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You can run, you can hide: The epidemiology and statistical mechanics of zombies
Open accessAbstract
We use a popular fictional disease, zombies, in order to introduce techniques used in modern epidemiology modeling, 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.
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Written from the paper’s full text.
TL;DR
The paper uses zombie outbreaks as a fictional model system to illustrate epidemiological modeling and critical phenomena, from a solvable well-mixed SZR model to stochastic lattice simulations and a continental US-scale outbreak. It shows the 2D single-person-per-site zombie model lies in the percolation universality class and provides a quantitative US zombie-risk simulation.
Key findings
- The well-mixed SZR model admits a closed-form analytical solution; the final state depends on the sign of \(P = Z_0 + (1-\alpha)S_0\), with zombies winning for \(\alpha < 1\) and humans winning for \(\alpha > 1\).
- In stochastic simulations, the extinction probability for a supercritical outbreak is \(P_{\text{ext}} = \alpha\); for \(\alpha = 0.6\), population 200, and one initial infected, zombies won only 40% of runs.
- On a 2D square lattice with one person per site, the zombie model has critical point \(\alpha_c = 0.43734613(57)\), follows percolation scaling with \(\tau = 187/91\), and has fractal dimension \(d_f = 1.89(4)\), compared with the exact percolation value \(91/48 = 1.895833\).
- In the US-scale simulation using 2010 Census data, with \(\alpha = 0.8\), \(\beta = 3.6 \times 10^{-3}\)/h, and \(\mu = 0.0914\)/h, most of the US population becomes zombies within the first week; after 28 days the greatest risk is in northeastern Pennsylvania, while after 7 days risk concentrates around populous metropolitan areas.
Methods
- Deterministic SZR compartmental ODE model with susceptible, zombie, and removed states, nondimensionalized and solved analytically; compared with a density-dependent SIR model.
- Stochastic simulations using the Gillespie algorithm and, for the US scale, the Next Reaction Method; branching-process analysis for extinction probability.
- 2D square-lattice simulations with periodic boundary conditions, finite-size scaling, power-law cluster-size distributions, and box-counting fractal dimension; US simulation on a 1500 × 900 grid built from 2010 Census block data, with an added exposed state and zombie diffusion.
Limitations
- The paper does not include a formal limitations section. It states that real-world SZR-like dynamics would require refinement because medical personnel are only a small fraction of susceptible hosts, and it notes late-time computational inefficiency from simulating zombie diffusion.
- Evident from the text: parameters are chosen from popular-culture depictions and simple physical estimates rather than empirical zombie data; the US lattice uses binary closing to add artificial connectivity; humans are assumed not to move; and the outbreak model is explicitly fictional.
Model-written summary. Check it against the paper before relying on it.
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Claims the paper makes
0 of 5 cited sources checked · 0 supported · 0 flagged
- Findingreasoning
If zombies win in the SZR model, the final zombie population is Z∞ = Z0 + (1 - α)S0.
“If zombies win we have Z ∞ = Z 0 + (1 -α)S 0 .”
Quote in the paperSubstantial claim - Findingreasoning
P = Z + (1 - α)S is a constant of the SZR dynamics.
“so P is a constant of the dynamics.”
Quote in the paperSubstantial claim - Findingown data
For α = 0.6 with a population of 200 and one infected individual to start, the zombies win only 40% of the time.
“For the parameter settings used in Figs. 1 and 2, namely α = 0.6 with a population of 200 and one infected individual to start, the zombies win only 40% of the time.”
Quote in the paperNot a substantial claim - Findingown data
In stochastic dynamics, it is not always guaranteed that a supercritical (α < 1) outbreak will take over the entire susceptible population.
“Another implication of stochastic dynamics is that it is not always guaranteed that a supercritical (α < 1) outbreak will take over the entire susceptible population.”
Quote in the paperSubstantial claim - Findingreasoning
If α < 1, the SZR system always flows to a final state composed entirely of zombies and no humans, where P denotes the number of zombies that survive.
“and the system will always flow to a final state composed of entirely zombies and no humans, where P denotes the number of zombies that survive.”
Quote in the paperSubstantial claim
- Findingreasoning
The sign of P governs whether there will eventually be humans or zombies in the final state of the SZR model.
“Given the analytical solution, it is clear to see that the sign of P governs whether there will eventually be humans or zombies in the final state.”
Quote in the paperSubstantial claim - Findingreasoning
Unlike SIR, the SZR model admits an analytical solution, assuming R(0) = 0.
“Furthermore, unlike SIR, SZR admits an analytical solution, assuming R(0) = 0, and with Z 0 ≡ Z(0),S 0 ≡ S(0):”
Quote in the paperSubstantial claim - Findingreasoning
If α > 1, humans are more effective at killing zombies than zombies are at biting humans.
“If, however, α > 1, humans are more effective at killing zombies than zombies are at biting humans.”
Quote in the paperSubstantial claim - Findingreasoning
If α > 1, with enough zombies in the initial state, the zombies can still convert all humans before the humans kill all the zombies.
“With enough zombies in the initial state, we can still convert all of the humans before they have time to kill all of the zombies.”
Quote in the paperSubstantial claim - Findingown data
The number of zombies at the end of a stochastic simulation is not fixed.
“Additionally, the number of zombies we end with is not fixed, as shown in Fig. 3.”
Quote in the paperSubstantial claim - Findingreasoning
If humans win in the SZR model, the final susceptible population is S∞ = S0 - Z0/(α - 1).
“And if humans win S ∞ = S 0 - Z 0 α -1 .”
Quote in the paperSubstantial claim - Findingreasoning
The SZR dynamics is an exponential decay or increase in the ratio of humans to zombies χ = S/Z.
“Here we see that the dynamics is simply an exponential decay or increase in the ratio of humans to zombies χ = S/Z.”
Quote in the paperSubstantial claim - Findingnone given
The SZR model has a stable configuration when either the human or the zombie population is defeated (S = 0 or Z = 0).
