Silence Alerts

Joined a team that uses Nagios heavily. One of the issues we have is if the site connectivity goes down we suddenly receive a flood of alerts. I was wondering if there is a good way to silence all site related alerts if the particular connection goes down?
If there is not a good way to build that logic in echatrandom echatspin, what do you all recommend?
If there is not a good way to build that logic in echatrandom echatspin, what do you all recommend?