The continuing flood of patients into hospital emergency rooms has presented a daunting challenge to policymakers like Mr. Cuomo who are trying to predict on the fly not only where the crisis might be headed, but also when New York might be able to return to a semblance of normalcy. The governor said on Monday that schools and nonessential stores would remain closed at least through April 29.

The governor’s staff, in attempting to divine the course of the outbreak, has been using statistical models created by the Institute for Health Metrics at the University of Washington, which has received funding from the Bill and Melinda Gates Foundation, as well as predictions generated by McKinsey & Company and Weill Cornell Medicine.

Much of the uncertainty comes from the fact that the statistics themselves are far from solid indicators.

The number of hospitalizations, for instance, depends partly on admission standards. Some overwhelmed hospitals are sending home people who are in less dire shape, but whom they would admit in normal circumstances. There are also indications that Covid-19 deaths are being undercounted — especially those who die of the illness at home, rather than a hospital. And studies have shown that many people never even know they have been infected, one reason the governor spent much of his time in front of the camera scolding those New Yorkers who found the outdoors too inviting to resist.

“Frankly, there has been a laxness on social distancing, especially over this past weekend,” he said. “Now is not the time to be playing Frisbee with your friends in the park. Now’s not the time to go to a funeral with 200 people.”

“I understand how the religious services can help with the grieving process,” he said. “But, as a society, the risk is too great.”

To ensure that downward trends continued, officials in New York City announced that they were closing all dog parks and dog runs because people using them were not practicing adequate social distancing. City officials also promised to crack down on funerals where people were ignoring separation measures.