It all depends how you set up the alert, it will trigger according to how you set up the trigger criteria. e.g once a transaction fails more than 2 times in a row. it will stay in that state up until the transaction is passing again. You can then setit up so that it sends out a follow up email to the alert recipients informing them that the transaction is no longer failing.
Also I am not at my desk at the moment to check but you can acknowledge alerts, and the alerts are stored for a while. I am not sure exactly how long but you can set that as well.