They're being referred to as "the sick", short for "the sick people". There's a garden path sentence: "The old man the boat". At first it reads as "the old man [does something]" but then "the boat" means that makes no sense. It's actually "The old [people] man the boat", where "man" is a verb that means "to operate".
"The sick are in need of care." In other words, sick people are in need of care.
Yes, in this usage, "aegri" is masculine plural nominative.
There's a good, accessible book on pandemics in the Roman Empire: Kyle Harper, The Fate of Rome (Princeton, 2018). https://press.princeton.edu/books/hardcover/9780691166834/the-fate-of-rome