Holiday/Weekend Mode activates a special animation, denoting that the creature is "absent" and "on holiday" . While active, stats no longer drain, and the neglect counter is disabled. This can then be deactivated, returning the creature back to the screen, and allowing stats to drain once more and neglect to be counted when the game is closed.
"Neglect" is a feature to encourage regular 'logins', assuring the user checks in on their creature regularly to maintain their own self care. Entirely adaptable based on opinion, but perhaps neglect starts to take place after 48 hours. Each following 24 hours, the creature loses 1HP. This is then increased by 2HP every day (not 24 hours, but specifically day if possible, though 24 hours is fine) the user does log in. When the creature is below half HP, the creature is "sick" (animation? - further, sick versions of existing animations?), and "aging" is paused until brought back to full. Perhaps run HP on a total of 10? When suffering "neglect", the creature will comment upon return, showing concern for the user, but happy that they have returned.
Animations, fluff etc don't count as part of this ticket; they're just musings.
Look into how best to incorporate the "neglect" feature, as well as a Holiday Mode to pause neglect.