James Madison is one of the key players in Act II of the musical Hamilton (which is streaming on Disney+ in movie form), and he's notable for his bouts of coughing and use of a handkerchief, which nods to his real life illnesses and ailments. After the introductory number, Madison doesn't appear again in Act 1 of Hamilton, with actor Okieriete Onaodowan instead playing Hercules Mulligan. He is, however, mentioned, as one of the figures who writes the Federalist Papers alongside Alexander Hamilton (with Madison penning 29 of them).
He enters the musical proper in Act 2, serving as a key ally of Thomas Jefferson from "What'd I Miss" onwards, as they work together to take down Hamilton and defeat his plans for the nation's finances. With the focus more on Hamilton's battles against both Jefferson and Aaron Burr, James Madison isn't given too much of the spotlight, but he is an important part of things. One detail many notice about his appearances on stage is that he's often coughing and carries a white handkerchief with him, but why?
The coughing, handkerchiefs, and even some of the lyrics in Hamilton's songs are all in reference to James Madison's real life health issues. Madison picked up a variety of illnesses throughout his life, and was known to appear sickly, which in part came from his stature - at 5'4", he is the shortest President of the United States in history, and he often weighed less than 100lbs. In 1772, after the death of a close friend, Madison suffered from a number of seizures that were categorized as being epilepsy, although have since been re-diagnosed as epileptoid hysteria (via HealthGuidance). At this time, he also had depression and hypochondria.
Perhaps more telling in terms of Hamilton's references is the malaria that Madison contracted in in the late 1780s - at the same time that he and Jefferson were butting heads with Alexander Hamilton over America's financial policy. These cabinet debates are ingeniously framed in Hamilton as rap battles, and it's in "Cabinet Battle #1" that the musical most directly references Madison's ill-health, with Lin-Manuel Miranda rapping "Madison, you're mad as a hatter, son, take your medicine / Damn, you're in worse shape than the national debt is in" just as Madison has to take a seat while having yet another coughing fit (which can also be prominently seen in his exchange with Jefferson in "What'd I Miss"). More than just good wordplay, it's a fitting commentary on Madison's illnesses around this time in reality.
Madison would continue to suffer from illnesses throughout his years - including more attacks of malaria - but he lived a rather full and long life considering. He became President after Jefferson, being elected to office in 1809 and stepping down in 1817, and lived after that until 1836, although he had fevers, arthritis, and was largely bed-ridden by the end of his life. His illness, much like the man himself, isn't the focus of the musical, but it's yet another way that Hamilton somewhat slyly references the history it's telling.