Yes, it is hard because early on the number of living will far outweigh the number of zombies. You are telling me that hundreds of criminals can't somehow figure out how to kill zombies? They just need to push them to the ground and stomp their head in with their foot, not so hard. Everyone has seen a zombie movie in their lifetime and knows that you have to go for the head. Or you don't think one person would shout "Hey go for the head" Even if you were stupid you could figure it out. Let's say there was a population of 500 inmates. Only 6 survived? All the rest are zombies?
Also the zombies on the TV show don't just bite and move on. They bite, take down their kill, and then start feasting. They don't just go from person to person just to bite to infect them. They are going for the kill, and ripping all their guts out.
So yes it is hard to imagine a good scenario where you have a prison full of skinny inmate zombies.
You're assuming that the only way that the zombie plague could have take over the prison, (or every where really) is via a single infected person biting another person, that bites another person, etc.
Let's assume this is a virus (or bacteria, or prion, some sort of external pathogen). Clearly the immune system is unable to clear it; everyone is infected. Since everyone is infected, it obviously doesn't *require* direct transmission into the bloodstream via a bite.
So what was the initial mortality rate for the virus? Even the most lethal hemorrhagic fevers (Ebola, etc), don't have a 100% mortality rate. Was the virus so lethal it killed, in days, 70% of the people it infected? What if it was only 50%? Where is the tipping point? At what point does society begin to shut down...is it 40%? 30%? 20%? What mortality rate statistic would you have to hear before you made one panicked run to the grocery store and tried to hunker down until the crisis passed?
What happens when no one goes to work because they're afraid of what they're seeing on TV? 50% of the power generation in the US is produced by coal plants, which on average have a 20 day supply of fuel. Assuming people go to work to maintain the plant. Grocery stores turnover their food in 2-3 days...assuming truck drivers are willing to deliver the food.
It's pretty clear (assuming the TV producers have though these things through) that things went to hell pretty quickly. There is still food on the shelves. There is gas available to use.
There are also dead bodies in vehicles that don't show any sign of having reanimated. People are dead in their houses with no visible trauma (in beds, etc), so not everyone reanimates.
Here's one easy way to explain the state of the prison when they find it:
The pathogen has two methods of propagation; airborne and fluid-transmission. The airborne spread is via people sneezing and coughing. The pathogen settles in the lungs, producing inflammation and pneumonia (and more sneezing and coughing). Let's posit an incubation period of 36-48 hours, followed by rapid onset of symptoms. Hosts are contagious within 12 hours of initial infection. With air travel, that's probably enough time to spread it broadly enough that there is no easy way to quarantine it (if you want to really make it infectious, we could always give it a natural reservoir; maybe it infects, but does not kill, birds for instance).
The pathogen kills roughly 40% of infected hosts via acute pneumonic processes, causes mild to severe respiratory distress in another 40%, while roughly 20% are infected but asymptomatic. Due to a novel process of infection, humans have little to no immune defense against the initial infection, resulting in an effective infection rate of 100%.
Primary infection is localized to the lungs and mucous membranes. In a limited number of cases (call it 20%), the pathogen crosses the blood-brain barrier and induces meningitis. When this occurs, the disease is inevitably fatal; post-mortem, the deceased reanimate.
The infection is not cleared by the immune system in any known cases. Asymptomatic carriers and those recovered from the mild symptomatic version carry a small but noticeable pathogen load. Post-mortem, most (if not all) carriers will reanimate.
Secondary infection can occur from the bites (or potentially scratches) of the reanimated deceased. In this event, the pathogen is introduced directly into the bloodstream, quickly progressing to the meningitis form of the disease, followed rapidly by death and reanimation.
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TL;DR:
Zombie plague caused by a highly-contagious airborne pathogen that kills 40% of the infected, a portion of whom reanimate. Sick guard goes to work, coughs on co-workers, inmates. Rinse, repeat.