The number of new coronavirus cases in the Bay Area has more than quadrupled over the past four days since local laboratories started testing more people for illness, and public health officials expect the numbers to keep climbing dramatically over the coming days and weeks.

There were only three local cases of COVID-19 — the illness caused by the new virus — between Jan. 31, when the first Bay Area resident tested positive, and Thursday, when local labs began testing. In the four days since, 11 more people have tested positive — seven in Santa Clara County alone.

Similar scenarios are occurring in other states, most clearly in Washington state where six people have died in two local outbreaks that have sickened more than a dozen, mostly connected to a nursing home near Seattle.

Many of the new cases in the Bay Area and elsewhere — including the first cases that sparked the Washington outbreak — are in individuals with no known exposure to the virus, meaning they were infected presumably because it is circulating in the community.

Federal health officials last week broadened the criteria for who can be tested, and pledged to make more test kits available. As of Monday, California was testing people in ten 10 laboratories — including a state lab in Richmond and the Santa Clara County public health laboratory — and had more than 1,000 test kits, Gov. Gavin Newsom said at a news conference in Oakland. More labs are expected to start testing by the end of the week.

Public health officials in San Francisco said Monday that they too are beginning to test local residents in the city’s lab.

“For San Francisco, it is not a matter of if, but when,” said Dr. Grant Colfax, director of the San Francisco Department of Public Health. “We expect to have confirmed cases of the new coronavirus in San Francisco and are preparing for community spread of the virus.”

The total number of cases nationwide passed 100 on Monday as new diagnoses popped up around the country with the introduction of local testing. Until late last week, almost everyone who tested positive in the United States was known to have been infected outside the country — most of them were evacuees from a single cruise ship in Japan that was the site of a large outbreak.

But the situation has shifted. Now, about a third of all cases in the country are from community spread of the illness, according to public health reports.

That increased testing would lead to more cases isn’t surprising. But infectious disease experts said the fact that so many new cases were reported so fast probably means the virus has been circulating in parts of the country for a while. In fact, genetic sequencing comparing the first cases in Washington to the most recent suggests just that.

“The virus has been here for a while, we just haven’t seen it,” said Dr. John Swartzberg, an infectious disease expert at UC Berkeley. “It’s probably been flying below the radar. We’re still in influenza season. We’re still in the season of upper- and lower-respiratory syndromes. I suspect a lot of those cases may very well have been COVID-19 but we weren’t testing for it.”

Before Thursday, all testing for the new coronavirus was done by the Centers for Disease Control and Prevention at its laboratories in Atlanta, and it would often take a week or longer to get results. Because of limited facilities and resources, the criteria for testing was very narrow: It applied only to people who had recently traveled to China, the epicenter of the outbreak, or who had close contact with someone who’d already tested positive.

But last week, the first case of unknown origin was reported — in a Solano County woman who was tested only after repeated requests from the doctors treating her at UC Davis Medical Center in Sacramento. She had been hospitalized for more than week by the time her positive results came back on Wednesday. By then, more than 200 health care workers had been exposed, and at least three people who had contact with her have also tested positive.

After that case, the CDC agreed to expand testing protocols. Now, anyone who is seriously ill and for whom doctors can’t come up with another diagnosis can be tested.

In San Francisco, the city will not be conducting “on demand” testing of anyone who asks for it, Colfax said. Testing will be performed after health care providers, local public health officials and the CDC determine a patient meets specified criteria.

Coronavirus: What to do With the number of confirmed coronavirus cases in the U.S. climbing each day, many are wondering: What should I do if I have coronavirus-like symptoms, such as fever, cough and trouble breathing? It’s likely a cold or the flu, but here are recommendations from public health officials and infectious disease experts on what to do if you are experiencing symptoms: If you don’t think your condition is urgent, call your doctor or an urgent care clinic. Be prepared to answer questions about your recent travel history. Try to call ahead before showing up in case they need to redirect you to another medical center or emergency room, if necessary. If you believe your symptoms are urgent and you need to go to the ER, try to call ahead and let them know you’re coming “so that they’re prepared to isolate (you) to keep their other patients and community safe as well,” said Dr. Kathleen Jordan, an infectious disease specialist and chief medical officer at Dignity Health Saint Francis Memorial Hospital in San Francisco. “They’ll meet (you) at the door and make sure (you’re) isolated immediately.” If you are sick, put on a face mask before going out in public to protect others from potentially contracting what you have. A regular surgical mask should be sufficient — this is to prevent droplets from your coughs or sneezes from getting onto other people. You alone cannot take a coronavirus lab test; the decision to get you tested has to be made by your health care provider, in conjunction with the local health department. — Catherine Ho

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The expanded testing is important, but “it still doesn’t allow for the broad-scale kind of surveillance that we’re going to be needing in the coming weeks,” said Dr. Eric Blank, chief program officer with the Association of Public Health Laboratories.

“We obviously know that this bug has been in the U.S. for a while,” he said. “We need to answer questions like, how extensively has the virus come into the country? Where is it?”

Two weeks ago, the CDC announced that it planned to start widespread surveillance of coronavirus in several U.S. cities, including San Francisco. The intention was to tap into existing influenza surveillance networks and test potentially tens of thousands of samples. But it’s not clear when that surveillance would begin, as much-needed diagnostic test kits are only now reaching local labs.

Blank said that he hopes private manufacturers — and not just the CDC — will start producing kits, which would allow more far-reaching testing and surveillance by public and private labs. Then health care providers like UCSF, Stanford or Kaiser Permanente could test their own patients and help form a clearer picture of the spread of illness.

“The current tests that are positive represent the tip of the iceberg,” said Dr. Stephen Parodi, an infectious disease expert at Kaiser Permanente’s Solano County medical centers. “These individuals clearly were infected from somebody.“

In other words, for every seriously ill person in a hospital with COVID-19, there could be someone in the community with mild illness, or no symptoms at all, who’s never been tested.

”If you want to start defining who has a cold and cough and flu and who has COVID-19, it’s incumbent we have that testing available,” Parodi said.

San Francisco Chronicle staff writers Sarah Ravani and Dominic Fracassa contributed to this report.

Erin Allday is a San Francisco Chronicle staff writer. Email: eallday@sfchronicle.com