Zombie Apocalypse Michael Barnes
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Zombie Apocalypse
Michael Barnes
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When Zombies Attack!
Philip Munz Ioan Hudea Joe Imad Robert J. Smith�
Authors:
Modeling of an Outbreak of Zombie Infection
"When Zombies Attack!: Mathematical Modelling of an Outbreak of Zombie Infection", by Philip Munz, Ioan Hudea, Joe Imad and Robert J, Smith?.
In Infectious Disease Modelling Research Progress, eds. J.M. Tchuenche and C. Chiyaka, Nova Science Publishers, Inc. pp. 133-150, 2009. ISBN 978-1-60741-347-9.
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Models Considered
Basic • Latent Infection • Quarantine with Latent Infection Treatment with Latent Infection • Impulsive Reductions
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Zombies vs.�Infectious Disease
• Zombie model has two non linear terms • Infections depend on �
# of humans and # of zombies�
• Deaths depend on both �# of humans and # of zombies
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Model #1�The Basic Model
Note: Birth and death rates are ignored as the attack is assumed to take place over a short period of time
Classifications
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System of Equations
where
Probabilities per time period
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Equilibrium
Since, birth rate = death rate = 0 this is a closed system
thus, under equilibrium conditions
and
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Equilibrium
Two Equilibrium Positions Doomsday �Disease-Free �
From the first equilibrium equation
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Stability of Pendulum
Lectures 6 & 7 discussed stability of equilibrium positions for single variable pendulum equations
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Stability of �Zombie Attack
The stability of equilibrium positions for �multi variable zombie attack equations
depends on
Eigenvalue of Jacobian matrix
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Defining the Jacobian
so taylor expansion
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Significance of Eigenvalues
and similarly for
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Jacobian of�Basic Model
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Disease-Free Equilibrium�
We are interested in the sign of the �eigenvalues of this matrix so...
Since the characteristic equation has a root with a positive real part this equilibrium is unstable.
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Doomsday Equilibrium�
Since the characteristic equation has no roots with a positive real part this equilibrium is stable.
Again we want the sign of the eigenvalues
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Basic Model Conclusion Humans cannot coexist with zombies
Zombies will overtake Humans
No Matter what
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Model #2:�Latent Infection and Treatment
• Changes from Basic Model • The zombie condition may be “cured”
• Gestation period when infected
• Probabilities per time period
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Latent Infection and Treatment
Due to the , there is now a possibility for coexistence
Setting these equations equal to 0 and solving for Z yields
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Coexistence
Since the quadratic equation has all positive coefficients there are no positive eigenvalues
This equilibrium is stable, humans and zombies can coexist
Ignores birth and death rates
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Graphs Zombie attack with cure
Coexistence
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