Well after the part of the script that alerts the user, place this ...
... and on delete.php put your code that removes the alert from a database and put this at the end of the page ...
All of this basically just opens a popup window that has a page that will delete the alert from the database and it then closes the same popup so no one knows. (It all happens in less than a second)
I haven't tested the idea but it should work.
Hopefully this helps.