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Coronavirus in New York came mainly from Europe, studies show.

New research indicates that the coronavirus began to circulate in the New York area by mid-February, weeks before the first confirmed case, and that it was brought to the region mainly by travelers from Europe, not Asia.

“The majority is clearly European,” said Harm van Bakel, a geneticist at Icahn School of Medicine at Mount Sinai, who co-wrote a study awaiting peer review.

A separate team at N.Y.U. Grossman School of Medicine came to strikingly similar conclusions, despite studying a different group of cases. Both teams analyzed genomes from coronaviruses taken from New Yorkers starting in mid-March.

The research revealed a previously hidden spread of the virus that might have been detected if aggressive testing programs had been put in place. On Jan. 31, President Trump barred foreign nationals from entering the country if they had been in China — the site of the virus’s first known outbreak — during the previous two weeks.

Viruses invade a cell and take over its molecular machinery, causing it to make new viruses. An international guild of viral historians ferrets out the history of outbreaks by poring over clues embedded in the genetic material of viruses taken from thousands of patients.

In January, a team of Chinese and Australian researchers published the first genome of the new virus. Since then, researchers around the world have sequenced over 3,000 more. Some are genetically identical to each other, while others carry distinctive mutations.

C.D.C. issues new back-to-work guidelines for essential workers.

The Centers for Disease Control and Prevention published new guidelines on Wednesday detailing how essential employees can go back to work even if they have been exposed to people infected by the coronavirus, provided they do not feel sick and follow certain precautions.

Those employees can return if they take their temperature before heading to their workplaces, wear a face mask at all times and practice social distancing while on the job, Dr. Robert Redfield, the C.D.C. director, said at the White House briefing. They should not share headsets or other objects that touch their faces, and they should not congregate in break rooms or crowded areas, he said.

Dr. Redfield said that employers should send workers home immediately if they developed any symptoms. He also said they should increase air exchange in their buildings and clean common surfaces more often. The goal, he said, was to “get these workers back into the critical work force so that we don’t have worker shortages.”

The new guidance appears to blend earlier advice. Last week, the C.D.C. recommended that even healthy Americans wear masks in public after data showed as many as 25 percent of people infected with the virus were asymptomatic, at the urging of the White House, businesses, workers and others to kick-start the idled economy.

Dr. Deborah Birx, the White House’s coronavirus response coordinator, and other government experts suggested at the briefing that the strict measures being taken by Americans to stem the spread of the virus may be leveling new cases in areas like New York, Detroit, Chicago and Boston.

In New York, ‘the bad news is actually terrible.’

New York, the hardest hit state in America, reported its highest number of coronavirus-related deaths in a single day on Wednesday, announcing that another 779 people had died. That brought the virus death toll to 6,268 in New York State, which Gov. Andrew M. Cuomo noted was more than twice as many people as the state had lost in the Sept. 11, 2001, attacks.

“I went through 9/11,” he said at his daily briefing. “I thought in my lifetime I wouldn’t have to see anything like that again — nothing that bad, nothing that tragic.”

The number of hospitalizations had fallen in recent days, he said, suggesting that social distancing measures were working to flatten the steep curve of the virus’s spread, at least for now. The rates depend not only on the number of new arrivals but also on hospital admission standards.

“If we stop what we are doing, you will see that curve change,” Mr. Cuomo warned.

Then he pivoted to a more somber tone. “The bad news isn’t just bad,” he said. “The bad news is actually terrible.”

New Jersey also had a record number of deaths in the past day: Gov. Philip D. Murphy said that 275 people had died there, up from 232 on Tuesday. More people have died in New York and New Jersey — a total of 7,772 — than in the rest of the United States combined.

Mr. Cuomo said that the staggering death toll could continue to rise even as hospitalization rates were falling, because it reflected people who had been on ventilators for long periods of time.

He expressed reluctance to offer a timeline on when social gatherings could begin again, when he was asked about New York’s theater industry, which will remain shuttered until at least June. “I wouldn’t use what Broadway thinks as a barometer of anything,” he said.

New York State now has more confirmed cases than any single country in the world outside of the United States.

The virus might not fade in warm weather, scientists warn.

The homebound and virus-wary across the Northern Hemisphere, be it Mr. Trump or cooped-up schoolchildren, have clung to the possibility that the pandemic will fade in hot weather, as some viral diseases do.

