In the era when people traveled by sailing ship and steamer, illnesses usually burned themselves out before boats reached shore, a new study finds.

On Dec. 22, 1874, the H.M.S. Dido arrived in Fiji from Sydney, Australia, carrying about 200 people and an invisible payload. A king of Fiji and his son, who were on the ship, were infected with measles. When they debarked, they started an epidemic that killed 20,000 people in Fiji — up to one-fourth of the population — who had no immunity to the disease.

But in those days, when people traveled by sail or steam, such events were the exception rather than the rule. A new report, published last week in The Proceedings of the National Academy of Sciences, uses mathematical models to show how viruses had to beat very long odds to be transmitted across the sea. Most often, the study found, infectious diseases burned themselves out on board before ships ever docked.

In the contemporary world, it is expected that new diseases and older infectious menaces will spread almost instantly around the globe, as happened with Covid-19. But where was the inflection point? Elizabeth Blackmore, a doctoral student at Yale, and James O. Lloyd-Smith, an ecologist at the University of California, Los Angeles, set out to find the moment when viral transmission started to change.

John McNeill, a historian at Georgetown University who was not involved in the study, said Ms. Blackburn’s use of sophisticated mathematical modeling “has achieved something here that no historian or anybody else has been able to do before — to quantify likelihoods of transmission.”

Kyle Harper, a historian at the University of Oklahoma who was also not involved in the study, said the work “breaks new ground.”

Ms. Blackmore said that she and Dr. Lloyd-Smith thought of the idea to look at shipping when she was working on her master’s degree. They learned that the first reports of smallpox outbreaks of smallpox in California were not until 1806 and 1838. And smallpox was first reported much later elsewhere in the Pacific.

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