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The issue here is determining whether it is your internet service that is at fault, or whether something you are doing on your network, or perhaps your router itself, is killing the router's links to the broadband.
You do have to reboot the modem every time you change the device that is directly connected to it. That's just the way it works.
You'll have to test your connection and that means for a while running using just one computer connected direct to the modem, forgetting the router and rest of the network. If that is disconnecting then you can truly start to suspect the modem / broadband service.
Virgin Medai don't "support" routers, so if you suspect a fault with their service, you mustn't mention the router word if you call Broadband Tech Support, and mustn't have it connected. It's not that you aren't allowed to use a router, just Virgin Media won't help you if there are problems with it use.
If the modem to PC connection run over a period of time remains stable then you need to start looking at the router setup, perhaps the ethernet leads in use, and most importantly what types of traffic your network is generating that could be overwhelming something.