“Unlike a traditional disease (e.g., as modeled by SIR), for the zombie model, we have a stable configuration when either the human or the zombie population is defeated (S = 0 or Z = 0).”
Quote in the paperSubstantial claim - Findingreasoning
The final dead zombie population R remains small for all values of α.
“The final dead zombie population R remains small for all values of α; for extremely virulent zombies α 1, very few will be killed by the humans before all of the humans are converted, while in the other extreme few zombies are created so there are few to be killed.”
Quote in the paperNot a substantial claim - Findingreasoning
The SZR model's interactions are density dependent, in the sense that the rate at which humans are converted to zombies and zombies are killed depends on the total count of zombies and humans in the site.
“Notice that these interactions are density dependent, in the sense that the rate at which we convert humans to zombies and kill zombies is dependent on the total count of zombies and humans in this site.”
Quote in the paperSubstantial claim - Findingown data
For the SZR model, the dynamics is fairly independent of population size once the population size gets above around 100 individuals.
“Looking at the SZR results in the top row, we can see that the dynamics is fairly independent of population size once the population size gets above around 100 individuals.”
Quote in the paperNot a substantial claim - Findingreasoning
There are certain regions-those surrounding populous metropolitan areas-that are at greater risk of being overrun by zombies after seven days.
“Here you can clearly see that there are certain regions-those surrounding populous metropolitan areas-that are at a greater risk.”
Quote in the paperSubstantial claim- The authors state this as an assumption: Humans do not move during the zombie outbreak.
- The authors state this as an assumption: Zombies behave like a random gas inside a cell, so the probability that a zombie will cross a cell boundary is roughly one-fourth of the zombies within v t of the edge moving across that edge in a small amount of time.
- The authors state this as an assumption: Zombies undergo random motion inside a cell at 1 ft/s and interact with a human anytime they come within 100 ft.
- Findingown data
Simulating zombie outbreaks on fixed lattices shows qualitatively different behavior for small α and large α.
“Simulating zombie outbreaks on fixed lattices, there is qualitatively different behavior for small α and large α.”
Quote in the paperSubstantial claim - Findingown data
Even four months in, remote areas of Montana and Nevada remain zombie free.
“Even four months in, remote areas of Montana and Nevada remain zombie free.”
Quote in the paperNot a substantial claim- The authors state this as an assumption: Humans do not move during the zombie outbreak.
- The authors state this as an assumption: Zombies behave like a random gas inside a cell, so the probability that a zombie will cross a cell boundary is roughly one-fourth of the zombies within v t of the edge moving across that edge in a small amount of time.
- The authors state this as an assumption: Zombies undergo random motion inside a cell at 1 ft/s and interact with a human anytime they come within 100 ft.
- Findingown data
When α is very small on a fixed lattice, zombies seem to grow until they infect the entire lattice.
“When α is very small, the zombies seem to grow until they infect the entire lattice.”
Quote in the paperSubstantial claim - Findingreasoning
After several weeks, the map exhibits stronger anisotropy, as the zombie front is influenced by large inhomogeneities in population density.
“After several weeks, the map exhibits stronger anisotropy, as we spread over larger geographical areas and the zombie front is influenced by large inhomogeneities in population density.”
Quote in the paperNot a substantial claim- The authors state this as an assumption: Humans do not move during the zombie outbreak.
- The authors state this as an assumption: Zombies behave like a random gas inside a cell, so the probability that a zombie will cross a cell boundary is roughly one-fourth of the zombies within v t of the edge moving across that edge in a small amount of time.
- The authors state this as an assumption: Zombies undergo random motion inside a cell at 1 ft/s and interact with a human anytime they come within 100 ft.
- Findingown data
The median population of an occupied site is 13.
“The mean population of the occupied grid sites is 420; the median population of an occupied site is 13.”
Quote in the paperNot a substantial claim - Findingown data
For zombies on a two-dimensional lattice, the critical point occurs at α_c = 0.43734613(57).
“For zombies on a two dimensional lattice, this critical point occurs at α c = 0.43734613(57); the resulting integrated cluster size distribution is shown at the top of Fig. 7.”
Quote in the paperSubstantial claim - Findingreasoning
Density-dependent SIR dynamics is very strongly population dependent.
“Notice that density-dependent SIR is very strongly population dependent.”
Quote in the paperSubstantial claim - Findingown data
After 28 days, it is not the largest metropolitan areas that suffer the greatest risk, but the regions located between large metropolitan areas.
“After 28 days, it is not the largest metropolitan areas that suffer the greatest risk, but the regions located between large metropolitan areas.”
Quote in the paperSubstantial claim- The authors state this as an assumption: Humans do not move during the zombie outbreak.
- The authors state this as an assumption: Zombies behave like a random gas inside a cell, so the probability that a zombie will cross a cell boundary is roughly one-fourth of the zombies within v t of the edge moving across that edge in a small amount of time.
- The authors state this as an assumption: Zombies undergo random motion inside a cell at 1 ft/s and interact with a human anytime they come within 100 ft.
- Findingown data
The three curves are for very similar α, leading to excellent precision in identifying the critical point.
“Notice that the three curves are for very similar α-leading to excellent precision in identifying the critical point.”
Quote in the paperSubstantial claim - Findingreasoning
Observed extinction probabilities for 1000 Gillespie runs of a population of 10^4 individuals at various values of α agree with the expectation that they go as α.
“The observed extinction probabilities agree with the expectation that they should go as α, here shown as the dashed line.”
Quote in the paperSubstantial claim - Findingreasoning
In California, the region near Bakersfield in the San Joaquin Valley is at the greatest risk.
“For instance, in California it is the region near Bakersfield in the San Joaquin Valley that is at the greatest risk as this area will be overrun by zombies whether they originate in the San Francisco area or the Los Angeles-San Diego area.”
