mathematical modeling of a zombie outbreak · mathematical modeling of a zombie outbreak j. m....

MathematicalModeling of a

ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel

Equilibria andstability

Stability of thepredator-preymodel

EpidemiologicalmodelsZombie-ism as aDisease

Sphere of Influence

Otherdirections

References

Mathematical Modeling of a ZombieOutbreak

Jean Marie Linhart

[email protected]

2012

1


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Zombies!

Zombies are the walking undead. Slowmoving ex-human horrors, they feast onhuman flesh and turn humans intozombies.According to MSNBC, Zombies areworth $5 billion in today’s economy.(Ogg [2011])Humans vs. Zombies (HvZ) is played oncollege campuses throughout the US (atTAMU since 2009) and on sixcontinents. (HvZauthors [2012])Even the CDC has emergencyprepardness plans for the ZombieApocalypse. (CDCAuthors [2011])

2


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Predator-Prey Hypotheses

Hypotheses:

Zombies, Z , are walking undead; feed on human fleshHumans, S (for susceptibles), are preyZombies kill or zombify humans via mass-actioninteractionZombies bite humans to create more zombies(mass-action)Zombies are subject to decay

3


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Predator-Prey Differential equations

Lotka-Volterra predator-prey equations (Lotka [1910],Volterra [1931]):

dSdt

= πS − βSZ

dZdt

= αSZ − δZ

These are autonomous differential equations, they do notdepend on t , only on S and Z .Seemingly simple, they have no analytic solution, and canonly be approximated numerically.

4


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Human and Zombie populations over time.

Parameters:π = 0.01, β = 10−4, α = 5× 10−5, δ = 0.01Initial conditions: S0 = 500 and Z0 = 1

0 500 1000 1500 2000 25000

500

1000

1500Population vs. time

time

Popu

lation

HumansZombies

5


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Phase plane for the predator-prey equations

Using the chain rule:dZdt

dtdS

=dZdS

Parameters:π = 0.01, β = 10−4, α = 5× 10−5, δ = 0.01

0 50 100 150 200 250 300 350 400 450 5000

50

100

150

200

250

300

350

400

450

500

Humans

Zom

bies

Phase Plane Plot of Predator−Prey Equations

6


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Questions!

Aside from plotting a phaseplane how can I tell howchanging the initial conditions will affect the solution toa system of differential equations?Does this type of model always give this type of cyclicbehavior?

7


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Changes in initial conditions: Equilibria andStability

An equilibrium solution x(t) = x0 for an autonomousdifferential equation x ′ = f (x) occurs when f (x0) = 0.

Example: x ′ = 0.1x( x

10− 1)(

1− x20

)What is a stable equilibrium? Change the initialconditions a little bit, and the model goes to theequilibrium in the long-term.

0 5 10 15 20 25 30 35 40 45 50

0

5

10

15

20

time

x(t)

Equilibria and Stability

8


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Changes in initial conditions: Equilibria andStability (cont.)

−5 0 5 10 15 20 25−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1dx/dt = f(x)

x

f(x)

Stable when f ′(x0) < 0.

9


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Equilibria and stability in higher-dimensionalsystems

Simplest case: ~x ′ = A~x where ~x is a 2-d column vector, A isa 2×2 matrix, we find the eigenvalues of A:

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x

y

+ real eigenvalues

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x

y

− real eigenvalues

10


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Equilibria and stability in higher dimensionalsystems (cont.)

~x ′ = A~x where ~x is a 2-d column vector, A is a 2×2 matrix,we find the eigenvalues of A:

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x

y

+ and − real eigenvalues

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x

y

purely complex eigenvalues

11


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Equilibria and stability in higher dimensionalsystems (cont.)

~x ′ = A~x where ~x is a 2-d column vector, A is a 2×2 matrix,we find the eigenvalues of A:

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x

y

+ real part, complex eigens.

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x

y

− real part, complex eigens.

The bottom line: negative real eigenvalues or negative realparts means stability.

12


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Stability for nonlinear autonomous systems

Nonlinear autonomous system ~x ′ = ~F (~x); ~F (~x0) = ~0

Find the eigenvalues of the Jacobian matrix:

J(~x0) =

∂F1

∂x1(~x0)

∂F1

∂x2(~x0)

∂F2

∂x1(~x0)

∂F2

∂x2(~x0)

If all real parts are negative, we have hyperbolic equilibriaand stability.If we add zero eigenvalues, we must analyze the equationsto determine stability.

