VACAVILLE, Calif. — In the parking lot of a big-box store, Rick Lodwick tossed a jumbo pack of sanitizing wipes into the back of his car.

“I’m middling alarmed,” Mr. Lodwick, an engineer, said as he listed the provisions he bought on Thursday after learning that a woman from his county, Solano, was believed to be the first person in the United States to test positive for the coronavirus without having a known connection to others with the illness.

“When I heard it was here, I thought, We’re going to have trouble,” Mr. Lodwick said.

A day after the authorities announced the case of the woman, who they say may be the first person infected through community transmission in this country, residents of Solano County had a wide range of reactions, from apathy to alarm.

Officials in the county, a partly rural community between San Francisco and Sacramento, declared a health emergency. An investigation was started into all contacts the sick woman might have had with friends, neighbors and medical workers. And people shopping in Solano County’s strip malls and neighborhoods had many questions. Would their children be safe at school? Were there places they should avoid?

“It’s scary,” said Adrian Obando, a stay-at-home father who was watching his daughter turn cartwheels on the lawn of a park near NorthBay VacaValley Hospital, where the woman was treated before she tested positive for the coronavirus.

Still, Mr. Obando said that he was worried, but that he did not think there was much he could do. He said he could see with his children how easily colds and flus spread. “They get sick all the time,” he said.

The woman, whose identity was not released by state and local officials in a series of news conferences about the case, was the 60th person in the United States to test positive for the virus, which has killed 2,800 people and sickened 82,000 more worldwide.

But the woman’s case is different, the authorities have said: It is the first in which health officials have not been able to determine how the patient got sick. That, officials said, might be a first sign of spreading within the United States, where all of the previous cases had been related to travel to Wuhan, China, the center of the outbreak; to relatives who had been abroad; or to a cruise ship that had become infected.

Dr. Bela T. Matyas, the public health officer in Solano County, said officials had begun monitoring dozens of people with whom the woman had interacted in recent weeks — including relatives, co-workers and others she might have had contact with through her job — and had asked them to quarantine themselves at home. Those who experience symptoms, Dr. Matyas said, will be tested for the coronavirus and be asked to further isolate themselves. The officials would not say where the woman worked or what sort of work she did.

This county, which is home to both Bay Area commuters and military personnel stationed at Travis Air Force Base, also has other connections to the coronavirus. Hundreds of Americans who were traveling when the outbreak began unfolding have been housed on the base as part of a federally mandated quarantine, and several have tested positive for the virus. But on Thursday, health officials emphasized there was no indication that the woman had come in contact with anyone at the base, nor had she traveled recently.

Outside Travis Air Force Base on Thursday, there were few signs of concern. Pilots in flight suits greeted one another with handshakes at a nearby fast-food restaurant. No one wore masks during the lunch rush.

“To me it’s just another flu,” said Beverly Garske, who was taking a walk a few minutes from the base’s entrance.

Having a possible case of community transmission occur in her county made the virus feel “close to home,” she said. But she shrugged off the idea that she might become exposed.

“My husband and I don’t get out much,” Ms. Garske said. “If we do go out, usually it’s to go fishing.”

Nern Lewis, who lives minutes away from Travis Air Force Base, said she was not concerned. “I mostly just stay in my house,” she said, “so I’m not afraid.”

But as Ms. Lewis sat in the sunshine in front of her ranch house on Thursday morning, she said some of her relatives had grown more nervous about the virus. The new case was a troubling mystery.

She had been planning to return to her native Thailand to visit her sister but was now putting off the trip.

“It’s not a good time for it,” Ms. Lewis said. “You might not be able to come back.”

The woman who has the coronavirus was initially treated in Vacaville and then taken by ambulance to the University of California, Davis Medical Center in Sacramento, where she was still being treated as of Thursday night.

At the hospital earlier in the day, some workers seemed deeply concerned about possible exposure.

“I’m worried how this will affect my baby,” said Vickie Poncalo, a cashier at the hospital’s cafe who is seven months pregnant.

“Should we be wearing a mask and worrying that people are walking around here?” she asked. “Should I even be here?”

In the gift shop, Madeline Daniels said she was wondering about all of the possible ways the virus might spread.

“Money is the dirtiest thing,” she said.

Dr. Peter L. Beilenson, the director of Sacramento County’s Department of Health Services, said officials were bracing for a larger outbreak in Northern California.

“There’s almost assuredly going to be a significant number of people testing positive,” Dr. Beilenson said.

Thomas Fuller reported from Vacaville, and Nicholas Bogel-Burroughs from New York. Julie Checkoway contributed reporting from Sacramento.

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