Amid assurances that the ship was disease free, hundreds of elated passengers disembarked. Some went sightseeing, visiting beaches and restaurants and getting massages. Others traveled on to destinations around the world.

One, however, did not make it much farther than the thermal scanners at the Kuala Lumpur airport in Malaysia. The passenger, an American, was stopped on Saturday, and later tested positive for the coronavirus.

On Sunday, with passengers already headed for destinations on at least three continents, health officials were scrambling to determine how a big a problem they now have — and how to stop it from getting bigger.

“We anticipated glitches, but I have to tell you I didn’t anticipate one of this magnitude,” said Dr. William Schaffner, an infectious disease specialist at Vanderbilt University Medical Center.