There only study available for diagnosis of Down Syndrome is Karyotyping.
There are triple ,Quadruplet marker studies and amniocentesis
done during the first trimester of pregnancy
to confirm Downs Syndrome.
I can understand your anxiety but such kids do have poor immunity and do have many problems related to heart ,GIT ,Respiratory ,ears,thyroid
etc and hence are prone for recurrent infection.
I would advise you to give him a good healthy diet,get him vaccinated for optional as well as compulsory vaccine
and lastly follow up at regular intervals with his doctor.