But the National Academy of Sciences, in a public report sent to the White House, has said, in effect: Don’t get your hopes up. After reviewing a variety of research reports, the panel concluded that the number of studies, of varying quality of evidence, simply do not offer a clear forecast of what will happen to the spread of the coronavirus in the summer.

The report cited a small number of well-controlled laboratory studies that show that high temperature and humidity can diminish the ability of the virus to survive in the environment. But the report noted the studies had limitations that made them less than conclusive.

“Given that countries currently in ‘summer’ climates, such as Australia and Iran, are experiencing rapid virus spread, a decrease in cases with increases in humidity and temperature elsewhere should not be assumed,” the report stated.

Homes for disabled people experience a surge in coronavirus cases.

As the coronavirus preys on the most vulnerable, it is taking root in New York’s sprawling network of group homes for people with special needs. As of Monday, 1,100 developmentally disabled residents in New York State have tested positive for coronavirus, and 105 have died, state officials said, a death rate far higher than in the general population.

Separately, a study by a large consortium of private service providers found that residents of group homes and similar facilities in New York City and surrounding areas were 5.34 times more likely than the general population to develop Covid-19 and 4.86 times more likely to die from it.

A caregiver on Staten Island, who said about 50 colleagues had tested positive, described the challenges faced by those remaining on the job.

“One of the individuals here is positive, and his behavior is to get up, to pace, and he wants to give me a hug, shake my hand,” the caregiver said, asking that his name not be used because he was not authorized to speak.

“They have a hard time realizing that they need to be isolated, and the psychologists aren’t coming out and talking to him,” he added. “We don’t have training for this. We’re just learning on the fly.”

Alabama treatment guidelines are ruled discriminatory.

The growing number of coronavirus cases has raised interest in state guidelines outlining who should be prioritized for lifesaving medical treatments in an emergency. But certain standards that Alabama had on the books are discriminatory, the federal health department’s Office of Civil Rights said on Wednesday.

Alabama’s criteria, contained in a 2010 document that set out the state’s guidance for rationing ventilators in an emergency, suggested that doctors consider withholding advanced treatment based on patients’ intellectual disabilities, with “profound mental retardation” and “moderate to severe dementia” weighing against them. The guidelines also referred to age as a potential category for exclusion, which raised questions of age discrimination, according to the review.

The state has agreed to remove all links to the document on its website and not include the contested guidance in its plan for responding to the coronavirus outbreak, the civil rights office said.

The quick resolution to a complaint meant that the state would not be subject to a lengthy investigation that might result in a potential loss of federal funds. “It sends a message to other states that they need to review their crisis standards-of-care policies to make sure that they are fully compliant,” Roger Severino, the civil rights office’s director, said in a statement.

Dr. Scott Harris, Alabama’s state health officer, said in a statement: “All people deserve compassion and equal respect, and with this in mind, the allocation of care cannot discriminate based on race, color, national origin, disability, age, sex, exercise of conscience or religion.”

Days before Easter, Kansas legislators overturn an order limiting religious services.

Republican legislators in Kansas on Wednesday rescinded an order intended to slow the spread of the coronavirus by limiting the size of church services, even as the number of cases in that state continued to rise.

The senate president, Susan Wagle, a Republican, said most people were aware that the virus was highly contagious and wanted to limit its spread, “but don’t tell us we can’t practice our religious freedoms,” according to The Wichita Eagle.

The move, days before Easter, came after Gov. Laura Kelly, a Democrat, signed an executive order banning gatherings of 10 or more people at religious institutions.

She warned that Kansas was approaching its projected peak infection rate in the coming weeks and “the risk for a spike in Covid-19 cases through church gatherings is especially dangerous.”

Ms. Kelly said on Wednesday that Kansas had 1,046 cases of the coronavirus and 38 deaths. At a news conference, she denounced lawmakers for reversing the order, calling it a “shockingly irresponsible decision that will put every Kansan’s life at risk.”

White House’s $250 billion request is inadequate, Pelosi says.

Speaker Nancy Pelosi of California said on Wednesday the White House’s request for a quick, $250 billion infusion for loans to small businesses would not pass the House without additional funds for hospitals and states, and other changes demanded by Democrats.

“The bill that they put forth will not get unanimous support in the House — it just won’t,” she said in an interview on NPR, pointing to, among other Democratic requests, a proposal to reserve half of the loan program for businesses owned by farmers, women, people of color and veterans.

