Mathematical Modeling of a Zombie Outbreak · Mathematical Modeling of a Zombie Outbreak J. M....
Transcript of 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
2012
1
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
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
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
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
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
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
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
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
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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
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
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
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
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
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
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
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.
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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
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
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
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
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
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.
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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
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.
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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
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
]
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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
Analysis of predator-prey model (cont.)
Jacobian matrices at equilibria:
J(0,0) =[π 00 −δ
]
J(δ
α,π
β
)==
0 −βδααπ
β0
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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
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?
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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
Criticisms of the Predator-Prey model
Humans are not prey. We fight back.We need to look at another model!
17
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
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).
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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
Basic Zombie Model
-
�
αSZ
ζR
βSZS Z R- -π
?
δS
dSdt
= π − βSZ − δS
dZdt
= βSZ − αSZ + ζR
dRdt
= αSZ − ζR
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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
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
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
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).
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Sphere of influence results
Start with 500,000 humans.
34
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
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
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
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
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
Other directions
Inwhi (Andy) Cho, Aron Hughes, Michael Marek:zombie decayMaria Troyanova-Wood: two-climate model withmigrationStochastic modelsAgent-based models
37
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
Other directions
Zombie meta-study with Humans vs. Zombies data frommultiple campuses.
2011 Texas A&M HvZ data:
38
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
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
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
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
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
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
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