The Mathematics of Brooks’ “World War Z”

I’m not entirely sure how I feel about Max Brooks’ novel World War Z. While the novel is written mainly as narrative exposition, a style I readily enjoy from years of history classes, I am not a fan of the viral zombie. I am also deeply troubled by a plot device employed in the story, but more on that later.

The story follows an unnamed narrator as he recounts the various reactions of different nation states as the global community is threatened by the zombie apocalypse. The reactions vary, and humanity survives, but the world is forever changed, forever paranoid. Further, there’s an element of anti-intellectualism to the work that’s a bit unnerving. 

In South Africa, a controversial plan is devised. It is an old plan, one conceived of during the apartheid reign as a means of insuring the survival of the white ruling class in the event of an uprising. The plan is the brain-child of the seemingly inhuman Paul Redeker, an emotion-eschewing recluse who has, since the start of the outbreak, revised his original survival strategy to accommodate the zombie war scenario.

Two things trouble me about Redeker’s solution. First, its callous treatment of other human being. Sure, it buys South Africa the time they need to survive, but it treats others as a means to an end, rather than as ends in their own right. In short, it violates the Categorical Imperative.

Second, I am skeptical of the viability of the Redeker plan in that it calls for the creation of a closed system, a valley walled by mountains where the survivors will wait out the pandemic. As soon as I read this, my “Closed System Alarm” went off.

In physics and mathematics, closed systems are systems that are… well… closed. That is, all of the energy, mass, and information that the system starts out with will be all of the energy, mass, and information that the system can ever have. Further, in a closed system, entropic forces are amplified. That is, without increasing amounts of energy and/or information added to the system, matter tends toward greater and greater disorganization.

When I considered what might happen if a person were to become infected after the system is closed off (assuming no strategy to rid the area of zombies is fool-proof,) then a modeling problem springs to mind. If there’s one set of problems math students rely on the most, it’s the ordinary differential equation (ODE.) One of the more interesting species of ODE is known as the Lotka-Volterra Predator-Prey Model or, more simply, the L-V equation.

The basic L-V equation is summed up by considering a small forest inhabited by rabbits (the prey) and wolves (the predators.)  As wolves hunt rabbits, wolf populations grow while rabbit populations shrink. As rabbit populations shrink, wolves starve to death. As wolf population shrinks, rabbit population grows. As rabbit population grows, surviving wolves have more to eat and so wolf populations recover and the cycle repeats itself ad infinitum. Ideally, the rabbit and wolf populations begin the cycle with an abundance of rabbits and a rate of rabbit-eating that doesn’t deplete the wolves’ food supply too rapidly. The system will tend toward a cyclic stasis, a changing equilibrium that can be captured in what’s known as a phase portrait.

Wolves vs Rabbits
              Predator-Prey Model 

The loops above show the way one population increases as the other population decreases. Time, as a dimension, is suppressed and indicated by the arrows (that set of arrows is known as the “vector field”) which suggest motion.

Now the problem here is that unlike wolves, zombies need no human prey to live. Their numbers are only diminished when they are killed. Thus, infecting humans only causes the zombie population to increase while the human population decreases. But unlike rabbits, humans fight back. So the safe zone must tend toward one extreme or the other: all zombies and no humans, or all humans and no zombies. There is no egg-shaped orbit in phase space here, there’s an arc toward one axis or the other and that’s got to be the way it is.

So the model above doesn’t really treat the situation in question, but it serves as a springboard for considering the problems faced by those trying to implement the Redeker plan. Suppose that a second, nonhuman vector emerged and entered the safe-zone? Suppose the zone wasn’t completely purged of zombies before it was sealed off? What’s the minimal number of zombies the zone can support before the human population within is assuredly doomed? It was said that the Redeker Plan considered all factors, but clearly it couldn’t.

The plan is deemed successful and is emulated around the world. Yet the problems inherent in the plan stem from a lack of information: How many zombies are in your neighborhood and will that mean the death of us all?

So the offensive nature of the Redeker Plan caused me to contemplate differential equations. Further, it caused me to contact a friend of mine with a PhD in mathematics in order to discuss the problem. There may be a paper in the works….

Shriek into the Void...

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