NEW YORK >> The steep rise in coronavirus deaths in New York seems to be leveling off in a possible sign that social distancing is working in the most lethal hot spot in the U.S., the governor said today — a trend that appears to have taken hold more convincingly in hard-hit Italy and Spain.

Gov. Andrew Cuomo warned that this is no time to relax the restrictions aimed at keeping people from getting too close to one another, and he ordered bigger fines against violators.

“The numbers look like it may be turning. `Yay, it’s over!’ No, it’s not. And other places have made that mistake,” he said.

Stocks rallied on Wall Street and around the world on the news out of the U.S. and in Europe, where deaths and new infections appear to be slowing not only in some of the most severely stricken countries but in the Netherlands and Germany as well. The Dow Jones Industrial Average was up nearly 1,300 points, or 6%, by early afternoon.

Britain’s outbreak was headed in the opposite direction, as the country reported more than 600 deaths Sunday, surpassing Italy’s daily increase for the second day in a row. And 55-year-old Prime Minister Boris Johnson, who was infected last month, was hospitalized in what his office described as a precaution because of persistent symptoms.

More than 10,000 people have died of the virus in the United States, and it leads the world in confirmed infections at around 350,000.

The number of dead in New York state climbed past 4,700, and the death toll in New York City closed in on the 2,753 lives lost at the World Trade Center on 9/11.

The state has been averaging just under 600 deaths per day for the past four days. As horrific as that number is, the roughly steady daily totals were seen a positive sign. Cuomo also reported that the number of new people entering hospitals daily has dropped, as has the number of critically ill patients who require ventilators.

At the same time, he ordered schools and nonessential businesses to remain closed until the end of the month, announced he is doubling the fines for rule breakers to $1,000, and lambasted New Yorkers for being out in parks over the weekend.

“That is just wholly unacceptable,” he said. “People are dying. People in the health care system are exposing themselves every day to tremendous risk walking into these emergency rooms.”

Elsewhere around the world, Italy’s day-to-day increase in new COVID-19 cases dipped again, for the lowest one-day increase in early three weeks. The country, ravaged by the virus, also saw a drop for the third straight day in intensive care beds occupied by infected patients.

Italy still has, by far, the world’s highest coronavirus death toll — over 16,500 — but the pressure on intensive care units in the north has eased so much that the region is no longer airlifting patients to other regions.

Nursing coordinator Maria Berardelli at the hard-hit Pope John XXIII hospital in Bergamo said that while the numbers of new patients had eased up a bit, hospital staff members were still pulling long, difficult shifts.

“There has been no reduction in the work,” Berardelli said. “There have been fewer admissions to the emergency room, but our intensive care units are still full, so the activity hasn’t been reduced.”

In Spain, deaths and new infections dropped again Monday. The health ministry reported 637 new deaths, the lowest toll in 13 days, for a total of over 13,000 dead. New infections were also the lowest in two weeks.

Emergency rooms in the hard-hit Madrid region of 6.6 million were returning almost to normal a week after scenes of patients sleeping on floors and in chairs.

Patients awaiting treatment in Madrid-area ERs went down Monday to 390 cases, one-tenth of the arrivals last week, the regional government said. The number of people being treated for coronavirus in intensive care stabilized at about 1,500 for five straight days.

Transport, Mobility and Urban Affairs Minister José Luis Ábalos said the figures show Spain is entering “a new phase of the battle.”

“This new phase does not mean we can let down our guard. We are assessing the measures that we will need to adopt,” Ábalos said.

It was a message repeated by leaders around the world: Any gains could easily be reversed if people do not continue to abide by lockdown rules.

Louisiana health officials reported 68 more coronavirus-related deaths, the state’s biggest jump since the outbreak began. In all, the state where New Orleans hosts millions of tourists yearly had about 480 reported deaths and over 13,000 infections.

A report from a federal watchdog agency found that three out of four U.S. hospitals surveyed are already treating patients with confirmed or suspected COVID-19. And some places, like Maryland, Virginia and Washington, D.C., are predicting the peak won’t come until May or well into the summer.

Worldwide, more than 1.3 million people have been confirmed infected and over 70,000 have died, according to Johns Hopkins University. The true numbers are certainly much higher, because of limited testing, different ways nations count the dead and deliberate under-reporting by some governments.

The virus is spread by droplets from coughs or sneezes. For most people, the virus causes mild to moderate symptoms such as fever and cough. But for some, especially older adults and the infirm, it can cause pneumonia. Over 270,000 people have recovered worldwide.

Austria and the Czech Republic both began openly discussing how to ease some of the crippling restrictions. Austria’s chancellor said the plan is to let small shops and garden centers reopen next week, limiting the number of customers inside, and the rest on May 1. The Czech government is proposing an end to the ban on travel abroad as of April 14 and the reopening of small stores.

In Asia, Japan’s prime minister said he will declare a state of emergency for Tokyo and six other prefectures as early as Tuesday. Infections are soaring in the country that has the world’s third-largest economy and its oldest population.