The study found that opioid-related spending on doctors was most highly concentrated in counties in the Northeast; the Midwest had the lowest concentration.

Areas with large numbers of payments and high overdose rates included four cities in Virginia — Salem, Fredericksburg, Winchester and Norton — as well as Cabell County, W.Va., which has one of the highest overdose death rates in the nation. Lackawanna County, Penn., which includes Scranton, also ranked high in both measures, as did Erie County, Ohio.

The authors said they were particularly struck by the fact that the number of marketing interactions with doctors — such as frequent free meals — was more strongly associated with overdose deaths than the amount spent.

“Each meal seems to be associated with more and more prescriptions,” Dr. Hadland said. He added that while pharmaceutical company payments to doctors seem to have started dropping, the practice of companies buying meals for doctors “remains alive and well.”

The study noted that while some states have sought to limit the total amount drug companies spend promoting their products to doctors — New Jersey, for example, recently adopted a new regulation limiting such spending to $10,000 per doctor, per year — what may matter more is for states or health systems to limit the number of interactions.

“I think what seems to be less important is the amount of money spent,” Dr. Hadland said, “compared with the number of interactions.”

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The study linked information from 2013 to 2016 across three national databases: the Open Payments database, which includes all payments made by pharmaceutical companies to physicians, which companies are required to report under a section of the Affordable Care Act; drug overdose data from the Centers for Disease Control and Prevention; and C.D.C. data on opioid prescriptions dispensed at pharmacies.