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The more dire assessments reflect the decisions of governors across the country to ease social-distancing measures even as the number of new cases holds steady and, in some cases, is even rising.

Indiana, Iowa, Kansas, Minnesota, Nebraska, Tennessee and Texas have all allowed some businesses to reopen despite seeing increasing cases, according to a New York Times database.

The projections, based on data collected by various agencies, including the Centers for Disease Control and Prevention, and laid out in an internal document obtained Monday by The New York Times, forecast about 200,000 new cases each day by the end of May, up from about 30,000 cases now. There are currently about 1,750 deaths per day, the data show.

That was not the only forecast of more carnage. Another model, closely watched by White House officials, raised its fatality projections on Monday to more than 134,000 American deaths by early August from Covid-19, the disease caused by the coronavirus. The model, from the Institute for Health Metrics and Evaluation at the University of Washington, more than doubled its previous projection of about 60,000 total deaths, an increase that it said partly reflects “changes in mobility and social-distancing policies.”

“There remains a large number of counties whose burden continues to grow,” the Centers for Disease Control and Prevention said.

Fifteen children, many of whom had fallen ill the coronavirus, have recently been hospitalized in New York City with a mysterious syndrome that doctors do not yet fully understand but that has also been reported in several European countries, health officials announced on Monday night.

Many of the children, ages 2 to 15, have shown symptoms associated with toxic shock or Kawasaki disease, a rare illness in children that involves inflammation of the blood vessels, including coronary arteries, the city’s health department said.

None of the New York City patients with the syndrome have died, according to a bulletin from the Health Department, which described the illness as a “multisystem inflammatory syndrome potentially associated with Covid-19,” the disease caused by the virus.

Reached late Monday night, the state health commissioner, Dr. Howard A. Zucker, said state officials were also investigating the unexplained malady.

The syndrome has received growing attention in recent weeks as cases have begun appearing in European countries hit hard by the coronavirus.

Last week, an alert was sent to general practitioners in London warning that “there has been an apparent rise in the number of children of all ages presenting with a multisystem inflammatory state requiring intensive care across London and also in other regions of the U.K.”

The chief medical officer for England, Chris Whitty, told reporters that a link with the coronavirus was “certainly plausible.”

Asked about the British reports, Dr. Maria Van Kerkhove, a World Health Organization scientist, told reporters last week that the inflammatory syndrome “seems to be rare.”

“There are some recent rare descriptions of children in some European countries that have had this inflammatory syndrome, which is similar to the Kawasaki syndrome, but it seems to be very rare,” she said.

Pediatricians in France, Italy and Spain have also reported dozens of cases of children presenting symptoms of toxic shock or Kawasaki syndrome, but doctors have said that it is too early to link them to Covid-19.

In the northern Italian town of Bergamo, one hospital saw 20 cases in April alone, and in four Parisian hospitals, 20 children have been hospitalized with inflammatory heart conditions. Spain has recorded a few dozen cases, and Switzerland and Belgium have reported a handful.

“And to parents,” she added, “if your child has symptoms like fever, rash, abdominal pain or vomiting, call your doctor right away.”

While the worst-hit parts of the United States have seen new infections recede and hospitalizations drop after strict social-distancing measures were put in place, the country is still in the firm grip of a pandemic.

For every indication of improvement in controlling the virus, new outbreaks have emerged elsewhere, providing a steady, unrelenting march of deaths and infections.

Julie Bosman, Mitch Smith and Amy Harmon report that any notion that the coronavirus threat is fading away appears to be magical thinking, at odds with what the latest numbers show.

Coronavirus in the United States now looks like this: More than a month has passed since there was a day with fewer than 1,000 deaths from the virus. Almost every day, at least 25,000 new coronavirus cases are identified, meaning that the total in the United States — which has the highest number of known cases in the world with more than a million — is expanding by 2 to 4 percent daily.

Rural towns that one month ago were unscathed are suddenly hot spots for the virus. It is rampaging through nursing homes, meatpacking plants and prisons, killing the medically vulnerable and the poor, and new outbreaks keep emerging in grocery stores, supermarkets or factories, an ominous harbinger of what a full reopening of the economy could bring.

Gov. Gavin Newsom of California said on Monday that the state would allow some stores to reopen on Friday, and that, if individual counties desired, they could relax restrictions further as long as they took precautions.

Mr. Newsom, a Democrat, said that among the businesses that could reopen on Friday, with modifications, were clothing stores, bookstores, florists and sporting goods stores, as well as manufacturing businesses that supply those shops.

The announcement was a cautious step toward removing some of the most severe restrictions that California had placed on everyday life. Dozens of states — led largely by those with Republican governors — have undone restrictions meant to slow the spread of the coronavirus.

“This is a very positive sign, and it’s happened only for one reason,” Mr. Newsom said at a news conference. “The data says it can happen.”

Store owners will be allowed to open for pickup on Friday only if they alter their workplaces, and they must enforce social distancing. Mr. Newsom added that more details about the required modifications would be released on Thursday.

The governor also said that if local health officials and county governments certified that they were ready to restart further, they would be able to open restaurants and other hospitality-sector businesses, with modifications. The counties would have to submit plans to the state health agency, and they would be made publicly available.

