YARE, Venezuela—Jean Pierre Planchart, a year old, has the drawn face of an old man and a cry that is little more than a whimper. His ribs show through his skin. He weighs just 11 pounds.

His mother, Maria Planchart, tried to feed him what she could find combing through the trash—scraps of chicken or potato. She finally took him to a hospital in Caracas, where she prays a rice-milk concoction keeps her son alive.

“I...