38
Zombie Apocalypse: An Epidemic Model REDay 2013 Cait Witkowski Brian Blais

Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

Embed Size (px)

Citation preview

Page 1: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

Zombie Apocalypse: An Epidemic Model

REDay 2013 Cait Witkowski

Brian Blais

Page 2: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

Overview

• Our interest

• Zombie basics

• Epidemic (SIR) model

• Munz et al., 2009

• Modifications

Page 3: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect
Page 4: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

What is a zombie?

• “Undead”

• Eat human flesh

• Infect healthy

• Difficult to kill (destroy brain)

Photo credit: thewalkingdeadstrream.net

Page 5: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

What is a zombie?

Main dynamics:

1. How you become a zombie

Sick, Bitten, Die

2. How you get rid of zombies

Cure, Death

Page 6: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

Epidemic Model

• Outbreak

• Spread

• Control

• Vaccination

Page 7: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

Epidemic Model

S I R

Susceptible Infected Recovered

Total Pop

Photo credit: http://whatthehealthmag.wordpress.com

Page 8: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

S I

Page 9: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

S

I

(change in quantity S)= -(constant)

Page 10: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

S

I

(change in quantity S)= -(constant)

S’= -θ

Page 11: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

S

I

(change in quantity S)= -(constant)

S’= -θ

I’= +θ

Page 12: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

Epidemic Model

I’ = - ζI

R’= +ζI

I R ζI

Page 13: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

Epidemic Model

S’ = -βSI

I’ = +βSI

S I βSI

Page 14: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

Epidemic Model

S’ = -βSI

I’ = +βSI—ζI

R’= +ζI

S I R βSI ζI

Page 15: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

Epidemic Model

Example of SIR dynamics for influenza Sebastian Bonhoeffer, SIR models of epidemics

Page 16: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

Epidemic Model

Modifications

•Death rates

•Latent periods (SEIS)

•Ability to recover (SEIR)

•Ability to become susceptible again (SIRS)

Page 17: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

“When Zombies Attack!: Mathematical Modelling of an Outbreak Zombie Infection”

Munz, Hudea, Imad, Smith (2009)

Goals:

•Model a zombie attack, using biological assumptions based on popular zombie movies

•Determine equilibria and their stability

•Illustrate the outcome with numerical solutions

•Introduce epidemic modeling with fun example

Page 18: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

“When Zombies Attack!: Mathematical Modelling of an Outbreak Zombie Infection”

Munz, Hudea, Imad, Smith (2009)

Conclusions:

•Disastrous outbreak unless aggressive tactics

•Collapse of civilization

Page 19: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

“When Zombies Attack!: Mathematical Modelling of an Outbreak Zombie Infection”

Munz, Hudea, Imad, Smith (2009)

Great idea, but…

•Models don’t match any film

•All results depend on poor model assumptions

•No data is used

•Wrong parameters given

Page 20: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

Munz Model

S Z R βSZ ζR

αSZ

Susceptible

Zombie

Removed

S’ = -βSZ

Z’ = βSZ + ζR

R’=—ζR

Page 21: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

Munz Model

If zombies are killed, they are soon recycled into the zombie population

S Z R βSZ ζR

αSZ

Zombies never die

Page 22: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

Munz Model

We can find no movies where the ζR term is a reasonable assumption

S Z R βSZ ζR

αSZ

Page 23: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

Reproduce results

• Before starting our models, replicate theirs

• Parameters are drastically wrong

Based on paper:

β = .0095

⍴ = .005

ζ = .0001

Based on model:

β = .0028

⍴ = 5

ζ = 5

Page 24: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

Latent Outbreak

Munz equations and parameters

Munz paper

Page 25: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

Latent Outbreak

Munz paper

Munz equations, but our fixed parameters

Page 26: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

Quarantine

Munz paper

Munz equations and parameters

Page 27: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

Quarantine

Munz paper

Munz equations, but our fixed parameters

Page 28: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

Upgrades

βSZ αSZ

S I Z X ρI

Shaun of the Dead Walking Dead Zombieland 28 Days Later Resident Evil

Page 29: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

Upgrades

S Z R X

αSZ

ζR

βSZ

δS

Night of the Living Dead Dawn of the Dead Day of the Dead

Page 30: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

Data Collection

• Watch films, pause to take population count whenever zombies are in the scene

• Record time (within film’s world) versus population increase

Page 31: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

Shaun of the Dead

Photo credit: http://shaunofthedead.com

Data points based off film

Page 32: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

Shaun of the Dead

S’ = -βSZ

Z’= βSZ - αSZ

X’= + αSZ

Page 33: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

Night of the Living Dead

Photo credit: http://everyview.com

Page 34: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

Night of the Living Dead

Photo credit: http://everyview.com

S’ = -βSZ

Z’= ζR - αSZ

R’= βSZ - ζR

X’= + αSZ

Page 35: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

Night of the Living Dead

Photo credit: http://everyview.com

Permanently removed

Temporarily removed

Page 36: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

50000 Simulations

Photo credit: http://everyview.com

Z X α

S R β

Page 37: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

Joint Distribution

Photo credit: http://everyview.com

β

α

Page 38: Zombie Apocalypse: An Epidemic Model - Bryant Universityweb.bryant.edu/~bblais/pdf/zombies_presentation_apr2013.pdf · What is a zombie? • “Undead” • Eat human flesh • Infect

Conclusions

• Zombie infection would likely be disastrous, but not inevitable as Munz et al. (2009) suggestions

• Data are necessary to make reasonable models and parameter estimations