Not really sure why airplane mode was being suggested here... Doesn't make much sense to me.
If you are getting the same messages on a regular (hourly?) basis I wonder if your friend could switch off their phone for an hour or two and see if it still happens? If it doesn't then it's their phone and if it does then it's presumably something in the network.
Mind you, you probably have better things to be doing if you're travelling/on holiday?
🙂Are either of you using an iPhone by any chance? Just wondering if it might be another Apple/iPhone/iMessage oddity?
Hope this helps.