it is easier to talk about 'a public health problem', than to be able to define this in a SMART way. There are many approaches to problem analysis, and the choice of approach will depend on the professional culture. The common elements in problem analysis include: 
Policy makers aim to create conditions and practices that achieve certain strategic targets. In public health such a strategic target could be "protection from childhood diseases", and policies to achieve this can include childhood vaccination programmes. In real life, obstacles can occur that obstruct achievement of the targets, for example low vaccine coverage.
Policy makers may worry about determinants of low vaccine coverage, and may ask for "research to find answers that address this problem". Since this is not a very SMART request, we need to get a clear and concise idea of what exactly is the request and then to agree with the policy maker on a specific question that is researchable, and that can be answered by a study.
During your interview with the policymakers you may discover that their main concern is about loss of confidence in vaccines among parents, which leads them to reject vaccination for their children. In that case you can propose to answer the question "what factors influence parents' decision to vaccinate their children?". Do you see how much more specific (and researchable) this question is compared to "find answers that address the problem of low vaccine coverage"? The policymaker may tell you "yes, but that is what I meant, isn't it?". This may be true, but at least now you know specifically in which direction to aim your study. Then you may move to the next step, which is....
The question "what factors influence parents' decision to vaccinate their children?" may seem already SMART enough to start a study, yet there may be a number of pitfalls that we want to avoid.
What do we know of the vaccination system? How is vaccination offered, who does that, and what sources of information (regarding any aspect of vaccination) do parents have? Do schools require vaccination of their pupils? etc.
In other words, we need to understand the key elements of how vaccines are promoted, offered, adminstered, financed, monitored in our population, in order to enable us to decide if we can then study that factors as possible determinants for the decision of parents to vaccinate children. This will also allow you to .....
Can each of these factors be changed by our policy makers or decision takers? If a factor cannot be changed, then it may not be of practical use to spend resources to study the influence of this factor on the outcome, even though it may still be interesting to the researchers. If our study fully depends on resources that we get from policy makers, we need to avoid that we study only what we think is interesting, but it mainly needs to enable them to do something with the results.
Never assume anything ! Always verify in a direct dialogue by asking questions.
Once we have restricted the aim of our study to specific objectives and questions we can address, then we move to the next step.....
Choosing a study design depends a lot on the questions that you want to answer and on the budget available. Yet before looking even at study design, we may need to take a bigger step back and decide what type of research we may want to apply.
This is what we mostly and traditionally do in field epidemiology. We perform systematic observations of what happens in society (empirical) and analyse the observation results in an objective, controllable and reproducable way.Cohort studies, case control studies and other analytical study designs belong to this class of research. Most of this research is quantitative or semi-quantitative: results are expressed in numbers.
The limitation of this type of research is that it usually requires strict definitions of hypotheses in advance. If we are unaware of the major determinant of the outcome of our interest, then this type of research will not allow us to find it. We only find what we define to look for.
With interpretative research you explore the 'experiences of the study subjects'. What are their ideas, experiences, hopes, fears? This type of research does not only look at 'the bare numbers', yet is usually qualitative in nature. It is often used in antropolohy and sociology.
In our example of the interest of policy makers in determinants of parents' decision to vaccinate their children, we may decide to start with interpretative research, to get an overview of concerns and triggers for parents. Perhaps in a second stage, empirical-analytical studies can quantify the effect of those determinants (though this may not even be a necessary step, and it depends on the possible solutions that policy makers consider).
An example of methods used in this class of research is case study design . Case studies are used in many sciences, including medicine. Though epidemiologists are often reluctant to extrapolate findings from limited number of cases to the whole population, single case studies can increase knowledge in complex areas enormously. For example the description of mr Phineas Cage, a mid 19 century Americam railroad foreman who lived for 12 years with a whole in his head and a large portion of his brain's frontal lobes destroyed, gave essential neuro anatomical insights.
Sometimes we just need to recognize when it is relevant to publish observations on a single individual.
This third class of research is not explicitly quantitative nor qualitative. Any type of data collection method may be used, with exception of 'experimentation'. Usually the researchers aim to contribute to processes in society that promote emancipation of groups. Their view is not only critical towards society, yet also towards the own study results.