The two beaches, Mr. Newsom said, had “put together an outstanding plan to reopen” with modifications.

The 9:30 a.m. hearing will be unusual, with a measure of social distancing enforced. Few lawmakers normally choose to sit through the entirety of a confirmation hearing, but senators have been given time slots to come to the hearing room. No more than six lawmakers are supposed to be in the room at once, a dynamic that could affect the rhythm of the questioning.

This time, Mr. Ratcliffe seems likely to receive a warmer reception, at least from Republican lawmakers who control the Senate. Senators want a permanent director to oversee the 17 agencies that compose the intelligence community as they wrestle with how to make sense of the origins and impact of Covid-19, both at home and abroad.

Pfizer and the German pharmaceutical company BioNTech have announced that their possible coronavirus vaccine has begun human trials in the United States. If the tests are successful, the vaccine could be ready for emergency use in the country as early as September.

The two firms are jointly developing a vaccine candidate based on genetic material known as messenger RNA, which carries the instructions for cells to make proteins. By injecting a specially designed messenger RNA into the body, the vaccine could tell cells how to make the spike protein of the virus without actually making a person sick.

The technology has the advantage of being faster to produce, and tends to be more stable than traditional vaccines. But no vaccine made with this technology for other viruses has reached the global market.

Pfizer and BioNTech injected the first volunteers with their vaccine candidate, BNT162, last month in Germany. The shot was given to 12 healthy adults, although the trial will eventually expand to 200 participants. In the United States, the drug companies plan to test the vaccine on 360 healthy volunteers for the first stage of the study, adding up to 8,000 volunteers by the end of the second stage.

Given the need to quash the virus, vaccine makers are racing to speed up their timelines for development, a process that typically takes years. Pfizer and BioNTech say they hope to have several million doses by September, if everything goes well with these human trials.

“We need to think differently, we need to think faster,” said Dr. Mikael Dolsten, Pfizer’s chief scientific officer. “If we get hit with a second wave of coronavirus infections in October at the same time as the flu, things will be much worse than what we’ve already experienced.”

At the same time, the global search for a vaccine was bolstered by a fund-raising conference on Monday organized by the European Union.

Nations from around the world — including Australia, Canada, Japan, and Norway — pledged $8 billion to fund laboratories that have promising leads in developing and producing a coronavirus vaccine.

On a recent evening in Rutherfordton, N.C., Beth Revis drove to the parking lot of a closed elementary school and connected to the building’s free Wi-Fi. Then, she taught a two-hour writing class from her driver’s seat.

Ms. Revis held a flashlight to her face with one hand and a selfie stick with her smartphone attached in the other, looking at the device to speak to her students.

Getting the internet in her area had always been a headache, Ms. Revis said. “But during the pandemic,” she said, “it has turned from a mild inconvenience to a near impossibility.”

For Ms. Revis and many others across the country, parking lots have been a digital lifeline during the pandemic. Instead of spending hours in restaurants, libraries and cafes, people without fast internet access at home are sitting in lots near schools, libraries and stores that have kept their signals on.

“I hope that there is a lesson learned from this,” said Gina Millsap, the chief executive of the Topeka and Shawnee County Public Library in Kansas. “Broadband is like water and electricity now, and yet it’s still being treated like a luxury.”

One in four Americans do not have high-speed internet access at home, according to the Pew Research Center, either because it is too expensive or because they live in a rural area with limited service. Some use their smartphone data plans for high-speed internet access, but those plans are often insufficient to handle work from home and distance learning.

A few days a week, a woman arrives at the Metropolitan Plant and Flower Exchange — a bunker along Route 17 North in Paramus, N.J. They know her by her hospital scrubs.

She picks up her standing order — yellow daffodils — and brings them with her to work at Hackensack University Medical Center.

They are not for her office or for co-workers. She carries them out back and walks into a parking garage. There are now three long trailers there, with loud motors powering their refrigerators. Inside each trailer are bodies in bags, stacked on shelves three high — coronavirus victims awaiting pickup.

The woman’s name is Tanisha Brunson-Malone, and she is a forensic technician at the hospital’s morgue who performs autopsies and oversees funeral home pickups of patients who have died. And she has been entering each trailer, walking the aisles between rows and placing a flower on each new body bag.

Ms. Brunson-Malone’s gesture is all but invisible, seen by only some colleagues and the funeral home workers who arrive to claim bodies. Her flowers are for the dead alone, a fleeting brush with dignity and decorum on the way from one sad place to another.

Before, she would open up the trailers each day and see how many people were dying alone. So she decided to give them a more dignified send-off. She said she spends $100 a week on flowers.

“I was kind of like their voice,” she said, “because they were voiceless.”

Reporting was contributed by Sheryl Gay Stolberg, Eileen Sullivan, Jesse McKinley, Joseph Goldstein, Elian Peltier, Marc Santora, Julian E. Barnes, Nicholas Bogel-Burroughs, Nicholas Fandos, Cecilia Kang, Matina Stevis-Gridneff, Lara Jakes, Michael Wilson, Julie Bosman, Mitch Smith, Amy Harmon, Knvul Sheikh and Matt Stevens.