The Undead Counterargument: When More Zombies Means Fewer Outbreaks

In the mathematics of outbreaks—zombie or otherwise—intuition says that more infectious agents should mean larger epidemics. Ramping up the transmission rate ought to make pandemics more likely, not less. And for most models, this holds true.

But a team of mathematicians from the University of Bergen and Chalmers University of Technology has discovered something strange hiding inside one of epidemiology's classic frameworks: a world where adding more zombies can prevent a zombie apocalypse.

The finding emerges from the Zombie Infection Model (ZIM), a mathematical structure that researchers Stein Bethuelsen, Erik Broman, and Samuel Modée have subjected to the first rigorous analysis since physicists introduced it as a thought experiment. Their conclusion, published on the arXiv preprint server, upends a basic assumption about how epidemics spread—and offers a lesson in why the gap between "obvious" and "proven" matters.

The Science

Epidemiologists have long used simple models to understand how diseases move through populations. The classic SIR model tracks individuals as they move through three states: Susceptible, Infected, Recovered. A susceptible person catches the disease from a sick neighbor; an infected person eventually recovers and gains immunity. The mathematics is tractable, the dynamics are well-understood, and for over a century, the model has informed public health decisions worldwide.

The Zombie Infection Model takes the SIR framework and twists it into something more adversarial. Here, an infected person isn't passive; they're a zombie. They don't recover on their own. Instead, they attack susceptible neighbors—and the neighbors fight back.

Concretely: each node in a network (representing a person, a city, a computer) can be in one of three states. A susceptible node becomes infected at a rate proportional to how many infected neighbors it has—specifically, at rate $\lambda$ per infected neighbor. But here's the twist: an infected node is killed by its susceptible neighbors at a rate proportional to how many susceptible neighbors it has—at rate 1 per susceptible neighbor.

The parameter $\lambda$ captures the "bite rate"—how aggressive the zombies are. Higher $\lambda$ means zombies bite harder and spread faster. Lower $\lambda$ means they're sluggish, easily beaten back by the living.

At first glance, you'd expect that cranking up $\lambda$ would always make zombie outbreaks more likely. Bethuelsen, Broman, and Modée set out to prove exactly this. What they found instead was far stranger.

What They Found

The researchers began with trees—connected networks with no loops, like a family genealogy chart. On these simple structures, their intuition held: if you increase $\lambda$, the probability of an infinite zombie outbreak definitely increases (Bethuelsen et al., 2026). More formally, there's a coupling between the two processes—one with ${\lambda }_1$ and one with ${\lambda }_2>{\lambda }_1$—such that the higher-$\lambda$ process always has at least as many zombies at every point in time.

The same monotonicity holds for the initial conditions. If you start with more zombies, you never end up with fewer.

But trees are special. They have no cycles. Real networks—friendship graphs, airline routes, the internet—are full of loops, shortcuts, and redundant connections. When the researchers expanded their analysis to bounded-degree graphs with cycles, the monotonicity broke apart.

Theorem 2.3c) delivers the counterintuitive blow: there exist finite, connected graphs where increasing the bite rate actually decreases the probability that a given node becomes infected. A zombie outbreak that would happen with ${\lambda }_1$ fails to happen with ${\lambda }_2>{\lambda }_1$. This isn't a numerical quirk or a negligible effect—it holds for the zombie outbreak event itself, the probability that the infection spreads indefinitely.

Critical Bite Rate Transition

Label	Value
790,205	790,205 nodes
1,658,421	1,658,421 nodes
17,448,501	17,448,501 nodes
35,616,869	35,616,869 nodes
61,002,692	61,002,692 nodes

At λ=2.26 and λ=2.27, zombie clusters grow indefinitely without reaching the simulation boundary. At λ=2.28 and above, they do reach the boundary, indicating finite outbreaks.

Escape Ratio vs Bite Rate

Label	Value
2.26	2.26
2.27	2.27
2.28	2.28
2.29	2.29
2.30	2.3

Escape ratio serves as a proxy and upper bound for survival probability on Z^2, showing the critical threshold appears near λ≈2.28.

Why does this happen? The key lies in the kill mechanism. When a zombie attacks a susceptible neighbor, there's a chance the susceptible fights back and kills the zombie instead. More zombies in an area means more attacks—but it also means more opportunities for the living to destroy the undead. A small horde of efficient zombies might sweep through a neighborhood, but a massive swarm of slightly faster zombies might exhaust itself against a population that's fighting back with proportionally greater force.

