WASHINGTON (Reuters) - Five drug companies are now increasing production of the vaccine for the   H1N1   swine flu , and 10 million more doses are expected next week, Health and Human Services Secretary Kathleen Sebelius said on Saturday.

President   Barack Obama   on Friday expressed frustration about the slow pace of production of the vaccine, which has resulted in just 26.6 million doses as of Friday, far below earlier estimates of 40 million by the end of October.

Sebelius said those initial estimates were based on "overly optimistic" predictions by the five contracted vaccine makers for the U.S. market -- MedImmune, a unit of AstraZeneca, Sanofi-Aventis, Australia's CSL, GlaxoSmithKline and Novartis.

But production was now increasing and vaccine doses were being shipped seven days a week, Sebelius told CNN.

"The good news is that we have, as of yesterday, 26.6 million doses out and around the country. We are expecting another 10 million doses next week," Sebelius said. "So the vaccine is beginning to roll in larger volumes. And it's being distributed as quickly as it comes off the line."

"It's being shipped overnight. We're getting it from producers seven days a week," she added.

HHS initially estimated that 20 million doses would roll out every week, but the companies are currently producing only about 10 million doses a week.

The latest count shows 114 children have been killed by the virus in the United States since April, during a time when there is usually virtually no influenza, according to the U.S. Centers for Disease Control and Prevention.

CDC researchers estimated this week that as many as 5.7 million people in the United States have been infected so far, with at least 1,300 deaths. The flu has been reporting in 48 states, an unprecedented level.

David Axelrod, Obama's senior adviser, told National Public Radio in an interview aired on Saturday that the administration "overpromised" the vaccine based on the companies' assurances.

But he said the problem was abating every day: "We believe that that is improving on a daily basis, and we're going to have an ample supply in very short order."

Sebelius told CNN there was now a good mix of the nasal vaccine and the nasal mist available after earlier shortages, and the government expected to make sufficient vaccine doses available "over the next several months."

The United States still planned to participate in an 11-nation program to donate   H1N1   vaccine to developing countries, but only after the priority population in the United States had been vaccinated, Sebelius said.

"The first priority is to get the vaccine to the American people," Sebelius said. "That's always been the plan. It continues to be the plan."

But she said vaccinations were also critical in developing countries and refugee camps, where hundreds of thousands of people could die as a result of the flu, Sebelius said.

(Reporting by   Andrea Shalal-Esa , editing by Anthony Boadle)