Concerns about density were also at the forefront as New York officials discussed the spread of the virus in increasingly alarmed tones. New York City is now among the worst hot spots in the world: The city now has more coronavirus cases per capita than Italy, the world’s epicenter of the virus outside of China, where it originated.

Gov. Andrew M. Cuomo said more than 20,000 people throughout New York state had tested positive for the virus so far, and 157 had died. More than 2,600 remained hospitalized.

Hospitals across New York City and surrounding areas reported increasing numbers of cases as administrators announced new restrictions on visitors, and workers warned about shortages in protective equipment. Mr. Cuomo announced plans to send hundreds of thousands of masks, gloves and gowns to health care facilities, and said the Jacob K. Javits Convention Center in Manhattan would be repurposed into four “emergency hospitals.”

But he said that initial measures to control the spread of the virus were not working, especially in New York City, where people had been gathering in parks over the weekend and not staying far enough away from each other.

He said he was still awaiting a plan from the city to prevent residents — especially young people — from getting too close, perhaps by imposing more controls on public spaces and opening some streets to pedestrians.

“I touch this table — the virus could live here for two days. You come tomorrow, I’m gone, you touch that spot,” Mr. Cuomo said. “In New York City, all that density, a lot of people are touching a lot of spots, right? Park bench, grocery counters. Just picture the city in daily life.”