Collocations for asleep - adj.

be, lie, seem
The baby lay peacefully asleep in its pram. | All the houses seemed asleep.

drop, fall
I fell asleep almost immediately.

remain, stay

deeply, fast, heavily, sound
The children were all sound asleep in bed.

almost, half, nearly
At the end of the afternoon they were exhausted and half asleep.

peacefully | still