Quote in the paperNot a substantial claim- The authors state this as an assumption: Humans do not move during the zombie outbreak.
- The authors state this as an assumption: Zombies behave like a random gas inside a cell, so the probability that a zombie will cross a cell boundary is roughly one-fourth of the zombies within v t of the edge moving across that edge in a small amount of time.
- The authors state this as an assumption: Zombies undergo random motion inside a cell at 1 ft/s and interact with a human anytime they come within 100 ft.
- Findingreasoning
The area with the greatest one-month zombie risk is northeastern Pennsylvania.
“The area with the greatest one month zombie risk is northeastern Pennsylvania, itself being susceptible to outbreaks originating in any of the large metropolitan areas on the east coast.”
Quote in the paperSubstantial claim- The authors state this as an assumption: Humans do not move during the zombie outbreak.
- The authors state this as an assumption: Zombies behave like a random gas inside a cell, so the probability that a zombie will cross a cell boundary is roughly one-fourth of the zombies within v t of the edge moving across that edge in a small amount of time.
- The authors state this as an assumption: Zombies undergo random motion inside a cell at 1 ft/s and interact with a human anytime they come within 100 ft.
- Findingreasoning
Infestations on the coasts, with higher population density, have spread farther than those near the center of the country.
“Infestations on the coasts, with their higher population density, have spread farther than those near the center of the country.”
Quote in the paperNot a substantial claim- The authors state this as an assumption: Humans do not move during the zombie outbreak.
- The authors state this as an assumption: Zombies behave like a random gas inside a cell, so the probability that a zombie will cross a cell boundary is roughly one-fourth of the zombies within v t of the edge moving across that edge in a small amount of time.
- The authors state this as an assumption: Zombies undergo random motion inside a cell at 1 ft/s and interact with a human anytime they come within 100 ft.
- Findingown data
For the chosen parameters, most of the United States population has been turned into zombies by the first week.
“for the parameters we chose, most of the United States population has been turned into zombies by the first week”
Quote in the paperNot a substantial claim- The authors state this as an assumption: Humans do not move during the zombie outbreak.
- The authors state this as an assumption: Zombies behave like a random gas inside a cell, so the probability that a zombie will cross a cell boundary is roughly one-fourth of the zombies within v t of the edge moving across that edge in a small amount of time.
- The authors state this as an assumption: Zombies undergo random motion inside a cell at 1 ft/s and interact with a human anytime they come within 100 ft.
- Findingreasoning
The transition observed in the S population is largely independent of μ, except on the very small end.
“The transition observed in the S population is largely independent of μ, except on the very small end.”
Quote in the paperSubstantial claim - Findingown data
The fractal dimension of the clusters near the critical point is d_f = 1.89(4).
“We find a fractal dimension d f = 1.89(4), compared to the exact percolation value of d f = 91/48 = 1.895833.”
Quote in the paperSubstantial claim - Findingreasoning
As δS → 0, the SZR and SIR models are the same with α = μN/S0.
“Here as δS → 0, the two models are the same with α = μN/S 0 , another indication that the density dependent SIR model's virulence is dependent on population size.”
Quote in the paperSubstantial claim - Findingnone given
The only stable configuration for the SIR model is when there is no infected population (I = 0).
“Notice again that while the only stable configuration for the SIR model is when there is no infected population (I = 0), the SZR model is stable when either the humans or zombies are depleted (S = 0 or Z = 0).”
Quote in the paperSubstantial claim - Findingown data
With α chosen precisely at the critical value, the lattice model shows a giant component with fractal structure.
“With α chosen to be precisely at the critical value, we indeed see a giant component with fractal structure (Fig. 6).”
Quote in the paperSubstantial claim - Findingown data
The constructed population lattice has many empty grid sites, especially throughout the western United States.
“You will see the presence of many empty grids, especially throughout the western United States.”
Quote in the paperNot a substantial claim - Interpretationreasoning
A small error in α_c can produce large shifts in the resulting fits for τ and d_f.
“A small error in α c can produce large shifts in the resulting fits for τ and d f , demanding efficient programming and fast computers to achieve a definitive answer.”
Quote in the paperSubstantial claim - Interpretationreasoning
For the density-dependent SIR model on inhomogeneous population lattices, the most important parameter governing whether a particular site has a breakout infection is the population of that site on the lattice.
“When we move to inhomogeneous population lattices this means that for the density dependent SIR model, the most important parameter governing whether a particular site has a breakout infection is the population of that site on the lattice.”
Quote in the paperSubstantial claim - Interpretationreasoning
This corresponds to a low population density of ∼11 people/km^2.
“This corresponds to a low population density of ∼11 people/km 2 , again agreeing with our intuition.”
Quote in the paperNot a substantial claim- The authors state this as an assumption: Zombies behave like a random gas inside a cell, so the probability that a zombie will cross a cell boundary is roughly one-fourth of the zombies within v t of the edge moving across that edge in a small amount of time.
- The authors state this as an assumption: Zombies undergo random motion inside a cell at 1 ft/s and interact with a human anytime they come within 100 ft.
- Interpretationreasoning
At the critical point, the probabilities of outbreaks of size s are related to the probabilities at size s/B by some factor f: P(s, αc) = f P(s/B, αc).
“In particular, we expect at the critical point to find the probabilities of outbreaks of size s to be related to the probabilities at size s/B by some factor f : P (s,α c ) = f P (s/B,α c ).”
Quote in the paperSubstantial claim- The authors state this as an assumption: If the probabilities and form of mutual triggering are the same at each scale, then many properties of outbreaks would be the same after rescaling sizes by a factor of B.
- Interpretationreasoning
Because zombies do not fly airplanes, the zombie model is closer in spirit to the spread of certain agricultural infestations, where the disease spreads across a lattice of sites along the two-dimensional surface of the Earth.
“Zombies do not fly airplanes, so our model is closer in spirit to the spread of certain agricultural infestations, where the disease spreads across a lattice of sites along the two-dimensional surface of the Earth (although not in those cases where pathogens are transported long distances by atmospheric currents).”
