HTTP health check flapping - potential offenders?

First increase your monitor timings, from Interval 5 secs to 10 secs, timeout 16 secs to 31 secs see if that helps.

Second get a tcpdump if you can see what is happening to the connection. It is possible the connection is not being closed correctly so web server (or something on the network) stops allowing any more active connections. Have seen this on older Linux Redhat servers that just run out of available TCP ports from the F5's Self IPs.