On a recent early fall morning, hope was in short supply.

My first patient was a regular. Mr. D was a man in his 50s with diabetes. He had been living on Denver’s streets for most of the past five years, two of them with a walker in tow. Without stable housing and reliable access to insulin, he’d come to the hospital that morning with another limb-threatening infection.

I examined the telltale ulceration on the...