How much background information should I provide in a paper if it is a follow-up to a previous publication of mine? I have recently published a paper in which I have identified a problem and developed a potential methodology for solving it. In that first paper, I included a fairly extensive literature review of the subject so that I could support my proposed methodology. The primary goal of that review section was to introduce and elaborate on all of the variables that would need to be accounted for in the model.

The paper that I am currently writing is a follow-up to that original publication where I have gathered the data I need to test the methodology. In this paper, do I need to re-write the sections that introduce all of the elements of my model (the literature review in the first paper), or is it acceptable to state that this is a follow-up to a previous publication and refer the reader back to the initial paper? 

Of course, I will need to provide enough information in the second paper so that a reader will be able to understand the model. However, if I can simply provide an abbreviated explanation of the background and the elements that make up the model by citing my original paper, that would save me a great deal of work. It also would likely improve the second paper, since I will not have to spend several pages explaining things that I have previously established. However, I wanted to check to see if it would instead be wiser to just re-word the material that is present in the first paper.
I think you need to assume that the reader cannot access your first paper for some reason. If the second contains enough description without it, then that would be acceptable in most fields to refer to the previous work for more details/justification.
However, I would caution against you being the only judge of whether you have enough detail as you are likely too familiar with your own work. If you have a colleague who has not read the first article, that could read a draft for you to give their opinion, I would recommend it.