We've been using those red scrubbies as well and man we are always chasing threads in screens. It's not bad when they are new, but they break down quickly. I've tried a couple other brushes and I don't like the way they clean.
The search continues.
I had that issue practically from day 1. Little pieces of the scrubber will break off in the screen constantly. I switched to sponges, which work great for me (I use grout/tile sponges from home depot).