I'm not sure about how other programs work, but at my school, the MPH was the primary degree. Within the MPH there were several different educational tracks: epidemiology, health care admin., environmental health and statistics. Although there was some overlap (I had to take 2 stats courses and one course in each of the other areas (environmental health and health care admin.) each track had a specific list of requirements. I know some programs offer the MHA, but I doubt that such a degree would include enough epidemiology to be really useful.
The other route to clinical epi is via residency after med school. There are no fellowships but there are Public Health residencies. A portion of the residency is spent getting an MPH.
Most physicians who are clinical epidemiologists have their medical degree (DO/MD) and either a MPH or PhD. There are a few (mostly old timers) who have just the medical degree, however, in todays marketplace you'll need the extra training. With a medical degree and just the MPH in epi, you can do just about anything in the field you want (CDC, state epidemiologist, biotech, pharmaceuticals, academia, etc.) the PhD is not usually a requirement. Hope this answers most of your questions. 😀
Neurogirl DO, MPH