The Zombie Apocalypse

26 Pages·2015·0.24 MB·English

The Zombie Apocalypse
Maranda Meyer
Final Differential Equations Project
Math 274
Flathead Valley Community College
Spring 2015
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Abstract
The zombie apocalypse can be modeled as a nonlinear, autonomous, system
of ordinary differential equations, and is considered a disease model or an interacting
speciesmodel. Usingqualitativeanalysisandgraphicalsolutions,theequilibriumofthe
human population, zombie population, and the removed populations are determined
and classified. Furthermore, the effectiveness of the elimination of the entire global
human population due to zombie intervention is found using the most recent census
data, for an initial population of only one zombie with varied rates of infection.
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1 Introduction
Gear up and get ready for the end of the world. Ravenous zombies are coming, and
humanity needs to prepare, if we plan on surviving. Clearly, the first step to combating
the living dead is to create a mathematical model to determine the effects on the human
population. First the nature of zombies must be understood.
Zombies are resurrected human corpses, that have only the desire to devour human
flesh. The inevitable, initial cause for the creation of zombies is yet to be determined, but
will most likely be due to a virus, exposure to radiation, or genetic modification. In this
concept, I will be elucidating the highly contagious viral model. The transmission of the
infection, via bites or scratches, will lead to the conversion of the human population into
the zombie population. Also, for unknown causes, all uninfected humans also encompass a
dormant version of the virus that activates upon death, which will supplement the zombie
population. Additionally, zombies do not decay over time, nor do they reproduce or die
naturally, due to the virus, but can be terminated by intensive brain damage (eg. gun
shot to the head), beheading, or disintegration. Lastly, it is assumed that zombies will
not entirely devour a human, meaning that all infected humans will shift to the effective
zombie population, not the removed population.
Before the initial outbreak, it is assumed that humans can reproduce and die naturally,
thus having a natural reproductive rate. However, the earth has limited resources and
space, therefore it can only sustain a specific maximum population. After initial outbreak,
it is assumed that the natural human reproduction rate does not change. There are three
outcomes at contact between humans and zombies: zombie infects the human, the human
kills the zombie, or the human escapes. Most importantly, there is no cure to the virus.
This model is a non-linear, autonomous, system of differential equations. These equa-
tions are the rates of change of the human, zombie and removed populations. It can be
classifiedasamixofaninteractingspeciesandadiseasemodel. Thefollowingwilldescribe
what constitutes each population.
Firstly, humans reproduce and die naturally, according to the logistic model, and they
can be infected by zombies transferring this portion of the human population into the zom-
bie population. Secondly, zombies are, again, created by infected humans; however, basic
human nature has taught us that humans will fight back in attempts of survival and will
kill a percentage of the zombie population. Furthermore, a percentage of the naturally de-
ceasedhumanswillcomebackaszombies. Since,humansarerationalbeings,wecanchoose
to eliminate ourselves before turning, and this will be considered a part of the naturally
deceased human population. For the purposes of this model it is assumed that half of the
naturally deceased humans will turn into zombies. Lastly, under these circumstances, the
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removed population only consists of the eradicated zombie population, and the percent of
the human population that chose to eliminate themselves, which again for the purposes of
thismodelwillbeconsideredastheotherhalfofthenaturallydeceasedhumanpopulation.
Moreover, certain parameters and initial values are fixed, while others are variable.
From the most recent census data, the fixed parameters comprise of the human birth rate,
death rate, net reproductive rate, carrying capacity, and the initial global population and
are,1.870%,0.789%,1.081%,10billion,and7.174billion,respectively. Variableparameters
will comprise of the rate of human infection, rate of zombie eradication.
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1.1 Assumptions
1. All populations must be greater than, or equal to zero.
2. Human population is not fixed.
3. Humans reproduce and die naturally.
4. There are limited resources and space on earth, and humans have a maximum capac-
ity, meaning they have a logistic growth.
5. The natural human reproduction rate does not change after zombie intervention.
6. Zombies can infect humans via transmission of the virus, converting them to the
zombie population.
7. There is no cure to the infection.
8. Zombies do not reproduce or die naturally, nor do they decay over time.
9. Zombies can be killed by humans adding to the removed population.
10. Zombies will not entirely consume a human, meaning every infected human will add
to the effective zombie population.
11. Humans have a dormant version of the virus that will cause them to become infected
after their natural death.
12. Half of the humans that die naturally will add to the zombie population.
13. Half of the humans that die naturally will add to the removed population.
14. There are three outcomes at contact, zombie infects human, human kills the zombie,
or human escapes.
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2 Definition of Variables
2.1 Variables
• H = human population
• Z = zombie population
• R = removed population
2.2 Implied Parameters
• a = proportionality constant for modified death rate
• c = proportionality constant for modified birth rate
2.3 Fixed Parameters
• b = natural human birth rate: 1.870%
• d = natural human death rate: 0.789%
• r = net natural reproductive rate: 1.081%
• K = human population carrying capacity: 10 billion
2.4 Variable Parameters
• α = human infection rate
• β = zombie eradication rate
2.5 Initial Value Problem
• H(0) = 7.174 billion
• Z(0) = 1
• R(0) = 0
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3 Derivation of Differential Equations
3.1 Human Population
Before zombies are included in the human population equation, how the human popu-
lation grows must be determined.
