They're busy slamming known cheaters with permanent bans, and some just jump right back in after paying for another account. This is 'free money' for the zombie MMO, so SOE doesn't mind too much.
Dear #H1Z1 cheaters: Sorry some of you suddenly can't play anymore, your donation and tears are greatly appreciated. More bans inc.— Jimmy Whisenhunt (@jimmywhis) February 2, 2015
Whisenhunt asks anyone who believes they've come across a cheater should report the user via H1Z1hacker (at) soe.sony.com. Sony acknowledges handing out bans is just a short term fix for now, but it will give them some breathing space to set up counter-measures for the future.