Officials say it's still too early to determine how the disease spread from La Gloria into a global health emergency.
"It's a new virus, new virus combination, it does transmit from person to person and we already know it causes fatalities so we already have all the makings of a possible pandemic," Irwin Redlener of Columbia University's Mailman School of Public Health said.
But Dr. Nancy Cox of the CDC has said she believes the earliest onset of swine flu in the United States in this current outbreak happened March 28.
A quarter of the 50 million doses of Tamiflu stockpiled by the U.S. government has been released and the Obama administration has declared a public health emergency to free up the medicine and federal help to the states who need it.
But pharmacies in several states have been flooded with phone calls from concerned customers.
"Our first phone calls were doctors asking if we had Tamiflu," New York City pharmacist Yvonne Zampitella said. "They were prescribing it for their patients and family members."
Symptoms of the swine flu are similar to the regular flu, health officials say, including aching muscles, fever and fatigue. The virus appears to be responsive to medication.
"These drugs do not kill the virus, they help prevent its replication and therefore help reduce the symptoms, but they have to be taken within 48 hours so people have to recognize they have a serious illness, get to a doctor and start treatment," ABC News' Dr. Tim Johnson said on "World News" Monday.
"But we should not be telling people to go out and buy these drugs for use as preventive measure. We need to reserve drug for actual cases and outbreak."
The Associated Press contributed to the reporting of this story.