been tending to it from the past 3 weeks.
Me and my children have not contracted the disease. I am going on vacation for 4 weeks and want to leave him in the care of my housemaid. she expresses concerns that it will be transmitted to her neighbour's cats. is it likely?

Help Please