One interesting thing to look at, even if only 50% of the world's population dies, you're looking at some 3.7 billion bodies. The average human weight is about 135 pounds, so the remaining people would have to cope with about 5x10^11 pounds of dead people. That is a lot of bodies that will decay very quickly, especially in the more humid parts of the world.
Dealing with the bodies of the deceased was a huge issue during the Black death in the mid 1300's, and has been a problem whenever massive amounts of people die. In World War 1, people would live in trenches made out of the bodies of the recently killed as there was no point in trying to bury them, and this has been seen in wars and after famines since then. Even with current technologies would be almost unfathomable. Besides burning corpses, huge swaths of land (probably close to cities where populations are the most dense) would have to be devoted to burying, or even just piling bodies. It would be so overwhelming that the efforts to remove this mass of decaying bodies wouldn't be worth it, and much of the 20-50% of surviving people would move far out into the country side where the quantity of dead is more manageable.
On a three month timeline, the areas that the small communities of survivors would be living in now would be focused mainly on getting food and staying alive, as most of modern infrastructure would be gone. Cities would slowly be over run with carnivorous animals and eventually vegetation. Almost certainly some people would remain in the cities, deciding that living with the dead is worth whatever food and shelter remains. It would take decades, even centuries, for anything resembling a recovery.