A Dallas apartment where the first person diagnosed with Ebola in the United States had stayed is finally getting a thorough cleaning, days after the diagnosis left four people quarantined there with soiled towels and sweat-stained sheets from the Ebola patient.

After some delays, the first of three phases to clean the apartment began Friday afternoon. While the process will take days, at least sheets and towels that Ebola patient Thomas Eric Duncan used have been removed.

Also out are the four people -- the partner of the Ebola patient Thomas Eric Duncan, her 13-year-old son and her two 20-something nephews -- who'd been stuck there for days.

They had been ordered to stay inside the apartment until October 19. By that point, enough time should have passed to determine if any of them contracted Ebola or if they're in the clear.

Judge Clay Jenkins, director of the county's Homeland Security and Emergency Management, said Friday that Duncan's partner slept last night on a couch pillow on the living room floor. But now she and the others have a new place to call home in the meantime: a private 4-bedroom residence in Dallas, which was arranged with the help of someone in the local faith-based community.

"They were very glad to be going and very pleased when they got there," Jenkins said. "(One of them) got a big grin on his face, and he said, 'I wish somebody would get me a basketball.' So we're going to see if we can't take care of that."

The move added to a busy day for the quartet, who until then had gotten little more than brief stops from a health official and were told not to go more than a few steps outside. On Friday, they were visited by a hazardous materials team wearing masks, boots and yellow productive, as seen in photos tweeted by Dallas city spokeswoman Sana Syed. A large tarp was laid out, and several oil-drum like containers were on site.

Some have criticized the fact that it took so long to start the process, given that health officials announced three days ago that Duncan had Ebola. At that time, four people Duncan had stayed with in the Texas city were ordered not to leave the apartment because of possible lingering effects of Duncan, from his clothes to toilets to silverware.

Additionally, the Dallas hospital where Duncan is being treated has come under fire for how it handled his first visit there eight days ago.

Duncan's partner, identified only as Louise, told CNN's Anderson Cooper that hospital medical staff were twice told that Duncan, who was suffering with fever and abdominal pain, had recently arrived from West Africa -- key information that could have been a tipoff for Ebola, yet was never properly relayed. Instead, he was released with an antibiotic only to come back by ambulance even sicker on Sunday.

Dr. Anthony S. Fauci, the director of the National Institute of Allergy and Infectious Diseases, acknowledged to reporters Friday that "there were missteps" in Dallas as to how Duncan's case was initially treated. But he also insisted that "there were a lot of things that went right and are going right."

And Fauci stressed that, while there may be a case here and there, what's happened with Ebola in West Africa -- where more than 3,400 people have died and public health infrastructures have become overwhelmed -- won't happen in the United States.

"Our health care infrastructure in the United States is well-equipped to stop Ebola in its tracks," Fauci said.

(CNN)