Enjoy the fruits. They are pretty easy to propagate from the runners. I think I was told to leave them attached until they take root. We have given away a fair few before. I don't mind if some spread out of their beds.
We also have a weed that looks like a miniature strawberry plant. It has similar leaves and also sends out runners, but it has really deep roots. It's all over the lawn and I just have to try and keep it out of the beds.
Oh okay good to know.. oh I've seen those little plants too.. if they are small and look like strawberries I think I know what you are talking about.