2015

Zombies Epidemic Model

The basic models for a zombie epidemic system consists of 3 basic classes. The first is S, which is

the susceptible people. The second is the Z group which represents the zombies. The last is the R which

are the removed group. The susceptible group can go to the removed group by dying from natural

causes, not interaction with the zombie group (parameter ). In the removed class there are people

that have died from natural causes which can be resurrected as zombies (parameter ). There are also

zombies that have been killed in the removed class. The susceptible group can be turned in to zombies

from interaction with the zombies like being bitten or scratched (parameter ). Zombies can be killed by

removing the head or destroying the brain and then would be moved to the removed group (parameter

). The zombies can only come from 2 sources; the people that died from natural causes in the removed

group and people in the susceptible group that were bitten or scratched by zombies. The birth rate is

assumed to be constant in the basic model ().

The basic model becomes

S = - SZ S

Z = SZ + R SZ

R = S + SZ R

The S equation is the changes in the susceptible population it goes down from interaction with

zombies when the susceptible people are turned and when susceptible people die from natural causes.

The Z equation is the changes in the Zombie population, it goes up from interaction with the susceptible

group when they bite or scratch them and also from the removed group when the bodies are

resurrected. It goes down when the susceptible group kills a zombie by destroying the brain or

removing the head. The R equation is the changes in the removed population it goes up when zombies

are killed and when the susceptible people will die from natural causes but it goes down when a body is

resurrected into a zombie.

In the model we designed we added the group X which is when the zombies or anyone in the

removed group is killed completely, without the ability to come back. The removed group now consists

of people who have died from natural causes that can be resurrected. The zombies no longer go to the

removed group when killed they would go to the killed group X. The removed group can also go to the

killed group if the body decomposes or if someone destroys the brain of the body. The S and Z groups

are still the same in our new model. We also added that the zombies can die from not eating anything

for a certain period of time (starvation) and they would go to the killed group if that happened. Our

new model looks like this:

The equations become:

S = -cSZ dS

Z = cSZ + bRZ gZ aSZ

R = dS bRZ yR

X = gZ + aSZ + yR

The variables:

a is humans kill a zombie

b is zombies resurrects a body

c is zombie kills a human

d is human natural death

y is dead body is destroyed or decomposes

g is zombie starve

In this model, in the long run, everything would go to the permanently killed group if there was not a

high enough birth rate to keep the human race alive. We can examine what happens in a short time

period with this model so we dont have to worry about birth rates.

Computer Simulations:

This simulation shows that the zombies (red line) would take over the susceptible (black line) in a very

short time with the permanently killed group (blue) going up very quickly. In order for the susceptible to

eradicate the zombies they would have to kill them off very quickly before the first third of this time

window.

This is a simulation over a longer time showing that the zombies will slowly die out because they dont

have any food source, since the susceptible population was eradicated very quickly.

References

Munz, Philip, Ioan Hudea, Joe Imad, and Robert J. Smith? "When Zombies Attack!: Mathematical

Modelling of an Outbreak of Zombie Infection." Infectious Disease Modelling Research Progress

(2009): 133-50. Nova Science Publishers, Inc. Web. 20 Feb. 2015

Apendix METHOD RK4 STARTTIME = 0 STOPTIME=10 DT = 0.02 {Equations} d/dt(S) = -c*S*Z - d*S d/dt(Z) = b*R*Z - a*S*Z + c*S*Z - g*Z d/dt(R) = d*S - y*R - b*R*Z d/dt(X) = a*S*Z + y*R + g*Z {Inital Values} init S = 1500 init R = 1 init Z = 20 init X = 5 {Parameters} b = 0.9 a = 0.001 d = 0.0018 c = 0.001 d = 0.001 y = 0.3 g = 0.001

Longer run time to show the dying off of zombies METHOD RK4 STARTTIME = 0 STOPTIME=2000 DT = 0.02 {Equations} d/dt(S) = -c*S*Z - d*S d/dt(Z) = b*R*Z - a*S*Z + c*S*Z - g*Z d/dt(R) = d*S - y*R - b*R*Z d/dt(X) = a*S*Z + y*R + g*Z {Inital Values} init S = 1500 init R = 1 init Z = 20 init X = 5 {Parameters} b = 0.9 a = 0.001 d = 0.0018 c = 0.00154 d = 0.001 y = 0.3 g = 0.001