A full moon makes all ponies crave bellyrubs, and anybody who gives them one also becomes a pony. After the full moon is over, the infected remain as ponies but aren't infectious or rub-hungry until the next full moon. The type of pony depends entirely on the person—a unicorn will not always make more unicorns. The effects of the full moon will not be influenced by cloud cover, line of sight, or other obstacles, and the for the sake of the uninfected, a full moon lasts from dusk til dawn. Changelings will be unable to disguise themselves as anything other than an infectious species during this time, due to the amount of concentration required to change to such a different body shape.
A bellyrub counts as rubbing or scratching a pony's underside (for simplicity's sake, it's still the underside if they roll onto their back). A simple poke won't be contagious on its own, but good luck resisting the urge to rub that belly afterwards. Same with resting your hand on the belly but not moving it is also not contagious on its own, but it is even more difficult to resist the ensuing urge to bellyrub—don't tempt fate. A pony cannot use magic to directly force the requisite bellyrub motions. Ponies attempting to scratch their own bellies will find it doesn't sate the itch, even if they use more than just their hooves to do so. Resistant humans that are partially changed can scratch their own bellies without further changes, but it won't sate their itch, even if they still have fingers. Bellyrubs from other infected are just as unsatisfying.
Roughly 5-10% of humans are immune. No amount of bellyrubs will convert them. Resistance to the disease occurs in about 20% of humans, but all it really does is limit how much changes at a time—with less resistance meaning more changes and more resistance meaning fewer changes—and once infected, they follow the same rules regarding the full moon. If you want to stay partway between human and pony, pray you get just the changes you want and then try to avoid giving bellyrubs for the rest of your life, because your human parts will still be vulnerable. A body part will change into its analog (hands and feet into hooves, arms into forelegs, legs into hindlegs, etc.) but in the case of horns, wings, or similar parts, they will grow in when the part of the body they would be attached to changes. This means centaurs will not be possible, though satyrs are a possibility as long as they avoid further infection. Anthro is never the end result.
Possible species include batponies, unicorns, pegasi, earth ponies, zebras, changelings, and crystal ponies.