The disease of the Red Death is fictitious, meaning it doesn't really exist. It came from a short story written by Edgar Allan Poe called "The Mask of the Red Death". Poe describes it as causing sharp pains, and sudden dizziness, and then profuse bleeding at the pores leading to death within half an hour. It's highly likely that the disease was "inspired" by tuberculosis or consumption, as it was known back then. Poe's wife Virginia was suffering from the disease at the time the story was written.
The red death has been open to endless interpretations and Poe resisted in defining it too precisely. Others have suggested that it's actually Bubonic plague or the Black death, which is what I tend to believe. It has also been suggested that the Red Death is not a disease or sickness at all but something else that is shared by all of human kind inherently. You should read the story. It was also made into a movie starring Vincent Price in 1964 - a classic.