Running on the stairs is a good way to work on strength. We recommend the following exercises:
Run up, touching every step. Use a forefoot strike. Use a more bouncy stride and lift your knees.
Repeat the exercise, touching every other step. You are going to work on your strength.
Jump up the steps 3 by 3 with your feet together, landing on your forefeet.
Alternate power hiking (fast walking) and running
During these exercises, you should run faster than during competitions, while remaining in your comfort zone. Remember to come back down slowly to recover.