LONDON — The anti-vaccine movement has come for the pets.

A spreading fear of pet vaccines’ side effects has prompted the British Veterinary Association to issue a startling statement this week: Dogs cannot develop autism.

The implicit message was that dog owners should keep vaccinating their pets against diseases like distemper and canine hepatitis because any concerns that the animals would develop autism after the injections were unfounded.

The warning has a long tail. It grew out of an anti-vaccine theory that rippled across the United States and Europe as networks known as “anti-vaxxers” claimed that childhood vaccinations could cause autism. The belief, promoted by some celebrities like the television personality Jenny McCarthy, who says her son has autism, spurred many parents to begin boycotting traditional vaccines.

The theory gained prominence in 1998, after a study published in the medical journal The Lancet purported to show a link between autism and the measles-mumps-rubella vaccination. It caused a firestorm in health circles and among parents, resulting in a significant drop in vaccination rates for children in Britain.