Every organism is plagued with its own infectious diseases, many of which are specific to the species. Hamsters and gerbils don't get infected by exactly the same pathogens. With pets and other domestic animals, many of these microbes have the capacity to jump to humans and cause a zoonotic infection,
Reindeer are no different.
Some reindeer infectious diseases, which likely impact the ability to engage in reindeer games, include:
🎄 Reindeer Pest: high mortality infection with Clostridum septicum
🎄Necrobacillosis: severe hoof infection
🎄Foot and Mouth Disease
🎄Lyme Disease
🎄Pasteurellosis
There also other myriad infections, including nematodes, that can be found in reindeer populations which have implications for reindeer meat intended for human consumption.
None of these diseases, however, is known to cause severe erythema of the nose so Rudolph's affliction will remain a mystery.