Welcome and thanks for posting your query on HCM.
As per your query you are baby sitting one year kid having some skin issues and now your son got similar kind of problem.
It appears from your query that the 1 year old baby has "atopic dermatitis". This is a totally non contagious disease and do not cause any infection to the people surrounding the affected person.
Even then a simple logical thing is that if there would have been contagious nature of the disease then it would had affected you or the mother of the baby first.
So rest assured regarding any possibility of contagious nature of the problem.
Wish good health for your family,
Dr Sanjay Kumar Kanodia
(MD- Dermatology & STD