Daniel Kim Shelby Hassberger
Taylor GuffeyHarry Han
Lauren MorganElizabeth Morris
Rachel PatelRadu Reit
ZOMBIFICATION!
BackgroundOriginated in the Afro-Caribbean spiritual belief
system (a.k.a Voodoo) Modern Zombies follow a standard:
Are mindless monsters Do not feel pain Immense appetite for human flesh Aim is to kill, eat or infect peopleFast-moving
Problem StatementDevelop a mathematical model illustrating what
would happen if a Rage epidemic began at the primate facility at Emory University.
Identify an optimal, and the most scientifically plausible strategy for keeping the spread of zombies under control.
ClassificationFive classes
Susceptibles (S): Individuals capable of being infected Immune (I): Individuals incapable of being infectedZombies (Z): Infected, symptomatic individualsCarriers (C): Infected, asymptomatic individualRemoved (R): Deceased (both infected and
uninfected) incapable of being resurrected*
One-minute infection rate Infections:
Immunity exists Only Susceptible humans can become Zombies or Carriers Asymptomatic Carriers and Zombies can infect Susceptibles The Removed cannot be infected and resurrected Every Susceptible has the same chance of becoming infected
(regardless of demographics) Means of Removal:
Zombies can die of starvation Immune and Carriers can be eaten Susceptibles cannot be eaten, only infected
Heterochromia Iridium determines Carrier class Carriers < 250,000
Rage Virus Assumptions
Birthrate = Deathrate Constant Population (Closed System)
The United States is modeled as an equally distributed population, without geographic divisions
Jurisdiction is restricted to the United States, so strategies can only be implemented within the U.S.
General Model Assumptions
Basic Model
dqZ
I C
S Z R
αSZ + g1SC
βSZ + g2SC dqZ
bIZ
bCZ
Assumptions• Susceptibles can become a Zombie through infection by a Zombie or a Carrier• Zombies are infected, symptomatic individuals• Some Susceptibles may never be infected• b is the rate at which one Zombie will defeat (in this case infect) one individual in dayFormulaS Z = βSZ + g2SC
Valuesb = 0.0017g2= 2.00 E -5
I C
S Z R
αSZ + g1SC
βSZ + g2SC dqZ
bIZ
bCZ
Assumptions• Susceptibles can become a Carrier through infection by a Zombie or a Carrier• Carriers are infected, asymptomatic individuals
Valuesa = .000001g1= 1.67 E -8
FormulaS C = αSZ + g1SC
I C
S Z R
αSZ + g1SC
βSZ + g2SC dqZ
bIZ
bCZ
Assumptions• Zombies decease by starvation • If (I+C) is less than 1 million, zombies die at their natural death rate dq • Of the human population, only Immune and Carriers are factors because they can be eaten• Flesh is the equivalent to food, thus Zombies can die in 3 days from starvation
Valuesdq= 0.033
FormulaZ R = dqZ/(I+C)
I C
S Z R
αSZ + g1SC
βSZ + g2SC dqZ
bIZ
bCZ
Assumptions• b is the rate at which one Zombie will defeat (in this case eat) one Carrier in a day
Valuesb= 0.0017
FormulaC R = βCZ
I C
S Z R
αSZ + g1SC
βSZ + g2SC dqZ
bIZ
bCZ
Assumptions• b is the rate at which one Zombie will defeat (in this case eat) one Immune in a day
Valuesb= 0.0017
FormulaI R = βIZ
Basic Model Equations
S’ = -βSZ - αSZ - g1SC - g2SCZ’ = βSZ + g2SC – dqZC’ = αSZ + g1SC - βCZR’ = βCZ + βIZ + dqZI’ = -βIZ
Basic Model Plot• Susceptibles quickly turn• Zombie population grows sporadically; then Zombies die off• Immune population dies• Removed grows exponentially, and then stabilibizes• Doomsday Scenario
Model With Quarantine
dqZ
I C
S Z R
αSZ + g1SC
βSZ + g2SC
Q
bIZ
bCZ
(dqZ)
qZ(I+C+S)
dqQ
Assumptions
Formula
Valuesq=
Z Q = qZ(C+I+S)
• Immune, Carriers, and Susecptibles all quarantine Zombies at the same rate• Quarantine Zombies cannot escape
I C
S Z R
αSZ + g1SC
βSZ + g2SC
Q
bIZ
bCZ
dqZ
qZ(I+C+S)
dqQ
Assumptions
Formula
Valuesdq= 0.033
Q R = dqQ
• Zombies die in quarantine from starvation
Model With Quarantine EquationsS’ = -βSZ - αSZ - g1SC - g2SCZ’ = βSZ + g2SC + qZ(C+I+S) – dqZC’ = αSZ + g1SC - βCZR’ = βCZ + βIZ + dqZ + dqQQ’ = qZ(C+I+S) - dqQI’ = -βIZ
Model With Quarantine Plot
• The Susceptible Population drops but and then stabilizes• The Immune Population drops but then stabilizes• The Zombie Population grows but is captured and dies out• Removed population grows exponentially, then stabilizes • Humans Survive
Model With Cure
dqZ
I C
S Z R
αSZ + g1SC
βSZ + g2SC
dkZ(I+S+C)
bCZ
bIZ
dqZ
Assumptions
FormulaZ I = dkZ(I+S+C)
Valuesdk=• A cure turns a Zombie into an
Immune
Model With Cure Equations
S’ = -βSZ - αSZ - g1SC - g2SCZ’ = βSZ + g2SC - dqZ – dkZ(C+I+S)C’ = αSZ + g1SC - βCZR’ = βCZ + βIZ + dqZI’ = -βIZ + dkZ(C+I+S)
Model With Cure Plot
• Zombie Population grows, but decreases as they are being cured. However they continue to attack and they eventually starve to death• Susceptible Population is turned•The Immune Population slightly grows as more zombies are cured but eventually dies out•Removed grows exponentially, then stabilizes
Model With Extermination
dqZ+ kZ(I+C+S)
I C
S Z R
αSZ + g1SC
βSZ + g2SC dqZ + k(I+C+S)
bIZ
bCZ
Assumptions
FormulaZ R = dqZ + k(I+C+S)
• The extermination starts after 35 days • All Immunes, Carriers, and Susceptibles are armed
Valuesdq=k=
Model With Extermination Plot
• Zombie Population dies out• Susceptible Population survives at about 50% of original population•The Immune Population slightly decreases •Removed grows exponentially, then stabilizes
Model With Extermination
EquationsS’ = -βSZ - αSZ - g1SC - g2SCZ’ = βSZ + g2SC – dqZ – k(I+C+S)C’ = αSZ + g1SC - βCZR’ = βCZ + βIZ + dqZ + k(I+C+S)I’ = -βIZ
Choosing the ModelAlgorithm: Categories are from a scale of 1-10
0.5 x Number of People Survived + 0.3 x Practicality + 0.2 x Morality behind Treatment < 10
No Treatment: 0.5(0) + 0.3(10) + 0.2(4)= 3.8Quarantine: 0.5(2) + 0.3(5) + 0.2(7)= 3.9Cure: 0.5(0) + 0.3(2) + 0.2(10)= 2.6Extermination: 0.5(6) + 0.3(7) + 0.2(2)= 5.5
Conclusions• Model with Extermination is optimal
• Chose because:1. Most people survived2. Most realistic of the treatments3. Ranked low on morality, but time of crisis
References*will enter later
Questions..?