"Homeland Security Secretary Janet Napolitano echoed the warning and called it "very likely that we will see more serious presentations of illness and some deaths as we go through this flu cycle."

"In a normal seasonal influenza cycle, we would anticipate, across the United States, 35,000 deaths," Ms. Napolitano said. "Of course, this is not a seasonal flu, but I think it would not be unexpected that there will be some more severe illness and some deaths as we proceed forward...." 

Some border-control groups and Rep. Eric Massa, a New York Democrat who serves on the Homeland Security Committee, are calling on the Obama administration to close the southern border to prevent the outbreak from turning into a pandemic.

"The public needs to be aware of the serious threat of swine flu, and we need to close our borders to Mexico immediately and completely until this is resolved," Mr. Massa said.