You Can Run, You Can Hide: The Epidemiology and Statistical Mechanics of Zombies

Abstract: We use a popular fictional disease, zombies, in order to introduce techniques used in modern epidemiology modelling, and ideas and techniques used in the numerical study of critical phenomena. We consider variants of zombie models, from fully connected continuous time dynamics to a full scale exact stochastic dynamic simulation of a zombie outbreak on the continental United States. Along the way, we offer a closed form analytical expression for the fully connected differential equation, and demonstrate that the single person per site two dimensional square lattice version of zombies lies in the percolation universality class. We end with a quantitative study of the full scale US outbreak, including the average susceptibility of different geographical regions.

• The paper derives an analytical SZR model that identifies the critical termination rate needed to prevent complete zombie dominance.
• The paper employs Gillespie dynamics to show how stochastic fluctuations can lead to early outbreak extinction in small populations.
• The paper uses lattice simulations to link zombie propagation with percolation theory, revealing scale-free behavior at critical thresholds.

Insights into the Epidemiology and Statistical Mechanics of Zombies

This paper employs the popular fictional concept of zombies as a provisional framework for elucidating methods in modern epidemiology and statistical mechanics. It explores several model variants, ranging from continuous-time dynamics to precise stochastic simulations of a hypothetical zombie outbreak across the continental United States. Key contributions include an analytical solution for a fully connected differential equation model and evidence situating the individual-based two-dimensional square lattice model within the percolation universality class.

Analytical Perspectives on the $SZR$ Model

The central aspect of this study is the $SZR$ model, an adaptation of the classical $SIR$ (Susceptible-Infected-Recovered) framework in epidemiology. The $SZR$ model categorizes the population into susceptibles ($S$), zombies ($Z$), and removed ($R$) entities. The model introduces parameters $\beta$ and $\kappa$, governing the transition dynamics via interactions such as zombie bites and terminations by humans. The derivation of an analytical solution highlights conditions under which the entire population would inevitably transition to zombies unless hindered by a sufficiently high termination rate ($\kappa$). This model contrasts with traditional $SIR$ models in permitting a stable configuration where either humans or zombies are entirely extinguished.

Stochastic Simulations and the Role of Randomness

Building on the analytical foundations, this paper employs Gillespie dynamics to account for stochastic fluctuations, particularly significant in small population scenarios. The authors demonstrate that even within seemingly virulent outbreaks ($\alpha<1$), stochastic dynamics can result in the early extinction of the infection. The extinction probabilities and mean population outcomes emphasize the sensitivity of outbreak trajectories to smaller population density and random perturbations—remarkably underscoring the difference between deterministic and stochastic modeling in epidemiological studies.

Percolation Universality and Critical Phenomena

Through lattice-based simulations, this work investigates the critical behavior exhibited at the phase transition thresholds of zombie propagation. Establishing parallels with the percolation universality class, this study extends the understanding of how zombie outbreaks can scale across spatial dimensions. The phase transition analysis reveals scale-free behavior at criticality, corroborated by the alignment of outbreak size distributions with power law characteristics expected from percolation theory.

Large-Scale Simulation of a Zombie Outbreak

Expanding the scope, the paper simulates a zombie outbreak across the United States, introducing additional complexity through a latent state and zombie movement to capture realistic dynamics. This simulation, informed by parameters inspired by popular culture, provides insights into the spatial and temporal dynamics of outbreak propagation. The results indicate that densely populated regions or those nestled between populous cities face higher risks during protracted outbreaks, informing hypothetical tactical responses in managing such fictional scenarios.

Implications and Future Directions

The theoretical frameworks and models explored in this paper serve as an educational conduit for demonstrating sophisticated concepts in epidemiology and statistical physics. While grounded in fictional premises, the methodologies and findings present opportunities for broader applicability in modeling real-world contagion scenarios. Future explorations might refine these models by integrating more complex movement and interaction patterns, exploring universality classes further, or adapting these techniques for analogous studies in actual epidemiological contexts. This cross-disciplinary approach underscores the potential of integrating unconventional models to enhance our understanding of disease dynamics.