Because "sick" and "syg" are adjectives in both languages. Think of it more like "The doctors treat the sick ones", in Danish you wouldn't write the "ones", just leave it with the adjective. I know you wouldn't in English either, but just a good way to remember it
I see. I expected it to be more like in German: "Die Ärzte behandeln die kranken (adjective) Patienten (noun), but "Die Ärzte behandeln die Kranken (noun)". (I thought "the sick" was a noun in English, too.)
In English we might say "the sick", as if it's a noun, but the "people" at the end is definitely implied, making it an adjective for the implied noun. But I think you could make an argument either way about "sick" counting as a noun in this context.