They said Wednesday that they believed 12 to 18 people had direct contact. On Thursday morning, a spokeswoman for Dallas County Health and Human Services said it was thought that 80 people had contact directly with Mr. Duncan or secondarily with his direct contacts. Then in an afternoon news conference, Dr. Thomas R. Frieden, the director of the Centers for Disease Control and Prevention, said that health workers were assessing 100 people — including hospital workers and emergency medical technicians — to determine whether they had been exposed. That number does not include secondary contacts, a spokesman for the agency said.

The woman who spoke to CNN said Mr. Duncan had been sweating profusely and suffering from diarrhea. He also vomited in the apartment complex parking lot, according to local health officials. Like all others exposed to Mr. Duncan, who is listed in serious condition, the woman has not shown Ebola symptoms.

The failure to sanitize his sheets and towels also revealed a broader problem in handling materials possible infected with the virus. Hospitals say they face a major challenge disposing of waste generated in the care of Ebola patients because two federal agencies have issued conflicting guidance on what they should do. As a result, hospitals say, waste may pile up and they cannot get rid of it.

Those who are discovered to have had direct contact with Mr. Duncan will be monitored for three weeks, receiving twice-a-day visits from health officials who will take their temperature and inquire about other symptoms. “Our approach really is to cast a wide net,” Dr. Frieden said.

The four family members who are living in the apartment — the woman, one of her children and two nephews — are among a handful who have been directed by the authorities to remain in isolation, after what officials said was a failure to comply with an order to stay home. Texas health officials hand-delivered orders to residents of the apartment at the Ivy, a complex north of downtown, requiring them not to leave their home and not to allow any visitors inside until their incubation periods have passed.

Image The Texas Health Presbyterian Hospital in Dallas, where Thomas E. Duncan first sought treatment last week but was sent home. Credit... Richard Rodriguez/European Pressphoto Agency

The orders, known as communicable disease control orders, are permitted under the state’s health code. Violations could result in criminal prosecution or civil court proceedings. Police cars were stationed at the complex Thursday to keep reporters out and the family in. Officials said they were arranging to have food and other necessities delivered to the apartment.

President Obama called the mayor of Dallas, Mike Rawlings, on Thursday from Air Force One, officials said, pledging whatever federal resources were necessary to treat Mr. Duncan and make sure the disease does not spread.

State and federal officials sought to reassure local residents that they were doing all that was necessary to contain the virus. But the area remained somewhat on edge, and some parents kept children home from four schools where several students had been removed because of their contact with Mr. Duncan.

Mike Miles, the superintendent of the Dallas Independent School District, said that attendance at the schools had been about 86 percent, down from about the usual 96 percent. Attendance at other schools showed no change. Nurses and other health workers were making the rounds of classrooms to calm students and check for symptoms.

Students arriving Thursday at Emmett J. Conrad High School, down the block from the Ivy complex, said the removal of an exposed student earlier in the week had sent shocks through the classrooms and hallways.