JERUSALEM—Israel on Sunday began offering a COVID-19 booster to children as young as 12, and its prime minister said a campaign that began a month ago among seniors has slowed a rise in severe illness caused by the Delta variant. Announcing the decision, top Israeli health officials said the effectiveness of the second dose of the Pfizer–BioNTech vaccine waned six months after administration, making a booster necessary. “The third dose brings us to the level of protection achieved by the second dose, when it was fresh,” said Sharon Alroy-Preis, head of public health at Israel’s Health Ministry. “That means, people are 10 times more protected after the third vaccine dose,” she told a news conference, where the expanded booster drive was announced. Those eligible for the third shot can receive it provided at least five months have passed since their second jab—a timeframe shorter than an eight-month interval in effect …