WASHINGTON (Reuters) - The family who shared an apartment with a Liberian man who died of Ebola in Texas is showing no signs of illness, while the dog of a nurse who contracted the deadly virus is healthy and being cared for, Dallas's mayor said on Tuesday.

Thomas Eric Duncan's girlfriend, her 13-year-old son and two nephews in their 20s had been living with Duncan before he was admitted to a Dallas hospital on Sept. 28.

"So far no signs of the virus in any of them," Dallas Mayor Mike Rawlings said on CNN. "We check them twice a day, and everybody's healthy."

Rawlings said none of the other people being monitored after contact with Duncan, who died Oct. 8, have gotten sick. There is a 21-day incubation period for the virus that has killed at least 4,400 people, predominantly in West Africa.

Federal health officials are working around the clock at Texas Health Presbyterian Hospital, Rawlings said, as the U.S. Centers for Disease Control and Prevention tries to determine how 26-year-old nurse Nina Pham became infected while caring for Duncan in an isolation ward where he was treated for 11 days.

Pham's dog, Bentley, a 1-year-old King Charles Spaniel, was taken from her apartment Monday night and is being cared for in isolation, said the mayor, who called the nurse a hero.

"That dog was very important to her. We want to make sure that dog is healthy as can be at this point and being taken care of at this point," Rawlings said.

He said Pham was sent a video of her pet, "so hopefully that buoys her up a bit."

Officials have said they do not know how the virus infected Pham, who was wearing protective gear while caring for Duncan. She is the first person known to have contracted Ebola in the United States.

Dr. Brett Giroir, who was appointed by Texas Governor Rick Perry to lead a task force on infectious disease preparedness, said on CNN that every person at the Dallas hospital who had any contact with Duncan was interviewed by the CDC and local health authorities and assessed for risk.

He would not say how many hospital workers were on that list. Those who had real contact with Duncan were being actively monitored by the CDC, he said.

Ebola, which can cause fever, vomiting and diarrhea, spreads through contact with bodily fluids such as blood or saliva.

The infection of the Dallas nurse is the second known to have occurred outside West Africa since the outbreak that began in March. It follows that of a nurse's aide in Spain who helped treat a missionary from Sierra Leone who died of the virus.

Texas Health Presbyterian has been criticized for not admitting Duncan the first time he went to the hospital.

"He should have been identified as an Ebola patient and put in isolation," Giroir said. "There will clearly be lessons learned from this incident."

(Reporting by Doina Chiacu; Editing by Jonathan Oatis)