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Noah Abdullah hasn’t immunized his 4-year-old son, Michael, saying that he’d read vaccines might be “no good” and that he’d “rather do natural things” to strengthen his child’s immune system.

“I need to see more information before I start shooting him up with stuff,” Mr. Abdullah said.

Donna Mosley said her 3-year-old grandson also did not have his vaccinations, though she wishes he did. His mother is afraid the shots could cause autism, she said, and his father’s Muslim beliefs have made him “totally against it.”

The two boys attend Sister Clara Muhammad Elementary School in Harlem, a small school where most children had a religious exemption to immunization in the last school year, according to city health department data.

As the measles outbreak deepens in New York City, health authorities have been focusing on schools affiliated with ultra-Orthodox Judaism, because those are the only city schools within which measles transmission has occurred so far. But immunization data, reported annually by every school to the state, suggests that reluctance to vaccinate in New York is much more widespread .