Once upon a time there was a monastery in which lived several monks. One day a monk from another place came to the monastery, gathered all the monks and said to them:

"Dear friends, i'm afraid i have bad news. I've been told that some of you are infected with a mortal disease. Whoever is infected has a red dot behind the neck and the disease is not contagious. I won't tell who is infected and i ask you to do the same. So, from now on, i ask you not to communicate with each other in any kind of way. I also know that this monastery doesn't have any kind of mirror so you won't be able to see if you have the red mark. But being brilliant logicians as you are i propose the following: each day you will meet by the afternoon where you can look at each other and at night you will each go to your private rooms. If you are positive you are infected you leave the monastery during the night. I'm confident about your success. Farewell my friends!"

And so the monks did!
- In the 1st day the monks gathered in the afternoon and at night went to their rooms and none left the monastery
- In the 2nd day the monks gathered in the afternoon and at night went to their rooms and none left the monastery
- In the 3rd day the monks gathered in the afternoon and at night went to their rooms and all who were infected left the monastery

How many monks were infected? What was their reasoning?

Post your solution with explanation here!