Fewer than 200 people in San Francisco and three other Bay Area counties had tested positive for the coronavirus when the region was told to shelter in place, but cell phone data show that residents were quick to surrender to the orders and stay home, according to a federal report released Monday.

That early adherence may have helped protect the region from experiencing a far worse outbreak, public health experts have said.

The same report, which compared public health responses in four metropolitan areas in the U.S., showed that even though New Yorkers also were obedient to stay-home orders, the outbreak there was already so far advanced that it rapidly escalated — from roughly 5,000 cases on March 20 to nearly 50,000 just a week and a half later.

New York City now has more than 100,000 cases. The Bay Area has about 5,200.

The paper published by the Centers for Disease Control and Prevention compared New York City, Seattle and New Orleans with a subset of the Bay Area, including San Francisco, Alameda, San Mateo and Marin counties. The Bay Area was the first to issue aggressive physical distancing policies, and it’s also had the smallest outbreak of the four regions studied.

But the CDC report showed that in all four metropolitan areas, the pace of the local outbreak slowed as mobility decreased — in other words, as more people stayed at home.

It’s premature to say for sure that the Bay Area’s early sheltering in place is the reason its case counts haven’t climbed as quickly as in other places, according to the CDC. But the report adds to increasing evidence that the Bay Area, so far, has dodged a public health crisis.

For all four regions, “you can see the decline in mobility, and two weeks later you can start to see some of the declines in the growth rate,” said Kathleen Ethier, lead for the CDC’s community intervention task force, and senior author of the paper.

“We didn’t connect those things statistically,” she said. “But it’s heartening to see that we may be having an impact on disease.”

Public health officials have looked at several metrics to determine whether people are actually obeying shelter-in-place orders, and if that in turn is slowing down the spread of disease. Groups that use cell phones to track people’s movement have shown substantial decreases in foot traffic in San Francisco, for example.

Orbital Insight, which uses satellite imaging and mobile phone data to track population density, has produced heat maps that show far fewer people congregating in certain parts of San Francisco — especially the Financial District — after the shelter in place took effect.

The CDC study used data from SafeGraph, which aggregates location reports from mobile devices; the San Francisco data come from about 164,000 devices, or about 3.6% of the population. The CDC looked at the percentage of people staying within 500 feet of their home each day.

In February, roughly 20% of people in San Francisco were staying home. The same was true for the other three metropolitan areas.

But Bay Area residents began staying close to home much earlier than everyone else — both at an earlier date, and earlier in the course of the regional outbreak. By March 10 — a few days before the Bay Area’s regional shelter-in-place orders were issued — about a third of residents were staying home. A day later, more than half were staying home.

When the sheltering orders were issued, a few dozen cases had been reported in the San Francisco-Oakland region; there are now about 2,500 cases.

About a third of New York City residents also were already staying home shortly before shelter-in-place orders were issued on March 20, and half were staying home the day after. But the outbreak there was already far along by then.

Eventually more New Yorkers were adhering to shelter orders than people in San Francisco — about 58% compared to 53% — but the damage may have been done, as the virus was spreading freely in the East Coast community by then, infectious disease experts have said.

The trends weren’t quite as dramatic in New Orleans and Seattle, but they demonstrated that the more people stayed at home, the more their local outbreak slowed down, Ethier said.

Dr. Erica Pan, the Alameda County health officer, said the results did not surprise her, though she was pleased to see evidence that Bay Area residents quickly understood the importance of sheltering in place and behaved appropriately.

“People are generally staying at home and staying socially distant and generally following the guidelines. I’m really appreciating how the community has come together for this,” she said. “We understand that it’s a dramatic intervention and it’s having a huge social and economic impact.”

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