The World Health Organization said it could be days before scientists are even able to reliably determine how big a threat the new virus poses.
Officials in New Zealand and Israel confirmed a handful of cases, and said other people were still suspected of infection with the H1N1 virus, which is suspected of killing 152 people in Mexico.
The number of confirmed infections in the United States almostas the Obama administration said it was responding aggressively, as if the outbreak would spread into a full pandemic.
New Zealand's health minister says his country has 11 confirmed cases of swine flu - the first in the Asia-South Pacific region.
Tony Ryall told reporters in the capital of Wellington that "New Zealand's status is we have 11 confirmed cases'' of swine flu and 43 suspected cases.
Those infected were members of a group of students and teachers from a single school that reported having fevers and other flu-like symptoms on their return from a recent visit to Mexico.
He said swine flu was confirmed by laboratory tests on samples from three of 11 students from the group and "on that basis we are assuming" the eight others are also infected.
An Israeli hospital, meanwhile, confirmed the country's first case of swine flu and said the patient had fully recovered.
Hospital officials said the 26-year-old patient recently returned from Mexico. Laniado Hospital's medical director said Israeli Health Ministry laboratory tests confirmed swine flu. Dr. Avinoam Skolnik said he didn't know whether the strain was the same one that appeared in Mexico.
Skolnik said Tuesday the patient was in "excellent condition." No other details were immediately available.
No new cases of the disease were reported in Mexico or the United States overnight. Thus far, the only deaths confirmed to have been caused by the virus (20 as of Tuesday morning) have been in Mexico, posing one of the most urgent questions for health officials.
The World Health Organization (WHO) upgraded its global alert on Monday to an unprecedented level-4. The highest ranking is 6, which indicates a full-blown pandemic.
WHO flu spokesman Gregory Hartl said Tuesday the alert was raised because evidence showed swine flu passing from human to human - without any contact with infected animals. He also said scientists suspect U.S. swine flu patients may have transmitted the virus to others in the United States.
Confirmation would indicate the new flu strain is spreading beyond those travelers returning from Mexico.
The global health agency says so far, most people confirmed with swine flu were infected in Mexico. But Hartl said the source of some infections in the United States, Canada and Britain was unclear.
Hartl said two of the main challenges facing the world's health authorities are figuring out how efficiently the virus spreads within a population, and.
He said it was likely the virus would continue to spread around the world in the short term, as efforts to limit cross-border travel had failed to halt the disease.
Hartl said the WHO was recommending that all countries drop their travel bans and warnings, saying, "it didn't work." He said the economic cost of restricting peoples' movement around the world would be greater than the cost, in terms of public health, of trying to stop the spread of a virus which had already crossed so many borders and was spreading indigenously.
"Border controls don't work. Screening doesn't work," he said at a news conference at the Geneva headquarters of the United Nation's public health agency.
No airport quarantine units have been activated in the U.S., reports CBS News correspondent Thalia Assuras. Customs officials are using only "passive surveillance," - simply watching for symptoms and questioning suspect passengers.
Other countries have taken more dramatic measures, reports Assuras. Russia, Hong Kong and Taiwan have sought to quarantine people showing symptoms, and Japan is using heat sensors to detect flu-like temperatures in airport arrival halls - though their reliability is questionable.
European Union officials warned citizens Monday against traveling to the United States or Mexico.
Hartl also said the WHO had still not determined the source of the swine flu.
Mexican and American health officials are urgently trying to zero in on the origins. One potential lead, reported CBS News correspondent Hari Sreenivasan, are the massive industrial hog farms that have sprung up in Mexico in recent years - some operated by U.S. companies such as Virginia-based Smithfield Foods.
They deny being the source and say they're cooperating with health officials. But Sreenivasan reported that just last year the nonprofit Pew Charitable Trusts warned that hog farms could become breeding grounds for new strains of the flu.
"The warm conditions and the close proximity of animals being able to pass viruses back and forth and to the human workers," said Bob Martin of the Pew Environmental Group. "It's a situation ripe for the development of a novel virus."
As for the disease's mysterious fatal exclusivity to Mexico, Hartl said it remained just that, a mystery. "We don't understand why the disease has been more severe in Mexico," Hartl said. He suggested it may be due to other flu-season illnesses already being carried by the hardest-hit populations, weaker immune systems in the area, or a failure by medical officials to identify and treat the illness quickly enough, as it was still unrecognizable at the time.
Hartl said it could still be days before WHO scientists were able to determine exactly how virulent this strain of H1N1 swine flu is - whether it passes easily between people of different age groups and fitness levels particularly. This assessment, he said, would help the world body decide whether to elevate the risk level beyond the current 4.
Monday evening, New Jersey health officials said they had identified five probable cases of swine flu in people who recently traveled to Mexico and California.
The New Jersey Department of Health and Senior Services said Monday that all have mild forms of the flu and none has been hospitalized. The department is arranging for confirmatory testing at the CDC.
Also, 14 schools in Texas, including a high school where two cases were confirmed, will be closed for at least the next week. Some schools in California and Ohio also were closing after students were found or suspected to have the flu.
The CDC is releasing 11 million doses of the stockpiled anti-viral drug Tamiflu to affected areas, reports CBS News correspondent Kelly Wallace. New York City pharmacists report a run on the drug, which requires a doctor's prescription. But the CDC fears that doctors giving it to patients who don't really need it may cause shortages for those who are sick.
Ariana Drauch, a swine flu victim from St. Francis Prep, told Wallace her family can't find it anywhere.
"We called every drug store in Queens, New York, everywhere," Drauch said. "And there is no Tamiflu available."
Swine Flu News Worldwide: