The first case of Ebola diagnosed within the U.S. is prompting calls for heavy travel restrictions between the U.S. and those West African countries hardest hit by the outbreak — and one advocate is even warning against the possibility of “Ebola tourism” by patients seeking better care here.

Thomas Eric Duncan, the first diagnosed case, remained in isolation in a Texas hospital Thursday after having traveled from Liberia last month, leaving health officials to try to track down up to 100 persons he may have come in contact with. Hawaii officials, meanwhile, had put a patient in isolation, fearing another Ebola case there, but by Thursday afternoon had dismissed those concerns.

Still, Mr. Duncan’s case has raised thorny questions about U.S. preparedness and the steps the country is prepared to take to try to stem the spread of the deadly virus.

For now, the administration is rejecting calls for a visa ban for West Africans. “I don’t believe that’s something we’re considering,” a State Department spokeswoman told reporters. Health officials have described the Texas case, in which Mr. Duncan exhibited symptoms but was released from a hospital for two days, as a fluke misstep.

Calls for a travel ban extend back to the summer, when the disease first started to spread in Liberia, Sierra Leone and Guinea. Rep. Alan Grayson, Florida Democrat, was one of the earliest to propose restrictions, calling for a 90-day ban on travel from Ebola-touched countries to the U.S.

“If they’d instituted the travel ban when Alan Grayson, of all people, demanded it, [Duncan] wouldn’t be here,” said Mark Krikorian, executive director of the Center for Immigration Studies, who said a ban could have prevented the scrambling health officials are now doing to try to track Mr. Duncan’s movements and see whom he might have infected.

Mr. Krikorian said the decision to restrict travel should come from President Obama himself, but said top U.S. officials appear reluctant to use the country’s borders as a security mechanism to keep potential bad actors out.

Mr. Krikorian raised the notion of “Ebola tourism,” in which someone who fears he or she has come into contact with an infected person quickly schedules a trip to the U.S., where top-notch care is available.

“That’s not something we should be encouraging or allowing,” Mr. Krikorian said.

U.S. officials are now trying to verify the details of Mr. Duncan’s story, including denying having touched anyone with Ebola when he was questioned by airport screeners.

Mr. Duncan’s half-brother told CNN on Thursday that his relative did not come to the U.S. seeking treatment, saying he didn’t have any symptoms and didn’t know he had the deadly virus until after he was here.

Authorities are trying to track down those who may have come in contact with Mr. Duncan during his travels and his time in Texas.

This was reportedly Mr. Duncan’s first trip to the U.S., but he has extensive family here.

Screening out health risks was long a part of the U.S. immigration system, dating back to the Immigration Act of 1891, which excluded “persons suffering from a loathsome or a dangerous contagious disease.”

Today, the law gives the administration the power to refuse entry to foreigners with a “communicable disease of public health significance.” A decade-old executive order specifically lists Ebola as one of those diseases with the potential to cause a pandemic.

But experts question whether it’s possible, or even appropriate, to try to screen out infected persons at the border.

Relying on screening at airports is difficult, said Laurie Garrett, senior fellow for global health at the Council on Foreign Relations, who said during the previous SARS and swine flu outbreaks that uninfected travelers were quarantined while infected travelers got through.

“I think the idea that you can stop a virus at the airport or before it gets on an airplane is as much an antique as the notion that Ellis Island will screen out all disease among the immigrants,” she told reporters.

For now, the efforts to screen out would-be carriers from among travelers to the U.S. seems scant.

Airlines are running passengers through thermal detectors or taking their temperature, and are denying seats to those that have a temperature. But Mr. Duncan didn’t have a fever when he left Liberia and was allowed to board.

Homeland Security officials said they have well-tested procedures that have shown “positive results” in the past in minimizing risks to public health.

U.S. Customs and Border Protection said its personnel are trained to observe and question passengers, looking for “general overt signs of illnesses.” The agency has also posted signs at airport inspection areas and has been distributing a fact sheet to travelers entering the U.S. from affected countries, telling them the signs to watch for if they become sick.

African countries have been far less timid.

International SOS, a travel security firm, lists more than a dozen countries that have imposed some sort of ban, ranging from closure of land borders to outright prohibitions on entry for anyone that has visited an infected country in the previous weeks.

In the U.S., the team responding to Mr. Duncan’s case in Dallas has faced questions over its handling — including having released the patient after a first visit.

Now officials have ordered the woman Mr. Duncan was staying with, her child and two of her nephews to remain quarantined at home. And officials finally sent a contractor to collect and dispose of the sheets soaked with Mr. Duncan’s sweat, which the woman complained had been left at the house for days.

Five school-age children are being monitored at home because they interacted with Mr. Duncan prior to his hospitalization on Sunday. The revelation spooked parents of fellow schoolchildren, some of whom kept their own children home.

Mike Miles, superintendent of the Dallas Independent School District, said the children will be hooked into a program that lets them continue their education at home.

While Mr. Duncan is one of just a handful of Ebola patients in the U.S. — the others were diagnosed overseas and brought back here on a special plane — the danger of an outbreak has raised all manner of public health questions.

One analysis from 2013, published by the Global Policy Journal, even raised the question of whether al Qaeda-inspired terrorists in West Africa could harness the Ebola virus as a weapon.

“The increase in natural outbreaks in the region, coupled with a possibility of a terrorist group recruiting experts to acquire the virus and to prepare it to use as a bioweapon, should lead policymakers to consider the risk of a deliberate outbreak,” Amanda M. Teckman, an administrative assistant at Seton Hall University, who earned her master’s degree from the university’s school of diplomacy and international relations, said in the analysis.

“This prospect is worthy of consideration, particularly in East Africa due to the history of terrorist attacks by different groups in the area; the potential for these groups to obtain Ebola in the field; the lack of political capacity in the region and global will to develop a vaccine; and the pathogen’s natural occurrence in the region,” Ms. Teckman wrote.

• Guy Taylor contributed to this report.

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