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As eligibility for the coronavirus vaccines expanded to all D.C. adults on Monday, Mayor Muriel E. Bowser (D) made clear that the city will keep its focus on equity, despite demands from residents and businesses to open access as widely and quickly as possible.

“We have a system that we know is equitable and is getting the vaccine out,” Bowser said in response to a question about the health department telling pharmacies not to dole out appointments on their own.

The request, made last week, has frustrated independent pharmacy owners, who get doses directly from the federal government and created their own sign-up systems to allocate them.

The city says it has been successful in reaching many of the people in greatest need of the vaccine. Last week, officials announced they had offered appointments to every senior citizen, essential worker and adult with a qualifying health condition who had registered for a shot.

And government and nonprofit leaders worry that expanding appointment options for others who have just become eligible could leave out a larger population that is in need of protection but ill-informed about how to register.

About 181,000 people remain registered in the city’s system, D.C. Health Director LaQuandra Nesbitt said. Starting this week, the city will offer 20 percent of its appointments to senior citizens on that list, 30 percent each to essential workers and people with medical conditions, and 20 percent to the general population. Half the shots in each category will be reserved for residents of certain prioritized Zip codes.

If there are not enough people in categories such as senior citizens or essential workers, those appointments will go to other groups, including the general public.

The city gets about 25,000 doses from the federal government most weeks, though its allotment of Johnson & Johnson doses has varied widely and the drugmaker is providing dramatically fewer doses across the country this week.

At that rate, the city could offer an appointment to every currently registered person by the end of May, or sooner if a large number of residents drop out of the registration portal because they have been vaccinated in other states or by hospitals and other independent medical providers.

But Bowser cautioned Monday: “For all those 181,000, there are thousands who have not yet registered at all.”

Nonprofit health clinics in the city that serve mostly poor Black and Hispanic patients are supposed to help reach those who have trouble accessing the city sign-up portal.

Short on staff, space and vaccine doses, however, the clinics are vaccinating only a tiny portion of low-income people who need shots. Seiji Hayashi, a medical director of Mary’s Center, one of the largest clinics, said that as of early April, Mary’s Center had vaccinated 2,400 of its 60,000 patients.

Kelly Sweeney McShane, president of the nonprofit Community of Hope (which has 14,000 patients and can vaccinate about 400 people per week), praised the health department’s emphasis on equity over speed alone.

“What we are finding is that there are a lot of our patients who want the vaccine, are eligible and would strongly benefit from the vaccine, and there’s a lot of barriers for them to figure out ‘how do I get it?’ ” she said.

“We could open the door and say, ‘One thousand people, walk in the door.’ They would likely not be 1,000 in our patient population that have inequitable access.”

The city has tried different equity strategies, such as adding the priority Zip codes and offering vaccine clinics in partnership with churches and the public housing agency.

The current approach: a centralized registration system, which asks residents to register once and then await an email offering them a vaccine appointment. An algorithm sorts among people of varying priority levels.

That system has never been the only game in town. Hospitals have always been separate, offering appointments to their own patients and other residents. So are the nonprofit clinics. But Giant and Safeway, initially the only commercial pharmacies receiving vaccine doses in the District, have made all their appointments through the website.

In recent weeks, the federal government began supplying doses directly to many more pharmacies in the District, including large chains like Walgreens and CVS, as well as independent businesses.

And many of them started scheduling appointments on their own.

“We’re a national chain. We’ve got stores in 50 states. [Our own website] is the best way for us to work as a company,” CVS spokeswoman Amy Thibault said. She said almost all appointments CVS has offered at its seven D.C. stores that are providing vaccines have been claimed rapidly. Only seniors, health-care workers and teachers are eligible.

But other stores with their own scheduling systems were told on a conference call with D.C. Health last week to stop scheduling their own appointments.

“We believe the most equitable approach to getting the vaccine out is centralized scheduling,” Nesbitt said.

Michael Kim, the pharmacist-owner of Grubbs Pharmacy on Capitol Hill, said he spent time designing a website for booking appointments that worked well when he received his first two vaccine allotments from the federal government: 400 doses one week and 500 a week later. He had been hoping to become a vaccine provider for months.

After the call with the health department, Kim stopped scheduling appointments on his own website. Now he is in limbo, he said. He has been seeking information from the city about getting patients through the city system, with no answers.

When he gets another shipment of doses, he doesn’t know how he’ll use them.

“I don’t feel that D.C. Health always has the right answers or the right methods of doing things,” said Kim, who said the city also briefly ordered him to stop when he first started doing coronavirus testing at his pharmacy. “It’s just crazy. I feel like they are so insecure in their own selves that they have to flex their muscles and say, ‘Hey, we’re the boss.’ They’ll frame it as in the best interest of the public, but it comes down to just their mind-set. They want to control what the businesses are doing.”

Asked whether the health department has the authority to compel businesses to choose the city’s system for distributing doses that come from federal programs, Nesbitt did not answer directly. Instead she noted that the city’s system has reported a much better rate of using doses than federal programs have in the District.

The District made a long-awaited change to its registration system on Monday: The website is now available in seven languages, written by professional translators. Previously, non-English speakers had to use a Google translate button.

The District reported 73 new coronavirus cases and three deaths from the virus Monday. Virginia reported 1,310 new cases and 14 deaths, and Maryland reported 985 new cases and 15 deaths.