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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