The 1451 Review (Volume 1) 2021

Lattice End-States for Different Values of αα
Figure 8.
From left to right: lattice end-state for super-critical outbreak (αα = 0.1), outbreak at the
critical value (αα = 0.83) and contained outbreak (αα = 0.95). White points are humans,
black points are zombies and grey points represent removed zombies. The simulations
were carried out with 10,000 individuals on a 100 x 100 grid. The patient zero zombie is
placed in the centre of the lattice.
In the supercritical outbreak case zombies dominate the entire lattice, with large
connected regions demonstrating how effectively they were able to infect the lattice.
While there are numerous patches of the removed state, these are small and mostly
isolated indicating a poor fight back by the human population. Significantly, there are
surviving humans in the end-state which is a notable deviation from the previous
models where either humans or zombies are eliminated. Unfortunately for these
survivors their existence is very much a post-apocalyptic one, with only 234 surviving
humans (2% of the initial population) limited to single sites.
In the contained case, large populations of humans untouched by the spread of the
disease are present, with over 5000 survivors in total. These patches dominate the
lattice and represent areas where humans have been particularly effective in fighting
zombies. A non-trivial number of zombies are still present, mostly concentrated in the
centre of the grid due to the disease originating from this area. However, for each
zombie they are matched, if not exceeded by the number of defeated zombies that
surround them. In each case, we observe ‘emergent behavior’ as the agents on the
lattice interact to form complex fractal patterns, despite their individual interactions
being governed by simple rules.
Small World Inhomogeneous Lattice Approach
Utilising an inhomogeneous lattice is an alteration of the simulation that results in
another step towards realism. The previous approach results in an extremely high
population density that favours the density dependent zombie disease. Furthermore
the ‘barriers’ of dead zombies that protect humans seem far-fetched, as zombies are
often presented as adept at breaking down or circumventing physical impediments in
film and literature.
Zombie Progression Across the Inhomogeneous Lattice
The critical value case presents a far more balanced picture between the three subpopulations.
Although the number of zombies is still large, the number of terminated
zombies is very similar. The parity between these states demonstrates how even the
conflict between human and zombie is at the critical value. The most pronounced
difference between the supercritical and critical values is the greater number of
surviving humans, with significant communities observed at the edges of the lattice.
Numerous pockets of human survival are also observed towards the center of the
lattice, a common feature of these pockets is that they are surrounded by terminated
zombies. This draws parallels to the forest fire model, where fire breaks and burned
trees are used to prevent the spread of a fire, dead zombies are used as a morbid
protective barrier for humans.
Figure 9.
The progression of the zombie infection in the small world inhomogeneous lattice. Left
image is the initial lattice with 199 susceptible agents (light grey circles) and 1 zombie
(black circle). Middle is an intermediate state of the lattice; at this point the susceptible
population have failed to contain the outbreak. Right is the final state of the lattice
comprised entirely of zombies and removed agents (dark grey circles). The grid was 20
x 20 in size and αα = 0.6.
The inhomogeneous lattice corrects for this by introducing empty lattice nodes
that allow the human population to be more spread out, areas of greater density are
also possible as multiple individuals can inhabit one point. Zombies are now able to
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