Quote in the paperSubstantial claim- The authors state this as an assumption: At criticality, an outbreak of size Bs is formed by what would have been B smaller outbreaks of size s which happened to trigger one another, and these in turn are formed by what would have been B outbreaks of size s/B.
- Premisenone given
Percolation theory predicts τ = 187/91 in two dimensions.
“Percolation theory predicts τ = 187/91 in two dimensions, and we test that prediction in the bottom part of Fig. 7.”
Quote in the paperSubstantial claim - Interpretationcitation
The zombies in the films are 1.25 times more effective at biting humans than the humans are at killing the zombies.
“This means that the zombies in the films are 1.25 times more effective at biting humans than the humans are at killing the zombies.”
Quote in the paperSubstantial claim - Study design and conductreasoning
The process of choosing Z-S bonds and applying probabilities β/(β + κ) and κ/(β + κ) matches the stochastic dynamics of the zombie model operating on the lattice.
“This process matches the stochastic dynamics of our zombie model operating on the lattice.”
Quote in the paperSubstantial claim - Premisecitation
The SIR model has been demonstrated to undergo a phase transition.
“The SIR model has been demonstrated to undergo such a phase transition, and we expect the zombie model does as well.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [18]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Interpretationreasoning
The two transitions in SIR couple differently to the population of infected and susceptible.
“The two transitions in SIR couple differently to the population of infected and susceptible.”
Quote in the paperSubstantial claim - Interpretationown data
The population dependence for lower population sizes in the SZR model is an effect of stochasticity.
“The population dependence for lower population sizes is an effect of the stochasticity.”
Quote in the paperNot a substantial claim - Interpretationown data
The formula (36) quantifies an emergent scale invariance at α_c: the properties of epidemics of size s are rescaled versions of the properties at a smaller scale s/B.
“This formula quantifies an emergent scale invariance at α c : the properties of epidemics of size s (here the probability) are rescaled versions of the properties at a smaller scale s/B”
Quote in the paperSubstantial claim - Interpretationown data
The single person per site zombie infestation, near the critical virulence, will on long length scales develop spatial infestation patterns that are well described by two-dimensional percolation theory.
“We conclude that the single person per site zombie infestation, near the critical virulence, will on long length scales develop spatial infestation patterns that are well described by two-dimensional percolation theory.”
Quote in the paperSubstantial claim - Interpretationown data
The zombie model on the two-dimensional lattice is in the percolation universality class.
“The clear agreement convincingly shows that the zombie model on the two dimensional lattice is in the percolation university class.”
- Also stated in: § VI. CONCLUSION · boosted
“We investigated the critical behavior of the single person per site two-dimensional square lattice zombie model and demonstrated it is in the percolation universality class.”
Quote in the paperSubstantial claim - Also stated in: § VI. CONCLUSION · boosted
- Interpretationown data
The empty grid sites disconnect the east and west coasts in a clearly artificial pattern.
“This disconnects the east and west coasts in a clearly artificial pattern-our zombies in practice will gradually wander through the empty grid points.”
Quote in the paperSubstantial claim - Interpretationreasoning
In the limit important for extinction, namely early times with very large populations, the SZR model and SIR are effectively the same, since the population of susceptibles is nearly constant.
“In precisely the limit that is important for studying the probability of an extinction event, namely at early times with very large populations, the SZR model and SIR are effectively the same, since the population of susceptibles (S) is nearly constant.”
Quote in the paperSubstantial claim - Interpretationown data
The qualitatively different lattice behavior for small and large α suggests evidence of a phase transition.
“This suggests evidence of a phase transition.”
Quote in the paperSubstantial claim - Premisecitation
The same extinction probability (P_ext = μ = R0^-1) is observed for the SIR model.
“This same extinction probability (P ext = μ = R -1 0 ) is observed for the SIR model [1].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [1]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- The quote only partly states this claim. 1
- It may state two separate claims as one. 15
- Probably not a substantial claim. 6
The quote only partly states this claim.
FindingreasoningFor an α < 1 simulation in the limit of large populations, the probability of extinction is P_ext = κ/(β + κ) = α.
“The probability of extinction is just given by our dimensionless inverse virulence α.”
Quote in the paperSubstantial claimIt may state two separate claims as one.
FindingreasoningUnlike the SZR model, the SIR epidemic is self-limiting and there will always be some susceptibles left at the end of the outbreak, no matter how virulent the disease is.
“Unlike the SZR model, here we see that no matter how virulent the disease is, the epidemic will be self-limiting, and there will always be some susceptibles left at the end of the outbreak.”
Quote in the paperSubstantial claimIt may state two separate claims as one.
FindingreasoningThe SIR model does not admit a closed form analytical solution, but a parametric solution can be found.
“The SIR model does not admit a closed form analytical solution, but we can find a parametric solution by dividing the first equation by the third, revealing”
Quote in the paperNot a substantial claimIt may state two separate claims as one.
Findingown dataWhen α is large on a fixed lattice, zombies do not spread very far and are always defeated by their neighboring humans.
“When α is large, the zombies do not spread very far, always being defeated by their neighboring humans.”
Quote in the paperSubstantial claimProbably not a substantial claim. The paper’s conclusions may not depend on it.
Findingnone givenThe mean population of the occupied grid sites is 420.
“The mean population of the occupied grid sites is 420; the median population of an occupied site is 13.”
Quote in the paperNot a substantial claimIt may state two separate claims as one.
FindingreasoningThe relevant virulence parameter μ for the density-dependent SIR model has a population dependence, unlike the SZR α parameter which depends only on disease constants β and κ.
“Unlike the α parameter for SZR which depends only on our disease constants β,κ, the relevant virulence for the density dependent SIR model (μ) has a population dependence.”
