Shia had been unknowingly killing her male descendants before meeting Misha.
Shia was a demon that needed to feed on lifeforce to survive, at least in while in the human world. During the visions of the past we learned that the best source of lifeforce for a demon is from their own kind. When visiting his dying great grandfather, Kotarou's aunt told him the family's "not blessed when it comes to males". Now, when Shia first appeared all we knew was that she was looking for something that she only had the vaguest idea of what it was, but cat she was with seemed to be steering her toward it. She met Kotarou, and the cat noted he might have been what she was looking for. Shia then drained his lifeforce, and he would have died were it not for Misha.
Put all this together and it looks like "Nya" (who we later learned knew everything from the beginning) was purposely leading Shia to her descendants instead of to her husband himself, who she would then kill out of hunger, not knowing their real connection. It was only after meeting Misha did it stop.