13


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Analysis of the Predator-Prey model

Equilibria:

dSdt

= πS − βSZ = 0 =⇒ S = 0 or Z =π

β

dZdt

= αSZ − δZ = 0 =⇒ Z = 0 or S =δ

α

We have two equilibria: (0,0) and(δ

α,π

β

)Jacobian matrix:

J(S,Z ) =

[π − βZ −βSαZ αS − δ

]

14


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Analysis of predator-prey model (cont.)

Jacobian matrices at equilibria:

J(0,0) =[π 00 −δ

]

J(δ

α,π

β

)==

0 −βδααπ

β0

15


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Analysis of Predator-Prey model (cont.)

At (0,0)λ = π or λ = −δ

This is a saddle point; it is not really of interest to us.

At ( δα ,πβ ),

λ = ±i√δπ

we get periodic circulation.

Since this result is independent of choice of the parametersα, β, π, δ, we know this behavior is generic.

Now, can anyone think of any criticisms of this model?

16


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Criticisms of the Predator-Prey model

Humans are not prey. We fight back.We need to look at another model!

17


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Epidemiological models for a Zombie outbreak

Zombie disease models were popularized in 2009 with thepublication of “When Zombies Attack” in Infectious DiseaseModelling Research Progress. (Munz et al. [2009])

They work with three classes:

Susceptibles, S, or uninfected humans.Zombies, Z , the walking undead (normally I forinfected)Removed, R, the dead (normally recovered).

18


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Basic Zombie Model

-

�

αSZ

ζR

βSZS Z R- -π

?

δS

dSdt

= π − βSZ − δS

dZdt

= βSZ − αSZ + ζR

dRdt

= αSZ − ζR

19


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Model Results: doomsday

Parameter values π = 0.0001, ζ = 0.0001, δ = 0.0001

Both started with all humans, no zombies.

First plot α = 0.005 < β = 0.0095; zombies more effectiveSecond plot α = 0.01 > β = 0.008; humans more effective

0 2 4 6 8 10 12 14 16 18 200

100

200

300

400

500Zombies More Effective

time

Pop

. (th

ousa

nds)

SZR

0 1000 2000 3000 4000 5000 60000

100

200

300

400

500Humans More Effective

time

Pop

. (th

ousa

nds)

20


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Equilibria of the basic Zombie model

As originally written, the equations don’t permit anequilibrium because the birthrate π is a constant. We willconsider a short time-frame where birth and death rates areset to zero. π = δ = 0

dSdt

= −βSZ = 0

dZdt

= βSZ − αSZ + ζR = 0

dRdt

= αSZ − ζR = 0

Two equilibria: (N,0,0) (all humans) and (0,N,0) (allzombies; doomsday).

21


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Stability for the basic Zombie Model

The system of equations and Jacobian matrix for the system

dSdt

= −βSZdZdt

= βSZ − αSZ + ζR

dRdt

= αSZ − ζR

J =

−βZ −βS 0βZ − αZ βS − αS ζαZ αS −ζ

22


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Stability for the zombie-free equilibrium

The eigenvalues of the Jacobian matrix at (N,0,0):

λ = 0,−(ζ + [α− β]N)±

√(ζ + [α− β]N)2 + 4βζN2

This will have a root for λ that is positive, and therefore thezombie-free equilibrium is not stable.This is generic behavior; it does not depend on parametervalues aside from that they are positive and α > β.

23


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Doomsday equilibrium

Eigenvalues of the Jacobian matrix at (0,N,0):

λ(λ+ βN)(λ+ ζ) = 0

λ = 0, λ = −βN, λ = −ζ

A zero eigenvalue does not guarantee stability in general.

Looking at the ODEs, it is stable, and the behavior isgeneric, since it doesn’t depend on the values of theparameters.

What criticisms would you make of this zombie model?

24


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Criticisms of this zombie model.

Allowing the dead to spontaneouslycome back as zombies rigs the gameagainst us. (All my students)A corpse cremated to ash should not beable to come back as a zombie. (All)If zombies are animated corpses,shouldn’t they be subject to decay?(Cho, Hughes, Marek)If zombie-ism is a disease, then deadshould mean dead. (Many)Constant birthrate is not used in modernpopulation modeling. πS (Malthusianmodel) is more common. (All)

25


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Basic Disease model

?

