Limiting error messages sent to adminEmail

[millipedia]

Recently we had a site being hammered (probably by bots) which resulted in the MySQL database running out of connections.

It happened overnight so I woke up to a couple of thousand emails sent to the email address set in config->adminMail along the lines of 'SQLSTATE[HY000] [2002] Connection refused'

I managed to battle the bots and sort the error out, but now I've got my monthly MailGun invoice for the transactional emails ... and it's over $2! (normally it stays within the free tier).

Hopefully we can scrape together the money to cover the bill ... , but it got me to wondering if there's a way of throttling those admin error messages. I wouldn't want to lose them entirely but it wasn't useful to have so many emails saying the same thing - and if they'd gone to an inbox I wasn't checking then it could have ended up with much worse bill.

I know that TracyDebugger in production mode sends errors messages and then requires you to clear a flag before you receive more.
Is there the equivalent for adminEmail? Or is there a way to disable the admin email error messages so I can just rely on TracyDebugger?