“Why wouldn’t you give them an avenue to participate?” she asked.

On Wednesday evening Mr. Trump called on Congress to approve more money for the loan program “this week, as soon as possible.”

“I think we have a pretty good understanding with the Democrats, hopefully it’s going to be bipartisan,” he said. “We do not have time for the partisan games, we don’t want that, the obstruction, or totally unrelated agendas.”

In a joint statement earlier, Ms. Pelosi and Senator Chuck Schumer of New York, the minority leader, said they wanted to add $100 billion for hospitals, community health centers and health systems — in part to shore up testing and the distribution of critical safety gear for health workers on the front lines — as well as $150 billion for state and local governments and a 15 percent increase in food assistance benefits.

It was unclear whether Republicans would agree to the additions, although some lawmakers warned against doing anything that could delay an infusion of cash that both parties agree is badly needed for small businesses.

A jail in Chicago is now the largest-known source of U.S. infections.

The Cook County Jail in Chicago, a sprawling facility that is among the largest jails in the nation, has emerged as the largest-known source of U.S. virus infections, according to data compiled by The New York Times.

At least 353 cases can be linked to the jail — more than have been connected to the U.S.S. Theodore Roosevelt; a nursing home in Kirkland, Wash.; or the cluster centered in New Rochelle, N.Y.

The Cook County Sheriff’s Office, which operates the jail, said 238 inmates and 115 staff members had tested positive as of Wednesday.

The outbreak appears to confirm the concerns of many health officials, who warned that America’s overcrowded and unsanitary jails and prisons could be a major source of spread. Those warnings prompted the authorities across the country to release thousands of inmates to try to slow the infection, save lives and preserve medical resources.

Still, hundreds of diagnoses have been confirmed at local, state and federal correctional facilities — almost certainly an undercount, given a lack of testing and rapid spread — leading to hunger strikes in immigrant detention centers and demands for more protection from prison employee unions.

In Cook County, officials released hundreds of inmates early — all of whom had been convicted of nonviolent crimes like drug possession and disorderly conduct. Judges are continuing to examine the cases of each inmate to determine if bonds can be lowered for certain people. That would allow dozens, perhaps hundreds, more people to be released, officials say.

In New York City, jails like Rikers Island are also seeing infection rates grow exponentially. City and state officials have promised the mass release of inmates. But many say they are not moving quickly enough, putting inmates, staff and the city at risk.

Seismographs reveal that stay-at-home orders have quieted the planet.

Seismometers may be built to detect earthquakes, but their mechanical ears hear much more. Even the everyday hum of humanity — people moving about on cars, trains and planes — has a seismically detectable heartbeat.

But as billions of people have been instructed to stay home to try to curtail the pandemic’s spread, the roar of urban life has turned into a whisper all over the world. Today, in cities large and small, the thumping pulse of civilization is now barely detectable on many seismograms.

“It did make the scale of the shutdowns a bit more real to me,” said Celeste Labedz, a graduate student in geophysics at the California Institute of Technology.

In person, you can see only your neighborhood’s dedication to remaining home. With seismometers, Ms. Labedz said, you can see the collective willingness of millions of the world’s urban dwellers to hunker down.

London is no longer buzzing. Paula Koelemeijer, a seismologist at Royal Holloway, University of London, said the seismometer in her suburban house was clocking a 20 to 25 percent reduction in average weekly noise, compared with the week before Britain began its lockdown.

Noise levels on some seismic stations in Los Angeles have dropped to below half of what they normally are, Ms. Labedz said.

Claudio Satriano, a seismologist at the Paris Institute of Earth Physics, detected a 38 percent drop in the average daytime noise in his city.

Scientists are now able to better hear the planet’s natural tectonic soundtrack. With the volume of humanity reduced, “we can detect smaller earthquakes, just like how it’s easier to hear a phone ring in a library than at a rock concert,” Ms. Labedz said.

California will spend $1.4 billion on personal protective equipment.

Gov. Gavin Newsom of California said on Wednesday that the state would spend $1.4 billion on personal protective equipment for medical personnel, supermarket workers, employees of the Department of Motor Vehicles and any other “front-line employees walking the streets.”

One day after announcing that the state had reached agreements to buy 200 million masks a month from factories in Asia, Mr. Newsom and other officials described a broad effort that includes the purchase of gowns, face masks and other equipment.

