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Reports of new cases have fallen significantly around the country since July; they are now flat in 26 states and falling in 15 others. But in nine states, cases are still growing, and in some, setting records — especially in the Midwest.

Iowa, Kansas, Minnesota, North Dakota and South Dakota all added more cases in a recent seven-day stretch than in any previous week of the pandemic. Together, they reported 19,133 new cases in the week ending Sunday, according to a New York Times database — 6.4 percent of the national total, though the five states are home to only 4 percent of the population. In each, some of the biggest surges in new case numbers have come in college towns.

The Dakotas, which had made it through the summer without suffering the big increases seen in some other parts of the country, have both recently set single-day case records. On Saturday, South Dakota added 425 new cases and North Dakota added 374, their worst days yet. Grand Forks, home of the University of North Dakota, has one of the highest per capita growth rates in the country.

Iowa’s recent outbreak, so serious that it prompted Gov. Kim Reynolds to shut down bars and nightclubs in six counties, has been most pronounced in college towns like Ames and Iowa City, which reported greater numbers of new cases per capita over the past two weeks than any other metro area in the country.

The situation is similar in Kansas: Lawrence, the home of the University of Kansas, has seen one of the steepest recent surges. Local officials in Lawrence-Douglas County announced last week that nine fraternity and sorority houses at the university were under quarantine, and adjusted those orders as test results came back.

The rise in cases in Minnesota is not as severe as in neighboring Iowa or the Dakotas, but a growing number of counties in the state have more than 10 percent of their tests coming back positive. In a visit to St. Paul on Sunday, Dr. Deborah L. Birx, the White House coronavirus response coordinator, called that a “worrisome” trend and warned that the state may have to increase precautions against the virus as the fall and winter approached, particularly in rural communities.

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Where cases are highest per capita

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France is also surging, as are parts of Eastern Europe, and cases are ticking up in Germany, Greece, Italy and Belgium, too, but in the past week, Spain has recorded the most new cases on the continent by far — more than 53,000. With 114 new infections per 100,000 people in that time, the virus is spreading faster in Spain than in the United States, more than twice as fast as in France, about eight times the rate in Italy and Britain, and 10 times the pace in Germany.

Spain was already one of the hardest-hit countries in Europe, and now has about 440,000 cases and more than 29,000 deaths. But after one of the world’s most stringent lockdowns, which did check the virus’s spread, it then enjoyed one of the most rapid reopenings. The return of nightlife and group activities — far faster than most of its European neighbors — has contributed to the epidemic’s resurgence.

Now, as other Europeans mull how to restart their economies while still protecting human life, the Spanish have become an early bellwether for how a second wave might happen, how hard it might hit and how it could be contained.

“Perhaps Spain is the canary in the coal mine,” said Prof. Antoni Trilla, an epidemiologist at the Barcelona Institute for Global Health, a research group. “Many countries may follow us — but hopefully not at the same speed or with the same number of cases that we are facing.”

The median age of sufferers has dropped to around 37 from 60. Asymptomatic cases account for more than 50 percent of positive results, which is partly because of a fourfold rise in testing. And the health institutions feel much better prepared.

Epidemiologists aren’t certain why it arrived so soon.

Explanations include a rise in large family gatherings; the return of tourism in cities like Málaga; the decision to return responsibility for combating the virus to local authorities at the end of the nationwide lockdown; and a lack of adequate housing and health care for migrants. The surge has also been blamed on the revival of nightlife, which was reinstated earlier and with looser restrictions than in many other parts of Europe.

Greece, which has experienced a spike itself in cases, issued on Friday a new directive temporarily suspending all passenger flights to the Catalonia region in Spain.

Credit…Alexandra Hootnick for The New York Times

Despite being at high risk for developing Covid-19, a large number of doctors, nurses and other health care workers may be going undiagnosed after they become infected, according to a new report released on Monday by researchers from the Centers for Disease Control and Prevention. The findings raise concern that these health care workers could unknowingly spread the infection within the hospital.

