I'm Troubled by Vermin since various days ago, my value was around 19 now 11, using the normal ways ("A bad case of rattus faber:*") is taking me hours to reduce them to 0. Is there a way to do it more efficiently?
As you reduce your Troubled by Vermin quality, you will unlock more options to decrease it further. It will also, due to the way qualities and change points work, decrease faster the lower the quality is. The options in your lodgings are the only way to decrease Troubled by Vermin, though.
Buy a Talkative Rattus Faber from the Bazaar, in the Nassos Zoologicals shop, for 80 pence. It will open up a new storylet to complete the entire Troubled by Vermin storyline in an action or two in your Lodgings.
Of course, this means that you'll be bypassing a fair chunk of content. Personally, I regret having taken this course of action; I did it around Troubled by Vermin 17 or so, which means I missed quite a bit. Keep in mind that you can ignore this storyline for as long as you want, and it will have no ill consequences. If you're bored of it, come back later.