Mr. Duncan is believed to have traveled “every so often” to the U.S. from Liberia in years past to visit family members, and this was to be his longest visit yet, according to an acquaintance of the sick man. He arrived on this latest trip Sept. 20, traveling through Brussels from Monrovia, to visit a Liberian woman who lives in Dallas, according to a friend of the woman. He started feeling unwell four days after his arrival, and was hospitalized in an isolation unit with suspected Ebola on Sunday.

Now, a 10-person team from the federal Centers for Disease Control and Prevention, along with local and state health officials, are trying to track down everyone Mr. Duncan may have come into contact with in the four days between when he first started feeling sick and when he was finally isolated in the hospital. Officials said Wednesday they are monitoring at least a dozen people, but CDC spokesman David Daigle said that number is likely to grow as officials try to find every person who might possibly have come into contact with the sick man. “That’s active surveillance—running down every nook and cranny,” he said.

A law-enforcement vehicle on Wednesday sits near the main entrance to the Ivy Apartments in Dallas, where the Ebola patient was reportedly staying before he was hospitalized. Associated Press

Officials at Texas Health Presbyterian Hospital Dallas are facing questions about why they let Mr. Duncan leave their emergency department last Friday, when he first visited the hospital, even though he complained of fever and told a nurse that he had recently come from Africa, two alarm bells for the Ebola virus.

Mr. Duncan told a nurse—who was asking questions on an Ebola “checklist” that the hospital had implemented—that he had come from Africa, said Mark Lester, an official with the Dallas hospital’s parent company, Texas Health Resources.

But Dr. Lester said that the information obtained by the nurse was “regretfully” not fully communicated to the entire clinical team who cared for the patient during his first visit. Mr. Duncan reported having abdominal pain in addition to fever during that visit, according to Edward Goodman, an epidemiologist with the hospital. Mr. Duncan returned to the hospital Sunday by ambulance and was then recognized as a possible Ebola case and placed in isolation, Dr. Goodman said.

That miss is now “a wake-up call” for health-care facilities around the country, said Nathaniel Hupert, a disease and disaster-preparedness expert and associate professor at Weill Medical College of Cornell University. “There will probably be more cases as the epidemic increases” in West Africa, he said. “We’re going to have to be vigilant probably for the coming year in this country.”

Health officials diagnosed the first case of Ebola in the United States on Tuesday, bringing a disease that has killed more than 3,000 people this year in West Africa to the middle of America. (Photo: AP)

Members of the Liberian community in Dallas said Mr. Duncan had come to Dallas to visit a local Liberian woman. Mr. Duncan is the father of one of her children, said John Zuo, who said he is a friend of the woman.

The five children have been ordered to stay at home while they are monitored, but they haven’t been quarantined, said Christopher Perkins, the Dallas County Health and Human Services Medical Director/Health Authority. Mr. Daigle of the CDC said the children were being kept out of school to make it easier to monitor their health.

None of those suspected of coming into contact with Mr. Duncan has exhibited symptoms of Ebola, officials indicated. Health authorities will visit all contacts in person every day to take their temperatures and to ask about possible symptoms, Mr. Daigle said.

The five children attended school earlier this week, according to Mike Miles, the superintendent of the Dallas Independent School District. He emphasized the students haven’t shown any symptoms of Ebola, and so are believed to have posed no threat to their fellow students. He said additional health-care professionals will be on site at the schools to monitor students, and extra custodial staff had been dispatched to clean the schools.

“It’s business as usual,” Mr. Miles said.

The CDC has said that passengers on the flights that Mr. Duncan took to the U.S., through Brussels, aren’t at risk, because Ebola patients are only infectious when they develop symptoms of the disease.

A spokeswoman for Brussels Airlines, the only carrier that serves Monrovia from Brussels and one of the few international airlines that still flies to the Liberian capital, said: “We cannot exclude the fact that this passenger had been on board, but we cannot confirm it.”

At the Ivy Apartments, where the patient was reportedly staying before he was hospitalized, the gates were closed and a guard said the complex’s managers had no comment. Several residents said they hadn’t been notified about the Ebola case, or contacted by government health officials.

How does the U.S. contain the threat of the Ebola outbreak? Vanderbilt University’s infectious diseases expert Dr. William Schaffner discusses on Lunch Break with Tanya Rivero. Photo: AP

 Bodily fluids present an infection risk to medical workers treating patients with ebola. What gear do they wear? WSJ’s Jason Bellini has #TheShortAnswer.

David Blakeney, a 50-year-old retiree who lives there, said he wasn’t concerned.

“There’s a better chance of getting West Nile virus than Ebola,” he said.

Tom Frieden, the CDC director, said Tuesday that the strong infection-control and public-health measures in place in the U.S. would keep the virus from spreading.

On Wednesday, White House spokesman Josh Earnest reinforced that comment. “It is our view that we have the medical infrastructure that is necessary to try to treat this individual that does have Ebola in a way that doesn’t pose a significant risk to other patients in the hospital, to the doctors and nurses who will be caring for that individual and certainly doesn’t pose a significant risk to the broader community,” he said.

Hospitals around the U.S. have taken steps to improve screening for possible Ebola patients. Mount Sinai Health System in New York has had three suspected cases of Ebola in recent weeks, said Brian Koll, executive director for infection prevention. Precautions in place start with alerting all triage nurses, receptionists and even security guards that anyone who comes in with a fever and has traveled to Africa recently should be put into isolation immediately. “We don’t let them sit in the waiting room,” Dr. Koll said.

Staffers wearing protective gear take a more detailed history, check for diarrhea, breathing problems, muscle aches or other symptoms, he said. They run blood tests to evaluate red and white counts, salts and liver function. If the hospital’s infectious-disease specialists think the patient is at high risk for Ebola, they contact the New York City Department of Health, which can run a test for the disease and return results within six hours.

Dr. Koll said Mount Sinai conducts drills covering everything from what rooms will be used for isolation to how to quickly don gloves, gowns and masks to how blood samples are transported.