what the base rate is

There is a not insurmountable but a pretty large problem here. Rates for which groups? There are a LOT of relevant subgroups (sex, age, ethnicity, social group, geographic group, current medical conditions, previous medical conditions, diet, etc.).

If you just want an overall picture, CDC publishes mortality and morbidity tables, I believe, which should supply you with some sort of base rates.

Medical diagnostic expert systems exist and do reasonably well, but they are not trivial.

On a practical note, the doctors' guild is likely to take a luddite position towards this X-/