You can train stamina. Click on the training menu on the left side, and select "team" training and "stamina". The training update is on Friday each week. I've trained stamina two weeks, and most of my players have gone up both weeks.
Before I trained stamina, my players were rotating in a lot. A fresh 2nd stringer was better than a tired starter. Now my starters are playing longer.
In one game before I trained stamina, my players were swapping in and out, and the other team was stuck with their starters - I assume because of one of the substitution options. I really pulled away in the 2nd and 3rd quarters, but then the other team came back in the 4Q because by then all my players were getting tired.
I don't know if it was a stamina effect, but my team scored 17,17,17 and 32 by quarter tonight.