Your question doesn't specify how well-adapted the proposed research is to the particular public-health professor you've contacted. I think this is a key issue. Let me explain:
Most academics have broad training, some expertise across a wide range of areas, and then are true masters (in some cases, the top ones in the world) in a much smaller range of topics. I considered at this point giving examples from my own experience to indicate what sort of things which are "up my alley" that would get me to respond quickly rather than to be interested in principle but respond only much later or never, but this ran into the following issues: (i) the examples would be meaningful only to mathematicians with some experience in number theory, which I think the OP is not, (ii) unfortunately every time I came up with what I thought was a description of "my alley", I could then think of someone who sent me work or questions in that area to which my response was inadequate or even non-existent! But I think there is a lesson here: when you send a paper (or a question or an idea) to anyone, it is always a bit of a crapshoot: you may well get absolutely no response, and the reasons for this may have absolutely nothing to with you or your work. In fact history is filled with examples of much more famous mathematicians rejecting -- or worse, ignoring -- much more important work.
The moral of the above paragraph is this: you should cast a wider net. If there is one faculty member that you feel is the perfect person to get involved with, then you may as well contact them first, and the job of your message ought to be to convey to them why they are a perfect match for the project at hand (and that the project at hand may be worthy of their time). Even so, your perfect match may not see it that way or may simply be too busy to give you the attention you deserve. (Again, apologies to all the people who sent me good work or asked me good questions over the years to which I did not respond. I definitely respond to some of the correspondence I receive; I do not always have time to respond to all of it; and I sometimes make what looks in retrospect to be bad choices about what to respond to. In particular, something promising that I don't have the time to think about at the moment tends to get put on the "answer later" shelf, but my algorithm for addressing correspondence does not have a step which ensures that the answer will actually come later.) It is okay to write twice to someone when you think it is important, and the repetition can be distressingly effective: as I wrote elsewhere on this site, unfortunately for some types of correspondence I implicitly assume that if I only hear about it once then it cannot be so important. This gets me in trouble from time to time, but not as often as it should.
More likely though there will be several people who could help you with your idea. The classiest move is to contact each of them individually and explain why their expertise is well-tailored to your idea. The chance that you hear back from any one person increases dramatically this way. And then you can get started with the person who you hear back from. In some cases this may result in getting emails later on from people who were interested but didn't have time to reply. Those, as they say, are the breaks.