There are 31 new cases across the state, bringing the total to 173, Cuomo said. More than half are in Westchester County, where New Rochelle is located.

“New Rochelle, at this point, is probably the largest cluster of these cases in the United States,” Cuomo said.

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The radius emanates from the Young Israel of New Rochelle synagogue, which has been ground zero for infections in the area.

People living and working inside the zone can go about their day and visit businesses such as grocery stores, and movement won’t be restricted, the governor said, but large gatherings will be discouraged. “You are not containing people,” he said. “You are containing facilities.”

“This is not an exclusion or quarantine zone,” New Rochelle Mayor Noam Bramson told reporters in a Tuesday afternoon news conference. Several thousand people live within the one-mile radius. People can freely enter and leave.

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“The heaviest burdens” of the virus fell on the congregants of Young Israel, Bramson said. “There has been a great commitment to the common good” from members who self-quarantined, he added.

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The outbreak in New Rochelle was traced to one family before the illness spread. Adina Lewis Garbuz, the wife of the lawyer at the epicenter of the area’s coronavirus outbreak, published a public Facebook post on Friday in which she described the moment she learned that her 50-year-old husband, Lawrence Garbuz, was diagnosed with coronavirus.

“When I first heard that Lawrence was positive, I immediately recognized there was going to be pandemonium all around us,” she added. “Other than offering what I could to stop this from spreading further, all I and my family care about is that my husband/their father get better. We shuttered the windows, turned off the internet and together stayed strong and in good spirits.”

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She, their son and their daughter also have tested positive for the virus. She said the family is focused on her husband, who was in critical condition, since she and her children have not experienced symptoms more severe than a “slight cough.”

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New Rochelle Schools Superintendent Laura Feijóo said the order will affect three schools — an elementary school, a middle school and a high school. She reiterated that the district did not want to close but was doing so in response to the governor’s office directive.

“We believe students are safest in schools and are eager to reopen as soon as possible,” Feijóo told parents in an email Tuesday. “It is inevitable that one of our students or staff will contract the virus. What is in our control is to be ready, calm, decisive and responsive to any and all circumstances which may arise.”

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More than 5,500 students, half of the city’s school district, will be out of school from March 11 through March 25, Feijóo said. The three schools — New Rochelle High School, Albert Leonard Middle School and Ward Elementary — will be provided 2,000 Chromebooks on loan from the district so students can complete summer break-like assignments.

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More than 2,800 students at those schools who typically receive free or reduced-fee lunches will receive delivered meals. The district was also working to see if it could provide care for students with working parents.

Hand sanitizer was being delivered to the other schools that remain open. The National Guard will assist in disinfecting the schools.

At 3:30 p.m. Tuesday, a steady stream of high school students walked home from school for what would be the final time in the upcoming two weeks. One teen said he was “happy” school had closed but had a lot of homework to do.

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After the students passed, the foot traffic vanished from the sidewalks along North Avenue in downtown New Rochelle.

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When the National Guard arrives in New Rochelle, it will distribute thousands of meals and assist with cleaning, Bramson said. The one-mile zone will not be militarized or have checkpoints. The guard had not arrived yet, he said, and he did not know when it would.

The city does not have the resources to deliver hot meals to students who rely on school breakfasts and lunches. Nursing and retirement homes in the city were refusing visitors, and, he said, seniors in New Rochelle would receive cold meals. The senior center was closed.

The news was met with confusion and concern for those within the zone, who have said the transmissions had taken a toll even before the new order.

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Michael Weissman, owner of Mikey Dubb’s Frozen Custard, said most restaurant owners in the area have experienced about a 30 percent decrease in customers since the synagogue was shut down.

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He said that Tuesday, he was dropping off bags of frozen custard for the Jewish holiday of Purim, ringing doorbells and leaving them on doorsteps.

“People acted quickly and responsibly and seem to be staying in quarantine,” he said. “You have to live your life and be cautious.”

Weissman said store employees are scrubbing the store with Clorox and using gloves, and he expects that orders through meal delivery services will increase but that foot traffic could be down.

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“I get the occasional customer who’s very anxious,” he said. “Right now, a lot of people are trying to stay positive and continue as normally as they can.”

Vincent Arminio, a construction superintendent, lives within the one-mile containment zone.

His son’s elementary school closed today for the next two weeks. “That’s going to be a challenge,” Arminio, 38, said. He was worried about child care — he had to work, as did his wife, a schoolteacher in a neighboring district that remains open. “I’m not going to be quarantined from my bills,” he said.

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Arminio’s mother-in-law, who has respiratory problems, lives with his family. She has been staying inside. Some restaurants have closed, and customers have vanished from the Acme supermarket where his wife shops. “We’re stocked up."

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Cuomo urged calm after declaring the containment zone by stressing the reality of those most affected — older people with health problems. Washington state and New York have a near-identical number of confirmed infections, he noted, but no deaths in New York have been reported. The deaths in Washington state have centered on a nursing home.

Only 14 people in 173 confirmed cases are hospitalized in New York, Cuomo said.