Well... Let me start off by saying I'm not entirely familiar with this game. In fact, today I learned pretty much all I know about it! That being said, we came up with two strategies to deal with the Dylanlings.
First, we can move our cats around the house. This'll work better the more rooms you have, but it doesn't eliminate the issue entirely. From what I've heard, if you have 2 rooms and an attic in the game, doing this will only stall Big Dylan for 9 to 11 minutes, and that's if you keep doing it all the time!
Second of all, Caine has been trying to make cats of his own. These are good for crowd control, and - quite important for our purposes - immune to Big Dylan, lowering the chances of cats around it getting infected. Caine can also enter the game himself, if the situation calls for it.
Third of all, while not necessarily a strategy on it's own, Maisie over there - girl with the pink hair - has a crystal ball, and strangely enough she seems to be able to talk to the people of Boon County. The downside to that is, obviously, this is just the first step towards doing something else, and that's assuming you don't accidentally get in touch with someone who's, well, not quite in touch with the state of the world.
Now, the important thing is that Big Dylan WILL try to overwhelm us. Intimidation is his modus operandi. So just remember to not panic and think what the best course of action is. Alright?