The Marine Hospital Service wasn’t a “public health” service. It was for taking care of disabled and ill federal beneficiaries. Until 1878, it was concerned with treating members of the military. The first exercise of general public health powers seems to be in the Quarantine Act of 1878, but that was directed specifically at preventing vessels with infected people from entering US ports or informing relevant state and local officials if they did: https://www.loc.gov/law/help/statutes-at-large/45th-congress.... This was focused on infectious disease screening of Ellis Island immigrants. Maybe by this point you can say that public health is within the federal government’s jurisdiction insofar as it’s an adjunct to the government’s power over the border, which is fair. But because the government’s power over the border is plenary, you can shoehorn many things that would otherwise be purely state issues into that.

The reorganization into the US Public Health Service was in 1912, and that’s when it picked up general authority to study infectious diseases. But that era isn’t really relevant to what’s constitutional.