Network Aware Viruses – eg. Welchia

Do you need to open your Internet browser/some Internet application to open yourself up to the threat of viruses such as Welchia?
While I am aware of the patches that are available, occasionally even patched machines seem to get variants of such viruses.
If there is an Internet connection available to a machine through a network, and none of the machines attempt to access the Internet in any way then will the virus still propogate?
I would have thought it would need to see that a device exists before it sent itself, but would the keepalives on the modem be enough to possibly spark it off?