In the Player's Handbook, disease is mentioned in the Petrified condition's description ('the creature is immune to poison and disease'), but there is no Diseased condition to be found, nor is disease even listed in the book's index. Is the idea that every disease is unique and so there is no 'general' condition to cover it?
Based on the appearance of disease in the Otyugh entry (HotDQ supplment, p. 14) and the publication history of D&D, it's a fair bet that diseases will be defined in an ad hoc manner rather than systematised. Only a couple editions define diseases in a "disease system" way, and the editions that 5e is giving hat-tips to don't do that, instead opting to cover diseases by giving them simple, unique game and fictional effects defined where relevant.
We've already seen this ad hoc methodology in the Player's Handbook as well, with the spell contagion defining its own diseases. The trend seems to point firmly toward not prescribing a core mechanic for diseases and the overall game's design not needing one anyway, instead leaving them to be defined by their definition — much like spell effects are, in fact.
Based on that, we can expect new diseases to be simple and largely defined in three places:
- The DMG (pp.256-257)--includes a few "standard" diseases that are common, may recur across a number of monsters and adventures, and are recommended for baselines from which to create homebrew diseases
- The entries for monsters that inflict diseases other than those covered by the DMG already
- Adventures that feature unique or new diseases (in line with the expected DMG's "diseases do what the designer/DM say" guidelines), except where already defined by the MM or DMG
This is how disease was treated in many editions. It allowed for simplicity and clarity of effects without descending into tangles of rules interactions, and gave DMs flexibility in creating new diseases to suit homebrewed monsters and settings. It worked very well, and is in-line with the general return to simple rules stated simply that D&D 5e is showing.
Pushing disease into the DMG, and defining them in an ad hoc way instead of within a known "disease mechanic" or system, also allows for a wider variety of play styles to be supported: disease effects can be shared with players if the DM's aim is, e.g., for them to be a mechanical challenge or puzzle; or kept hidden and only revealed through narrative description if the DM's aim is, e.g., for diseases to be frightful unknowns that require seeking medical help to diagnose and cure.
At this time, we do not know what general mechanics disease will fall under. It appears quite possible that no general disease mechanics will be defined.
Diseases are not, to my knowledge independently enumerated in the monster manual (though I'll recheck this when I get home this evening), so mostly likely these will be completely within the purview of the Dungeon Master's guide (again if enumerated at all independent of monsters that inflict them).
However, numerous creatures can inflict ongoing effects that are called disease, or are very similar to diseases. For example, a classic disease/curse inflicting character, the mummy, has a rotting fist attack that inflicts a curse which prevents the healing of hit points and penalizes maximum HP on a daily basis until such time as the curse is removed.
Here's an example of an actual disease from another disease riddled classic the Otyugh (from the HOTDQ supplement)
If the target is a creature, it must succeed on a DC 15 Constitution saving throw against disease or become poisoned until the disease is cured. Every 24 hours that elapse, the target must repeat the saving throw, reducing its hit point maximum by 5 (1d10) on a failure. The disease is cured on a success. The target dies if the disease reduces its hit point maximum to 0. This reduction to the target’s hit point maximum lasts until the disease is cured. (p 14)
At this point, rather than the stage system of 4e, or whatever mechanics that other previous editions used, disease is a saving throw keyword. The most common effect seems to be reducing HP maximums. (Hat tip to Ace Calhoon for pointing out a good freely available example)
So far it appears as you suspect, that most diseases/curses will simply fall under the purview of the monsters that inflict them and the mechanics will need to be evaluated individually. However, again, I would not at all be surprised to see a section on disease in general in the DMG.