- We honestly do not know
- The ancient left behind did not necessarily have an infection.
So firstly that is a plot hole that we will never likely be able to fill. The series was cancelled (which I am happy to report I saw some mention of a new movie adaptation of some sort in the works...)
But just because she had some "contagion" does not mean that she had an infection that was deadly or dangerous to her fellow ancients. The only reason that everyone got infected from her is because humanity had never dealt with "that" before. It is possible that what she had was the equivalent of the common cold to the ancients. She was able to heal it pretty easily and showed no real signs of self danger. If anything she was just a carrier.
As has already been mentioned the plague the ancients were combatting was the Wraith; remember that we believe the wraith are evolved from ancients that were infected by the wraith/iratus bug. That plague spread across their new galaxy and the ancients ascended to escape it... (basically said screw you to two universes and then of course punished anyone that tried to help out...)
As far as the ancient left behind anything is possible. We know that the ancients just up and decided to move "along from our region of space long ago." As far as we know it was not out of fear or anything, just simple curiosity.
It is interesting that she was enclosed within the block of ice that sustained her. To know why she was left behind we would need to know that. Whether that is some form of cryo-containment or just an innate ability the ancients possessed, similar to Teal'k being able to survive underwater for a short time, we do not and will likely never know. Without knowing the circumstances behind why she was frozen in the first place we won't ever have a chance of answering that question.