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This was published in the New England Journal of Medicine, involving 5220 children in 15 medical centers across the world. Notably it was a randomized-controlled trial, with half the children getting a sham injection, and the others the real thing.
The flu jab also decreased fever, doctor visits, school absence, and parent absence from work. There were also no difference in the serious side effects in both groups.
Vaccines work by tricking our immune system to form immunity to them. The vaccines contain the same coating as the real virus, but without sickness-causing ability. So they are like decoys. It has helped eradicate smallpox, and much decreased polio in large parts of the world. The flu virus, however, changes its coat every season, thus requiring yearly vaccination against the new strains.
New York City has now decreed mandatory flu vaccination for its preschool and day care centers. This should greatly reduce the sickness caused by the flu virus.