Quote in the paperSubstantial claimProbably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
Findingown dataThe paper's fractional fluctuations are less than 0.1%, while the experimental results vary over several orders of magnitude.
“Notice the small vertical scale: our fractional fluctuations are less than 0.1%, while our experimental results vary over several orders of magnitude.”
Quote in the paperNot a substantial claimIt may state two separate claims as one.
Findingown dataIn the early stages of the outbreak, while the population is roughly homogeneous, the zombie plague spreads out in roughly uniform circles, where the speed of infection is tied to local population density.
“In the early stages of the outbreak, while the population is roughly homogeneous, the zombie plague spreads out in roughly uniform circles, where the speed of the infection is tied to the local population density.”
Quote in the paperSubstantial claim- The authors state this as an assumption: Humans do not move during the zombie outbreak.
- The authors state this as an assumption: Zombies behave like a random gas inside a cell, so the probability that a zombie will cross a cell boundary is roughly one-fourth of the zombies within v t of the edge moving across that edge in a small amount of time.
- The authors state this as an assumption: Zombies undergo random motion inside a cell at 1 ft/s and interact with a human anytime they come within 100 ft.
It may state two separate claims as one.
Findingown dataAfter four weeks, much of the United States has fallen, but it takes a very long time for the zombies to diffuse and capture the remaining portions of the United States.
“After four weeks, much of the United States has fallen, but it takes a very long time for the zombies to diffuse and capture the remaining portions of the United States.”
Quote in the paperSubstantial claim- The authors state this as an assumption: Humans do not move during the zombie outbreak.
- The authors state this as an assumption: Zombies behave like a random gas inside a cell, so the probability that a zombie will cross a cell boundary is roughly one-fourth of the zombies within v t of the edge moving across that edge in a small amount of time.
- The authors state this as an assumption: Zombies undergo random motion inside a cell at 1 ft/s and interact with a human anytime they come within 100 ft.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
Findingown dataThe most populated grid site is downtown New York City, with 299,616 individuals.
“The most populated grid site is downtown New York City, with 299,616 individuals.”
Quote in the paperNot a substantial claimIt may state two separate claims as one.
Findingown dataThere is a transition in the susceptible population near α = 1, corresponding to where the continuous dynamics would show a sharp boundary; in the stochastic model the boundary is blurred due to stochasticity.
“We can clearly see a transition in the susceptible population near α = 1 corresponding to where our continuous dynamics would show a sharp boundary.Here the boundary is blurred, again due to the stochasticity.”
Quote in the paperNot a substantial claimIt may state two separate claims as one.
Findingown dataFor α > α_c (when the zombies lose), the large outbreaks are suppressed below the power law, and for α < α_c (when the zombies often win), the largest outbreaks bend up.
“Notice that for α > α c (when the zombies lose) the large outbreaks are suppressed below the power law, and for α < α c (when the zombies often win) the largest outbreaks bend up.”
Quote in the paperSubstantial claimIt may state two separate claims as one.
InterpretationreasoningThis greater risk is partly because those regions have lots of individuals who could potentially serve as patient zero, and partly due to the rapid spread of zombies in those areas.
“This is partly because those regions have lots of individuals who could potentially serve as patient zero, and partly due to the rapid spread of zombies in those areas.”
Quote in the paperSubstantial claim- The authors state this as an assumption: Humans do not move during the zombie outbreak.
- The authors state this as an assumption: Zombies behave like a random gas inside a cell, so the probability that a zombie will cross a cell boundary is roughly one-fourth of the zombies within v t of the edge moving across that edge in a small amount of time.
- The authors state this as an assumption: Zombies undergo random motion inside a cell at 1 ft/s and interact with a human anytime they come within 100 ft.
Probably not a substantial claim. The paper may give no reason why it holds. It may state two separate claims as one.
InterpretationreasoningMedical personnel represent only a small fraction of all susceptible hosts, so a refinement to an SZR-type model would be required to account for their role.
“(Medical personnel, however, represent only a small fraction of all susceptible hosts, so a refinement to an SZR-type model would be required to account for this.)”
Quote in the paperNot a substantial claimIt may state two separate claims as one.
Interpretationnone givenThe SZR model requires that susceptible hosts directly participate in the removal of zombie hosts from the infectious population, leading to runaway outbreaks as susceptible hosts are depleted.
“As noted, the SZR model requires that susceptible hosts directly participate in the removal of zombie hosts from the infectious population, leading to runaway outbreaks as susceptible hosts are depleted.”
Quote in the paperNot a substantial claimIt may state two separate claims as one.
PremisereasoningFor zombies, it is appropriate to model the bite parameter β as a rate per person per unit time, so that a zombie in a larger population is proportionally more effective at infecting and is killed proportionally faster.
“By claiming that zombies can be modeled by a single bite parameter β that itself is a rate per person per unit time, we are claiming that a zombie in a block with 5000 people would be one hundred times as effective at infecting new zombies as a zombie in a block with fifty people; similarly the zombie in question would be killed one hundred times faster.”
Quote in the paperSubstantial claim- Rests on an assumption: Zombies directly seek out hosts to infect, at which point the human and zombie engage in a duel to the death, making density-dependent interactions appropriate.
It may state two separate claims as one.
PremisecitationThe SIR model on a two-dimensional lattice with a single person per site falls into the percolation universality class, though details of its cluster growth can differ.
“The SIR model on a two-dimensional lattice with a single person per site falls into the percolation universality class [19], though details of its cluster growth can differ [20].”
Quote in the paperSubstantial claim0 of 2 sources checked- cites [19]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- cites [20]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
It may state two separate claims as one.
InterpretationreasoningThe holes of all sizes in the critical cluster reflect proximity to the threshold: the battle between zombies and humans is so evenly matched that one gets an emergent scale invariance in the survival patterns.
“This reflects the proximity to the threshold: the battle between zombies and humans is so evenly matched, that one gets an emergent scale invariance in the survival patterns.”
Quote in the paperSubstantial claimProbably not a substantial claim.