The report also found that workers who said they always wore a face covering, like a surgical mask or N95 respirator, when caring for patients had significantly lower rates of infection. However, many of these frontline workers reported shortages of personal protective equipment, and those workers also had higher rates of infection, according to the report.

Health care workers, particularly nurses, have raised concerns for months about not being protected adequately. The findings underscore the importance of regularly testing hospital employees for the virus and ensuring they are protected when they come into contact with patients. “Universal use of face coverings and lowering clinical thresholds for testing could be important strategies for reducing hospital transmission,” the researchers said.

The researchers looked at the results of antibody tests for 3,248 workers. Serum specimens were collected from early April to mid-June from frontline workers at 13 medical centers across the United States, including in California, Ohio, Maryland, New York and Tennessee.

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About 6 percent of the workers had antibody evidence of a previous coronavirus infection, the report said, and more than two-thirds of these individuals had not been previously diagnosed. Almost 30 percent were asymptomatic.

The findings suggest that some infections “are undetected and unrecognized,” possibly because some workers are asymptomatic or those with symptoms are not reporting them or being tested, the researchers said.

Workers who reported that they did not always wear a face covering had a 50 percent greater infection rate, according to the analysis. Some 6 percent of those who were masked had antibodies, compared with 9 percent of those who were not.

GLOBAL Roundup

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India and South Korea — two countries that have taken vastly different approaches to fighting the virus — are among the latest to release grim data about the pandemic’s economic impact.

The Indian economy shrank by nearly 24 percent in the second quarter compared with the same period a year earlier, the government reported on Monday. Even that figure — from a quarter in which Prime Minister Narendra Modi initiated one of the most severe lockdowns anywhere — may underestimate the damage, since India’s informal economy, involving millions of rickshaw drivers, day laborers and others, mostly lies beyond the reach of government statisticians.

Either way, it is the domestic economy’s most drastic fall in decades, and a much bigger contraction than in any other major country — including the United States, whose economy shrank by 9.5 percent over the same period.

India also has the world’s third-highest coronavirus death toll at 65,288, according to a New York Times database, behind only the United States and Brazil. And even as parts of its economy are reopening, new cases have exceeded 75,000 for each of the past five days.

Unlike India, the authorities in South Korea have mostly tried to fight the pandemic while keeping as much of the economy running as possible. But South Korea’s economy shrank by 2.7 percent year-on-year in the second quarter anyway, and its contraction of 3.2 percent from the previous quarter was the sharpest since the last three months of 2008, the country’s central bank said on Tuesday.

The biggest drag on South Korea’s economy was its falling exports, which plunged by 16.1 percent in the second quarter compared with the first as companies sold fewer cars, smartphones and other products abroad.

There was at least one bright spot: Domestic consumption was up 1.5 percent from the previous quarter, thanks to billions of dollars’ worth of free cash and coupons that the government has doled out to families.

But few analysts believe that South Korea can sustain that level of consumption amid the social distancing restrictions that were imposed to fight a recent surge in infections, including a requirement that restaurants close earlier than usual.

In other news from around the world:

  • Officials in Manila, the capital of the Philippines, allowed some businesses in the city, including gyms and barber shops, to partially reopen on Tuesday. The country has Southeast Asia’s largest confirmed outbreak in Southeast Asia, with more than 220,000 cases since the pandemic began and nearly 27,000 in the past week alone.

  • With the virus spreading quickly in Gaza, Israel and the Palestinian militant group Hamas agreed Monday night to ease up on bombarding each other. Israel agreed to let fuel flow back to Gaza’s power station, and a cash infusion from Qatar helped seal the deal. The virus has accelerated its spread in Gaza since last week, when Hamas officials reported the first cases of community transmission. As of Monday, there were 243 active cases of local spread and 37 among returning travelers held at quarantine facilities, according to the Hamas-run health ministry. Officials have reported three virus-related deaths in the past week and say tests are in short supply.