- - -

δS

S Z RαSZβSZπS

dSdt

= πS − βSZ − δS

dZdt

= βSZ − αSZ

dRdt

= δS + αSZ

The dead cannot become zombies; dead is dead. Zombies,however, are immortal unless beheaded or cremated byhumans.

26


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Disease results

Parameter values π = 0.0001, δ = 0.0001First plot α = 0.005 < β = 0.0095; zombies more effectiveSecond plot α = 0.01 > β = 0.008; humans more effective

27


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Stability of the disease model results

π = δ = 0, omit removed (dead) class since it doesn’tinfluence the others.

dSdt

= −βSZ = 0

dZdt

= βSZ − αSZ = (β − α)SZ = 0

Equilibria: (S,0) and (0,Z )

J(S,Z ) =

[−βZ −βS

(β − α)Z (β − α)S

]

28


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Stable equilibria for both doomsday and humansurvival

Humans win, eigenvalues (N,0), α > β:

λ =−(α− β)N ±

√(α− β)2N2 − βN2

Zombies win, eigenvalues (0,Z ), β > α:

λ =−βZ ±

√β2Z 2 + 4(α− β)Z

2

In both cases, eigenvalues are always negative. These areboth stable equilibrium.

29


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Student work

Fred Doe and Amanda Roehling (Doe and Roehling [2012]):

Arming the populace: armed citizen 5× as effective asunarmed.Result: if done quickly enough and humans effectiveenough, humans survive.If the populace cannot handle weapons, then usemilitary intervention: zombies were more effective thanhumans; military troops more effective than zombies.Results: depended on how many troops and whendeployed. A strong, fast, response was most likely to beeffective.

30


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Sphere of influence in a supernatural zombiemodel

Frances Withrow and Tyler Gleasman (Withrow andGleasman [2012]):

It is not possible for humans or zombies to interact witheveryone in one time step. Mass action becomes

αS

1 + κSZ

Observe

limS→∞

S1 + κS

=1κ

limS→0

S1 + κS

= 1

1/κ is the maximum number of susceptibles a zombiecan interact with in a time step.If the unit of time is a day, κ ≈ 1/500 seems like areasonable value.

31


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Sphere of influence model, cont.

Supernatural zombies. Zombies can bring corpsesback to life.Zombies can be beheaded or cremated and madepermanently dead.

-πS S?

β S1+κS Z

δS- D ζ D1+κD Z

- Z αS Z1+κZ- R

32


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Sphere of influence ODEs

dSdt

= πS − βSZ1 + κS

− δS

dZdt

=βSZ

1 + κS− αSZ

1 + κZ+

ζDZ1 + κD

dRdt

=αSZ

1 + κZdDdt

= δS − ζDZ1 + κD

33


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Sphere of influence results

Start with 500,000 humans.

34


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Sphere of influence stability

Short time scale π = δ = 0. I omit the removed class, sinceit doesn’t influence any of the other classes.

The zombie-free equilibrium has Jacobian:

J(S, 0,D) =

0 −

βS

1 + κS0

0βS

1 + κS− αS +

ζD

1 + κD0

0 −ζD

1 + κD0

Eigenvalues are given by

λ = 0 or λ =βS

1 + κS− αS +

ζD1 + κD

The latter is negative since −αS dominates.

I argue this is stable despite the zero eigenvalues, andgeneric behavior.

35


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Sphere of influence stability

The doomsday equilibrium has Jacobian (no dead becausethe dead eventually all become zombies):

J(0, Z , 0) =

−βZ 0 0

βZ −αZ

1 + κZ0 ζZ

0 0 −ζZ

Eigenvalues are given by λ = 0, λ = −βZ , λ = −ζZ .

This is stable and generic behavior

36


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Other directions

Inwhi (Andy) Cho, Aron Hughes, Michael Marek:zombie decayMaria Troyanova-Wood: two-climate model withmigrationStochastic modelsAgent-based models

37


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Other directions

Zombie meta-study with Humans vs. Zombies data frommultiple campuses.

2011 Texas A&M HvZ data:

38


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Open questions

Is it possible for humans and zombies to coexist? Canwe find a stable coexistence equilibrium?Can we find a model with a stable disease-freeequilibrium and an unstable doomsday equilibrium?How should we model the HvZ game?What can HvZ tell us about zombie models?

