U.S. health officials are no longer recommending that schools close because of swine flu.

The government last week advised schools to shut down for about two weeks if there were suspected cases of swine flu. Hundreds of schools around the United States have followed that guidance and closed schools.

Health and Human Services Secretary Kathleen Sebelius said Tuesday that the swine flu virus had turned out to be milder than initially feared. She says the government is changing its advice on closing schools.

Sebelius says parents should still make sure to keep sick children at home.

The tally of confirmed swine flu cases in the United States jumped Tuesday to 403 in 38 states, but officials said that's largely from catching up on a backlog of lab tests rather than a sudden spurt in new infections.

The new number, up from 286 on Monday, reflects streamlining in federal procedures and the results of tests by states, which have only recently begun confirming cases, according to the Centers for Disease Control and Prevention in Atlanta.

The state of Oklahoma confirmed its first case Tuesday, and Tennessee confirmed a second case.

The CDC confirmed four additional cases of swine flu in Maryland Monday afternoon, and Gov. David Paterson announced new numbers of confirmed cases in New York — the majority of them in New York City.

Paterson said New York's tally was up to 90, 73 of which were in the city.

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Beyond eating into the backlog, the new number also reflects that "we do think this virus is fairly widespread," said Dr. Anne Schuchat of the Centers for Disease Control and Prevention in Atlanta Sunday.

"Virtually all of the United States probably has this virus circulating now. That doesn't mean that everybody's infected, but within the communities, the virus has arrived."

Scientists are still gathering information on how severe the nation's at least 30 hospitalized cases are, she said. They are mostly older children and young adults, in contrast to ordinary flu, which tends to send the elderly and very young to the hospital, Schuchat said.

The only swine flu death in the U.S. is that of a Mexican toddler who was visiting Texas.

Tracking Swine Flu

Local authorities announced more school closings Sunday, including all 24 schools in a district west of Detroit after a high school student came down with an apparent case of the illness.

On Monday afternoon, acting CDC chief Dr. Richard Besser, said the CDC is taking another look at its advice about closing schools because of swine flu.

Although Besser did not give further details, the government now recommends that schools with confirmed cases of swine flu close for at least two weeks.

With swine flu, or the H1N1 flu as the government prefers to call it, authorities say it's spreading just as easily as regular winter flu. But they also say it doesn't seem to cause as severe a disease as it did in Mexico.

A big concern is whether the virus will return, perhaps harder, when regular influenza begins its march here. Flu season in the Southern Hemisphere is about to begin, and U.S. authorities will watch how the swine flu circulates there over the coming months as they prepare the first vaccine and then decide whether to order that large amounts of it be produced in the fall.

Production of regular winter flu vaccine is going full-tilt, "to make sure we kind of clear the decks," Health and Human Services Secretary Kathleen Sebelius said.

"We are testing the virus strain for H1N1 virus so that we're ready to go into production later, in a month or two, when we make sure that we have the right dosage and the right tests. So we'll be ready for both," she said.

Even if the swine virus doesn't prove as potent as authorities first feared, Besser said that doesn't mean the U.S. and World Health Organization overreacted in racing to prevent a pandemic, or worldwide spread, of a virus never before seen.

With a new infectious disease, "you basically get one shot, you get one chance to try to reduce the impact," Besser said. "You take a very aggressive approach and as you learn more information you can tailor your response."

It was just over a week ago that authorities learned the new flu CDC had detected in a few people in California and Texas was causing a large outbreak and deaths in Mexico, triggering global alarm.

"We didn't know what its lethality was going to be. We had to move. Once you get behind flu, you can't catch up," Homeland Security Secretary Janet Napolitano said.

The Associated Press contributed to this report.