My wife and I have just returned from the first two and a half legs of a QM2 world cruise. We were both in excellent health when we boarded this vessel but are now both suffering from coughs which we caught on QM2. I started coughing on 6th day of a 39 day cruise and although I paid to see the ships doctor the best he could do was offer a cough medicine, In Cape Town we did manage to get antibiotics which appeared to be working , the cough nearly cleared up but after a few further days came back with a vengeance, My wife's cough started about 4 weeks into the cruise and she had to visit a doctor in Melbourne as we were worried about the long flight home.
About 1800 passengers left the cruise in Cape Town, about 25% of these were coughing. Coughing on the ship quietened down for a few days and then returned.
Speaking to several long term Cunard cruisers it appears they either bring antibiotics and steroids from home or alternatively buy these in Tenerife which was our second stop.