In Illinois, Gov. J.B. Pritzker said on Monday that more than 2,200 locations and providers were already prepared to provide the more than 500,000 pediatric doses that the state will initially receive.

State health officials in Texas said on Monday that the federal government would initially allocate approximately 1.3 million pediatric doses to the state.

Mayor Bill de Blasio of New York City said on Thursday that shots would be available for children at city-run vaccination sites within 24 hours after federal clearance, and at locations like doctors’ offices and pharmacies by 48 hours.

“This is a moment parents have been waiting for, to know their kids will be safe,” Mr. de Blasio said at a news conference. “Now, New York City will be ready.”

The city has ordered 231,000 pediatric doses and is working with nearly 1,500 community pediatricians and family doctors to plan vaccination logistics and engage with patients, said Dr. Dave A. Chokshi, the health commissioner.

Much of the rollout of children’s shots is expected to fall on pediatricians and family physicians, who have relationships with patients and children. Many of those physicians, however, are also strained by staffing shortages and a long line of patients trying to book appointments delayed by the pandemic.

Dr. Sterling Ransone Jr., the president of the American Academy of Family Physicians and a physician in rural Deltaville, Va., said that he would keep his office open later on weekdays and on Saturdays to accommodate the demand for pediatric shots.

Some experts have warned, however, that the same inequities that plagued the vaccine rollout for adults earlier this year could hinder the rollout for children.

“We cannot see what we saw in the earliest stages of rolling out the vaccines for adults, in which advantaged persons and persons of means figure out a way to be first in line,” said Dr. James E.K. Hildreth, the president of Meharry Medical College, a historically Black institution.

He said that school nurses, churches and local health officials would be key in reaching some children and families who might not have insurance or access to pediatricians.

Black and Hispanic children are less likely to be tested for the virus but more likely to be infected, get hospitalized and die from Covid-19 than white children are, according to the Kaiser Family Foundation. Hospitalization rates in the 5-to-11 age group are three times as high for Black, Hispanic and Native American children as for white children, according to the C.D.C.

Vaccine hesitancy among all parents is another concern. A survey from the Kaiser Family Foundation released on Thursday found 27 percent of parents of 5- to 11-year-olds were eager to vaccinate their children right away, while a third said they would wait and see how the rollout went.

The uptake among adolescents has been slower than public health experts hoped: Pfizer’s vaccine became available to children 12 to 15 in May, but only roughly 40 percent of that age group is now fully vaccinated, compared to 69 percent of adults.

Daniel E. Slotnik contribued reporting.

The effort to vaccinate students will begin as many return from a midsemester break.

Case rates have fallen in recent days, but Britain is reporting an average of 40,700 new daily infections, according to a New York Times database, and deaths have increased by 32 percent in the past two weeks. About 68 percent of the population is fully vaccinated, and more than six million people have received a booster shot.

Britain lifted the bulk of coronavirus restrictions over the summer, though some nations moved more slowly than others. Nightclubs reopened in England in July but only opened in Northern Ireland on Sunday as social distancing restrictions were lifted.

Gov. Kim Reynolds of Iowa, a Republican who opposes government requirements for masks and vaccines, signed the bill into law on Friday, saying in a statement that “no Iowan should be forced to lose their job or livelihood over the Covid-19 vaccine.” Though Republicans have the majority in the Iowa House and Senate, there was bipartisan support for the bill.

Elsewhere, the Republican-controlled legislature in Tennessee approved a slate of bills early Saturday limiting the enforcement of Covid-19 protections, though lawmakers backed off provisions that would have prevented many businesses in the state from enforcing mask mandates, The Associated Press reported.

The legislation prohibits government entities and public schools, as well as many private businesses, from mandating vaccinations or requiring proof of vaccination, according to the Knoxville News Sentinel. After pushback from the business sector, however, the lawmakers agreed to carve out various industries from the ban on Covid-19 vaccine mandates, including many health care facilities and entertainment venues.

Under the new legislation, government entities will largely be prohibited from implementing mask mandates, unless they meet specific criteria. The changes limits public schools from requiring masks, except in extreme circumstances, according to the education news website Chalkbeat Tennessee.

In Florida, Gov. Ron DeSantis, a Republican who has been a strident opponent of federal vaccine mandates, on Friday called for a special session of the State Legislature in two weeks to consider protections for workers who could lose their jobs because of vaccination requirements, saying in a statement that a person’s “right to earn a living should not be contingent upon Covid shots.”

On Saturday, hundreds rallied outside the Statehouse in Topeka, Kan., expressing opposition to federal Covid-19 mandates as a legislative committee considered recommendations for the Republican-controlled legislative body, The Associated Press reported. The very name of the panel — the Special Committee on Government Overreach and the Impact of Covid-19 Mandates — signaled the political resistance to federally imposed requirements.

By contrast, the Illinois General Assembly, which is controlled by Democrats, raced on the final day of its fall session to make it harder for people to avoid Covid mandates. The legislature updated the state’s “conscience” law that was approved in the 1970s to protect doctors and other health care workers from having to provide abortions that conflict with their beliefs.

The bill, which will be signed into law by Gov. J.B. Pritzker, a Democrat, states that it is not a violation of the law for an employer to take any measures or impose any requirements that help prevent contraction or transmission of Covid-19.