  • Virus protests in Germany last weekend drew some 38,000 people of varied beliefs, including anticapitalists, esoterics and ordinary citizens angry at having to wear face masks — as well as 3,000 members of the far-right scene, hundreds of whom stormed the Parliament building waving the black, white and red flag of the pre-1918 German Empire that once inspired the Nazis. Germany has handled the virus crisis well, and polls suggest few Germans support the protests. But some officials worry that the far-right faction that opposes virus rules is trying to exploit the pandemic for political gain.

  • Prime Minister Justin Trudeau of Canada said on Monday that his government had arranged to buy 76 million doses of a vaccine now under development by Novavax, a biotechnology company based in Gaithersburg, Md., and 38 million doses of a different proposed vaccine from Johnson & Johnson. The Novavax vaccine is currently undergoing trials in the United States, Australia and South Africa. Canada, a country of 37 million people, announced earlier in the month that it would also buy 20 million doses of a Pfizer vaccine and 56 million doses of a Moderna vaccine.

  • The international airport in Ghana will reopen on Tuesday after being shut down since March, President Nana Akufo-Addo said in a nationwide broadcast late Sunday. Passengers must provide proof of a negative PCR test — the most widely used diagnostic test for the new coronavirus — within 72 hours of departure, in addition to paying for a second test upon arrival. Land and sea borders remain closed in Ghana, which has had more than 44,000 cases and 276 deaths, according to a Times database.

U.S. Roundup

Credit…Anna Moneymaker for The New York Times

President Trump heralded Stage 3 clinical trials beginning for a vaccine candidate under development by AstraZeneca at a White House briefing on Monday.

White House officials had previously singled out AstraZeneca’s potential vaccine in private discussions with congressional officials as a possibility for an emergency use authorization by the Food and Drug Administration in the fall. That is a time frame ahead of the Nov. 3. election.

Mr. Trump has made the search for therapeutics and a vaccine central to the way in which he talks about the pandemic. His administration has been widely criticized for its slow and disjointed response to the public health emergency, and the public response has taken a toll on his standing in the polls.

Mr. Trump’s remarks came just days after he retweeted a conspiracy theory that distorts data to suggest that only 9,000 people in the United States have died of the virus. The tweet rejected the data of his own administration — and attacked the very people he has put in charge of trying to stop the pandemic, among them Dr. Anthony S. Fauci and Dr. Deborah L. Birx.

“So get this straight — based on the recommendation of doctors Fauci and Birx the US shut down the entire economy based on 9,000 American deaths to the China coronavirus,” said the summary of an article by the hard-line conservative website Gateway Pundit that was retweeted by the president.

Twitter deleted one of the tweets that Mr. Trump reposted advancing this claim, replacing it with a message: “This Tweet is no longer available because it violated the Twitter Rules.”

The actual death toll is at least 183,400, and independent analyses have put the figure even higher — most likely around 200,000 — in part because some deaths have not been recognized as connected to infection from the virus.

George Rutherford, an epidemiologist at the University of California at Berkeley, said the message in the tweet was “clearly a huge disconnect” from the scientific reality.

“If anything,” Dr. Rutherford said, “the number of deaths has been underestimated.”

  • Hurricane Laura interrupted virus testing across Louisiana. Gov. John Bel Edwards has said that because it will take time to ramp the testing back up, officials must depend on other markers, like hospitalizations, to track the virus. There are already signs of trouble, he said, including increases in hospitalizations on Friday and Saturday. As of Monday evening, the state had reported 4,950 deaths and nearly 150,000 cases.

  • Data compiled by the American Academy of Pediatrics from the summer show that cases, hospitalizations and deaths from the virus have increased at a faster rate in children and teenagers than among the general public. The data, collected from May 21 to Aug. 20, varies from state to state, possibly obscuring differences in how the virus affects infants, young children and adolescents

  • In 35 states, voters can request ballots so close to Election Day that it may not be feasible for them to be received and sent back in time. Deadlines and other rules may change, but this interactive has the current breakdown of how much time voters will have in each state.

