An average of 146,000 people per day have been tested for the coronavirus nationally so far this month, according to the COVID Tracking Project, which on Friday reported 3.6 million total tests across the country. To reopen the United States by mid-May, the number of daily tests performed between now and then should be 500,000 to 700,000, according to the Harvard estimates.

That level of testing is necessary to identify the majority of people who are infected and isolate them from people who are healthy, according to the researchers. About 20 percent of those tested so far were positive for the virus, a rate that the researchers say is too high.

“If you have a very high positive rate, it means that there are probably a good number of people out there who have the disease who you haven’t tested,” said Ashish Jha, the director of the Harvard Global Health Institute. “You want to drive the positive rate down, because the fundamental element of keeping our economy open is making sure you’re identifying as many infected people as possible and isolating them.”

The researchers said that expanded testing could reduce the rate to 10 percent, which is the maximum rate recommended by the World Health Organization. In Germany, that number is 7 percent, and in South Korea, it is closer to 3 percent.