DALLAS — Health officials said on Friday that they had identified 10 people who are most at risk of contracting Ebola after coming into contact with an infected African man now in isolation in a Dallas hospital.

Among them, health officials said, are the four people who shared an apartment with the patient, Thomas E. Duncan, and medical workers who treated him. Another 40 people are being monitored daily but are considered at relatively low risk, officials said. No one has developed any symptoms of the disease.

For those who have been exposed to the virus, there is nothing to do but wait.

“There really isn’t anything that can be done to prevent Ebola from developing if infection has occurred,” said Beth P. Bell, the director of the National Center for Emerging and Zoonotic Infectious Diseases at the Centers for Disease Control and Prevention.

While officials said the risk of the disease spreading widely were minimal, they warned that others might fall ill. “There certainly is a possibility that some of the people who have been in close contact with the patient might get sick with Ebola,” Dr. Bell said.

The first signs of the illness often appear within eight to 10 days, but can take as long as 21 days.

Image Thomas E. Duncan at a wedding in Ghana in 2011. Credit... Wilmot Chayee, via Associated Press

Mr. Duncan arrived in Dallas from Liberia on Sept. 20. Ebola cannot be spread until a person is symptomatic, and Mr. Duncan told doctors he first felt ill on Sept. 24. As he became sicker, the amount of virus in his system increased, and so did the risk of contagion.

Anyone who came into contact with him between Sept. 24 and Sept. 28, when he was placed in isolation at the hospital, is considered at risk of infection.

The four people who shared an apartment with Mr. Duncan before he was hospitalized remained in quarantine on Friday, and Texas officials apologized to them both for their failure to communicate and for not moving more quickly to have the apartment cleaned of possibly infectious materials.

After much delay, the cleanup began at the Ivy Apartments Friday afternoon as workers in yellow protective suits scoured the property, taking any item that might harbor the virus, which can survive outside the body for several hours on hard surfaces and for several days in pools of fluid. At least two of the contractor’s trucks were parked outside of Apartment 614, where Mr. Duncan had been staying. The green trucks were brightly painted and had advertising slogans on them, including “Cleaning Guys, Haz-Mat Clean-Up” and “The Annihilator.”

The top elected official in Dallas County, County Judge Clay Jenkins, said he visited the family on Thursday night to apologize. He said that they were understandably scared and confused, and that he hoped to find them more comfortable accommodations.

“I want to see them treated as I would want my own family treated,” Mr. Jenkins said.

The disease centers and the Department of Transportation have different protocols for handling medical waste associated with diseases such as Ebola, which led to a greater delay in getting the apartment cleaned.