From before multiplayer with your friends was extremely different than what today's hosting looks like. Right now as for today, in Contagion we have a party system now. This party system maybe familiar with you if you played Team Fortress 2. The mechanics of this party system is pretty much similar and a lot easy than you can imagine. I'm gonna show you on how you'll be able to play with your friends. The reason why I made this guide is because I've been seeing a lot of ppl saying that they cannot play with their friends. I'll teach you all how you'll be able to play with your friends.

• 1. First things first you're gonna launch your game!
  2. Once you launch your game, you will be able to invite your friends!
  
  3. If your friends are in your party group . Click on HOST GAME
  
  4. Click LOBBY
  
  5. Once you click LOBBY , You can now set up the game however you like it.
  NOTE: Keep in mind depending on what your lobby setup PLEASE read the descriptions of what it's telling you. I can't stress this enough because like I'm telling you I've been seeing a lot of players complain that they can't play with their friends. I have not once encounter the problem with using Peer 2 Peer, (Personally I find it easier to use since I don't run into any problems with it whatsoever.)
  
  6. Now that you got your settings ready if you may now begin.
  NOTE: For your friends to join in, It will probably take a few minutes for them to actually get in your game. There should be a green JOIN GAME right next to the party system HUD, if you wanna use that as well since not all the time your friends will join in automatically. Use this command in console: cl_lobby_info that way it'll give you lobby information!

So this seems to be most common problem now a days but this game does use Steam API Network which the lobby setup says so. You could also look at Steam Support for more imformation about Steam API. [heres the link that will tell you some information based on it: https://steamstat.us ]
I will keep in mind that There's nothing that can be fixed if you wait for too long because Steam is all behind Peer 2 Peer. There's another option you can try to use Server Reservation there's a few servers that Blue Fang Solutions provided are Lobby Only! However I doubt that you could try that with Custom Maps because those Lobby Only servers don't provide any of that.

- Restart the game.
- Check windows Defender before launching game since it might affect P2P
- Check your firewall to see if it’s blocking contagion.exe
Note from Contagion Developer;

"If Windows Firewall is active (which it should be, never deactivate that), make sure contagion.exe is allowing outgoing connections. If not, no peer-2-peer connection can be made, since the firewall is basically blocking it (by default, like it should).

This usually happens when the user does not allow Contagion on the firewall when they launch it for the first time (basically clicking on deny on the popup)."

- Make sure both players are set to *Online* on Steam.
- There shouldn’t be ANY port forwarding as if Contagion doesn’t use port forwarding [which could effect p2p] unlike games like No more room in hell does.
- Check if VPN is on, if it is, Disable it.
- Check out the p2p settings that's next to your profile picture in the party hud and mess around with it to see if that fixes any issues.
- If there's a password in your campaign game or lobby check it by using sv_password and make sure your friends connect to the one that has given to you or create one yourself.

For Contagion, it does not have a lot of custom maps but in order to play a custom map, you need to Make sure you AND your friend both have the custom map subscribe from workshop or if you wanted to playtest your OWN map make sure you and your friend have the map in your map folder. contagion/maps Otherwise if you don't do this your friend might get an error saying map cannot be played or saying Missing map .

Thought I put some more information based on Peer 2 Peer if there's anymore thoughts that I'm not aware about:

https://psteamcommunity.yuanyoumao.com/app/238430/discussions/0/3057364366997118395/
https://www.reddit.com/r/HomeNetworking/comments/o611jk/peer_to_peer_connection_completely_severed/