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The C.D.C. expands the list of symptoms, and the W.H.O. warns of a long road ahead.

Even as governors across the United States proceeded cautiously with plans to allow businesses to reopen and other nations looked for ways to restart stalled economies, the workings of the virus continued to vex the scientific and medical community.

For weeks, most people in the United States have been told that they qualify for a test only if they have three symptoms associated with the disease: high fever, cough and shortness of breath.

As health experts have gained more experience with Covid-19, the disease caused by the virus, they are finding that many infected people have no fevers or that their fevers wax and wane over a period of weeks and are sometimes accompanied by chills.

The Centers for Disease Control has now expanded the list of symptoms to include repeated shaking with chills, muscle pain, headache, sore throat and a loss of taste or smell.

That differs from the guidance of the World Health Organization, which says that the most common symptoms are fever, dry cough and tiredness.

“Some patients may have aches and pains, nasal congestion, sore throat or diarrhea,” the W.H.O. says. “These symptoms are usually mild and begin gradually.”

The Oxford scientists say that with an emergency approval from regulators, the first few million doses of their vaccine could be available by September — at least several months ahead of any of the other announced efforts — if it proves to be effective.

Dr. Tedros Adhanom Ghebreyesus, the director of the W.H.O., told a virtual news conference in Geneva on Monday that the pandemic was far from over.

“We have a long road ahead of us and a lot of work to do,” he said.

In the seven weeks since the president promised that anyone who needed a test could get one, the United States has conducted about 5.4 million tests, far more than any other country, but still the equivalent of only about 1.6 percent of the total population.

A group of experts convened by Harvard University’s Edmond J. Safra Center for Ethics has called for five million tests a day by early June, ramping up to 20 million per day by late July.

Under growing pressure to expand coronavirus testing as states move to reopen their economies, President Trump unveiled a plan on Monday to ramp up the federal government’s help to states, but his proposal ran far short of what most public health experts say is necessary.

Mr. Trump’s announcement came after weeks of his insisting, inaccurately, that the nation’s testing capability was “fully sufficient to begin opening up the country,” as he put it on April 18. Numerous public health experts say that is untrue, and Mr. Trump’s plan may do little to fix it.

An administration official said the federal government aimed to give states the ability to test at least 2 percent of their populations per month, though the president did not use that figure and it was not in his written plan. Instead, Mr. Trump and other officials with him in the Rose Garden said the United States would “double” the number of tests it had been conducting.

“These were not complaining people. They had everything they needed. They had their ventilators, they had their testing,” Mr. Trump said on Monday after a call with governors. “We’re getting them what they need.”

In fact, governors have been complaining that they do not have nearly enough tests to give them the kind of information they need to make difficult decisions about reopening. They say they are competing with one another — and with other countries — for the components that make up the testing kits, including nasal swabs and needed chemicals.

Rather than one coordinated federal response, the Trump administration has been engaging on an ad hoc basis as states take the lead.

In some states, restaurants would remain closed. In others, they would be allowed to open with stringent new seating requirements. From the construction industry to entertainment, the rules of engagement varied depending on the state or city.

Gov. Greg Abbott of Texas announced that stores, restaurants, movie theaters and malls would be allowed to reopen with limited capacity on Friday. In Ohio, Gov. Mike DeWine unveiled a more incremental plan that would allow manufacturing work to resume and offices to reopen next week.

Arizona and Florida have stay-at-home orders that are set to expire on Thursday, but the governors of both states have been vague about their plans.

Organized like a PowerPoint presentation with roughly 25 slides, the tutorial describes what the virus is; how it is treated; how to avoid getting it; and the conditions laid down by the state — wearing masks, not congregating — for a safe work environment.

Each employee is supposed to fill out a short form attesting to having read the material. Vermont will not actually police compliance, although Gov. Phil Scott said that every company should appoint a safety officer to monitor work conditions.

“This is evolving, it’s not perfect, it’s not a flip of the switch, it is just something we have to provide, guidance, and there is going to be some social pressure as well to adhere to these procedures,” Mr. Scott told reporters on Monday. “I think guidance and education is the best policy.”

The state allowed people in solitary professions to return to work last week, including appraisers, property managers and attorneys who work alone.

It added manufacturing and construction businesses but limited the number of workers at any one location to five. Outdoor retailers like gardening centers and greenhouses can also reopen, but with a maximum of 10 people.

Even as governors moved to ease restrictions, business owners were often left trying to make sense of a cacophony of messages from President Trump, governors, county commissioners and mayors.

“I couldn’t sleep last night because I was so confused,” Jose Oregel said on Monday morning before reopening his barbershop in Greeley, Colo.

In a call with the nation’s governors, Mr. Trump’s growing impatience with the restrictions was evident.

He suggested that some governors should move to reopen their public schools before the end of the academic year.

“Some of you might start to think about school openings,” Mr. Trump said, according to an audio recording obtained by The New York Times. “The young children have done very well in this disaster that we’ve all gone through, so a lot of people are thinking about the school openings.”

At least one state was already moving forward with the possibility of reopening schools. Montana, which has among the fewest cases and deaths, will give schools the option to reopen starting May 7.

The officials say new Treasury Department rules that prohibit local governments from using their share of $150 billion provided last month for “revenue replacement” are impractical. Requiring that such assistance be confined to costs directly tied to the pandemic will be of limited help, they say, particularly in communities that have a low incidence of cases but have seen their revenue dry up because of the shutdown of the economy.

