When Zombies Attack!Based entirely on work by P. Munz, I. Hudea, J. Imad and R.J.

Smith?

Casey Briggs

University of Adelaide

May 24, 2011

Definitions

DefinitionA zombie is a reanimated human corpse that feeds on human flesh.

DefinitionIf a zombie’s head has been removed or brain been destroyed, thenwe say that the zombie has been defeated.

The Basic Model

Consider three basic classes:

• Susceptible (S)

• Zombie (Z)

• Removed (R)

Change in Susceptibles

Susceptibles die due to natural causes (that is, non-zombie related)at rate δ.

When a zombie meets a susceptible, they infect it at rate β.

Susceptibles are created at constant birth rate Π.

Thus, the rate of change of S is given by

S ′ = Π− βSZ − δS

Change in Zombies

A human can become a zombie after ‘losing’ an encounter withanother zombie, with parameter β.

Humans can also be resurrected from the dead, at rate ζ.

Zombies move to the removed class once they are defeated. Thishappens with parameter α after an altercation.

Thus, the rate of change of Z is given by

Z ′ = βSZ + ζR − αSZ

Change in Removed

Corpses are resurrected into zombies at rate ζ.

Zombies are defeated with parameter α after an encounter with ahuman.

Humans die of natural causes at rate δ after an altercation.

Thus, the rate of change of R is given by

R ′ = δS + αSZ − ζR

The Basic Model

So the basic model is given by

S ′ = Π− βSZ − δS (1)

Z ′ = βSZ + ζR − αSZ (2)

R ′ = δS + αSZ − ζR (3)

Simplifications

Let us assume that the zombie outbreak is happening over a shorttimescale. Then we can ignore birth and natural death rates, andthus Π = δ = 0.

S ′ = −βSZZ ′ = βSZ + ζR − αSZR ′ = αSZ − ζR

Steady States

To investigate the steady states of the model, we set thedifferential equations to zero.

0 = −βSZ0 = βSZ + ζR − αSZ0 = αSZ − ζR

From the first equation, either S = 0 or Z = 0.

If S = 0, then we have the equilibrium

(S̄ , Z̄ , R̄) = (0,N, 0)

That is, the doomsday scenario in which the entire population arezombies.

Steady States

If Z = 0, then we have the equilibrium

(S̄ , Z̄ , R̄) = (N, 0, 0)

That is, the entire population is human.

These equilibrium points show us that under this model,human-zombie coexistence is impossible

Numerical SolutionsInitially 500 susceptibles and 0 zombies

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SusceptiblesZombies

Figure: α = 0.005, β = 0.0095, ζ = 0.0001, δ = 0.0001

Numerical SolutionsInitially 499 susceptibles and 1 zombie

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Figure: α = 0.005, β = 0.0095, ζ = 0.0001, δ = 0.0001

Numerical SolutionsInitially 499 susceptibles, 0 zombies and 1 removed.

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Figure: α = 0.005, β = 0.0095, ζ = 0.0001, δ = 0.0001

A Revised Model

There is typically a period of time after a human gets bitten beforethey become a zombie (typically around 24 hours). We modify themodel to include an extra class of infected individuals (I ).

• Susceptibles now move to an infected class once bitten.

• Infected individuals can still die a natural death beforezombifying. If not, they become a zombie.

S ′ = Π− βSZ − δSI ′ = βSZ − ρI − δIZ ′ = ρI + ζR − αSZR ′ = δS + δI + αSZ − ζR

This model gives us the same coexistence result. As before,zombies take over the population, but the process takes aroundtwice as long.

Possible Zombie Remedies

How might we go about containing a zombie outbreak?

• Quarantine infected humans and zombies

• Find a cure for zombie-ism

• Impulsive eradication

Quarantine

We assume that quarantined individuals are removed from thepopulation and cannot infect new individuals while they remainquarantined.

• The quarantined area only contains members of the infected orzombie populations (entering from each class with some rate).

• There is a chance some members will try to escape, but if thatwere to happen they would be killed before finding freedom.

• Individuals killed in the manner above enter the removedclass, and can thus be resurrected later.

Quarantine

We again have the same two equilibria as before, and nocoexistence. But will quarantining control the outbreak?

It turns out that eradication depends critically on quarantiningthose in early stages of infection.

But we expect that quarantining a large proportion of infectedindividuals is unrealistic because of infrastructure limitations.

A Cure

Suppose we have a treatment that would allow a zombie to returnto their human form again.Assumptions:

• We no longer need the quarantine.

• The cure brings zombies to their original human formregardless of how they became zombies.

• The cure does not provide immunity.

A CureThis model now allows for the possibility of coexistence betweenzombies and humans (as you might expect).

When Zombies Attack! 145

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Figure 9. The model with treatment, using the same parameter values as the basic model.

Here, we returned to the basic model and added the impulsive criteria:

S� = Π− βSZ − δS t �= tnZ � = βSZ + ζR − αSZ t �= tnR� = δS + αSZ − ζR t �= tn

∆Z = −knZ t = tn ,

where k ∈ (0, 1] is the kill ratio and n denotes the number of attacks required until kn > 1.The results are illustrated in Figure 10.

0 2 4 6 8 100

100

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500

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1000

Num

ber

of Z

om

bie

s

Time

Eradication with increasing kill ratios

Figure 10. Zombie eradication using impulsive attacks.

In Figure 10, we used k = 0.25 and the values of the remaining parameters were

Impulsive Eradication

We could attempt to control the zombie population by destroyingthem at such times as our resources permit. This results in animpulsive effect.

S ′ = Π− βSZ − δS t 6= tn

Z ′ = βSZ + ζR − αSZ t 6= tn

R ′ = δS + αSZ − ζR t 6= tn

∆Z = −knZ t = tn

where k ∈ (0, 1] is the kill ratio, n is the number of attacks.

Impulsive Eradication

When Zombies Attack! 145

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&!!

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5

5

;07L,/3*,7

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Figure 9. The model with treatment, using the same parameter values as the basic model.

Here, we returned to the basic model and added the impulsive criteria:

S� = Π− βSZ − δS t �= tnZ � = βSZ + ζR − αSZ t �= tnR� = δS + αSZ − ζR t �= tn

∆Z = −knZ t = tn ,

where k ∈ (0, 1] is the kill ratio and n denotes the number of attacks required until kn > 1.The results are illustrated in Figure 10.

0 2 4 6 8 100

100

200

300

400

500

600

700

800

900

1000N

um

ber

of Z

om

bie

s

Time

Eradication with increasing kill ratios

Figure 10. Zombie eradication using impulsive attacks.

In Figure 10, we used k = 0.25 and the values of the remaining parameters were

Discussion

• Quarantining is unlikely to help unless it is extremelyaggressive.

• A cure would result in some human survivals, but coexistencewith zombies

• Sufficiently frequent attacks, with increasing force, will resultin eradication.

Conclusion

In the event of a zombie outbreak, we must act quickly anddecisively to eradicate them before they eradicate us.

Be alert, but not alarmed.

Further Reading

Philip Munz, Ioan Hudea, Joe Imad, Robert J. Smith?When Zombies Attack!: Mathematical Modelling of anOutbreak of Zombie InfectionInfections Disease Modelling Research Progress, 133–150,2009.