The Girl With All the Gifts by M.R. Carey (Mike Carey's pen name, of Hellblazer and Felix Castor fame)
(I'll keep this spoiler-free aside from revealing a very early and obvious facet of the world.)
I just finished reading it moments ago and it is now my SF&F book of the year. It's thrilling and very, very dark. The similarities with The Last of Us begin with the catastrophe which has rocked the book's world: a fungal, pseudo-zombie infection.
As similarities in setting and themes mounted, I had to look up some dates to see if one could have inspired the other. Though the book only came out in 2014, its short story incarnation, Iphigenia in Aulis, was published in 2012. So it's unlikely that either source drew from the other.
I highly recommend this book if you enjoy the game's world or Mike Carey's writing. The plots are wildly different, but The Girl With All the Gifts could have been a subversive DLC side-story set in the same universe. But please don't even glance at the plot synopsis if you decide to read it. It doesn't sell the book well and spoils the fantastic, creeping dread of the first third.
And if anyone else has read it, I'd love to hear your comparison between the two.