  • The Agriculture Department, under pressure from Congress and officials in school districts across the country, said on Monday that it would allow schools to provide free breakfast and lunch to any child or teenager through the end of 2020, provided funding lasts. Advocates for the poor hailed the announcement as an important step to ensure that more needy children are fed during the pandemic but said it still fell short.

  • The United States Open began in New York on Monday. After contact tracing, some players were put on tighter restrictions. These athletes did not test positive for the virus, but they had close contact with an infected athlete who has been withdrawn from the tournament.

  • Hundreds gathered in Massachusetts on Sunday to protest the state’s new requirement that nearly all students in the state receive a flu shot by the end of the year. The measure is believed to be the first such mandate in the nation for influenza vaccination and is intended to help prevent a combined wave of virus and flu infections that could overwhelm the health care system. But protesters complained that the mandate infringed on parents’ rights to make medical decisions for their children.

Hong Kong’s Beijing-backed coronavirus testing program began on Tuesday amid concerns about safety, privacy and the influence of the mainland Chinese government.

The program is open to everyone, and the local government has touted it as generous, vital aid from the central Chinese government. More than half a million of the city’s 7.5 million residents have already registered for it.

But some members of Hong Kong’s medical community have criticized the one-off voluntary tests as a waste of resources, saying they could create a false sense of security.

Another concern is that samples could be used for Beijing’s sprawling surveillance — a claim the government has denied.

Still others say it is preposterous that the local government is allowing citywide testing after using the virus to justify postponing citywide elections that had been scheduled for Sept. 6.

On Tuesday, Hong Kong’s leader, Carrie Lam, said that the government’s critics were trying to “cause worries and fears” about the program to scare people away.

“They don’t understand the details of the program, the procedures, the safeguards that we have put in place,” Mrs. Lam told reporters.

Ken Li, a tennis instructor who took the test Tuesday, welcomed the government’s plan.

“Getting tested is no doubt a better option,” Mr. Li, 50, said outside one of the city’s 141 swabbing stations. “Then people can isolate themselves if they’re infected.”

Also on Tuesday, a Hong Kong employee of Founder Securities, a mainland Chinese company, said that it had pressured its staff in the city to take the tests and required them to present their results, according to the Hong Kong Financial Industry Employees General Union.

The union said on Facebook that it was deeply concerned about the company’s order, which it said had been issued to all employees in Hong Kong. Founder Securities did not immediately respond to a request for comment.

Hong Kong’s third and most severe wave of infections, which peaked in July, appears to have gradually eased, with only nine new confirmed cases on Monday. The city’s schools are to resume in-person instruction on Sept. 23.

Credit…Michelle Mishina-Kunz for The New York Times

Hawaii, trying to head off a surge of the virus that hit in mid-August, will require visitors and residents arriving on the islands to register by computer with the state starting on Tuesday. The online registration, to be filled out before arrival, will ask travelers to provide their health status as well as their intended destination.

The information will be used to determine if all people arriving on the islands, whether tourist or resident, require additional health screening at the airport. It will also be used to ensure that those people maintain a strict 14-day quarantine.

During that two-week period, visitors and returning residents must check into the computer program each day, indicating their health status and that they are remaining in their hotel, home, condominium or other residence (any food must be delivered). If a person does not check in, the state will call to follow up, and if the person cannot be reached, it will dispatch police officers to follow up.

“If they can’t get ahold of you, they’ll start sending law enforcement,” Doug Murdock, the chief information officer for the state of Hawaii, said in an interview. He said that the information in the database could also be used by the police to crosscheck in the event that a visitor is pulled over for speeding, or caught up in some other situation involving law enforcement.

“All of the law enforcement, the counties and the attorney general have access to the database,” Mr. Murdock said. Those caught breaking quarantine can be fined up to $5,000, or imprisoned for up to one year.

