Three more cases in Santa Clara County were also announced, and the many developments represented a significant escalation of the situation in California, which has had at least 54 cases of the virus, the most of any state. Officials in Los Angeles County — with 10 million residents, the largest county in the nation — declared a state of emergency, warning that the number of confirmed cases was likely to rise.

“This is a call for preparedness,” Dr. Barbara Ferrer, the director of the Los Angeles County Department of Public Health, said at a news conference on Wednesday. “Know what you might do should we see an explosion of cases.”

While seeking to tamp down public anxieties, officials also warned that a worsening spread of the coronavirus could potentially lead to a grim new reality. Officials could order “extreme measures of social distancing.” Major sporting events could be played without spectators. Schools could shut down.

“This is the time for people to have a plan for what they would do should their child’s school need to close,” Dr. Ferrer said.

Dr. Jonathan Fielding, a public health expert at the University of California, Los Angeles, said that the number of cases in California, a large state that is a frequent entry point for travel from Asia, was likely to rise, particularly as officials increase their capacity to test and diagnose cases.

“What was unfortunately predictable has now come into direct line of consciousness for many people who were hoping it would not happen,” said Dr. Fielding, who previously worked as the public health director for Los Angeles County. He added, “It’s very likely that it’s going to get worse before it gets better.”

The 11 deaths in the United States are a small fraction of the more than 3,000 deaths that have been traced to the virus worldwide. Early estimates of the coronavirus death rate from China, the epicenter of the outbreak, have been around 2 percent. On average, seasonal flu strains kill about 0.1 percent of people who become infected.