The first homeless person to test positive for the coronavirus in San Francisco was living in a city shelter, officials announced Thursday, setting off a race to trace everyone the individual has recently interacted with, as officials tried to contain a potential outbreak among the city’s most vulnerable residents.

Health officials said they isolated the person — who was at the Division Circle Navigation Center — in a hotel room, where they were described as being in “good condition.” Officials said it was unclear where or how the person got the coronavirus.

Recent counts from state and federal sources have fewer than a dozen total homeless COVID-19 cases in California and fewer than 300 nationwide.

City officials said the shelter remains open, and that masks have been provided to all of the center’s residents and staff, with instructions to wear them at all times. Health officials are trying to trace every person the infected individual contacted in and outside of the shelter. The gender and age of the individual was not disclosed.

A doctor and health workers were also deployed to the shelter to conduct symptom and temperature screenings for all residents and staff, while a cleaning crew was being dispatched to disinfect the facility, which is located at 224 South Van Ness Ave. and houses about 180 people.

Health officials said any residents with symptoms will be tested and quarantined at hotel rooms. Other shelter residents will be moved out of the shelter to hotels “based on their risk of exposure and risk of disease,” which includes vulnerable people over 60 or with underlying conditions, they said.

Abigail Stewart-Kahn, interim director of the Department of Homelessness and Supportive Housing, said the city had planned for weeks in anticipation of the first coronavirus patient at a homeless shelter.

“The well being of our homeless neighbors is our top priority, and we will continue to work in partnership with our nonprofit service providers, (Department of Public Health) and the entire city to aggressively contain the virus and protect our community,” Stewart-Kahn said in a statement. “To the Division Circle Navigation Center community, I am here for you, HSH and DPH are here for you as is the entire city.”

Jennifer Friedenbach, director of the Coalition on Homelessness, called the development “very, very frustrating and scary.” Advocates for the homeless have urged city officials for weeks to move people from group living situations like shelters to hotel rooms, where they can quarantine if they must.

“This has been our fear. We are really scared,” Friedenbach said. “The folks who live at that Navigation Center are very vulnerable ... this was exactly why we had been pushing them to stop the practice of having them sleep one on top of the other.”

City homeless officials said they have been racing to set up overflow accommodations so they can cut the 2,000-person population in San Francisco shelters by about half. That would allow them to increase the physical distancing between beds to the recommended 6 feet, instead of the current 3 feet or less.

Trent Rhorer, director of San Francisco’s Human Services Agency who is in charge of emergency housing, said officials hopes to meet that goal next week.

The huge Moscone West convention center will be open as an overflow shelter to take transfers by Friday, he said, and it will have 394 beds. Two other sites “are in negotiation,” he said, and if they come online, they will bring the total to 850 beds.

The city is also planning to move vulnerable homeless people who are 60 years or older, or have underlying conditions, into hotels rooms.

Rhorer said he expected to have 2,000 rooms in 10 hotels available by Friday evening, but those will not just be for homeless quarantines. They will also house health care workers needing respite and the most vulnerable among 19,000 people living in SRO hotels and permanent supportive housing. Those types of housing often have shared kitchens and other common spaces, making isolation difficult.

Housing homeless people in hotel rooms requires extra staffing for case management and counseling, he said. Many chronically homeless people have mental or substance abuse issues, “and you can’t just give them a room and say, ‘here you go,’” he said. “They need help while they’re inside.”

Also Thursday, five city supervisors, including Hillary Ronen, whose district includes the Division Circle shelter, issued a statement saying they intend to introduce an emergency ordinance on Tuesday requiring “that at least 1,000 rooms be used for unhoused people currently in congregate settings like shelters.

“We cannot wait any longer,” they wrote. “We will mandate that the city lease an estimated 14,000 hotel rooms by the time the coronavirus is expected to reach peak infection on April 28.”

Ronen and several other supervisors held a news conference at the shelter Thursday afternoon to emphasize the urgency of their demand.

“We have the hotel rooms, we have the money, we have the staffing, why wouldn’t we do this?” Ronen said. “Why wouldn’t we do this right now and save thousands of lives?”

City planners have set a target of renting 4,500 hotel rooms during the crisis, with 1,000 of those set aside for health workers so they may rest and isolate themselves from their families.

Randy Quezada, spokesman for the city’s Emergency Operations Center, said the plan for now is to stick to that target, with the specific parameters regarding homeless.

“The city’s strategy is led by scientists and health experts, and for homeless people that means housing only the most vulnerable, those who need to isolate but can’t and those who have coronavirus. If the guidance we’re getting from science changes we can adapt, for but now that is the strategy. Science has to lead the way we make these decisions.”

National guidelines from the Centers for Disease and Control and Prevention advise that homeless people who don’t show signs of COVID-19 or who aren’t in a vulnerable category should remain in homeless camps, but be encouraged to keep 6 feet of space between them and supplied with sanitary gear. All counties in the Bay Area have been using those guidelines.

Rhorer estimated that securing and maintaining 14,000 hotel rooms could cost around $60 million a month in rent and staffing — at a time when the city is facing a huge budget deficit because of costs associated with the pandemic.

“A lot of the homeless on the street are not able to self-care in hotel rooms, and it’s potentially dangerous for them to be placed in there without support,” he said. “So they need staffing.”

Kevin Fagan and Alejandro Serrano are San Francisco Chronicle staff writers. Email: kfagan@sfchronicle.com, alejandro.serrano@sfchronicle.com. Twitter: @KevinChron, @serrano_alej