“We made a big, bold bet on a new strategy, and it is bearing fruit,” Mr. Newsom said.

As part of the plan, the state is contracting with a company to sterilize 80,000 N95 masks daily. The masks can be used about 20 times before being discarded.

Mr. Newsom also announced one of the state’s highest death tolls: 68 people died over the past 24 hours, bringing California’s total deaths to about 450.

Amid nationwide concern about the disproportionate number of nonwhite people who have died from the virus, Mr. Newsom said preliminary data did not show that trend in California.

He said Latinos made up 30 percent of the state’s cases and 29 percent of the deaths; Asians made up 14 percent of cases and 16 percent of deaths; and black residents made up 6 percent of cases and 3 percent of the deaths. But he offered the caveat that the state has ethnicity data for only about one-third of cases.

The S&P 500 is up 23 percent from its March low.

Wall Street resumed its rally on Wednesday. With a more than 3 percent gain, the S&P 500 is now up about 23 percent from its March 23 low.

The market has been steadily climbing since it hit that bottom, a rebound that began after the Federal Reserve and lawmakers in Washington took unprecedented steps to protect the world’s largest economy from a collapse amid the pandemic. Stocks are still down about 19 percent from their late February high.

To some extent, the recent gains reflect Wall Street’s fear of missing out on the rebound that many analysts predicted would eventually come.

“If you wait until the coast is clear you will have missed a huge part of the gains,” said Matt Maley, chief market strategist at Miller Tabak a trading and asset management firm. “And professional investors can’t afford to do that.”

For now, though, it is big money managers — not mom-and-pop retail investors — who are in on the action. Hedge fund traders and mutual fund managers have swooped into the market, driving sharp gains for blue-chip shares that have been battered by the market sell-off.

Still, the market’s recent optimism is set against a grim backdrop of economic and human catastrophe that continues to play out — and which threatens to undercut any rally at a moment’s notice.

A new report on weekly jobless claims on Thursday is certain to show millions more Americans are out of work. The two prior reports recorded more than 10 million claims for unemployment in late March.

Here are answers to common questions about the malaria drug Trump keeps pushing.

There is no proof that any drug can cure or prevent infection with the coronavirus. But in the face of an exploding pandemic with a frightening death toll, people are desperate for a bit of hope.

The drug that has received the most attention is hydroxychloroquine, which Mr. Trump has repeatedly recommended, despite warnings from his own health officials that there is little data to support its widespread use as a treatment against the virus.

With so many mixed messages, here are answers to common questions about the drug, including what it is, how it is being used, what studies show and what its potential side effects are.

Passover will be different this year.

For generations Jewish families have gathered for the first night of Passover to recount the 10 plagues from the Book of Exodus — frogs, pestilence, death — and to remember how God delivered the Jewish people from slavery in Egypt thousands of years ago.

Jews observed the Seder in the fifth century B.C. on the Egyptian island of Elephantine, and they observed it in 1943 as German troops liquidated the Warsaw ghetto. And on Wednesday in homes across the United States, families will once again light candles at the Seder table and ask why this night is different from all other nights.

Of course, with a literal plague in their midst, families cannot meet in person this year and may even tweak their Haggadahs — the text that is annually read aloud — to reflect the moment. But the power of Passover remains, perhaps even more so as a symbol of perseverance.

The Times asked families around the country to share reflections on the Passover story in this moment. Their words speak to the power of memory, the meaning of plague, and how crockpots and cookbooks can connect us with loved ones of generations past and future.

Food banks are being squeezed by rising hunger and dwindling resources.

Demand for food assistance in the United States is rising at an unprecedented rate, as millions of Americans find themselves out of work and school closures mean that many families who counted on them for free or subsidized meals need to turn elsewhere.

The surge in need is coming just as food banks face shortages of both donated food and volunteer workers.

It’s a nationwide phenomenon:

• At Food Bank for the Heartland in Omaha, the amount of food donated for March dropped by nearly half. The food bank typically purchases $73,000 of food in a month this time of year but has spent $675,000 in the past four weeks.

• In Washington State and Louisiana, the National Guard has been called in to help pack food boxes and ensure that the distributions run smoothly.

• Feeding America, the nation’s largest network of food banks, with more than 200 affiliates, has projected a $1.4 billion shortfall in the next six months alone.

