Assurances from US health officials that there is almost no chance of a US Ebola outbreak are doing little to allay Americans' fears, days after the first domestic case was diagnosed.

"We have seen a lot understandable concerns," Tom Frieden, director of the US Centers for Disease Control and Prevention said Sunday.

The patient, Thomas Eric Duncan, who traveled from Liberia to Texas in late September, was in critical condition Sunday at a hospital in Dallas.

A native of Liberia, Duncan arrived in the United States on September 20 without exhibiting any symptoms.

His illness was diagnosed last week -- but only after he was turned away from the hospital days earlier, potentially infecting several people he came into contact with while infectious.

Before announcing the American case last Tuesday, the CDC was receiving around 50 Ebola-related phone calls a day, Frieden told reporters during a phone conference.

Since then, the number has jumped to 800, he said.

US fear is evident in the difficulty Texas authorities had finding alternate living accommodations for four people being forcibly quarantined in the apartment where they and Duncan had lived.

It was also apparent in the near impossible search for a cleaning or maintenance service to decontaminate the apartment and take away a garbage bag that held the sheets that were on Duncan's bed when he was sick.

Ted Cruz, the conservative Republican senator from Texas, sent a letter to the Federal Aviation Administration suggesting flights from the West African nations hard hit by Ebola not be allowed to land.

Fear not taken 'lightly'

"To the people who are scared, we don't take lightly your fear, we respect it, we understand it, but we have to get our actions and our policies based on scientific evidence," Anthony Fauci, director of the National Institute of Allergy and Infectious Diseases, told "Fox News Sunday."

"We know if you do isolation, 'contact tracing' and appropriate treatment, you will not have an outbreak," he said, adding that there could still be other cases of infected passengers arriving on American soil.

US authorities have stressed daily that they are taking no chances.

"The first thing is make sure the American people understand how hard it is to contract Ebola and to understand that there is no country in the world better prepared than the United States to deal with this," Dan Pfeiffer, senior adviser to President Obama, told NBC's "Meet the Press."

In Dallas, 10 people who were in close contact with Duncan are being tested two times a day for signs of infection such as fever, until they have passed Ebola's 21-day incubation period.

A total of 46 others who were less exposed are being tested once a day. One of the individuals, who had been missing since Saturday, was located by police Sunday afternoon.

Frieden said that none of the people, including the 10 most at risk, so far have exhibited any sign of infection.

On Saturday, authorities held a plane from Brussels with 250 passengers onboard for more than two hours at the Newark, New Jersey airport after one traveler vomited. The CDC eventually determined that it was not due to Ebola.

A total of six Americans, three of whom are doctors, have been infected by Ebola in West Africa.

The World Health Organization says the outbreak has killed 3,439 people, of whom 2,069 died in Liberia.