The stat stamina decreases over time, generally around one level per one to two seasons (though you may see variation from player to player). You can train stamina as a team training, which will effect all players equally regardless of how many minutes they play in a week -- most players will gain a level of stamina every one to two weeks that you train stamina.
Your players *are* effected by how many minutes they play per week, via their game shape. A player who plays too little (less than 48 minutes per week), or too much (generally believed to be more than 90 minutes per week) will see their game shape fall to a lower level. The highest possible level of game shape is proficient, and you will notice a substantial difference in how a player plays with proficient game shape compared to respectable game shape.
Game shape is reset to respectable prior to the first league game of each season, and then is updated at each training update (which happens Friday mornings) based on the number of minutes played during the past week -- totaled across Saturday league games, Tuesday league games, and scrimmages/cup games.