39


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

Wrap-up

To prepare for a zombie outbreak, the best tactic we’veidentified is to get people trained with weapons.Keep the military prepared and teach children how toshoot and handle weapons from an early age.To protect your family: weapons, food, water, a remotehide-away.Keep those math students busy trying to improve themodels and save us all.

Thanks for coming and thanks for listening!

40


ZombieOutbreak

J. M. Linhart

Introduction

Predator-Preymodel




Sphere of Influence

Otherdirections

References

References

Shaun of the dead, 2004. URL http://www.imdb.com/title/tt0365748/combined. Edgar Wright (director).Krystal Fancey Beck. Horror girls – zombie, October 2008. URL

http://www.thezombified.com/blog/2008/10/horror-girls-zombie/.Kelli Bordner and Noah Bordner. Zombieville. MikaMobile, 2010. URL http://www.zombievilleusa.com/.

Videogame for iOS.Connie Burk. How zombies could save your life. URL http://farout.org/category/zombies/. illustrator not

identified.CDCAuthors. Zombie preparedness, 2011. URL http://www.cdc.gov/phpr/zombies.htm.Fred Doe and Amanda Roehling. Zombies. Technical report, Texas A & M University, 2012. Final Project, Math 442.Leonardo Finocci. Friendly zombie. WWW, 2010. URL

http://leonardofinocchi.deviantart.com/art/Friendly-Zombie-151356204?q=&qo=.Honeybadger. The girl’s guide to surviving the apocalypse, August 2012. URL

http://www.ggsapocalypse.co.uk/know-your-idols-32-lara-croft/.HvZauthors. Humans vs. zombies (hvz), 2012. URL http://humansvszombies.org/.A. J. Lotka. Contribution to the theory of periodic reaction. J. Phys. Chem., 14(3):271–274, 1910.P. Munz, I. Hudea, J. Imad, and R. J. Smith? When zombies attack! Mathematical modeling of an outbreak of zombie

infection. In J. M. Tchuenche and C. Chiyaka, editors, Infectious Disease Modelling Research Progress, pages133–150. July 2009. ISBN 978-1-60741-347-9. URLhttp://mysite.science.uottawa.ca/rsmith43/Zombies.pdf.

John C. Ogg. Zombies worth over $5 billion to economy, October 2011. URL http://www.msnbc.msn.com/id/45079546/ns/business-stocks_and_economy/t/zombies-worth-over-billion-economy.

Creator Unknown. URLhttp://theviewfromamicrobiologist.fieldofscience.com/2011/07/are-we-doomed.html.

Creator Unknown, 9 2012. URL http://www.ryanmews.com/wp-content/uploads/2012/09/zombie-cartoon-character-184x300.jpg.

V. Volterra. Variations and fluctuations on the number of individuals in animal species living together. In R. N. Chapman,editor, Animal Ecology. McGraw-Hill, 1931.

Frances Withrow and Tyler Gleasman. When zombies attack: Round 2. Technical report, Texas A& M University, 2012.Final Project, Math 442.

41

http://www.imdb.com/title/tt0365748/combined

http://www.thezombified.com/blog/2008/10/horror-girls-zombie/

http://www.zombievilleusa.com/

http://farout.org/category/zombies/

http://www.cdc.gov/phpr/zombies.htm

http://leonardofinocchi.deviantart.com/art/Friendly-Zombie-151356204?q=&qo=

http://www.ggsapocalypse.co.uk/know-your-idols-32-lara-croft/

http://humansvszombies.org/

http://mysite.science.uottawa.ca/rsmith43/Zombies.pdf

http://www.msnbc.msn.com/id/45079546/ns/business-stocks_and_economy/t/zombies-worth-over-billion-economy

http://www.msnbc.msn.com/id/45079546/ns/business-stocks_and_economy/t/zombies-worth-over-billion-economy

http://theviewfromamicrobiologist.fieldofscience.com/2011/07/are-we-doomed.html

http://www.ryanmews.com/wp-content/uploads/2012/09/zombie-cartoon-character-184x300.jpg

http://www.ryanmews.com/wp-content/uploads/2012/09/zombie-cartoon-character-184x300.jpg

mathematical modeling of a zombie outbreak · mathematical modeling of a zombie outbreak j. m....

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