The unusual confinement order was imposed yesterday after the family failed to comply with a request not to leave their apartment, according to Dallas County Judge Clay Jenkins.
Texas State Health Commissioner David Lakey said the order would help ensure the four can be closely watched, including checking them for fevers over the next three weeks.
"We didn't have the confidence we would have been able to monitor them the way that we needed to," he said.
The family will not be allowed to receive visitors, officials said.
Authorities were also concerned about the cleanliness of the home and hired a cleaning service, Lakey said.
"The house conditions need to be improved," he said.
A woman who lives in the apartment, Louise Troh, said she has been quarantined with her 13-year-old son and two nephews. "Who wants to be locked up?" she said in an interview with The Associated Press.
Troh said she was waiting for the Centers for Disease Control and Prevention to collect a bag of the bed sheets and towels Thomas Eric Duncan used.
Visitors from the American Red Cross were seen yesterday bringing food to the apartment door. The North Texas Food Bank said it sent three days of cereal, tuna, produce and other supplies.
Outside the apartment, the management of the 300-unit complex in northeast Dallas was passing out flyers about Ebola to neighbors. Private security guards and local sheriff's deputies blocked off the entrance to dozens of reporters.
Apartment manager Sally Nuran said employees were power-washing sidewalks and scrubbing common areas, though she believed Duncan had not visited most of the complex in his short time there.
Elsewhere, Texas health officials expanded their efforts to contain the virus, reaching out to as many as 100 people who may have had direct contact with Duncan or someone close to him.
None of the people is showing symptoms, but public-health officials have educated them about Ebola and told them to notify medical workers if they begin to feel ill, Erikka Neroes, a spokeswoman for the Dallas County Health and Human Services agency, said yesterday.
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