By the end of March, many U.S. health departments, including those in Pennsylvania and New Jersey, stopped trying to figure out where and how coronavirus patients had become infected.

Though testing remained limited, it was clear the virus had long been spreading beyond those who had traveled abroad and their immediate family members. With a mounting number of cases, the job of tracking who had been exposed to whom was just too big.

Now that the rate of new infections is declining in some parts of the country, talk of this practice, called “contact tracing,” is back. But it remains a task of staggering scope, with various public health experts predicting that thousands of tracers, perhaps hundreds of thousands, are needed to keep the virus in check.

Efforts to hire and train tracers are underway in California, Massachusetts, and New York, with two primary goals: For each new infection, identify that person’s contacts who may have been exposed, then warn them to self-isolate for 14 days.

In Philadelphia, Health Commissioner Thomas Farley said the city will not start contact tracing again until the daily number of new cases drops below 50 — for now, it is in the hundreds — though he welcomes a volunteer tracing effort that began this month at the University of Pennsylvania.

“Contact tracing is essential to be able to safely reopen society,” he said Wednesday. “We can use all the help we can get.”

The concept has been a powerful tool in the public health field for more than a century, enabling epidemiologists to get a handle on the spread of infectious diseases and interrupt the chain of transmission. But because the coronavirus is spread in many cases by people who have no symptoms, containing it is unusually challenging, said Joel Hersh, former director of the bureau of epidemiology at the Pennsylvania Department of Health.

So long as most people limit their interactions to immediate family members, tracking the potential contacts of infected people is manageable, Hersh said. But as society reopens, he said, imagine someone who goes back to work in one of the Comcast buildings in Center City, then starts to develop a dry cough a day later.

“You got on the elevator with maybe 25 people you don’t know," he said. “How do you determine who the contacts were who may have been exposed to you?”

The Penn volunteers have called hundreds of infected patients and their contacts since the effort began two weeks ago, said Carolyn C. Cannuscio, an associate professor of family medicine and community health at the university’s Perelman School of Medicine.

More than 60 people have been trained to make the calls, most of them students in public health, social work, nursing, and medicine, she said. So far, they are calling only patients treated within the Penn Medicine system and their contacts, but they are ready to expand the effort in coordination with the city Department of Public Health, she said.

Patient confidentiality is a top priority, Cannuscio said. One volunteer calls a patient to ask for the names and numbers of any contacts the person had up to 48 hours before developing symptoms. Those names and numbers — but not the name of the original patient — are then given to a second volunteer.

Upon receiving such a call, the contacts may of course be able guess the name of the patient who exposed them, but the tracer is not able to confirm it.

“Even if the person asks for that information, the caller from our team can honestly say, ‘I’m not aware,’” Cannuscio said.

When calling a patient’s contacts, the tracer tells them the date of the potential exposure and recommends that they restrict interactions with others for 14 days from that encounter.

The number of tracers needed is a matter of some debate, according to a report by the Johns Hopkins Center for Health Security. When calculated per 100,000 people, the numbers of tracers in use varies widely: 15 in Massachusetts; four in New Zealand; seven in Iceland; and 81 in Wuhan, China, where the pandemic emerged in December.

“When applied to the U.S. population, a New Zealand-like approach would mean a total of 13,000, and a Wuhan-like approach would mean more than 265,000 contact tracers in the United States,” the report authors wrote.

The need depends on other factors beside population size, such as housing density and how easy it is to reach people. Some populations, such as the homeless, require in-person visits.

A contact tracer need not have a college degree and would typically make $17 an hour, the Hopkins report said. That means the cost to hire 100,000 people for a year would exceed $3.5 billion.

Volunteers such as Penn’s may ease some of the burden, but the hope is to find government or philanthropic support to pay them so the effort will be sustainable, Cannuscio said.

Another approach is to supplement human efforts with smartphone apps that would identify a patient’s close contacts within the previous two days. Several research teams say they are developing such tools in a way that would maintain privacy.

Generally, calls from contact tracers are well received by patients and by those who may have been exposed, Cannuscio said. Patients in particular are eager to help.

During one recent call, a Penn patient offered to donate plasma — the liquid component of blood — in hope that the antibodies within it could be used to treat others, Cannuscio said.

“People who are sick don’t want others to suffer,” she said. “I think there’s probably some healing potential there.”