“It is clear the revenue loss is going to be coronavirus related; it is just that the expenditures are not specifically for coronavirus,” said Senator Sherrod Brown of Ohio, one of 46 senators in the Democratic caucus who signed a letter made public on Sunday urging the administration to revise the ban.

The Federal Reserve said on Monday that it would substantially expand its municipal lending program, allowing smaller cities and counties to sell their investment-grade debt to the central bank.

But some scientists express fear that Mr. Trump’s remarks could breathe life into a fringe movement that embraces the medicinal powers of a powerful industrial bleach known as chlorine dioxide, which the president did not mention specifically.

“For a lot of people, Trump represents an alternative to pointy-headed experts in white lab coats who speak a language we can’t understand,” said Dr. Alan Levinovitz, a professor at James Madison University who studies the relationship between science and religion. “When you feel existentially threatened by a deadly virus, and the president says you can take control of your health with a product in your kitchen cabinet, that’s incredibly empowering.”

The problem, of course, is that ingesting or injecting industrial bleach can be deadly. Chlorine dioxide destroys red blood cells, wreaks havoc on the digestive system and can cause severe damage to the liver and kidneys.

Minutes after a $310 billion aid program for small companies opened for business on Monday, the online portal for submitting applications crashed. And it kept crashing all day, much to the frustration of bankers around the country who were trying — and failing — to apply on behalf of desperate clients.

Some bankers were so irritated that they vented on social media against the Small Business Administration, which is running the program. Rob Nichols, the chief executive of the American Bankers Association, wrote on Twitter that the trade group’s members were “deeply frustrated” at their inability to access the system. Until the problems were fixed, he said, “#AmericasBanks will not be able to help more struggling small businesses.”

But for the second time in a month, the relief effort, called the Paycheck Protection Program, turned into chaos, sowing confusion among lenders and borrowers. A centerpiece of the government’s $2 trillion economic stimulus package, the program offers small companies — typically those with up to 500 workers — forgivable loans of up to $10 million. The S.B.A. is backing the loans, but customers must apply through financial institutions.

When the aid program first went live on April 3, the Treasury Department’s goal was to quickly steer money to the neediest businesses — hair salons, coffee shops, dry cleaners and others. But many large banks needed more time to set up their systems and held off for days on taking applications, leading to an outcry from borrowers who could not afford to keep waiting. Many were also furious that hundreds of publicly traded companies, as well as wealthy clients of some big banks, got access to those funds.

With tens of thousands of schools in dozens of states shuttered through the remainder of the academic year, an estimated 55 million students will be home for double the length of their normal summer vacations, if not longer.

“The stay-at-home orders and the social distancing are the right thing to do” to slow the spread of the virus, said Andrew G. Rundle, the lead author of a report in the journal Obesity on school closings and childhood weight gain. “But this six-month period or longer is doubling out-of-school time, and it’s magnifying or exacerbating all of the risk factors that we think about for summer weight gain.”

While focusing on the immediate effects of the pandemic is a priority, Dr. Rundle and his co-authors write, they also point out that when the illness relents, one of the lingering effects could be a worsening of the obesity crisis among children. Childhood obesity rates have been on the rise in America for the past four decades, with more than a third of all youth under the age of 19 classified as overweight or obese. Studies show that overweight children are much more likely to become overweight adults, and that puts them at a higher risk of developing heart disease, cancer and Type 2 diabetes.

New York and New Jersey, the two states hit hardest by the virus, have shown enough progress that their governors started on Monday to offer details on how reopening might go in the months ahead.

The optimistic signs came as both states announced one-day death tolls — 337 in New York, and 106 in New Jersey — that were less than half of their peaks.

Gov. Andrew M. Cuomo of New York suggested that some businesses in what he called low-risk industries like construction or manufacturing might restart in parts of the state after May 15. Much of the state, including New York City and its suburbs, would remain shut down longer, he said.

”I don’t think you’ll see us taking in each case identical steps, but I think you’ll see our steps harmonized,” Mr. Murphy said.

Mr. Murphy said schools in New Jersey might be able to reopen before the end of June. And Mayor Bill de Blasio of New York said he thought the city’s beaches could be partially open by the end of the summer.

Even as the virus’s spread appeared to be slowing, Mr. Cuomo cautioned that the economic effects of the outbreak had probably not reached their peak. With over a million New Yorkers out of work, food banks across the state are experiencing huge jumps in demand, he said.

Abortion through telemedicine is a quietly growing phenomenon, driven in part by restrictions from conservative states and the Trump administration that have limited access to abortion clinics.

Now, the coronavirus pandemic is catapulting demand for telemedicine abortion to a new level. With much of the nation under strict stay-at-home advisories, several states, including Arkansas, Oklahoma and Texas, have sought to suspend access to surgical abortions during the crisis.

A telemedicine program has been allowed to operate as a research study for several years under a special arrangement with the Food and Drug Administration. It allows women seeking abortions to have video consultations with certified doctors and then receive abortion pills by mail.

Over the past year, the program, called TelAbortion, has expanded to serve 13 states. Not including two recently added states, about twice as many women had abortions through the program in March and April as in January and February.

As of April 22, TelAbortion had mailed 841 packages containing abortion pills and confirmed 611 completed abortions, according to Dr. Elizabeth Raymond, senior medical associate at Gynuity Health Projects, which runs the program.

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Reporting was contributed by Pam Belluck, Michael Gold, Jack Healy, Shawn Hubler, Andrew Jacobs, Marc Santora, and Neil MacFarquhar.