“In the beginning, it felt like house arrest,” said Samuel Heilman, a New Rochelle resident whose family was among those ordered to self-quarantine on March 3. The families had all attended the same synagogue, Young Israel of New Rochelle, as a lawyer who was the first person diagnosed with the coronavirus in the community.

“In effect, it felt like we were being punished,” Mr. Heilman said. “But the punishment turned out to be a blessing in disguise. This is really a case of perspective.”

Although aggressive testing and confinement orders have shown promise in New Rochelle and elsewhere, including in South Korea, it may be too late to employ similar strategies in places like New York City, where the number of positive cases has overwhelmed the city’s ability to offer tests broadly or to trace the contacts of those infected.