The efforts by Hawaii are among the most advanced uses of technology by any state in the nation to try to screen visitors. Mr. Murdock said Hawaii had the advantage in this respect of being relatively inaccessible — requiring travel by boat or airplane. “Most states have a hard time controlling their borders,” he said. “It’s a little different from us.”

He said that Hawaii was not altogether trying to discourage travelers but that it would “enforce the quarantine.”

“If someone wants to come here and do online college for two weeks, that’s fine, or work from home. That’s OK, too.”

As it stands, Mr. Murdock said, “we probably arrest two or three people a day for quarantine violation,” before the introduction of the new technology. He said violators get caught when they are turned in by neighbors or wind up involved in a police matter, like through speeding. “This could lead to more arrests,” he said, adding: “We’re not doing it mainly for that. We’re doing it to encourage more compliance.”

After the 14 days, residents and visitors are free of the database, the state said. There are exemptions, too, for instance for health care workers.

Officials in Hawaii reported seven new coronavirus deaths on Monday, a single-day record for the state.

Credit…Jeenah Moon/Getty Images

With the Labor Day weekend approaching, Gov. Philip D. Murphy announced new guidelines on Monday for residents going out.

On Friday, New Jersey movie theaters and other indoor performance venues can reopen with limits for the first time since the middle of March, and restaurants can open for indoor dining at 25 percent of capacity.

All occupants must wear masks at theaters, “unless you’re pulling it down to put away a handful of popcorn,” Mr. Murphy said Monday afternoon. The occupancy of each room in the theater will be capped at either 25 percent of the capacity limit, or 150 people — whichever is less.

The governor also announced that he was lifting the limit on indoor gatherings to 150 people for certain events, including weddings, funerals and political rallies. The relaxed rules, he said, “does not mean by any stretch that we can let up our vigilance even one bit.”

“We know that this is a virus of opportunity, so let’s not give it any unforced opportunities,” he added.

The reopening of indoor dining comes about two months after Mr. Murphy canceled a scheduled restart as virus cases spiked in parts of the country that had relaxed rules on restaurants. Outdoor dining in New Jersey resumed on June 15.

The slow easing of restrictions has been a growing source of tension in New Jersey, a densely populated state that has had more than 15,900 virus-related deaths, the nation’s highest rate per 100,000 residents.

Last week, the governor said all health clubs could fully reopen for workouts starting Tuesday.

Indoor dining remains forbidden in New York City. Gov. Andrew M. Cuomo of New York on Monday acknowledged that New Jersey’s allowing indoor dining could further put restaurants in the city at an economic disadvantage and said that state officials were considering whether to allow indoor dining there.

But Mr. Cuomo said that he was still apprehensive, especially with city schools set to reopen soon. “I want as much economic activity as quickly as possible,” he said. But, he added, the state was trying to find a balance between that and public health concerns.

Credit…Jason Szenes/EPA, via Shutterstock

The U.S. Open is usually every bit as noisy and chaotic as New York City itself, with matches that sometimes start near midnight and 50,000 fans carousing into the wee hours.

This year, the tournament looks — and sounds — a lot different. It began on Monday in an unusually empty USTA Billie Jean King National Tennis Center in Queens. A grid of fans cheered remotely from screens that surround the court at Arthur Ashe Stadium.

As the first match of the tournament — played by Angelique Kerber of Germany and Ajla Tomljanovic of Australia — was underway, the loudest sounds were of screeching trains, from the Long Island Rail Road yard just beyond the tennis center’s walls, and planes flying out of nearby La Guardia Airport.

Ms. Tomljanovic, who lost, 6-4, 6-4, described the bizarre sensation of slugging through the most intense points only to have all that effort met with the sound of one coach clapping. “That’s usually when the crowd would erupt,” said Ms. Tomljanovic, who likes to look at the stands during her changeovers but had nothing to see but seats covered by tarps.

