Workers unload containers as officials prepare to empty the apartment of a health worker who was infected with the Ebola virus at Texas Health Presbyterian Hospital in Dallas, Texas, Oct. 13, 2014. Reuters

The second health care worker in Texas to contract Ebola lived alone and had no pets, Dallas Mayor Mike Rawlings said Wednesday. The unidentified woman tested positive for the deadly virus after caring for an Ebola patient at a Dallas hospital.

The woman is the second health care worker at Texas Health Presbyterian Hospital who has tested positive for the disease, according to the Texas Department of State Health Services. The woman reported a fever Tuesday and was immediately isolated at the hospital.

“We are preparing contingencies for more,” said Dallas County Judge Clay Jenkins in a Wednesday morning press conference. “And that is a very real possibility.”

Texas Health Resources Chief Clinical Officer Dan Varga defended the hospital's ability to contain Ebola. “A lot is being said about what may or may not have occurred for our colleagues to contract this disease,” he said. “Let’s be clear. We’re a hospital that serves this community incredibly well, and we have for nearly a half a century.... Our team's spirit is tried and tested.”

The worker was among the medical staff who took care of Thomas Eric Duncan after he was diagnosed with Ebola. Duncan, of Liberia, died last week. The first health care worker diagnosed with Ebola is nurse Nina Pham. She had a dog, who is now in quarantine.

According to the CDC, 76 people at Texas Health Presbyterian Hospital may have been exposed to Duncan after his second visit to the emergency room, and 48 more people, who may have been exposed before he was hospitalized, are being monitored.

The announcement of the latest Ebola case in the U.S. comes after National Nurses United, a nurses' union, claimed late Tuesday that the Dallas hospital did not have enough protective measures in place to deal with Ebola patients.