I. What is Public Health
A. Medical Research via Study of Populations
B. Prevention and Intervention in Populations
C. Common to all - statistically driven approach is used to discover information and to evaluate interventions

II. Are public health activities Pikuah Nefesh?
1. Pikuach nefesh is usually limited to situations with a significant level of danger as measured from the perspective of the individual.
Public health efforts often address behavior that is only marginally dangerous to any given individual, especially if only engaged in on an isolated occasion, but on the scale of a population or an entire lifetime represents a significant threat. Consider the eating of hot dogs, which is believed to contribute to an individual's risk of a number of cardiovascular diseases and cancers. Yet no evidence indicates that eating an occasional hot dog will cause such an outcome. In the language of halacha, this would be permitted on the grounds that "shomer pta'im hashem", any individual is free to take very remote risks and rely on Hashem to protect him. This is apparently outside the applicability our usual vigilance with regard to safek nefashot. There is a serious question as to whether given our current level of statistically based knowledge, we can say with certainty that a known number of people who engage in the activity will die, and thus at the population and policy level we should view the issue as one of safek nefashot or even vadai nefashot, and be required to drop everything and intervene.

The big question is how can we allow risk taking on the basis of Shomer Ptaim Hashem even in the case of remote risks- doesn't that countermand the imperative of pikuach nefesh even in the case of remote danger?

There are 2 basic approaches to understand Shomer Ptaim Hashem -
1.  It is dangerous, at a level that would be significant for pikuach nefesh if we were responding to an existing emergency, but we don't need to have that level of vigilance with regard to preventative measures. Philosophically, the emphasis here is on the shmira from Hashem. It's not somchin al hanes to rely on hashem to protect you personally when no emergency yet exists. This may be the meaning of the passage in yevamot which emphasizes Hashem's protection.
2.  Dashu Bei Rabim is statistical observation that its really not dangerous at all - in this view the same standard of statistical danger applies both to pikuach nefesh and shomer ptaim hashem. This may be the view of the passage in the gemara that says "kevan de'dashu bei rabim, shomer ptaim Hashem" - because we observe that many people do it and they don't get hurt, we have statistical proof that Hashem always protects them, so practically speaking it isn't dangerous. In today's scientific language we would probably just call it "not dangerous", because today we define danger/risk as a function of how often people actually get hurt doing something.
In other words, the notion of shomer ptaim hashem is only applicable where there is no statistical evidence of elevated risk, only a subjective perception of danger. For example, air travel versus car travel. In that view, pikuach nefesh could be applied to anything known to statistically raise the risk of death, provided the intervention were undertaken on behalf of people currently at risk.

(Most scientifically-oriented achronim agree that exceptionally hazardous behavior such as smoking crosses the threshold of shomer ptaim hashem and does become a pikuach nefesh issue. Even less dangerous behavior, if engaged in beyond certain thresholds, could approach this level - eating three hot dogs every day might approach a pikuach nefesh concern.)

2. Pikuach Nefesh is really limited to "Choleh Lefaneinu".
The Noda Beyehuda and most achronim who follow him (Tzitz Eliezer, Chatam Sofer, R. Moshe Feinstien) view pikuach nefesh as limited to efforts on behalf of a person who is currently afflicted, as opposed to medical training or research. For this reason the Noda Beyehuda opposed scientific research that will desecrate deceased human bodies, and R. Moshe wrote that going to medical school is not a valid reason to stop learning Torah (!) .

It is possible to debate the definition of lefanenu, they obviously don't need to be physically present, it is enough to know they exist. Today of course it would be trivial to get a list of the patients worldwide suffering from a particular condition, especially if we were offering a possible treatment for it. Most likely, today it means that the activity could save people currently afflicted with the condition, as opposed to bearing fruit only at some date in the far future when it could only save people who are currently healthy. Note that the Tzitz Eliezer (1915-2006) who fundamentally accepts the Noda Beyehuda's opinion, speaks as though practically speaking in the usual case we do have an afflicted person "lefaneinu".

To recap, most public health research and prevention activities would not fall under pikuach nefesh and thus we would not have the unequivocal imperative to engage in them or license to violate halacha on their behalf. Some public interventions might fall under pikuach nefesh, if they have the potential to save those currently afflicted. It is possible that "current affliction" should include those at the asymptomatic phase of the condition, which would broaden this more.

III. Given that it's not normally Pikuah Nefesh, what is it?
R. Moshe seems to say that studying medicine, given that it's not pikuach nefesh, is totally optional and not really important.
Tzitz Eliezer, by contrast, says that setting up kupot cholim is very important and commendable even if it is not pikuach nefesh.
Presumably even if it's not pikuach nefesh (because it's too indirect or because it's for conditions that are not fatal) it would still fall under the broad imperative of returning "aveidat gufo". We know from the gemara in bava metzia that systems were put into place to facilitate hashavat aveida, they didn't just say that if you happen to see an avaida, go find the guy somehow. Setting up those systems is presumably part of the process, whether formally a mitzva or not. Setting up public health systems and performing research would be comparable to that - it is preparation for hashavat aveidat gufo.
Of course chazal also spoke in many places about the special importance of working on tzorchei tzibbur, communal needs, of which health is obviously a paramount example.

IV. Applications
It might seem that if public health activities are not actually pikuach nefesh, this downplays their importance. It is true that this lowers their imperative, however, this also frees us to exercise more discretion. In pikuach nefesh we do not draw any distinction between different quality or duration of life, we just work at maximum capacity to save those whom we encounter first. This is a far cry from a public policy approach, where we do look at what interventions can save the most life and improve quality of life most. Because it isn't pikuach nefesh we have more license to choose to allocate resources in the most effective way. We couldn't choose to let an old person die to save a young person in the operating room, but given a fixed budget we can choose to research or prevent diseases afflicting young people because the total amount of years saved will be far greater, assuming an equal number of subjects in both groups. We can choose to screen for the more common disease, because this will save more people than screening for the less common disease. We can also choose to study a particular condition based on similar considerations.

Within saying that we can exercise judgement, though, there is a further question about whether one should only exercise "objective" judgement, or whether one may use subjective judgement as well. In a pikuach nefesh scenario, one would not be allowed to rescue one person from a fire instead of ten people from drowning, even if he is especially concerned about fire. But in a public health context this might be OK. The closest analog would be the allocation of tzedaka. There we find that there are some basic objective rules - prioritizing food over clothing, etc. But "subjective" values are also allowed or even mandated, like giving to one's relative first. I believe a strong argument can be made that when dealing with public money and public resources we should do our best to use "objective" criteria, but when an individual makes a donation or effort they are free to support the project of their choice.