By being so readily available, One Medical can reduce visits to an emergency room or an urgent care center, said Dr. Jeff Dobro, the company’s chief medical officer.

As primary care doctors become an “increasingly endangered species, it is very hard to practice like this,” he said.

Long-Term Lifelines

But more traditional doctors like Dr. Purifoy stress the importance of continuity of care. “It takes a long time to gain the trust of the patient,” he said. He is working with Aledade, another company focused on reinventing primary care, to make his practice more competitive.

One longtime patient, Billy Ray Smith, 70, learned that he needed cardiac bypass surgery even though he had no symptoms. He credits Dr. Purifoy with urging him to get a stress test.

“If he hadn’t insisted,” Mr. Smith said, “it would have been all over for me.” Dr. Purifoy’s nurse routinely checks on him, and if he needs an appointment, he can usually see the doctor that day or the next.

“I trust him 100 percent on what he says and what he does,” Mr. Smith said.

Those relationships take time and follow-up. “It’s not something I can do in a minute,” Dr. Purifoy said. “You’re never going to get that at a MedExpress.”