Suppose one of the astronauts was a carrier for a disease that did not show up in the medical testing prior to flying up to the ISS.
How would the outbreak of this contagion be dealt with?
NASA, ESA, and RSA follow very conservative policies towards sending a potentially infected person into space. All the blood tests and other screenings in the days, weeks, and months leading up to a launch would make it virtually impossible for an astronaut to carry a communicable disease up to the ISS without showing symptoms or being detected as an asymptomatic carrier (a Typhoid Mary), either of which would have the astronaut grounded.
However, astronauts have gotten sick in space. Obviously, motion sickness (or zero-g sickness) is a common problem, but it's not caused by an infection. It's more akin to seasickness, caused by a combination of your lunch constantly churning around in your stomach as you move, and conflicting inputs from sensory organs like your inner ear that are a gravity-dependent indicator of "up". For that, there are and always have been barf bags aboard spacecraft, of continually increasing sophistication. Antiemetic drugs are a newer additional measure, but as the primary side effect of most such drugs like phenergan is drowsiness, they're saved for the worst of the worst.
The first, worst, and only publicly-known episode of a communicable disease spreading aboard ship was Apollo 7, in which the mission commander, Wally Schirra, gave a rather nasty head cold to his two crewmates. The resulting breakdown in order aboard that craft (it was an eleven-day flight, the second-longest thus far and the first manned flight of the Apollo capsule, so there were several SNAFUs relating to the crew's supplies and accommodations on top of the illness) ultimately resulted in all three astronauts being permanently grounded (it was Schirra's last flight anyway, but Cunningham and Eisele had been in line for Moon landings).
Apollo 7 also made NASA flight doctors very skittish about infections; the original prime crew of Apollo 13 included Alan Shepard, but an ear infection caused NASA medical staff to switch his entire crew with Jim Lovell's. Ken Mattingly was famously scrubbed from the Apollo 13 crew three days before the launch due to exposure to measles. Ironically, he never got sick, while Fred Haise, as was popularized in the movie, brought up a non-contagious urinary tract infection on that same flight, and without any antibiotics in the medical kit to treat the infection, he was in considerable pain and suffering during most of the mission and for weeks after landing. It's not known whether the mission would have been scrubbed anyway due to his illness; by the time the worst of his symptoms manifested, the accident that crippled the Odyssey had already occurred.
I have not been able to uncover any other instances of astronauts contracting an infectious disease while en route to or aboard the station, but the ISS does have measures in place for illnesses and other ailments, similar to other U.S. space missions.
Here is a link to the Medical Operations section of the ISS "user guide". In it are the following tidbits:
The various procedures in this guide infer that the ISS has a small pharmacy aboard in its drug cabinet, containing several options each for analgesic pain relievers, antihistamines, decongestants, antiemetics, anti-diarrheals, sleep aids, etc up to the "hard stuff" like cortizone and albuterol steroids, opioid painkillers including hydrocodone and possibly morphine (it's mentioned in drug interaction warnings but no procedure says to administer it), antibiotics, antifungals, and even a local anesthetic (Lidocane/Xylocaine) and two anti-psychotics/anti-anxietals (Haldol and Diazepam/Valium), each of those in oral and injectable form. Procedures detail the dispensing of these drugs for a variety of symptoms and presentations including those associated with common communicable illnesses (sinus blockage, cough, vomiting/nausea, fever, headache, rash, etc).
As stays aboard the ISS are long-term, and 31 of the total 88 ISS crewmembers to date have been women, the medical kit includes such sundries as the anti-fungal drug fluconazole for yeast infections, and - hand to God - pregnancy tests. Apparently this made the news in 2001, when this medical procedures document was first leaked, as it represented NASA's first tacit acknowledgment that some of its astronauts might try to join the "200-mile-high club".
Similarly, due to the long duration of ISS missions, various conditions that would otherwise be treated very primitively with first-aid measures on a moon shot or Shuttle flight instead have procedures and supplies for a more lasting fix; these conditions include everything from broken bones and concussions to kidney stones and dental cavities. The kit includes tools and materials specifically to perform dental repairs such as crown replacements, cavity fillings, bracing/securing of cracked teeth, and extractions.