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Since the nation began distributing vaccines more than a month ago, it has moved far more slowly than officials hoped and has been stymied by widespread logistical problems. But West Virginia has stood out for its success in getting people vaccinated. About 9 percent of all West Virginians have received a first dose of the coronavirus vaccine, a larger segment than in every state but Alaska and double the rate of some. No state has given a larger share of its residents second doses, a crucial step to securing the best chance at immunity.

While many states are struggling to hand out the shots that the federal government has provided to them, West Virginia has given out 83 percent of its doses, by far among the highest. The patchwork system of distribution in the country is in its early weeks, and experts say operations may change significantly as vaccinations gear up further, but many states so far have struggled to give even half of their allotted vaccines. California and Rhode Island have used just 45 percent of their shots.

“West Virginia is about at the top of the charts,” said Dr. Mark McClellan, a former commissioner of the U.S. Food and Drug Administration. “We need to get more states to the point that they have the vaccination capacity of West Virginia.”

As many states struggle with logistical and bureaucratic challenges, leaving vaccines unused, West Virginia offers a remarkable example of a state that has successfully given out almost all of its shots. Yet it provides perhaps the clearest picture of a more fundamental problem that looms ahead for the country: Even the most efficient state vaccine operations do not have enough shots for all who want them.

“It’s not like it’s a cake walk, but we have efficiency,” said Dr. Clay Marsh, the state coronavirus czar. He estimated that West Virginia has the capacity to handle 125,000 doses a week, but is getting just 23,600. At the current pace, officials said, it could take up to five months to finish vaccinating people 65 and older, let alone younger people in the general population.

“We can push a lot more,” Dr. Marsh said. “We just don’t have anything to push.”

The race to vaccinate millions of Americans comes as the country confronts worrying new variants of the virus. The stakes are particularly high in West Virginia, where residents are among the oldest and least healthy in the nation. Two in 10 West Virginians are 65 or older, and nearly half of adults have a higher risk of developing a serious case of Covid-19.

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“People are dying every day,” said Albert L. Wright Jr., the chief executive of WVU Medicine, the state’s largest health care provider, which opened a mega-clinic in Morgantown last week. “We just realized, the only way out of this is to vaccinate our way out.”

While the rollout in West Virginia has not been without its frustrations for the many residents who are desperate to receive the vaccine, the state’s approach offers insight into what has worked.

Early on, the state got a significant head start because it initially opted out of a federal program to vaccinate people in nursing homes and other long-term care facilities.

While other states chose the federal plan, which partnered with Walgreens and CVS to inoculate people in nursing homes around the country, officials decided the idea made little sense in West Virginia, where many communities are tucked into the hills, miles from the nearest big box store, and about half of pharmacies are independently owned.

West Virginia created a network of pharmacies in the state, pairing them with about 200 long-term care facilities. As a result, West Virginia finished its first round of vaccinations at nursing homes last month, while many states were just getting started. By the end of this week, officials expect to have delivered a second round of shots to all nursing homes.

A growing number of governors and state health officials have voiced frustration with the speed of the federal program, which has been slow in part because of the sheer number of long-term care facilities nationwide. Some states, like Maine, have also begun looking to local pharmacies as a resource.

“Using your local partners and really having more control over where the vaccine is going, that’s what has been successful for West Virginia,” said Claire Hannan, executive director of the Association of Immunization Managers.

The approach in some ways reflects the scrappy outlook that has become embedded in West Virginia, a state that is used to being labeled in broad strokes by outsiders. After years of coming in at the bottom of various national rankings — highest rate of cigarette smokers, largest share of adults with multiple chronic conditions, among the highest prevalence of diabetes and obesity — there is a sense of proud defiance around the state’s vaccine plan.

“Our state motto is montani semper liberi — mountaineers are always free,” said Stephen New, a lawyer in Beckley, whose father, a former coal miner, is scheduled to get vaccinated at a local clinic this week. “There is a fierce sense of independence here that we don’t need to follow others.”

Central to West Virginia’s strategy, too, is putting the National Guard at the helm of vaccine operations.

“They are logistical experts,” said Jim Kranz, a vice president at the West Virginia Hospital Association, who described a military approach far different than typical government bureaucracies jammed up by memos and lengthy meetings. “The Guard just says, ‘I don’t need a written plan, this is what we’re doing.’”

In recent weeks, other states have increasingly been turning to the National Guard for assistance.

Inside a former drill hall on the National Guard’s gated campus in Charleston, a core state team of representatives from various agencies meets at a command center, working amid a labyrinth of computer screens, white boards and plexiglass dividers, in the shadow of an enormous American flag.

Officials in West Virginia say the in-person command center, which runs counter to pandemic-era Zoom meetings and may come with health risks, has also helped quickly resolve problems. For instance, a recent brainstorming meeting led to a swift change: To avoid a mix-up between Pfizer-BioNTech and Moderna vaccines, officials changed the colors of cards showing that a resident had been vaccinated — red for Moderna, blue for Pfizer.

“It’s a person at a table,” said Joe Peal, a retired colonel and the chief of staff for the vaccine task force. “We absolutely could not do it virtually.”

Officials say they have also learned what not to do — including not promising shots that they don’t yet have.

After a crowd of people in their 80s unexpectedly showed up outside the health department in Charleston one shivering winter night, officials there decided clinics could not be first-come, first-served. They required appointments, and to avoid later rescinding those appointments, as has happened in states like New York and Florida, appointments are not made until the county officials have that week’s allotment of vaccines on hand, sitting in their own freezer.

Some of West Virginia’s success also may partly be explained by characteristics of the state. Its population, 1.8 million people, is tiny compared to states like Texas, which are handling far more vaccines. West Virginia is also among only a handful of states that do not allow philosophical exemptions for vaccines for school, according to the Immunization Action Coalition, suggesting a culture where acceptance of vaccines may be more prevalent.

West Virginia’s rollout has by no means been without hiccups. And some of the problems have been similar to those in states with far worse records in distributing shots: Phone lines have been mobbed and many West Virginians have spent hours calling to get appointments without success.

The health department in Kanawha County, which includes Charleston, was so overloaded with phone calls last week, its phone system crashed and its Facebook page was flooded with complaints. “I’ve called 250+ times but still unable to get through,” one person wrote.

The distribution of vaccines — mainly to health care workers and older residents so far — also has been concentrated in urban centers, raising concerns about equal access in rural and more impoverished areas. Of the state’s 10 poorest counties, just one had a vaccine clinic in recent weeks, according to an analysis by the nonprofit newsroom Mountain State Spotlight.

But the main problem, officials say, is not logistical but a matter of supply: They need more vaccine.

Dr. Sherri Young, the health officer in Kanawha County, said one big reason phone lines are overrun is because the county cannot reliably schedule clinics weeks in advance. If there were more vaccine, she said, “people wouldn’t be as panicked.”

Pfizer and Moderna have pledged to deliver a combined 200 million doses for use in the United States by the end of March, with an additional 200 million doses to be delivered by the end of July. A third vaccine maker, Johnson & Johnson, could put millions more doses on the market if approved.

West Virginia had hoped to make the case that because it is moving through its vaccines, it deserved to get more, but so far states have been allocated doses based on population.

Jim Doria, 69, a retired epidemiologist, had gone months without seeing his grandchildren who live in Philadelphia and was eager to be among the first to get vaccinated after the state opened the system to people in his age group last week. He estimated that he called as many as 700 times.

“I won the lottery,” he said, after getting off a wait list.

But for every person like Mr. Doria, countless others were left disappointed. Statewide, West Virginia has given shots to around 70,000 of its senior citizens, with more than 250,000 left to go.