Elusion is the act of successfully hiding or escaping from someone. Your cat's elusion makes it hard to find her when it's time to take her to the vet.
When you avoid being found or caught, that's elusion. Playing hide and seek is all about elusion, as is being a successful prison escapee. Any situation that involves a person cleverly evading detection or capture can be described as elusion, though it's much more common to hear the verb elude, which came first. Its original meaning was "delude or make a fool of," from a Latin root, eludere, "escape from or make a fool of."