Alias Interfaces Timing Out

I have a linux RedHat 7.2 box acting as a router. This box has two NICs. NIC 1 is connected to the Internet with 3 IP addresses. The gateway IP from the ISP and two aliased routable IPs for two webservers. NIC 2 is connected to the internal network with 192.168.1.1 address. The problem is when there has not been traffic going through the physical NIC 1 going out to the Internet for a period of maybe a couple of hours the aliased IPs are completely inaccessible from the Internet. Hence noone can access the two websites unless someone within the 192.168.1.0 network is accessing the Internet or someone is pinging the router IP address from the Internet.

Has anyone experienced this before or know what the issue is?

Thanks.