“I’ve never seen anything like it,” said Stacy Dean, vice president for food assistance policy at the Center on Budget and Policy Priorities, a left-leaning research organization in Washington, D.C. She has studied food security for more than a quarter century. “People love the phrase ‘the perfect storm,’” she added, “but nothing is built for this.”

Union for food workers asks for ‘mandatory’ guidance to protect its members.

The nation’s largest union representing grocery store and pharmacy workers asked the Centers for Disease Control and Prevention on Wednesday to issue “immediate and mandatory guidance” to protect the workers, and others at food processing and meatpacking facilities.

In a letter to Dr. Robert Redfield, the C.D.C. director, the international president of the United Food & Commercial Workers International Union asked that the agency require stores to, among other things, limit the number of consumers who enter at any given time, allow employees to wash their hands at least once every 30 minutes and ensure that employees wear masks and gloves while on the job.

The union president, Anthony Perrone, also asked for the C.D.C.’s help improving safety conditions at food processing facilities — for example, by requiring them to provide protective gear to their workers.

“Given the direct health threat faced by these food workers,” Mr. Perrone said in the letter, “we believe it is absolutely critical that the C.D.C. take action before this growing crisis threatens our nation’s food supply.”

The food and commercial workers’ union represents over 900,000 members employed at grocery stores, pharmacies, and food processing and meatpacking plants. Several big supermarket chains have reported deaths of employees from Covid-19; they include workers at a Trader Joe’s in New York, a Giant in Maryland and a Walmart outside Chicago.

The C.D.C. did not immediately respond to a request for comment on the letter, and it was unclear if the agency could require such actions. Its guidance is generally not mandatory.

Some college students don’t want to pay in-person tuition for online classes.

Students at the University of Chicago are organizing a tuition strike, threatening to withhold their payments for the spring quarter if the school doesn’t give them a hefty discount.

That cry is being heard on other campuses as well, as students complain that online classes don’t measure up to the real thing and say they shouldn’t have to pay the full load for a subpar experience, especially at a time when more are facing financial uncertainties.

While a number of colleges are offering refunds of room and board charges, students in a number of schools are asking them to lower tuition as well.

At the New School in New York City, students have called for a boycott of online classes this week if the school didn’t refund part of their spring tuition. Students at Stanford, the University of California at Berkeley and New York University’s Tisch School of the Arts have all started online petitions calling for partial refunds.

In Chicago, an ad hoc group of undergraduate and graduate students is calling for the institution to cut tuition by half and eliminate fees for as long as the pandemic continues. The group has collected more than 1,400 signatures on a petition.

“Students are extremely vulnerable,” Julia Attie, a senior who is one of the organizers, said in an email. “Many are experiencing severe financial and housing insecurity.”

Undergraduate tuition for the spring term, which began this week, is more than $19,000 and is due on April 29. The total annual cost of attendance at the University of Chicago, including tuition, housing and other costs, is more than $80,000, one of the highest in the country. The university guarantees free tuition for families with incomes under $125,000.

The university said that it was already being forced to take austerity measures because of the crisis.

In an email to faculty and staff on Tuesday, Robert Zimmer, the university’s president, said it was pivoting from focusing on health issues raised by the virus to looking at financial issues. Mr. Zimmer said that the university expected a weakened endowment, reduced alumni donations and a higher demand for financial aid, all of which would take a toll.

Mr. Zimmer predicted that the financial effect of the pandemic was “likely to be as great as or even greater than in the financial crisis of 2008-09.”

W.H.O. leader warns that politicizing the pandemic is ‘playing with fire.’

Dr. Tedros Adhanom Ghebreyesus, the director general of the World Health Organization, made an impassioned plea on Wednesday for global solidarity, warning that politicizing the pandemic was “playing with fire” and that disunity and finger-pointing would result in “many more body bags.”

Dr. Tedros’s comments came a day after Mr. Trump falsely claimed that the W.H.O. had “missed the call” on the rising threat in China and threatened to withhold American funding for the organization, which exceeds $400 million annually. Secretary of State Mike Pompeo doubled down on that threat, saying at a news briefing on Wednesday that the United States was re-evaluating its funding to the organization.

“When there are cracks at the national level and global level, that’s when the virus succeeds,” Dr. Tedros said, though he did not cite Mr. Trump by name. “Please quarantine politicizing Covid. That’s the way if we want to win.”

He added: “We shouldn’t waste time pointing fingers. We need to unite.”

