A small measles outbreak in the Bay Area last year spread almost entirely among families who had chosen not to vaccinate their children — including two young boys whose mother lied to public health investigators about their immunization status — underscoring the gaps that remain in vaccination coverage in California, according to a report published Friday.

Only seven people were infected in the outbreak that started in Santa Clara County. One likely reason it was so contained is that vaccination rates statewide have been high ever since a 2016 law ended most options for families to opt out of immunizing children, public health authorities said.

But the cases also demonstrate that with a disease as highly infectious as measles, even small clusters of unvaccinated people can put communities at risk. And it’s alarming, infectious disease experts said, that some parents are so attached to anti-vaccination beliefs that they would undermine a public health investigation.

“On the one hand, yay, this outbreak only got so big. It was only seven people. But it could have been five, or three,” said state Sen. Richard Pan, D-Sacramento, who authored the legislation that ended most vaccination exemptions. “We had our public health workers trying to contain an outbreak, and people actively not wanting to help them. What this outbreak shows is we still have a problem.”

All those infected were otherwise healthy boys or men who ranged in age from 4 to 33. No one died or experienced serious complications. All but one had not been vaccinated.

Though small, the outbreak triggered a massive public health investigation that spanned two states and 10 counties. Hundreds of people exposed to the virus were contacted. Those who hadn’t been immunized were asked to get the vaccine or quarantine themselves.

The report, published by the U.S. Centers for Disease Control and Prevention, comes as several states are managing large measles outbreaks that have infected at least 160 people so far this year. California hasn’t had a major outbreak since 2015, when more than 100 people were sickened.

That incident led to Pan’s legislation, which did away with personal exemptions that had allowed families to choose not to vaccinate their children for any reason. The vaccination rate for the state climbed from about 90 percent to 93 percent after the law was implemented. Infectious disease experts believe that a rate of 93 percent creates so-called “herd immunity,” in which an entire community is protected.

Santa Clara County’s immunization rate is higher than the state average — about 97 percent — but as elsewhere in the state, there are still clusters of unvaccinated people. Some parents are finding ways to get medical exemptions for their children; statewide, such exemptions have tripled since 2016. Children who already had personal exemptions before the law took effect aren’t required to get vaccinated unless they change schools.

Measles and vaccinations Measles information from the CDC: www.cdc.gov/measles Vaccination rates by community in California: www.shotsforschool.org/k-12/how-doing California vaccination requirements: https://bit.ly/2NBbYOT

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Last year’s Bay Area outbreak seems to have been fueled by both groups of families, public health experts said. It demonstrates that there’s more work to be done to persuade parents that vaccines are necessary and safe, said Dr. George Rutherford, head of the division of infectious disease and epidemiology at UCSF.

“We’re down to the kernel of the problem. These are the remaining people who aren’t immunized,” Rutherford said. “California is way out in front of this, and there aren’t many places to look to for hints to close down these problems.”

The outbreak began with a Santa Clara County family that traveled to England in February 2018, according to the CDC report. At the time, England was in the middle of a large measles outbreak, and among the family members on the trip was a young man who was an organ transplant recipient, which made him particularly vulnerable to measles.

The parents had chosen not to vaccinate their children, but before that trip, they had agreed to give the son with an organ transplant a preventive therapy called immunoglobin: measles antibodies that are delivered intravenously. The therapy is safe for people with a compromised immune system, but provides only short-term protection against measles.

The couple’s other son, who was 15 at the time, was not vaccinated, and contracted measles. He was diagnosed on March 4, after he’d come home, and quarantined. But he’d been socially active in the four days before he was diagnosed, a period when people are known to be able to spread the infection. Public health authorities in Santa Clara County launched an investigation.

At the boy’s school, public health officials found one contact, another 15-year-old boy, who was not vaccinated and was quarantined. He developed measles three weeks later.

Also three weeks later, a 16-year-old boy in Santa Clara County was diagnosed with measles. Public health officials learned that he had been at a Boy Scouts event with the first patient. He hadn’t been identified earlier because the first patient had forgotten to report attending that event.

The infected 16-year-old, who had not been vaccinated, passed the virus to a 21-year-old man at a different scouting event. That man had been vaccinated, but because the vaccine is only up to 97 percent effective, a small group of people will remain vulnerable, public health officials said.

The last three cases were in a family that largely refused to cooperate with public health authorities. Among the patients was a 7-year-old boy who had gone to the same tutoring center as the first patient, and his 4-year-old brother. Both boys were unvaccinated, but when public health authorities spoke with their parents during the initial investigation, the mother lied about their immunization status.

Their cases were found only after the boys’ uncle, a 33-year-old man in Alameda County, was diagnosed with measles. The uncle, also not vaccinated, told public health investigators he thought his 7-year-old nephew was the source of his infection, but refused to reveal his identity.

“We did some detective work” and traced the nephew to the tutoring center, said Dr. George Han, deputy health officer with the Santa Clara County Public Health Department and author of the CDC report.

Contacted again by public health officials, the mother said the boys had never been vaccinated. Both had medical exemptions from a doctor located “hundreds of miles” from their home, Han said. He reported the doctor to state public health authorities.

“There are widely accepted reasons that you shouldn’t get vaccinated, such as you’re undergoing chemotherapy or you have received an organ transplant,” Han said. “When we looked at their medical exemptions, we didn’t see anything like that.”

Pan said the state should consider ways to prevent people from obtaining “questionable” medical exemptions. Some other states require exemptions be issued by the state public health department.

But Pan is more interested in getting at the source of the problem: the spread of misinformation, largely on social media, about vaccine safety and the health consequences of diseases like measles.

“I’m a big believer in the First Amendment. People can stand on a street corner and say what they want,” Pan said. “But we’re not obligated to help spread that misinformation. We have to start thinking about why people make the decisions they do, and how you get to the root of that.”

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