Another player, Cameron Norrie of Britain, said he had tried to focus during his match on all the people watching at home.

“At least I am giving them something to cheer about,” he said. “In the back of my mind, everyone was watching.”

Credit…William Widmer for The New York Times

Many dangers linger in Louisiana since Hurricane Laura, Gov. John Bel Edwards has warned: Downed power lines. Falling debris. Carbon monoxide poisoning from generators, the cause of more than half of storm-related deaths.

And the coronavirus. It has killed 400 times as many Louisianans as the storm, the governor noted, “just to put things in perspective, for whatever this may be worth.”

The storm interrupted virus testing across the state, so the new-case data “is not going to be robust over the next several days,” Mr. Edwards said at a news conference Sunday evening. It will take time to ramp the testing back up, he said, so officials must depend on other markers, like hospitalizations, to track the virus — and there are signs of trouble, the governor said, including increases in hospitalizations on Friday and Saturday.

Calcasieu Parish, which includes Lake Charles, one of the cities most affected by the storm, has reported 7,439 confirmed cases of the virus and 182 deaths since the pandemic began, according to a New York Times database. As of Sunday afternoon, Louisiana as a whole had reported 148,030 cases and 4,931 deaths.

“The public health emergency is real,” Mr. Edwards said.

Experts warned that the storm stood to worsen the coronavirus’s spread. Residents face more exposure while staying with relatives and friends or cleaning their property, or in other circumstances where it is difficult to maintain social distance. That may be especially true in the southwest, where hundreds of thousands of people lack power and may not see it return for days.

“If we think of Laura on top of Covid or Covid on top of Laura, it’s new and overwhelming,” Susanne Straif-Bourgeois, a professor at the L.S.U. Health Sciences School of Public Health who studies pandemics, told The New Orleans Advocate and Times-Picayune, which reported that the storm recovery provided fertile ground for new infections. “We’ve never had to deal with this before.”

Credit…Nancy Lane/Boston Herald, via Associated Press

Hundreds of protesters gathered outside the Massachusetts State House on Sunday to protest the state’s new requirement that nearly all students in the state receive a flu shot by the end of the year.

The measure, hailed by public health experts, is believed to be the first such mandate in the nation for influenza vaccination. It is intended to help prevent a combined wave of virus and flu infections that could overwhelm the health care system.

But protesters complained that the mandate infringed on parents’ rights to make medical decisions for their children, and called on Gov. Charlie Baker to rescind the order.

“Momma bears and poppa bears come out strong when their kids are threatened,” said Sean Courtney, who said he drove 90 miles from his home in West Springfield, Mass., to join the protest, and waved a flag that read “Don’t Tread on Me.”

He said the order was particularly unwelcome in Massachusetts because the state has such a high rate of voluntary vaccination. “We don’t understand the necessity of a mandate,” he said. “It’s not smallpox. It’s not freaking diphtheria.”

Near him stood three university students from Young Americans for Liberty.

“I don’t think the government should have the right to tell people what they can and cannot do,” said William Crowley, 21, an undergraduate at Boston University. “We were a country set up to protect the rights of individuals and hold back the government.”

Reporting was contributed by Reed Abelson, Peter Baker, Ellen Barry, José Bautista, Katrin Bennhold, John Branch, Choe Sang-Hun, Christopher Clarey, Troy Closson, John Eligon, Reid J. Epstein, Marie Fazio, Vanessa Friedman, Matthew Futterman, Michael Gold, J. David Goodman, Jeffrey Gettleman, David M. Halbfinger, Ethan Hauser, Jennifer Jett, Patrick Kingsley, Niki Kitsantonis, Dan Levin, Maggie Haberman, Eric Nagourney, Adam Rasgon, Matt Richtel, Campbell Robertson, Rick Rojas, Kaly Soto, Lucy Tompkins, Tracey Tully, Marc Tracy, Neil Vigdor, Sameer Yasir and Elaine Yu.