PremisecitationIt is often the case that there are several universality classes with critical exponents close to one another, as in the case of percolation.
“However, it is often the case that there are several universality classes with critical exponents close to one another as in the case of percolation [22].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [22]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Probably not a substantial claim.
Premisenone givenThe exact percolation value of the fractal dimension is d_f = 91/48 = 1.895833.
“compared to the exact percolation value of d f = 91/48 = 1.895833.”
Quote in the paperSubstantial claim
The paper relies on this without support.
The SZR model's population is fully connected and well-mixed, so any infected individual can infect any susceptible individual with equal probability.
“Until now, we have considered fully connected, well-mixed populations, where any infected individual can infect any susceptible individual with equal probability.”
The paper relies on this without support.
On the two-dimensional square lattice, infection spreads through nearest neighbor bonds only.
“The infection spreads through nearest neighbor bonds only.”
The paper relies on this without support.
Zombies directly seek out hosts to infect, at which point the human and zombie engage in a duel to the death, making density-dependent interactions appropriate.
“Zombies directly seek out hosts to infect, at which point the human and zombie engage in a duel to the (un)death.”
The paper relies on this without support.
Zombies do not fly airplanes.
“Zombies do not fly airplanes, so our model is closer in spirit to the spread of certain agricultural infestations, where the disease spreads across a lattice of sites along the two-dimensional surface of the Earth (although not in those cases where pathogens are transported long distances by atmospheric currents).”
The authors state this as an assumption.
At criticality, an outbreak of size Bs is formed by what would have been B smaller outbreaks of size s which happened to trigger one another, and these in turn are formed by what would have been B outbreaks of size s/B.
“Let us oversimplify to say that at criticality an outbreak of size Bs is formed by what would have been B smaller outbreaks of size s which happened to trigger one another, and these in turn are formed by what would have been B outbreaks of size s/B.”
The authors state this as an assumption.
If the probabilities and form of mutual triggering are the same at each scale, then many properties of outbreaks would be the same after rescaling sizes by a factor of B.
“If the probabilities and form of this mutual triggering is the same at each scale, then it would not surprise us that many properties of the outbreaks would be the same, after rescaling the sizes by a factor of B.”
The authors state this as an assumption.
Humans do not move during the zombie outbreak.
“We assume that the humans do not move, not only for computational efficiency, but because, as we will see, the zombie outbreaks tend to happen rather quickly, and we expect large transportation networks to shut down in the first days, pinning most people to their homes.”
The authors state this as an assumption.
Zombies behave like a random gas inside a cell, so the probability that a zombie will cross a cell boundary is roughly one-fourth of the zombies within v t of the edge moving across that edge in a small amount of time.
“we assume that the zombies behave like a random gas inside the cell, so that the probability that a zombie will cross a cell boundary is roughly 1 4 Z L 2 Lv t, that is, one-fourth of the zombies within v t of the edge will move across that edge in a small amount of time.”
The authors state this as an assumption.
Zombies undergo random motion inside a cell at 1 ft/s and interact with a human anytime they come within 100 ft.
“we similarly assume that the zombies are undergoing random motion inside the cell at 1 ft/s, and they interact with a human anytime they come within 100 ft.”
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Zombie movement equation omits three outgoing directions
MajorReaction (45) moves a zombie at rate μZi to each nearest neighbor. On the stated four-neighbor square lattice, total outgoing movement is 4μZi, whereas equation (41) subtracts only μZi.
“Żi = μ j Z j -μZ i ,(41)”
“i j : (Z i ,Z j ) μZ i --→ (Z i -1,Z j + 1). (45”
Figure caption gives the wrong power-law slope
MinorThe text derives a cumulative cluster-size law proportional to s^(2−τ), whose log-log slope is 2−τ. Figure 7's caption instead calls the slope 2τ.
“The fact that it forms a straight line on a log-log plot indicates that P s is a power law, and the slope is 2τ .”
“Integrating from s to ∞, the probability that a point belongs to a cluster of at least s in size (P s ) should at the critical point itself follow a power law: P s ∼ s 2-τ .”
Exposed-state equation omits new infections
MinorEquation (38) gives the exposed population only a loss term, but reaction (42) moves bitten humans into that state. The equation therefore cannot describe the stated infection reaction.
“Ėi = -νE i , (38”
“(S i ,E i ) βS i Z i ---→ (S i -1,E i + 1), (42”
Citations
Cites
Modeling Infectious Diseases in Humans and Animals
Keeling, M. J., & Rohani, P. (2008) · Princeton University Press eBooks · doi:10.1515/9781400841035 · Cited by 2,822
Efficient Exact Stochastic Simulation of Chemical Systems with Many Species and Many Channels
Gibson, M., & Bruck, J. (2000) · The Journal of Physical Chemistry A · doi:10.1021/jp993732q · Cited by 1,610
Sethna, J. P., Dahmen, K. A., & Myers, C. R. (2001) · Nature · doi:10.1038/35065675 · Cited by 1,218 · cited 1×
“This formula quantifies an emergent scale invariance at αc: the properties of epidemics of size s (here the probability) are rescaled versions of the properties at a smaller scale s/B [21]—the system is self-similar to itself at different scales.”
BackgroundSelf-citationOn the critical behavior of the general epidemic process and dynamical percolation
Grassberger, P. (1983) · Mathematical Biosciences · doi:10.1016/0025-5564(82)90036-0 · Cited by 649
Perspective: Stochastic algorithms for chemical kinetics
Gillespie, D. T., Hellander, A., & Petzold, L. R. (2013) · The Journal of Chemical Physics · doi:10.1063/1.4801941 · Cited by 331
Epidemic models and percolation
Cardy, J., & Grassberger, P. (1985) · Journal of Physics A Mathematical and General · doi:10.1088/0305-4470/18/6/001 · Cited by 231
Percolation and epidemics in a two-dimensional small world
Newman, M. E. J., Jensen, I., & Ziff, R. M. (2002) · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics · doi:10.1103/physreve.65.021904 · Cited by 152
Recent advances and open challenges in percolation
Araújo, N. A. M., Grassberger, P., Kahng, B., et al. (2014) · The European Physical Journal Special Topics · doi:10.1140/epjst/e2014-02266-y · Cited by 144 · cited 1×
“several universality classes with critical exponents close to one another as in the case of percolation [22].”
