There are several factors that determine how quickly a zombie disease would spread.
Type of pathogen. If we're going off of current zombie lore, the zombie-producing agent would most probably be a virus -- rather than another type of pathogen like a bacterium, fungus, or parasite. Viruses are pretty agile and can change and adapt more rapidly to their environment than other types of diseases. They are also incredibly hard to cure. There are many bacteria that are developing a resistance to current antibiotics, but that takes time. So if we're assuming the zombie pathogen is novel and NOT a mutation of an already-existing antibiotic-resistant bacteria like, say, multi-drug resistant tuberculosis, a virus has the best chance of spreading and spreading fast.
Treatment options. We're assuming it's a virus. Does an antiviral exist that is effective at killing it? Probably not. But if so, once we see that the virus is spreading, we can identify who has the disease or who could have been infected, and then treat them prophylactically.
Natural immunity. Does the body produce antibodies in response to the virus? If yes, we could potentially figure out a way to develop a vaccine. Because zombies are terrifying, I'll bet that the regulatory agencies at large would expedite the approval process to get it into production, too. But if the virus attacks the immune system directly (much like HIV), or morphs too quickly for our antibodies to keep up, then, well, that option's off the table.
Mode of transmission. How is the virus spread? In most zombie lore, it's the bite that spreads the virus from person to person, meaning that the pathogen is probably spread through saliva and/or blood -- a rather ineffective way to transmit the pathogen quickly (unless, of course, there's a long latent period -- see next point). A more effective way for a virus to spread far and quickly would be if it had multiple modes of transmission, and if it were airborne, in the water supply, or via some kind of third-party vector or object. Think of the flu. It can be passed on via saliva, touching inanimate objects (like doorknobs), or through droplets in the air. As a result, roughly five to twenty percent of the population get it every year. As pathogens go, it's quite efficient. So a zombie virus would need to spread kind of like that.
Latency. Like Andrew Gutsch said, how long it takes for signs or symptoms to present matters. If someone doesn't know they are sick, they won't take precautions to keep from spreading the virus to others, even if they are contagious. For a zombie pandemic to happen, the virus would need a pretty long latency period combined with a long period of contagiousness.
Period of contagiousness. Andrew mentioned this, too. If someone is contagious BEFORE symptoms ever start, that could lead to higher levels of infectivity. For a zombie virus to spread very quickly, people would probably need to be contagious pretty much from the point of infection. A real-world example would be HIV. People recently infected with HIV are highly contagious, as the virus is able to replicate before the body even realizes what's happening. That is partially why HIV was able to spread so quickly before we knew what was going on. It had a long period of contagiousness combined with a long latency period.
Infectiousness and pathogenicity. Just because a person is introduced to a pathogen doesn't necessarily mean that person will become infected (infectiousness) or will get the disease (pathogenicity). We're all exposed to myriad pathogens every day, but our primary immune system usually does a good job of fending them off, as long as we're pretty healthy, eat our vegetables, etc. For other pathogens, however, exposure almost always means infection. Measles is a good example. With a secondary infection rate of over 90 percent, it is one of the most infectious diseases still in circulation.
Where the disease originates. Unfortunately, this matters, too. If the disease appears in a country with excellent medical care and sound health infrastructure, it could probably be caught before it got out of hand. There would be systems in place for quarantine and isolation. If treatments are available, they would be administered promptly. A successful zombie virus would need to crop up in a place with limited medical services, but where the infection could spread to other cities, regions, and countries with relative ease. That is, a relatively large transit city in a developing country.
So in summary ...
- It would have to be a virus for which no treatment or vaccine options exist.
- It must be able to spread via multiple modes of transmission (including airborne, through inanimate objects - called "fomites" - and/or via some other vector like a rat, flea, or mosquito).
- Infected individuals should have a long time before they show symptoms but be able to pass on the virus almost immediately.
- If people are exposed to the virus, they must become infected.
- And the disease should originate in a place with limited access to medical care but high volumes of people in transit.
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