The mongols had the Black Plague first but Genghis Khan decided to use this as an advantage. He attacked the Europeans by cattapulting the dead infested bodies away into towns and trading cities. This sent Europeans fleeing.
The Mongols are coming!
As they fled to other parts of Europe, flees, that transmit the bacterial virus of the Black Plague onto rats, also went onto rats on boats and ships. This lead to the plague being transmitted to people on ships and spreading the plague when they hit land.
Captain! The rats are back on board again!
The Black Plague spread rapidly with infected population traveling around Europe. When they eventually found out about the existence of the plague, some infected people would be locked inside their homes with a red marking on their door to show they are infected.
There goes Jenkins.
Luckily, a few places in Europe were safe. For example, the city of Milan, Italy was completely surrounded by the plague but was never touched by it.
You are safe here, young one.
Many peasants couldn't afford doctors and medics back then. Even though their methods of healing the infected were inefficient, nobles kept paying for their service anyway.
I am sorry. The nobles pay me more for my service.