One Brooklyn Health and other hospital systems have also sought help from state officials. Gov. Kathy Hochul, who declared a health care emergency in late November, has authorized nursing students and out-of-state doctors to assist during the current surge. The state has also deployed 120 National Guard members to nursing homes and had directed federal teams to upstate hospitals experiencing their own crises.

The state has not sent staff reinforcements to downstate hospitals, Mr. Raske said.

Jill Montag, a spokeswoman for the state Health Department, did not dispute that, but she said the state was helping hospitals coordinate their efforts and had secured 50 ambulance teams for New York City from the Federal Emergency Management Agency; 25 are to arrive on Saturday.

The problem has been compounded for community hospitals by the way the Omicron wave has spread, starting in wealthier parts of Manhattan and then moving to low-income neighborhoods that rely on safety net hospitals. This week, the Covid positivity rates topped 40 percent in the South Bronx and parts of Brooklyn and Queens. About 37,000 new cases a day are being reported citywide on average.

Longstanding health care disparities mean patients in poor areas are also arriving at hospitals with more pre-existing conditions, and vaccination rates in these areas tend to be lower, contributing to more severe illness. In the Brooklyn ZIP code where Brookdale is, about 2,500 people tested positive for the virus last week, with a positivity rate of 43 percent.

Not all safety net hospitals say they are being stretched to the limit. The area around St. Barnabas Hospital in the Bronx had nearly the highest test positivity rate in the city this week, above 46 percent. But officials said they were dealing with the higher volume, even with 7 to 9 percent of the work force out with Covid-19 or caring for sick relatives.

Not all of the hospital’s intensive care bed are full, and of the 30 patients presently in intensive care at St. Barnabas, only 10 have Covid-19, said Dr. Edward Telzak, the chief of internal medicine. “Covid-19 is not overwhelming us,” he added.

It is different at the pediatric emergency room at Montefiore Hospital, a major nonprofit hospital in the Bronx. Julian Grant, a registered nurse, said the small emergency room was often packed with up to 80 patients, with as few as two nurses to help them.