Hosea Matthews may have been killed by Agent Milton, but it turns out he was also sick. Throughout the game, players can hear Dutch and other members of the gang discuss the fact that Hosea is suffering from what seems to be a certain type of lung disease.

What was Arthur's illness?

Arthur stumbles into a doctor's office at the end of chapter five and is told, in no uncertain terms, that he has tuberculosis. He curses the doctor, and it becomes clear that's he's still Arthur.

Can you stop Hosea from dying?

Because Lenny and Hosea die as part of enormous scripted cutscenes (and ones that are integral to how the rest of RDR2's story plays out, too), there's no action the player can take that changes the events of the game.