Critics said that he did not have the authority to take the law into his own hands and forgo arrests. But other police departments, fed up with arresting addicts and getting nowhere, saw the Gloucester approach as a promising way to address the epidemic of heroin and prescription pain pills, which together killed 47,055 people in 2014 nationwide — more than died in car accidents, homicides or suicides.

Since the program began, 391 addicts have turned themselves in at the city’s brick police station. About 40 percent are from the Gloucester area; the rest come from all over the country. All have been placed in treatment.

Just as surprisingly, 56 police departments in 17 states have started programs modeled on or inspired by Gloucester’s, with 110 more preparing to do so.

In addition, 200 treatment centers across the country have signed on as partners. In six months, Gloucester, which steers people to treatment but does not itself provide it, has developed a nationwide network of centers willing to provide beds and take referrals by the police, regardless of whether the addict has insurance.

“This has the potential to be a disruptive innovation that changes the picture of how we deal with the disease,” said David Rosenbloom, a professor of health policy and management at the Boston University School of Public Health, who has been analyzing data for Gloucester. And it is a measure of the widespread desperation to move beyond the war on drugs that so many have been willing to try it.