If I were writing about disease stages and I wanted to find a relevant style guide to help me with decisions such as whether to capitalize stage in a term like "Stage I," and whether to capitalize descriptive stage names such as "Clinical Latency," I would obtain and consult the American Medical Association's AMA Manual of Style: A Guide for Authors and Editors, which is now, I believe in its tenth edition. Unfortunately, access to the online AMA Manual of Style requires a subscription—and I have neither a subscription nor a copy of any print edition of the manual.
For what it's worth, The Chicago Manual of Style, fifteenth edition (2003) seems to favor lowercasing stage (and clinical latency) in connection with progressive diseases:
8.153 Diseases, procedures, and such. Names of diseases, syndromes, diagnostic procedures, anatomical parts, and the like are lowercased, except for the proper names forming part of the term. Acronyms and initialisms are capitalized.
[Relevant examples:] acquired immunodeficiency syndrome or AIDS; Alzheimer disease; Down syndrome; non-Hodgkin lymphoma
However, as Chicago itself notes (at 8.152), "Medical writers or editors [in the United States] should consult the American Medical Association Manual of Style or Scientific Style and Format [for detailed guidance on style questions in these areas]." If you're writing primarily for a U.S. medical audience, it makes sense to follow the preeminent style guide for that audience. Of course, as in dealing with other type of style question, you are free to take advice from whatever source you like unless you're under strict instructions to follow a particular style.