Supporters said that the original 1970s law was never intended to allow residents to use moral or religious objections to avoid vaccination. They said the additional language would close what they argued was a loophole.

Víctor Manuel Ramos contributed reporting.

A spokesman for the National Health Commission had said on Saturday that the latest outbreak was “still developing rapidly” and that the situation was “severe and complicated.” The newest round of infections, though small compared to outbreaks in many other countries, is relatively large for China, which has officially reported just about 97,000 cases since the start of the pandemic.

On the Chinese social media platform Weibo, where the news of the suspension was trending, some commenters whweo said they had already bought tickets expressed disappointment. But many comments expressed support for the measure and concern about photos of crowds at the park during the Halloween weekend. China’s commitment to a “zero Covid” policy — which has made it an outlier globally — has widespread support domestically, as it has allowed relatively restriction-free travel within the country.

Still, some experts have warned that the economic toll of repeated lockdowns and other strict prevention measures may eventually become too heavy. Throughout the pandemic, domestic tourism and consumption have suffered when new outbreaks are reported, as people seek to avoid becoming trapped in high-risk areas.

Claire Fu contributed research

New Yorkers across several boroughs, for example, have reported garbage pileups and significant delays in trash pickups in their neighborhoods.

Sanitation officials say the most severe delays have been seen across Staten Island and in parts of southern Brooklyn, including the Bay Ridge and Bensonhurst areas. Data shows that overall 311 complaints soared last week with more than 1,000 reports of uncollected trash and recycling on some days, as the local news outlet Gothamist reported. (The number had hovered between 100 and 250 through the rest of October.)

Mayor Bill de Blasio said this week that he believes some of the delays may be attributed to declines from workers in protest of the vaccination requirement.

“I’m assuming it is related to people expressing their views on this new mandate,” Mr. de Blasio said at a recent news conference. “You want a protest, go protest. But when you’re on the clock, you have to do your job.”

Similar problems were reported among some workers in other agencies as the vaccination deadline approached.

Fire officials said that an uptick in sick calls among employees toward the end of the week was responsible for the temporary closures of several fire stations across the Bronx and other boroughs.

“The excessive sick leave by a group of our firefighters because of their anger at the vaccine mandate for all city employees is unacceptable, contrary to their oaths to serve and may endanger the lives of New Yorkers,” the fire commissioner, Daniel A. Nigro, said in a statement.

Union leaders for both fire and sanitation workers denied reports of the changes in work patterns.

Still, the overall rise in the vaccination rate for municipal workers offered a measure of optimism that significant disruptions to government agencies and city life could be avoided as enforcement of the mandate begins on Monday.

Before the requirement was announced on Oct. 20, and an option to opt into weekly testing was removed, only about 60 percent of employees in the Fire and Sanitation Departments had received at least one vaccine dose. The Police Department’s rate sat around 70 percent.

Holdout workers were offered $500 as a bonus if they got vaccinated by the end of the workday on Friday. By that deadline, more than 75 percent of fire and sanitation workers had been vaccinated — along with nearly 85 percent of the Police Department’s 36,000 uniformed and 15,000 civilian employees.

Joseph Goldstein contributed reporting.

Mr. Putin called on the W.H.O. to expedite the vaccine registration process. “As soon as this is done,” he said, “we will be able to restore and restart the economy.” He said he would also like the Group of 20 to “address the problem of mutual recognition of vaccine certificates.”

Over the summer, many countries opened to international travel, but the patchwork of rules regarding which vaccines would be accepted led to confusion and frustration for travelers, especially those who had received vaccines that were not widely accepted.

Two vaccines made by China, Sinopharm and Sinovac, are on the W.H.O.’s emergency authorization list. Across Asia and South America, millions of people have received doses of those vaccines, and millions more have received doses of vaccines, like Sputnik V, that have been authorized by individual governments only.

On Saturday, Mr. Xi said China had provided more than 1.6 billion shots to the world and was working with 16 countries on manufacturing vaccines, according to a transcript published by the official Xinhua news agency, Reuters reported.

Mr. Xi expressed support for a World Trade Organization decision that waived intellectual property rights for Covid-19 vaccines, Reuters said, and he called for vaccine manufacturers to transfer technology to developing countries.

Both pills are designed to interfere with viral replication. If approved by regulators, both pills could be prescribed at the first sign of infection or exposure without requiring hospitalization.

Merck has already reported data from its Phase 3 trials that showed molnupiravir reduced the risk of hospitalization or death by half. Merck has submitted an application to the U.S. Food and Drug Administration to authorize its pill. European Union regulators said on Monday that they had begun a review of molnupiravir.

Meanwhile, Pfizer said in a statement that it had begun Phase 2/3 trials to evaluate the efficacy and safety of its pill.

France has ordered 50,000 courses of Merck’s pills to be delivered starting in the end of November, the health minister, Olivier Véran, said on Tuesday.

The Philippines said this week that it had procured 300,000 courses of molnupiravir. Malaysia and Singapore have also secured supplies of Merck’s pills.

South Korea’s health ministry said it planned to purchase enough antiviral pills for 404,000 patients in total, and to have supplies available starting in the first quarter of 2022. It said it would closely monitor the progress of clinical trials for pills under development at several companies, including Merck, Pfizer and Roche, as it considers its options.