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Louisiana Governor Bobby Jindal is calling for President Barack Obama to halt flights from those countries hardest hit by the deadly disease, just days after the first patient in the U.S. with a confirmed case of Ebola was admitted to a Dallas hospital.
"We should stop accepting flights from countries that are Ebola stricken. Even countries in Africa have cut back on or stopped accepting flights from countries with Ebola outbreaks," Jindal said.
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The patient, Thomas Eric Duncan, fell ill four days after arriving in the U.S. from Liberia on Sept 20. He connected through Brussels before arriving in the country.
The Centers for Disease Control has said halting flights will not help to prevent the spread of the disease.
Bethune said some passengers, on the other hand, will opt to stay off flights.
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"Some of the discretionary traffic is going to go away because they're afraid," he said. "If [the outbreak] increases dramatically, you'll see a dramatic decrease in travel."
—CNBC's Meg Tirrell contributed to this report.