The Zombie Apocalypse - Flathead Valley Community College

The Zombie Apocalypse
Maranda Meyer
Final Differential Equations Project
Math 274
Flathead Valley Community College
Spring 2015
1
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
species model. Using qualitative analysis and graphical solutions, the equilibrium of the
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.
2
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 equations are the rates of change of the human, zombie and removed populations. It can be
classified as a mix of an interacting species and a disease model. The following will describe
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 zombie 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 deceased humans will come back as zombies. Since, humans are rational beings, we can choose
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
this model will be considered as the other half of the naturally deceased human population.
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%, 10 billion, and 7.174 billion, respectively. Variable parameters
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 capacity, 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 population 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 proportionality is known as the death rate, ”d”.
H 0 = bH − dH
This can be rewritten as,
H 0 = (b − d)H
Where the net natural reproductive rate is,
r =b−d
So,
H 0 = 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 constant 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 0 = (b − CH)H − (d + aH)H
This can be rewritten as,
H 0 = (b − d)H − (a + c)H 2
The carrying capacity, ”K”, is set to,
K=
b−d
r
=
a+c
a+c
7
Thus,
r
K
So the human population differential equation can be written as,
a+c=
H 0 = rH − (
r
)H 2
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,
H 0 = bH − dH − (
r
)H 2
K
It is important to show that the birth rate cannot be a negative value, as this is impossible. If it were able to be a negative value, it would contribute to the death rate through
the term − Kr H 2 .
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,
a+c=
0.01081
r
=
= 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
8
Thus the inequality holds true and the parameter, ”c”, cannot contribute to the death
rate, so the equation is,
r
H 0 = bH − (d + ( )H)H
K
That the following term contributes to increase of the human population, as total births,
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 population 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 decreases 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 0 = 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 0 = αHZ
Furthermore, zombies can be killed by humans. The zombie population decreases proportionally to the contact between zombies and humans, where the constant of proportionality is the zombie eradication rate ”β”.
Thus,
Z 0 = 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 0 = αHZ − βHZ + (d + ( )H)H)
2
K
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3.3
Removed Population
The zombies that are killed by humans turn into the removed population. They remain
permanently deceased.
So,
R0 = 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. This means half of the humans that die naturally will shift into the removed population.
Therefore, the removed population is finalized as,
1
r
R0 = βHZ + (d + ( )H)H)
2
K
3.4
Overall System of Differential Equations
The overall system of differential equations is,
r
)H)H − αHZ
K
1
r
= αHZ − βHZ + (d + ( )H)H)
2
K
1
r
= βHZ + (d + ( )H)H)
2
K
H 0 = bH − (d + (
Z0
R0
10
4
Qualitative Analysis
4.1
Analyzing Equilibrium for All Populations
This analysis will determine the equilibrium for all three of the populations. Equilibrium is when the rate of change of each population is zero. By setting each differential
equation to zero, the nullcline equations are resolved. Subsequently, the equilibrium points
are then found by solving all of the combinations of the systems of the nullcline equations.
Next, the equilibrium can be linearized by setting the values of the equilibrium point into
the jacobian. And lastly, from the jacobian, the type of equilibrium can be classified.
4.1.1
Solving the System of Nullclines
By setting each differential equation equal to zero the nullclines can be determined.
H-nullclines
r
H 0 = bH − dH − ( )H 2 − αHZ
K
r
2
0
H = rH − ( )H − αHZ
K
r
0 = H(r − ( )H − αZ)
K
Z-nullclines
d
r
Z 0 = αHZ − βHZ + ( )H + (
)H 2
2
2K
d
r
0 = H((α − β)Z + ( ) + (
)H)
2
2K
R-nullclines
d
r
)H 2
R0 = βHZ + ( )H + (
2
2K
d
r
0 = H(βZ + ( ) + (
)H)
2
2K
1. H-nullcline: H = 0
2. H-nullcline: r − ( Kr )H − αZ = 0
3. Z-nullcline: H = 0
11
r
4. Z-nullcline: (α − β)Z + ( d2 ) + ( 2K
)H = 0
5. R-nullcline: H = 0
r
6. R-nullcline: βZ + ( d2 ) + ( 2K
)H = 0
4.1.2
Equilibrium Points
After solving each combination of the systems of nullclines, the only valid equilibrium
point was found to be,
(H, Z, R) = (0, Z, R)
Some of the other equilibrium points were just subsets of the above equilibrium with specific
values of Z, but they do not need to be included as the above equilibrium contains these
points. Furthermore, some of the equilibrium points found were invalid for this model,
as some of the values for Z found were negative, and according to assumptions all the
populations must be greater or equal to zero. Also, one equilibrium point was found to be
undefined, and cannot be used.
This equilibrium point indicates that all three populations will be at equilibrium whenever the human population is depleted (equal to zero) for any potential zombie population
and removed population.
4.1.3
Jacobian
The jacobian allows the nonlinear system equations to be linearized, which makes it
possible to classify the type of equilibrium at the equilibrium point, by determining the
eigenvalues.
To determine the jacobian, the partial derivatives for each differential equation are
placed into a matrix as such,
r
)H 2 − αHZ
K
d
r
g(H, Z, R) = αHZ − βHZ + ( )H + (
)H 2
2
2K
d
r
h(H, Z, R) = βHZ + ( )H + (
)H 2
2
2K
f (H, Z, R) = rH − (
12
 ∂f
J(H, Z, R) =
∂H
 ∂g
∂H
∂h
∂H
∂f
∂Z
∂g
∂Z
∂h
∂Z
∂f 
∂R
∂g 
.
∂R
∂h
∂R

