Kaiser Permanente began “drive-through” testing for coronavirus in San Francisco on Wednesday, and public and private health care providers across the Bay Area also ramped up screening efforts, but no one knows how widespread the virus is, public health and infectious disease experts said.

It’s impossible to say how many tests are done daily in the Bay Area, because counties and private providers aren’t making that information public, in part because of patient privacy concerns.

Testing increased dramatically over the past week as federal authorities shipped more kits to public health laboratories, and as health care providers like UCSF and Stanford started testing on their own. Kaiser is sending tests to outside laboratories.

San Francisco’s Public Health Department can do several dozen tests per day, said Dr. Susan Philip, director of disease prevention and control. But the total capacity citywide increased over the past five days as more commercial laboratories, including Quest and LabCorp, plus university labs have come online.

The increase is “game-changing,” Philip said.

Testing has been a source of frustration for public health authorities battling coronavirus for weeks. Initially only the Centers for Disease Control and Prevention was able to test, but at the end of February, some state and county laboratories were given supplies to do their own testing.

Nationwide, 81 labs can now test for coronavirus. The CDC reports that several hundred samples are tested each day at these labs, and more than 10,000 samples were tested as of Tuesday. Most individuals provide more than one sample per test, so the number of samples is higher than the number of people tested.

Nineteen public health labs are testing in California, plus a few commercial and university labs.

On Tuesday, Gov. Gavin Newsom said the state had 7,675 tests available. After several new private labs are up and running, commercial testing is expected to have the capacity to test 5,000 to 5,500 individuals per day. As of Wednesday morning, 1,075 people had been tested statewide, according to the California Department of Public Health.

There were 190 confirmed coronavirus cases in California as of Wednesday afternoon, including 108 in the Bay Area. Four people in California have died because of the virus. Nationwide, more than 1,200 cases and 36 deaths were reported as of Wednesday afternoon.

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Public health officials agree that the actual number of people with COVID-19, the illness caused by the coronavirus, is much higher than the number of confirmed cases because there still aren’t nearly enough tests available. And for now, only patients who meet certain criteria are tested. Though federal officials have said that there will soon be enough tests to meet demand, Bay Area providers said that’s not close to happening.

“There is still not enough testing capacity for the health care providers who desire testing for their patients, which is in stark contrast to what the politicians are telling people,” said Dr. Karen Relucio, the Napa County health officer.

Kaiser, the Bay Area’s largest health care provider, raised tents outside a handful of facilities around the Bay Area, including a Geary Boulevard clinic, and invited some patients with doctor’s orders to drive up and provide samples for testing. The samples — obtained by a swab to the throat or nose — are processed at non-Kaiser labs.

The Kaiser drive-through sites are meant to prevent patients with respiratory symptoms from walking into facilities where they could potentially expose others to infection.

At public health laboratories, as testing capacity increased, officials said they were expanding — slowly — the criteria for who gets tested. In most places, only “high risk” patients are tested. That includes people with recent travel history to a place where the virus is widespread, or close contact with someone who has already been diagnosed with COVID-19.

More recently, the criteria were expanded to include people who are already very ill with a respiratory illness that has no other obvious cause. And at some places, including Stanford, certain people in high-priority jobs — such as health care workers or people who work with older or vulnerable populations — may get tested now.

“The criteria changes every day,” said Dr. Yvonne Maldonado, medical director for infection control at Lucile Packard Children’s Hospital at Stanford. “We really at this point don’t have the capacity to do wide-scale community testing for people who are just wondering if they’re infected.”

Decisions about who gets tested are based in part on a Bay Area-wide shift toward slowing down spread of the virus instead of trying to stop it altogether.

“To have the expectation that anyone who has any infectious disease is getting tested is just not realistic,” said Kelly Wroblewski, director of the infectious disease program with the Association of Public Health Laboratories. “You want people who are at the highest risk of severe illness to get tested, you want people who are exhibiting symptoms for the first time in a community to get tested. Those things get tested, instead of people who want some peace of mind.”

Solano County recently received its first testing kit from the CDC, which will allow testing of up to 800 people, said Dr. Bela Matyas, the Solano County health officer. He said the county also has access to commercial lab tests.

Even with the increased capacity, the county doesn’t plan to test everyone who shows symptoms — and that’s not a bad thing, he said.

“There comes a point where it’s not important to enumerate the number of cases,” Matyas said. “It’s important to handle people based on their observable symptoms. I would say most of the Bay Area is at that point now. All you need in order to get to that point is the recognition that you have community transmission of this disease.”

Erin Allday and Sarah Ravani are San Francisco Chronicle staff writers. Email: eallday@sfchronicle.com, sravani@sfchronicle.com