Peanut allergies are on the decline. A new study, published Monday, shows we may be on the way to eliminating the peanut allergies that have been problematic — and even deadly — for decades.

Health experts quickly took notice — and the resulting reversal in pediatric guidance has helped to push peanuts out of the No. 1 spot as the cause of food allergy for children under 3 in the U.S., ac...