WHEN ZOMBIE ATTACK Modeling an outbreak of zombie

WHEN ZOMBIE ATTACK! Modeling an outbreak of zombie attack Chan Pak Hei, Percy Chan See Wing, Ada Li Yin Yee, Lillian

OBJECTIVE Show an example of modeling – zombie attack See whether zombie will take over the world if there is a zombie outbreak See how fast do zombie take over our world Zombie apocalypse simulation

WHAT IS A ZOMBIE?

You all are not zombies!!! I am a zombie!

WHAT IS A ZOMBIE? They don’t infect human – not a zombie They dance like MJ – not a zombie Trailer of Night of the living dead (1968) They run instead of slow moving – not a zombie

DEFINITION Slow moving Cannibalistic (craving for human flesh) Bitten by another zombie (infected) Undead (slowly decaying)

MODELING – CLASSES OF ‘CREATURE’ S – Susceptible human Z – Zombie R – Removed elements

CONSTRUCTING THE MODEL… Population of Susceptible Human at time i = Si Population of Zombie at time i = Zi Population of Removed at time i = Ri Total population (Constant) = N = Si + Zi + Ri We assume the human don’t reproduce~! S Z R

HUMAN (ANDY) VS ZOMBIE (BENNY) Human wins Andy Benny Zombie wins Andy Benny

CONSTRUCTING THE MODEL a. SZ Z R Define: rate of a human wins = a For each susceptible: P(survive) = (a. N) * (Z/N) Number of zombie reduced from time i to time i+1: =(a. N) * (Z i /N) * S i = a. S i Z i Human wins Andy Benny

CONSTRUCTING THE MODEL b. SZ S Z Define: rate of a zombie wins = b For each zombie: P(turns a susceptible) = (b. N) * (S/N) Number of zombie reduced from time i to time i+1: =(b. N) * (S i /N) * Z i = b. S i Zombie wins Andy Benny

CONSTRUCTING THE MODEL R r. R Z Define: Rate of a removed turn a zombie = r Number of zombie increased from time i to time i+1: =r. R Hi, I come back again!

OTHER FACTORS Natural cause of death of susceptibles § E. g. Attack by vampire/ car accidents…

OTHER FACTORS Natural cause of death of susceptibles § E. g. Attack by vampire/ car accidents… Rate of a human dies in a non-zombie related death = d Reduction of susceptibles = d. S

ZOMBIE MODEL d. S S R a. SZ b. SZ Z r. R Assume non-zombie-related-death is insignificant.

FINAL ZOMBIE MODEL S b. SZ Z a. SZ r. R S i+1 = S i – b. S i Z i+1 = Z i + b. S i Z i + r. R i – a. S i Z i R i+1 = R i + a. S i Z i – r. R i When S i+1 = S i , Z i+1 = Z i , R i+1 = R i § Amount of each creature is stabilized, § The world reach an equilibrium R

WHEN THERE IS NO CHANGE OF POPULATION OF ANY KIND OF CREATURE… S b. SZ Z a. SZ r. R S i+1 - S i = – b. S i Z i+1 - Z i = b. S i Z i + r. R i – a. S i Z i R i+1 - R i = a. S i Z i – r. R i When S i+1 = S i , Z i+1 = Z i , R i+1 = R i § Amount of each creature is stabilized, § The world reach an equilibrium R

ZOMBIE MODEL AT EQUILIBRIUM S b. SZ Z 0 = – b. SZ 0 = b. SZ + r. R – a. SZ 0 = a. SZ – r. R a. SZ r. R R

E ARE IN A ZOMBIE WORLD!!!

DISCUSSION What conclusion can be drawn from 0 = – b. SZ 0 = b. SZ + r. R – a. SZ 0 = a. SZ – r. R Where a, b and r are not zero

DISCUSSION What conclusion can be drawn from 0 = – b. SZ 0 = b. SZ + r. R – a. SZ 0 = a. SZ – r. R Where a, b and r are not zero Either S or Z must be zero! At any equilibrium, either human or zombie die out. They cannot co-exist!

EQUILIBRIA 0 = – b. SZ 0 = b. SZ + r. R – a. SZ 0 = a. SZ – Rr At a doomsday equilibrium, zombies win (S, Z, R) = (0, Z i , 0) At a disease free equilibrium, human win (S, Z, R) = (S i , 0, 0)

DISCUSSION 0 = – b. SZ 0 = b. SZ + r. R – a. SZ 0 = a. SZ – Rr How long does it take to reach the equilibrium? That depends on the parameters a, b, and r

ZOMBIE APOCALYPSE SIMULATION Each of you will have a UNO card: Green side means you are a zombie (number doesn’t matter) Black side means you are a human Your card will be collected if you are removed (then you cannot interact with other unless you draw a zombie card from us) Assume P(removed -> zombie) = 0. 2

ZOMBIE APOCALYPSE SIMULATION People with odd number cards will interact with person sit right to you. If you two are not the same type § Please fight! § One of you will drawn a card from us § The color of card will determine who win in this battle. § Red card: zombie wins § Yellow card: human wins

WE WILL PLAY THE GAME FOR 3 TIMES First game: zombie is much stronger than human Second game: zombie is a bit stronger than human Third game: zombie is weaker than human 8 Red + 2 Yellow 6 Red + 4 Yellow 4 Red + 6 Yellow

FOR EACH TRIAL IN EACH GAME: To ensure every one is interacting with new player: After the 2 n+1 th round, players with odd no. card / reverse card move to your right. After the 2 n th round, players with even no. card/ +2 card move to your left.

0 1 2 3 4 5 6 7 Zombie >> Human Zombie < Human S S S Z R 2 0

SIR MODEL Zombie model is a variation of the SIR Model, but zombie disease cannot recover! Susceptible Infectious Recovered

ON-LINE DISCUSSION Do you like horror movie? Why or why not? Think of some ways to prolong your life in a zombie apocalypse. (You can find it from the internet or base on zombie movies you have watched. ) Do you think horror movies are mathematically correct? Why or why not?

HOMEWORK 1. Plot the number of human, zombie and removed according to model on p. 15 with parameters S=100, Z=2, R=0, a=b=0. 03, r=0. 2 up to 10 rounds of interactions. 2. What conclusion can be drawn from the plot? 3. (extra) Think of different parameters a, b, and r such that Z =0 and S >0

REFERENCE Original model of zombie modelling: http: //www. mathstat. uottawa. ca/~rsmith/Zombies. pdf SIR model for progress of an epidemic: http: //en. wikipedia. org/wiki/Compartmental_models_in_ epidemiology#The_SIR_model