The researchers proved an even stronger result in Theorem 2.5: there are graphs of bounded degree where the zombie outbreak probability is continuous in $\lambda$—but still not monotone. The function $\varphi (\lambda ,G,\Delta )$ giving the outbreak probability doesn't jump or spike; it flows smoothly from positive to zero as $\lambda$ increases. Yet that smooth flow crosses from "outbreak possible" to "outbreak impossible" somewhere in the middle.

Monotonicity by Graph Type

Label	Value
Complete graph	1
Regular tree	2
Integer lattice	3
General bounded-degree	4

Monotonicity holds on trees and complete graphs, but fails on general bounded-degree graphs with cycles.

Beyond these structural results, the researchers established several precise thresholds. If the bite rate satisfies $\lambda \le 1$, no zombie outbreak can occur on any graph—the living always have at least as much fighting power as the dead. For higher bite rates, the question of whether an infinite outbreak is possible depends on the graph's structure: it exists if and only if ordinary site percolation exhibits a nontrivial phase transition on that graph.

For specific topologies, the answers become concrete. On a complete graph (everyone connected to everyone), the final proportion of zombies converges to $\frac{\lambda -1}{\lambda }$ with probability $1-\frac{1}{{\lambda }^M}$, where $M$ is the initial zombie count—when $\lambda >1$. On regular trees, the critical threshold satisfies $\lambda (1-(\lambda /(\lambda +1){)}^{d-1})=1$ for degree $d$.

Why This Changes Things

The result isn't merely a curiosity about zombies. The Zombie Infection Model connects directly to the biased voter model and the Williams-Bjerknes tumor growth model—frameworks used to study opinion dynamics, cultural transmission, and cancer progression. The non-monotonicity discovered here applies to all of these.

More broadly, the paper illuminates a genuine challenge in mathematical epidemiology: the gap between what simulations suggest and what theorems guarantee. Numerical experiments on the two-dimensional integer lattice clearly indicate that higher $\lambda$ means larger outbreak probability. The math on more complex graphs says something subtler.

This matters for real-world modeling. When researchers fit models to data or use them to predict intervention effects, they often assume that increasing transmission makes outbreaks more likely—that's the foundation for vaccine strategies, quarantine policies, and behavioral interventions. The ZIM shows that this assumption, while reasonable, isn't mathematically forced. In networks with complex geometry, the relationship between transmission and outcome can flip.

The researchers frame their non-monotonicity results as what they consider the main contributions of the paper. They're rare in the literature on stochastic growth models—the researchers know of only one similar statement, in a different context. The proof required carefully balancing the competing forces of infection and removal, constructing graphs where the network structure creates feedback loops that invert the expected relationship.

What's Next

The paper leaves several questions open. The two-dimensional case remains unresolved: simulations strongly suggest monotonicity on ${\mathbb{Z}}^2$, but no proof exists. The critical values for the integer lattice are known only asymptotically; sharper estimates would connect the mathematical theory more tightly to numerical experiments.

For applications, the researchers' bounds on the outbreak threshold—showing that $\lambda \le 1$ guarantees extinction, and that high-$\lambda$ outbreaks exist iff site percolation percolates—give modelers something concrete to work with. The complete graph and regular tree results are exact, offering test cases for approximation schemes and simulation validation.

Beyond epidemiology, the ZIM's counterintuitive behavior invites reflection on complex systems more generally. When feedback loops exist—when infected individuals affect the environment that constrains them—simple intuitions can fail. The mathematics reveals that the relationship between cause and effect is not always what it appears.

For now, though, the most striking lesson may be this: in the right network, fighting back too hard against zombies can make them stronger. The living's advantage isn't a monotonic function of their defensive prowess. Sometimes, the best strategy is subtler than "more of everything."

The code and figures from the paper suggest simulations of vast size—clusters of millions of nodes, carefully tracking the boundary between survival and extinction. The transition appears sharp, occurring around $\lambda \approx 2.28$ for the two-dimensional lattice. But sharp isn't proven, and what the math guarantees differs from what the simulations show.

The gap between numerical evidence and mathematical proof is where this research lives. It's a reminder that in the science of outbreaks, intuition is a starting point—not a destination.