Starting with the Malthusian model, it is assumed that humans reproduce and die
naturally. The human population increases proportionally to the size of the population.
This constant of proportionality is known as the birth rate, ”b”. Additionally, the human
population decreases proportionally to the size of the population. This constant of propor-
tionality is known as the death rate, ”d”.
H(cid:48) = bH −dH
This can be rewritten as,
H(cid:48) = (b−d)H
Where the net natural reproductive rate is,
r = b−d
So,
H(cid:48) = rH
Furthermore, it is assumed that there is limited resources and space on earth, which
means there cannot be unlimited exponential growth. There is a maximum capacity called
the carrying capacity ”K”. This means we must move to the logistic model.
The birth rate will decrease proportionally to the size of the population, let this con-
stant of proportionality be ”c”. The death rate will increase proportionally to the size of
the population, let this constant of proportionality be ”d”.
Thus the equation becomes,
H(cid:48) = (b−CH)H −(d+aH)H
This can be rewritten as,
H(cid:48) = (b−d)H −(a+c)H2
The carrying capacity, ”K”, is set to,
b−d r
K = =
a+c a+c
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Thus,
r
a+c =
K
So the human population differential equation can be written as,
r
H(cid:48) = rH −( )H2
K
For the purposes of this model, the equation is most useful if the overall birth rate is
separated from the overall death rate because the overall death rate will be used in both
the zombie and removed differential equations.
Therefore the equation is best written as,
r
H(cid:48) = bH −dH −( )H2
K
It is important to show that the birth rate cannot be a negative value, as this is impos-
sible. If it were able to be a negative value, it would contribute to the death rate through
the term − r H2.
K
Therefore, it must be proven that,
b−cH ≥ 0
Or,
b ≥ cH
Since the initial population, H(0) = 7.174 billion, is less than the carrying capacity,
K = 10 billion, the maximum human population is 10 billion. We also know that the net
natural reproductive rate ”r” is 1.081%.
Therefore,
r 0.01081
a+c = = = 1.081×10−12
K 10,000,000,000
So at most, if a = 0,
c = 1.081×10−12
So the inequality where ”H” is the maximum population,
b ≥ cH
Becomes,
0.01870 ≥ (1.081×10−12)(10,000,000,000)
0.01870 ≥ 0.01081
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Thus the inequality holds true and the parameter, ”c”, cannot contribute to the death
rate, so the equation is,
r
H(cid:48) = bH −(d+( )H)H
K
Thatthefollowingtermcontributestoincreaseofthehumanpopulation,astotalbirths,
bH
And the following term contributes to the decrease of the human population, as total
natural deaths,
r
−(d+( )H)H
K
Now that it is understood what the differential equation for the natural human popula-
tion is, the effects of the interaction between humans and zombies can be added. Zombies
will infect a percentage of the human population at contact. The human population de-
creases proportionally to the contact between zombies and humans, where the constant of
proportionality is the infection rate ”α”.
Thus the differential equation for human population is finalized as,
r
H(cid:48) = bH −(d+( )H)H −αHZ
K
3.2 Zombie Population
As the human population decreases proportionally to the contact between zombies
and humans, where the constant of proportionality is the infection rate ”α”, the zombie
population increases by the same amount.
So,
Z(cid:48) = αHZ
Furthermore, zombies can be killed by humans. The zombie population decreases pro-
portionally to the contact between zombies and humans, where the constant of proportion-
ality is the zombie eradication rate ”β”.
Thus,
Z(cid:48) = αHZ −βHZ
Since humans have a dormant version of the virus that activates upon death, half of
the humans that die naturally resurrect and add to the zombie population.
Therefore, the zombie differential equation is finalized as,
1 r
Z(cid:48) = αHZ −βHZ + (d+( )H)H)
2 K
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3.3 Removed Population
Thezombiesthatarekilledbyhumansturnintotheremovedpopulation. Theyremain
permanently deceased.
So,
R(cid:48) = βHZ
Again, humans have a dormant version of the virus that will cause them to become
infected after death, but they can choose to eliminate themselves before turning into a
zombie, where half of the naturally deceased human population will have the ability to do
so. Thismeanshalfofthehumansthatdienaturallywillshiftintotheremovedpopulation.
Therefore, the removed population is finalized as,
1 r
R(cid:48) = βHZ + (d+( )H)H)
2 K
3.4 Overall System of Differential Equations
The overall system of differential equations is,
r
H(cid:48) = bH −(d+( )H)H −αHZ
K
1 r
Z(cid:48) = αHZ −βHZ + (d+( )H)H)
2 K
1 r
R(cid:48) = βHZ + (d+( )H)H)
2 K
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