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Given our world's response time and action to major outbreaks of viruses that have happened in the past, how quickly would this virus have to spread to have a full on apocalypse where government and armies have been destroyed.
It may not even be a matter of how quickly, how would it spread at all without a government or army successfully destroying it?
Anything that increases the period where people are carries before combing a "zombie". If the infection does not activate until after someone dies. Then they have all the time while they are alive to infect other people. The quicker it causes someone to transform from normal to zombie, the more likely governments will be to survive.
Making the infection vector air born(increase infection routes). If it can survive for long times in the air, or on a surface then you multiply the opportunities for infection to occur.
Super bug. Make the infection/virus immune to the standard disinfectants. If we can not sterilize it, we can not stop it. (Model it after the pryons in Mad Cow/Creutzfeldt–Jakob disease)
Super bug mod. Make it so that certain sterilization technics act as a catalyst. People take the standard actions to sterilize, and instead make the infection more virulent.
There is a steam game where you play as the infection. A couple plays through of that might lead to other ideas.
The most important requirement is a moderately long combination of incubation/asymptomatic infectiousness. I've not run across many descriptions of zombies which would allow one to pilot a jet or a ship across an ocean, so global spread requires travel by the asymptomatic.
If the turned zombie only becomes infectious once she is obviously a shambling, rotting, brain-craving revenant, stopping the spread of the condition is likely to be pretty effective. The more victims the condition can infect without being detected, the better it will spread.
Newly infected should turn within a few minutes at most
The reason for this is to remove any ability of newly infected humans to react. If it takes many hours for a newly infected to turn from patient zero, then the virus will be easy to contain.
Lets say those bitten by patient zero take 2 hours to turn. At first, people may seemingly be murdered by patient zero, but very soon police will be called and the injured will be seen in hospitals. Police officers will easily put down patient zero. Those seen in hospitals will turn, but will be quickly restrained. I would expect any trained physicians to quarantine new victims bitten in the hospitals, thus stopping the infection there. As for those 'murdered' by patient zero outside the hospitals, it will not take law enforcement long to put down the 'murder' victims rising from the scene.
However, if patient zero is in a major population center, and his victims turn in a few seconds, then the cascade of zeds will exponentially increase, completely wiping out whatever continent it was on.
As for how the infection will spread overseas, that has to be left to you. Perhaps a terrorist organization collects some of the victims and unleashes them against their victims. Perhaps one zed strays onto a cargo boat as it's leaving port to safety on a new continent.
"Proof (this is fiction of course)"
Note how every single zombie story with a long (hour or more) period between bite and turn focuses on survivor drama, and starts out with everything 100% apocalyptic from word go. None of these shows or movies ever show from patient zero.
Zombie Land - We get flashbacks to before the outbreak, but no detail about how it's spreading.
Dawn of the Dead - The MC wakes up, everything's in chaos, and she takes refuge in the mall.
Walking Dead - The story starts when Rick wakes up; after the apocalypse.
Left 4 Dead - Story starts after the outbreak.
Dead Rising - People turn slowly, and the outbreak is contained to one town.
The only one that shows me a plausible apocalyptic spread is WWZ. And the newly bitten turn in 8-12 seconds.
It really should spread quickly, but not so quickly that all the humans alive die out in a week. So maybe a year or 2 is good. Though 6 months is good, that would be very quick
