In the 14th century the disease began its spread in central Asia, and eventually made its way to Europe - where it was estimated to have killed between one and two thirds of Europe's population. In total the Black Death was estimated to have taken 75 million lives.
The immortal being Flint was in the Earth city then known as Constantinople when the Black Death came to the area. Rodents, rats to be specific, were carriers as they infested the city. (TOS episode: "Requiem for Methuselah")
In 2267, it was feared that Mickey, a rat that Hikaru Sulu had brought on board the Enterprise, was carrying the plague, and a shipwide search was begun for him. Kirk later deduced that this was a ruse on McCoy's part to prevent the crew from succumbing to space cafard. (TOS novel: Mission to Horatius)
In 2270, Commander Spock observed that if he had used Doctor Georges Mordreaux's time travel device to go back to Hikaru Sulu's favorite period in Earth history, he'd be regarded as a heathen freak and given two choices on how to die. One would be through blood poisoning sustained when cut by a sword, and the other would be the plague. (TOS novel: The Entropy Effect)