Tommy Thompson, who was in Brussels to meet European Union officials, said despite best efforts to contain severe acute respiratory syndrome, he expected it to cause deaths in areas that so far are unaffected.
"I am not confident at all. ... I do not think SARS is going to go away," Thompson said. "Even though it may level off now it could come back in the fall and then you can, I think, anticipate that you will have deaths in all the continents.
"The virus knows no borders whatsoever."
More than 7,800 people worldwide have been infected - mostly in Asia - since the disease first surfaced in southern China in November. At least 662 people have died.
The United States has reported 67 cases to the World Health Organization but no fatalities.
About 36 cases have been reported in Europe, with no fatalities.
Later Tuesday, a WHO spokeswoman said it was too early to tell whether SARS was a seasonal virus.
"We have only seen SARS for a couple of months and it is too early to know if it will establish a seasonal pattern," Maria Cheng said. "There is no evidence to tell one way or the other."
Thompson, whose department oversees the Atlanta-based Centers for Disease Control and Prevention, said he had allocated more than $20 million to set up an early warning system aiming to stop the spread of the disease.
WHO said over the weekend that most of the world's outbreaks of the disease are subsiding thanks in large part to strict isolation of patients in affected areas.
Thompson reiterated that people should not travel to affected areas until the virus was brought under control there.
"I would strongly recommend that you do not travel, unless it is absolutely necessary," he said.
And while SARS has cut a deadly path through China, Hong Kong and Taiwan, it hasn't affected Thailand much, except for the tourist trade, reports CBS' Christopher Glenn. The Thai government says fear of SARS caused tourism to plunge 50 percent last month. So it's offering a hundred thousand dollars to anyone who can prove he or she contracted the virus during a visit to Thailand.