What if instead of the dead reanimating and coming back to life, that a virus spread by bite that caused people to go insane and bite people. I imagine a mutated form of rabies. Something like that. But then those wouldn't really be "zombies" which are usually the undead, but now people see any human that bites/eats another human as a "zombie". It would just be an infected aggressive human being.
As for the more realistic virus/bacteria/parasite problems...ever seen the movie Contagion?
Martial law is very effective. Also, if the contagion was spread by bodily fluid, it would need to be able to sustain itself and continuously feed on something, without a proper source of nourishment that source will be the host which will die fairly quickly. If someone had a mutated form of rabies which caused that much of an issue, they wouldn't drink water, and would die within three to four days from the fever they would have from their immune system going haywire.
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I think it's already begun. Man has taken it into his own hands to initiate population control measures.
As natural methods of healing are being replaced with modern medicine, so will our dependence on it start to increase. Only those with enough money to afford the best healthcare will thrive while the rest of the world will be left to fend for themselves. Diseases are growing more resistant to vaccines and antibiotics, where soon enough those without access to healthcare won't stand a chance.
I haven't been the doctor in over 5 years and at least I can go out in public casually without catching anything, but in 10 years who knows if someone like me will even be able to step outside without the proper shots.