While some critics have called on Dr. Tedros to resign, he said he was not deterred.

“We will do everything we can to serve humanity,” Dr. Tedros said. “We’re not angels. We are human beings. So we make mistakes, like other human beings.”

The virus has found a foothold in rural America.

The virus has officially reached more than two-thirds of the country’s rural counties, with one in 10 reporting at least one death. Doctors and elected officials are warning that a late-arriving wave of illness could overwhelm rural communities that are older, poorer and sicker than much of the country, and already dangerously short on medical help.

“Everybody never really thought it would get to us,” said Grace Rhodes, 18, who is from Southern Illinois and is studying to become a nurse. “A lot of people are in denial.”

But many rural doctors, leaders and health experts worry that they will have fewer hospital beds, ventilators and nurses to handle any onslaught.

Virus illnesses and deaths are still overwhelmingly concentrated in cities and suburbs, and new rural cases have not exploded at the same rate as in some cities. But they are growing fast. This week, the case rate in rural areas was more than double what it was six days earlier.

Under fire for trades during the pandemic, senator says she will divest from individual stocks.

Seeking to move past allegations that she has tried to profit from the crisis, Senator Kelly Loeffler, Republican of Georgia, announced on Wednesday that she and her husband would divest from all individual stocks and move the money into mutual and exchange-traded funds.

Ms. Loeffler, a new senator, has faced weeks of attacks from her political rivals in both parties and scrutiny from the news media about stock trades worth millions of dollars made earlier this year in her name, before the pandemic roiled the financial markets. Ms. Loeffler’s critics questioned whether she and a handful of other lawmakers had used nonpublic information they had received from their jobs.

Ms. Loeffler continued on Wednesday to adamantly deny that, insisting that she had done nothing wrong, legally or ethically. The stock trades were all made by outside financial advisers who independently manage her investments, she said.

“I left the private sector to serve the people of Georgia, not make a profit,” Ms. Loeffler said.

Federal and local officials are concerned about cases in the D.C. region.

As cases keep rising in and around the nation’s capital, stories of residents not complying with social distancing guidelines have been prevalent. On ABC’s “Good Morning America” on Wednesday, Dr. Deborah Birx, the coordinator of the White House Task Force, said federal officials were “concerned about the metro area of Washington and Baltimore.”

As of Tuesday, there were 1,440 cases in Washington, and 27 deaths. The district’s latest data shows that nearly 60 percent of the dead were African-American people, though they make up about 46 percent of its population.

Mayor Muriel E. Bowser said that she was worried about the disproportionate impact the virus is having on black people — a concern that has also emerged in other places across the country.

The district’s stay-at-home order went into effect on April 1, nearly a month after its first case was confirmed, on March 7. Like other orders, it makes exceptions for grocery shopping, medical appointments, and “allowable recreational activities,” like walking and riding bikes.

Representative Eleanor Holmes Norton, the nonvoting delegate for the district, wrote to the acting director of the National Park Service on Tuesday requesting the closure of the Lincoln and Jefferson Memorials amid reports that they had been attracting crowds, making it hard to maintain social distancing.

“Closure would protect the public and NPS employees, including U.S. Park Police officers,” Ms. Norton’s letter said. “Federal agencies need to lead by example and do everything possible to flatten the curve.”

General Motors will send 30,000 ventilators to the federal stockpile.

After weeks of drama that included Mr. Trump’s unproven accusation that General Motors was trying to “rip off” the United States, the Department of Health and Human Services announced on Wednesday that the carmaker would provide 30,000 ventilators to the nation’s stockpile for $489 million by the end of August.

The first batch — 6,132 of the machines — will be delivered by June 1, after most of the peak demand is expected to have passed from the first wave of cases at hospitals. But even that initial number amounts to roughly two-thirds of what is now believed to be left in the stockpile after thousands of ventilators were sent to New York and other hard-hit cities.

In an early-morning statement, the secretary of health and human services, Alex M. Azar II, said the contract would be among the first during the crisis issued under the Defense Production Act, a Korean War-era law that essentially allows the United States to assure that it is the first customer in line — and that it can control the price it is being charged.

The formal contract comes two weeks after the White House pulled back from announcing what was intended to be a $1 billion contract for upward of 80,000 ventilators. Mr. Trump had accused the company of “wasting time,” and he also attacked Mary T. Barra, the company’s chief executive, with whom he had clashed last year over the closure of a G.M. facility.

