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They're slow and don't even have weapons.
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Which, IMO, is ridiculous. 20 years is too long for zombies to still be an issue.
Think about a typical Zombie story. The survivors are often killing multiple zombies per hour. Sometimes it's quiet and there are none, but sometimes it's frantic and it's tens of zombies per hour. Say it averages out to 1 zombie per hour, but only when you're out scavenging, so 10 per day. That's about 300 zombies per month, about 3500 per year, and that's without any real effort to hunt them down and eradicate them.
That's 35,000 per person over 10 years, 70,000 per person over 20 years -- and again, that's just casually encountering and killing 10 zombies per day, without making any real effort to eradicate them. At that rate, (casually killing any zombies they happen to encounter) it would take only about 23 people to clear the entire population of Manhattan (1.6 million) over 20 years. The population of Greater Tokyo is 37 million. At 10 zombies per day it would only take slightly more than 500 people to clear every zombie from the megacity over 20 years.
Now, just imagine you had a zombie-proof wall and someone whose job it was to go stab every zombie up against the wall. They could probably do 1-2 a minute, say 100 per hour, 1000 per day. Over 20 years that one person could personally handle 7 million zombies. Clearly, you'd also need to clear out and remove the bodies, but just in terms of culling the zombie population, it would be easy to do.
Even if zombies killed 99.9% of the population, they should be uncommon after a few months, and incredibly rare after a decade.
That's pretty much the situation in The Last of Us. Humanity has retreated behind walls. Zombies are mostly not a threat, but they exist outside of the encampments. You can live a life without fighting zombies, but if you need to travel for any reason, you're taking a risk. The biggest risk is from the tribes of people you'll encounter along the way though.
I like your detailed response but you do need to consider reckless people, mistakes and oversight. Encountering a horde with just 2 can become problematic.
Consistently killing 10 zombies every day for 20 years, my guess is you'd slip up sooner or later. So not killing them and trying to stay safe instead could be a better option.
They would still rot away before the 20 years are over though
Once they have a safe place, most people will kill zero a day. And the guy killing them on the walls would require them to come to the walls. Natural barriers should prevent that.
Which would mean that zombies are extremely rare after a few months, which isn't what we keep seeing in zombie fiction.