# Zombie Apocalypse Now¶

## Introduction¶

How could one not like zombie movies? It was about time that someone, Robert Smith? and his collaborators, shed some light on zombie attacks from a mathematical point of view (see ). Phil Munz (see ) applied a well-known coupled set of differential equations to this particular problem, that was independently developed by Alfred James Lotka and Vito Volterra in the late 19th/early 20th century. Although Volterra and Lotka were interested in (closed) eco-systems, their equations are ideally suited to model a world infested by zombies. Whenever there is a problem which is readily described by (coupled) differential equations, an analog computer is the ideal tool to tackle them as we will do in the following.

## Programming¶

The mathematicsl model itself is quite straightforward and consists of the following two coupled differential equations:

\begin{split}\begin{aligned} \frac{\mathrm{d}h}{\mathrm{d}t}&=\alpha h-\beta hz\label{equ_z1}\\ \frac{\mathrm{d}z}{\mathrm{d}t}&=\delta hz-\gamma hz-\zeta z\label{equ_z2} \end{aligned}\end{split}

$$h$$ and $$z$$ represent the number of humans and zombies, respectively. $$h_0$$ and $$z_0$$ are the initial conditions. The remaining parameters are

$$\alpha$$: Growth rate of the human population (birthrate).

$$\beta$$: Factor describing the rate at which humans are killed by zombies.

$$\delta$$: Growth factor of zombie population due to zombies killing humans and thus transforming them into zombies.

$$\gamma$$: Rate at which zombies are killed by humans.

$$\zeta$$: Normal “death” rate of the zombie population.

Figure 1 shows the resulting program for the two equations while figure 2 shows the program as implemented on an Analog Paradigm Model-1 analog computer. Zombie simulation on an Analog Paradigm Model-1 analog computer

## Results¶

Figure 3 shows a typical result obtained with the setup described above. The computer was run in repetitive mode with the time-constants of the integrators set to $$k_0=10^3$$. Additionally, only integrator inputs with a weight of $$10$$ were used, further speeding up the simulation by another factor of $$10$$. The IC-time was set to short, and the OP-time to $$60$$ ms. The oscilloscope was explicitly triggered with one of the TRIG-outputs of the CU to obtain a stable, flicker-free display.

Results of a typical zombie simulation.

The parameters were derived experimentally by playing with the various coefficients until a stable behaviour was obtained. The output shown was generated with $$h_0=z_0=0.6$$, $$\alpha=0.365$$, $$\beta=0.95$$, $$\delta=0.84$$ (very successful zombies, indeed), $$\gamma=0.44$$, and $$\zeta=0.09$$.