NEW YORK — This jampacked city, with its high-rises, brownstones and cheek-by-jowl single-family homes, is a ripe environment for the novel coronavirus that has killed more than 11,000 residents. That density also complicates a key strategy for alleviating the epidemic: testing.

In their initial response to the pandemic, city and state officials called for federal health officials to move more quickly on increasing testing capacity, seeking to identify those who had contracted the virus and isolate them to help stem the outbreak.

“Our single-greatest challenge is the lack of fast federal action to increase testing capacity and, without that, we cannot beat this epidemic back,” said Mayor Bill de Blasio.

However, when it became clear that they lacked sufficient diagnostic tests and needed to preserve personal protective equipment for health care workers to treat hospitalized patients, city officials shifted to calling instead for testing only the sickest people — often just those needing hospitalization — and encouraging anyone else feeling ill to stay home. This approach is in line with guidelines from the federal Centers for Disease Control and Prevention.

But other officials say wider testing is needed now. That call is coming especially from representatives of people in neighborhoods that are home to many first responders and people who continue to work during the shutdown because their jobs are considered essential.

The question of testing has been central to the efforts by the U.S. and other countries to minimize the pandemic. South Korea’s massive testing effort has been lauded as a powerful tool that helped reduce the spread of the virus. But the U.S. rollout of the initial tests was plagued with problems and delays, prompting frustrated calls for help from local officials and sharp criticism of President Donald Trump’s management of the crisis.

In New York, Gov. Andrew Cuomo has consistently pressed hard for tests, saying early last month that “we want to identify people, because [we] want to put them in a position where they’re not going to infect anybody else. … The more tests the better, the more positives you find the better because then you can isolate them and you slow the spread.”

But the viral outbreak in New York City has outpaced officials’ efforts to identify those who were infected. With the virus spreading through the city’s densely populated neighborhoods, city officials refocused testing priorities on those who required hospitalization while most people were under orders to remain in their homes or practice social distancing if they had to go out.

Shelter-in-place directives that keep people apart make sense for every community fighting the spread of the virus when the option of testing widely doesn’t exist. But keeping people off the streets is especially important in New York, whose 8.6 million residents are packed together more closely than those of any other major city, with more than 28,000 people per square mile, according to an analysis by Governing.com, an online platform for state and local government data.

For people with mild to moderate symptoms, or whose exposure to the virus is unknown, testing usually “won’t change what we do, but it will potentially expose them or others [to the virus] if they leave their homes to get tested,” said Dr. Theodore Long, vice president of ambulatory care at New York City Health + Hospitals, the public health care system that operates 11 hospitals and dozens of primary care clinics in the city’s five boroughs.

Campaigns used by other cities don’t work as well here. Drive-thru testing sites have limited utility, especially in Manhattan, where only about 1 in 5 people have a car. Testing at city parks or other outdoor venues could pose a significant risk of transmission because it would draw crowds, said Dr. Jeffrey Engel, former executive director of the Council of State and Territorial Epidemiologists, who is the organization’s senior adviser on COVID-19.

“We should just assume everyone’s infected,” Engel said.

About 80% of people who become infected with the novel coronavirus recover on their own. For New Yorkers who are worried about their symptoms, Health + Hospitals has a hotline staffed by clinicians who can help determine whether someone should come in for testing. So far they’ve handled more than 50,000 calls. Ninety percent of the time they advise people to stay at home, said Long.

The sickest patients are advised to go to one of the hospital emergency departments or urgent care clinics for assessment and possible testing, said Dr. Andrew Wallach, NYC Health + Hospitals’ chief medical officer of ambulatory care.

Meanwhile, tents outside the public hospitals have been set up to assess people with mild to moderate symptoms to spare emergency departments, said Long. The system isn’t perfect, though. At times it has led to long, snaking lines of people outside hospital tents braving inclement weather and not maintaining the recommended 6-foot gaps for proper social distancing.

But the current testing rules have left out crucial groups of people, some officials charge. Earlier this month, New York City Councilman Justin Brannan, U.S. Rep. Jerrold Nadler and six state legislators wrote to de Blasio and Cuomo demanding a testing site in the Bay Ridge neighborhood of Brooklyn.

The area is home to many nurses, EMTs and other front-line personnel whose jobs are deemed essential, such as sanitation workers and bank tellers, Brannan said.

“These folks have no choice but to go out and go to work every day,” Brannan said, where they risk spreading the virus or getting sick themselves. “It’s dangerous. These folks need testing.”

As of Tuesday, New York state had tested 526,012 people, nearly as many as California, Texas and Florida, the three most populous states, combined, according to the COVID Tracking Project, a crowdsourced compilation of the latest testing figures. That tally includes 118,302 people in New York City.

Hospitals recently expanded some of their testing protocols. NYC Health + Hospitals began opening up testing to all its workers, giving priority to employees who have symptoms of the virus or are in home isolation and those caring for COVID-19 patients.

Private hospital systems in the New York City area are also performing testing. NYU Langone Health, for example, has created several on-site testing areas by repurposing existing rooms that are large and well-ventilated, including a cafeteria, a library and an unused floor of a nursing home, said Dr. Andrew Brotman, executive vice president and vice dean for clinical affairs and strategy at Langone.

The health system is testing patients with COVID-19 symptoms who come to the emergency room as well as employees exhibiting symptoms, Brotman said.

Moving forward, “increasing testing will be a vital step to identify clusters of cases and stop the virus from spreading further — which will require isolating cases, tracing their contacts, and monitoring contacts under quarantine,” said Dr. Thomas Frieden, former CDC director and health commissioner of New York City. “To begin to release restrictions and open society back up, we need to expand our testing volume massively, and have an army of contact tracers trained and ready.”

Many experts agree. In his daily briefing last Thursday, Cuomo referred to large-scale rapid testing as “the bridge to the new economy and getting to work and restarting.”

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