But Mr. Trump was essentially ordering the company to do what it had already announced it was doing, even in the absence of a contract.

Here’s what you need to know about relief for small business.

The federal stimulus bills enacted in March, including a $2 trillion economic relief plan, offer help for the millions of American small businesses affected by the pandemic.

Cash grants. Low-interest loans. Payments to offset some payroll costs for businesses that keep or rehire workers. There are also enhancements to unemployment insurance and paid leave.

Here are the answers to common questions about these programs. And we will update this article as we learn more about the details.

More information on help, including details on the stimulus checks that many people will be receiving, can be found in our F.A.Q. for individuals about stimulus relief and our Hub for Help. If you have questions, or have applied for small business aid and can tell us how the process went, we’d love to hear from you.

How delays and unheeded warnings hindered New York’s fight against the virus.

New York City and the surrounding suburbs have become the epicenter of the pandemic in the United States, with far more cases than many countries have. More than 138,000 people in the state have tested positive for the virus, with nearly all of them in the city and nearby suburbs.

On Tuesday, Gov. Andrew M. Cuomo said that 731 more people had died of the virus, the state’s highest one-day total. The overall death toll in the state is 5,489.

Epidemiologists have pointed to New York City’s density and its role as an international hub of commerce and tourism to explain why the coronavirus has spread so rapidly there. And it seems unlikely that any response by the state or city could have fully stopped the pandemic.

State and city officials were hampered by a chaotic federal response from the onset, including significant problems with the expansion of coronavirus testing, which made it far harder to gauge the scope of the outbreak.

Normally, New York would get help from Washington, as it did after the Sept. 11 attacks. But President Trump in February and early March minimized the coronavirus threat and declined to marshal the might of the federal government when cases emerged in the United States.

The initial efforts by New York officials to stem the outbreak were also hampered by their own confused guidance, unheeded warnings and political infighting, a New York Times investigation found.

Desperation outweighs health concerns as people crowd Florida unemployment lines.

Near Miami, aerial videos showed hundreds of people packed closely together in lines to pick up unemployment benefit applications on Tuesday, a sign of the desperation among newly jobless Floridians and of the state’s embattled unemployment system.

The lines, in Hialeah, came as the state unemployment website was overwhelmed with submissions, leading people to seek paper forms amid a record number of unemployment claims in Florida and across the country. Nearly 10 million Americans have applied for unemployment benefits in the past two weeks alone.

The aerial videos recorded by WPLG, a local television station, show people clamoring for unemployment forms near a local library. Some applicants wore masks and gloves, but others went without protective gear.

As tensions rose among those in line, some shouted and others pushed in front of one another to get to a police officer who was handing out forms.

“We don’t want to turn this into a riot, into a shoving match,” Carl Zogby, a member of the Hialeah City Council, told WPLG.

Gov. Ron DeSantis, a Republican, for weeks resisted stringent social distancing measures, but ultimately ordered most of the state’s more than 21 million residents to stay at home beginning last week. He has faced mounting criticism over the difficulty Floridians have had filing for unemployment benefits.

Miami-Dade County officials said paper unemployment benefit applications would be available at 26 libraries beginning on Wednesday. In a statement, the county said the library system would be encouraging people to stand six feet apart with “informational signage and markings on the ground.”

Reporting was contributed by Reed Abelson, Peter Baker, Alan Blinder, Nicholas Bogel-Burroughs, Jonah Engel Bromwich, Audra D. S. Burch, Weiyi Cai, Emily Cochrane, Michael Cooper, Stacy Cowley, Elizabeth Dias, Caitlin Dickerson, Conor Dougherty, Julia Echikson, John Eligon, Nicholas Fandos, Lisa Friedman, Thomas Fuller, Robert Gebeloff, J. David Goodman, Abby Goodnough, William Grimes, Danny Hakim, Anemona Hartocollis, Adeel Hassan, Jack Healy, Danielle Ivory, Zolan Kanno-Youngs, Nicholas Kulish, Michael Levenson, Dan Levin, Patricia Mazzei, Colin Moynihan, Andy Newman, Jack Nicas, Richard A. Oppel Jr., David E. Sanger, Marc Santora, Charlie Savage, Dionne Searcey, Matt Stevens, Eileen Sullivan, Vanessa Swales, Sabrina Tavernise, Timothy Williams and Carl Zimmer.