Born in northeast Haiti with a heart defect, Brad Mertens Joseph is 6, has difficulty walking and is still in diapers.

His parents, accustomed to a dangerous 11-hour overnight bus ride to see cardiologists in the country’s violent capital, had finally found a solution to their son’s medical ailment, caused by a hole in his heart.

It involved open-heart surgery in Akron, Ohio, arranged by a nonprofit.

Those plans collapsed this week when President Trump issued an order banning people from a dozen countries, including Haiti, who don’t already have valid travel visas, from entering the United States

“When I heard that, I was really upset, and I wondered, ‘What are we going to do?’” the boy’s father, Dieudonné Joseph, said. “I was panicking, and I’m still panicking.”

The Josephs are among the many Haitians who are caught in the middle of Mr. Trump’s sweeping travel ban. From young professionals to medical residents to longtime visitors whose visas had lapsed, Haitians are bracing for the consequences of having a lifeline abruptly cut.

With its proximity to Florida, a long (often difficult) history with the United States, and grave political and social upheaval, Haiti has strong family and economic ties with its northern neighbor. People have businesses in both countries, and most middle-class Haitians have close relatives in South Florida or New York.