BackgroundCritical behavior of the susceptible-infected-recovered model on a square lattice
Tomé, T., & Ziff, R. M. (2010) · Physical Review E · doi:10.1103/physreve.82.051921 · Cited by 94 · cited 3×
“The SIR model on a two-dimensional lattice with a single person per site falls into the percolation universality class [3], though details of its cluster growth can differ [21].”
BackgroundMethodInfluentialThe Zombie Survival Guide : Complete Protection from the Living Dead
Brooks, M. (2003) · Cited by 62
Fu, J., Wu, S., Li, H., et al. (2014) · Journal of Computational Physics · doi:10.1016/j.jcp.2014.06.025 · Cited by 14 · cited 1×
“[5] Jin Fu, Sheng Wu, Hong Li, and Linda R Petzold.”
BackgroundBraaaiiinnnsss!: From Academics to Zombies
Robert, S. (2011) · Cited by 8
Mathematical Modelling of Zombies
(2014) · University of Ottawa Press eBooks · doi:10.1515/9780776621685 · Cited by 8
World War Z: An Oral History of the Zombie War
Sutton, J. (2010) · Naval War College review · Cited by 7
Bayesian Analysis of Epidemics - Zombies, Influenza, and other DiseasesPreprint
Witkowski, C., & Blais, B. S. (2013) · arXiv (Cornell University) · doi:10.48550/arxiv.1311.6376 · Cited by 7
(2011) · University of Ottawa Press eBooks · doi:10.1515/9780776619613 · Cited by 3
Núñez, F., Ravello, C., Urbina, H., & Pérez‐Acle, T. (2012) · arXiv (Cornell University) · doi:10.48550/arxiv.1210.4469 · Cited by 1
Keeling, M. J., Rohani, P., & Pourbohloul, B. (2008) · Clinical Infectious Diseases · doi:10.1086/591197
THE ZOMBIE SWARM: EPIDEMICS IN THE PRESENCE OF SOCIAL ATTRACTION AND REPULSION
Sander, E., & Topaz, C. M. (2014) · Mathematical Modelling of Zombies · doi:10.1515/9780776621685-017
Cartmel, A. (2014) · Mathematical Modelling of Zombies · doi:10.1515/9780776621685-001
Zombie Politics: Evolutionary Algorithms to Counteract the Spread of Negative Opinions
Hochreiter, R., & Waldhauser, C. (2014) · arXiv · doi:10.48550/arxiv.1401.6420
Flagged first, then by citations
Research works printed but not linked: 2
- P. Munz, 2009, Infect. Dis. Model. Res. Prog., 4, 133
- B. Mota, 2014, Bull. Am. Phys. Soc., 59, 1252P
Cited by
Sorted by citations
Statistical Mechanics: Entropy, Order Parameters, and Complexity
Sethna, J. P. (2021) · doi:10.1093/oso/9780198865247.001.0001 · Cited by 495
Self-citationOrder parameters, broken symmetry, and topology
Sethna, J. P. (2021) · doi:10.1093/oso/9780198865247.003.0009 · Cited by 32
Self-citationSkepticism and rumor spreading: The role of spatial correlations
Amaral, M. A., Dantas, W. G., & Arenzon, J. J. (2020) · Physical review. E · doi:10.1103/physreve.101.062418 · Cited by 16
Rumor propagation meets skepticism: A parallel with zombies
Amaral, M. A., & Arenzon, J. J. (2018) · Europhysics Letters (EPL) · doi:10.1209/0295-5075/124/18007 · Cited by 15
Periodic epidemic outbursts explained by local saturation of clusters
Gostiaux, L., Bos, W. J. T., & Bertoglio, J. P. (2023) · Physical review. E · doi:10.1103/physreve.107.l012201 · Cited by 10
How to stop undesired propagations by using bi-level genetic algorithms
Galiana, J., Rodrı́guez, I., & Rubio, F. (2023) · Applied Soft Computing · doi:10.1016/j.asoc.2023.110094 · Cited by 8
Updates on Rabies virus disease: is evolution toward "Zombie virus" a tangible threat?
Lippi, G., & Cervellin, G. (2021) · Acta bio-medica dell'Ateneo parmense/Acta biomedica · doi:10.23750/abm.v92i1.9153 · Cited by 7
“In keeping with this conjecture, an interesting simulation of an imaginary Zombie outbreak reveals that most of the US population would turn into Zombies within one week from appearance of the first case, whilst only some remotes zones in Montana and Nevada would remain infestation-free one month afterwards (40).”
BackgroundLibál, A., Forgács, P., Néda, Á., Reichhardt, C. J. O., Hengartner, N., & Reichhardt, C. J. O. (2023) · Physical review. E · doi:10.1103/physreve.107.024604 · Cited by 7
“[15], we have placed the model in a zombie framework, the model can be rephrased in terms of certain real world diseases such as HIV which, if left untreated, confer a lifelong ability to infect; however, under medical treatment from a health care provider, the infection rate can be reduced or dropped to zero, resulti…”
BackgroundMethodUnlocking the black box: teaching mathematical modeling with popular culture
Lofgren, E. (2016) · FEMS Microbiology Letters · doi:10.1093/femsle/fnw225 · Cited by 6
Critical properties of the SIS model on the clustered homophilic network
Santos, F. L., de Almeida, M. L., Albuquerque, E. L., et al. (2020) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2020.125067 · Cited by 5
Reanimating a Dead Economy: Financial and Economic Analysis of a Zombie OutbreakPreprint
Feinstein, Z. (2020) · RePEc: Research Papers in Economics · doi:10.48550/arxiv.2003.09943 · Cited by 5
“Though a simpler model than that of [1], we utilize the calibration taken in that model (from studies of the outbreaks in [22, 26]) in order to determine many of our epidemiological factors.”
