On Sunday, city and county officials said they had found a homeless man who was among those being monitored. The man rode in the ambulance that took Mr. Duncan to the hospital, before the vehicle was taken out of service and cleaned. Officials said the man, whom they identified as Michael Lively, had been monitored Saturday, was asymptomatic, and apparently went into hiding on Sunday before he was found.

Prosecutors here are considering whether to bring charges against Mr. Duncan following reports that he had contact with a woman who ended up dying of Ebola in Liberia and later lied about that contact before boarding a plane to the United States.

“We are actively having discussions as to whether or not we need to look into this as it relates to a criminal matter,” Dallas County’s district attorney, Craig Watkins, said on “Lone Star Politics” on NBC 5 on Sunday. “We’re working with all the different agencies to get to the bottom of it.” His spokeswoman said prosecutors had previously filed aggravated assault charges against people who were H.I.V. positive and knowingly had unprotected sex with others; and it was possible the same charge may apply in Mr. Duncan’s case.

At the apartment where Mr. Duncan and the others had been staying, about 20 containers were filled with linens and other items. The containers were being stored “in an extremely safe and secure location, and guarded by Dallas County Sheriff’s deputies,” Sana Syed, a city spokeswoman, said. The potentially contaminated medical waste that has been generated so far at the hospital, like bedsheets, gowns and gloves, has been incinerated.