With great haste and having only personal wellbeing in mind. Typically said of people who begin abandoning something or someone that is failing or about to fail. The expression refers to the popular notion that rats are the first to abandon a ship when it is sinking. I knew the business was doomed when employees started quitting en masse, like rats leaving a sinking ship. Like rats leaving a sinking ship, the disgraced athlete's corporate sponsors began pulling their endorsement deals one after another.