BackgroundMethodPassive and active field theories for disease spreading
Vrugt, M. T., Jeggle, J., & Wittkowski, R. (2023) · Journal of Physics A: Mathematical and Theoretical · doi:10.1088/1751-8121/ad4c31 · Cited by 2
“This can be justified by the short half life of exposed persons (about 30 min for Shaun of the Dead [118]).”
Critical properties of a vector-mediated epidemic process
Santos, F. L., de Almeida, M. L., Albuquerque, E. L., et al. (2019) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2019.122085 · Cited by 3
On the physical (im)possibility of lightsabers
Fillion‐Gourdeau, F., & Gagnon, J.-S. (2019) · European Journal of Physics · doi:10.1088/1361-6404/ab274a · Cited by 1
“This idea of using science-fiction themes in the classroom has already been exploited with great success by Munz and collaborators [17, 18] and others [19], who modelled a zombie attack based on biological assumptions coming from popular movies.”
BackgroundCorrelations, response, and dissipation
Sethna, J. P. (2021) · doi:10.1093/oso/9780198865247.003.0010 · Cited by 1
Self-citationRandom walks and emergent properties
Sethna, J. P. (2021) · doi:10.1093/oso/9780198865247.003.0002 · Cited by 1
Self-citationDo not let the dead bite! : different scenarios of the zombie epidemic reexamined
Tomczyk, W. (2015) · Jagiellonian University Repository (Jagiellonian University)
Towards a theoretically informed policy against a rakghoul plague outbreak
Kontopoulos, D. ‐. G., Kontopoulou, T., Ho, H., et al. (2017) · The Medical Journal of Australia · doi:10.5694/mja17.00792
Weiße Wanderer, Zombies, Parasiten und Statistik
Thompson, R. C. (2020) · Springer eBooks · doi:10.1007/978-3-662-61419-8_5
Sethna, J. P. (2021) · doi:10.1093/oso/9780198865247.003.0008
Self-citationSethna, J. P. (2021) · doi:10.1093/oso/9780198865247.003.0011
Self-citationPhase-space dynamics and ergodicity
Sethna, J. P. (2021) · doi:10.1093/oso/9780198865247.003.0004
Self-citationSethna, J. P. (2021) · doi:10.1093/oso/9780198865247.003.0012
Self-citationSethna, J. P. (2021) · doi:10.1093/oso/9780198865247.003.0005
Self-citationSethna, J. P. (2021) · doi:10.1093/oso/9780198865247.003.0003
Self-citationSethna, J. P. (2021) · doi:10.1093/oso/9780198865247.003.0006
Self-citationSethna, J. P. (2021) · doi:10.1093/oso/9780198865247.003.0007
Self-citationWhat is statistical mechanics?
Sethna, J. P. (2021) · doi:10.1093/oso/9780198865247.003.0001
Self-citationSethna, J. P. (2021) · doi:10.1093/oso/9780198865247.005.0001
Self-citationAlzubadi, H. (2024) · Applied Mathematics and Nonlinear Sciences · doi:10.2478/amns-2024-0847
Gros, C. (2024) · Springer eBooks · doi:10.1007/978-3-031-55076-8_11
Simulating pedestrian avoidance: The human-zombie game
Oriana, J. P., Patterson, G. A., & Parisi, D. R. (2024) · Physical review. E · doi:10.1103/physreve.110.024611
Auf ein Bier mit einem Zombieforscher
Blicker, L. (2025) · Nachrichten aus der Chemie · doi:10.1002/nadc.20254152831
Schmid, G. B. (2025) · Springer eBooks · doi:10.1007/978-3-662-70831-6_2
Hamiltonian structure and integrability of the susceptible-cleric-zombie-recovered epidemic model
Abdelhamid, H. M. (2026) · Physical review. E · doi:10.1103/cs87-l34c
Metadata
- Authors
- Alemi, A. A., Bierbaum, M., Myers, C. R., & Sethna, J. P.
- Affiliations
- Cornell University (US)
- Venue
- Physical Review E
- Publisher
- American Physical Society (APS)
- Published
- 2015-11-02
- Volume
- 92
- Issue
- 5
- Pages
- 052801-052801
- ISSN
- 1539-3755 (ISSN-L)
- 1550-2376
- OpenAlex
- W2111668503
- Semantic Scholar
- 9e46fff9414de07f2b91c0928e9c929f60d7fab7
- Cited by
- 40
- References
- 15
- Citations by year
- 2026: 1 · 2025: 2 · 2024: 5 · 2023: 3 · 2021: 19 · 2020: 4 · 2019: 2 · 2018: 1 · 2017: 1 · 2016: 1 · 2015: 1
- FWCI
- 1.30
- Type
- journal article
- Language
- en
- Open access
- hybrid
- doi.org
- Licence
- Locations
- Physical Review E (publishedVersion, cc-by)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Science Foundation (IIS–1247696)
- Cornell University
- Topics
- Physical Sciences › Mathematics › Applied Mathematics › Mathematical and Theoretical Epidemiology and Ecology Models
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Complex Network Analysis Techniques
- Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Theoretical and Computational Physics
- Keywords
- epidemic modeling, critical phenomena, stochastic epidemic simulation, percolation, susceptibility, square lattice, differential equation model, statistical mechanics
- MeSH terms
- Disease Outbreaks, Humans, Stochastic Processes, Models, Statistical, Epidemiologic Studies
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
A language model drew the claims from the paper on 2026-10-27, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).
A second model labelled each claim on 2026-10-27 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).
Statistics and arithmetic are recomputed, not judged.
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