PHOENIX, ARIZONA — Beatrice Herron, 73, clutched a flier offering low-cost cable TV, imagining herself settling into an apartment, somewhere out of the Arizona heat where,like others her age, she can settle into an armchair and tune into a television of her own.

Instead, the grandmother and former autoworker can be found most mornings in a food line, or seeking shade under the awning of a mobile street clinic. At night, she sleeps on a floor mat at a homeless shelter. She laments the odors of human waste outside and the thieves who have victimized her repeatedly.

“My wallet’s gone,” she said. “My purse was stolen.”

She hardly stands out from the dozens of seniors using wheelchairs and walkers at a complex of homeless shelters near downtown Phoenix, or from the white-haired denizens of tents in the surrounding streets — a testament to a demographic surge […]