−αH
0
r − ( 2r
K )H − αZ
J(H, Z, R) = αZ − βZ + d2 + ( Kr )H αH − βH 0 .
βZ + d2 + ( Kr )H
βH
0

Specifically, at the equilibrium point (0,Z,R) the jacobian becomes,


r − αZ
0 0
J(0, Z, R) = (α − β)Z + d2 0 0 .
βZ + d2
0 0
4.1.4
Classification
The classification of the equilibrium point is determined by the jacobian. Since the
jacobian has multiple columns of zeros, it is clear that the determinant will be equal to
zero. In this scenario, the eigenvalues can be determined to classify the equilibrium.


(r − αZ) − λ 0
0
J(0, Z, R) − λI = (α − β)Z + d2 −λ 0  .
βZ + d2
0 −λ
det(J(0, Z, R) − λI) = (r − αZ − λ)(λ2 )
The characteristic polynomial is,
λ3 − (r − αZ)λ2 = 0
λ2 (λ − (r − αZ)) = 0
Thus the eigenvalues are,
λ1 = 0
λ2 = 0
λ3 = r − αZ
Since the equilibrium occurs with two eigenvalues that are equal to zero, the equilibrium
point (0,Z,R) is classified as degenerate. This equilibrium is unstable.
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4.2
Analyzing Equilibrium for the Human and Zombie Populations
As seen in the previous analysis, the only time when all three populations are at
equilibrium is when the human population is zero. This raises the question if it is possible to
have both a sustained human population and a sustained zombie population in equilibrium.
This can be determined by only looking at the human and zombie populations and ignoring
the removed population.
4.2.1
Solving the Systems of Nullclines
By setting both differential equations equal to zero the nullclines can be determined.
H-nullclines
r
H 0 = bH − dH − ( )H 2 − αHZ
K
r
0
2
H = rH − ( )H − αHZ
K
r
0 = H(r − ( )H − αZ)
K
Z-nullclines
r
d
)H 2
Z 0 = αHZ − βHZ + ( )H + (
2
2K
d
r
0 = H((α − β)Z + ( ) + (
)H)
2
2K
1. H-nullcline: H = 0
2. H-nullcline: r − ( Kr )H − αZ = 0
3. Z-nullcline: H = 0
r
)H = 0
4. Z-nullcline: (α − β)Z + ( d2 ) + ( 2K
4.2.2
Equilibrium Points
After solving all of the combinations of systems of nullclines two valid equilibrium points
were found.
(H, Z) = (0, Z)
β − sα
b
(H, Z) = (2K(
),
)
2β − α 2β − α
14
where ”b” is the birth rate and ”s” is a constant which is equal to,
s=
d + 2r
2r
and ”d” is the death rate, and ”r” is the net reproductive rate.
Since we want to determine an equilibrium where both the human and zombie
populations are sustained, and that the human population is not equal to zero we can
solve the inequality,
H = 2K(
β − sα
)>0
2β − α
for the ratio between ”α” and ”β” that will yield a human population that is greater than
zero.
2K(β − sα) > 0
2Kβ − 2Ksα > 0
2Kβ > 2Ksα
β > sα
Remember,
s=
d + 2r
2951
=
2r
2162
So,
4.2.3
β > (
d + 2r
)α
2r
β > (
2951
)α
2162
Classification
Using mathematical software, the classifications for the equilibrium points for the human and zombie populations were determined.
For the equilibrium point,
(H, Z) = (0, Z)
15
It is determined to be a degenerate nodal sink.
For the equilibrium point,
(H, Z) = (2K(
β − sα
b
),
)
2β − α 2β − α
It is determined to be a nodal sink.
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5
Graphical Solutions
Using mathematical computer software, the system of differential equations can be
plotted against time in years. Allowing for the adjustment of the variable parameters ”α”,
the infection rate, and ”β”, the zombie eradication rate.
Since there are three outcomes at contact between zombies and humans (zombie infects
human, human kills zombie, and human escapes), the percent of the sum of the outcomes
must be 100%. So ”α” will be the percent chance that the zombie infects the human, and
”β” will be the percent chance that the human kills the zombie. This leaves the remaining
percent chance that the zombie escapes, that is not included as a parameter, as this would
just mean that the human survives and remains in the human population.
For the purposes of graphing, let’s assume that the chance that a human kills a zombie
is equal to the chance that a human escapes, as to have comparative numerical values for
the parameters ”α” and ”β”. The differential equations do not call for this, and any value
for ”α” and ”β” can be used so long as the sum does not exceed 100%.
However, for the following graphs it is assumed that,
1
β = (1 − α)
2
All of the following graphs will have the set initial conditions applied, and are as follows:
H(0) = 7.174 × 109
Z(0) = 1
R(0) = 0
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5.1
Graphs
Figure 1: α = 1 , β = 0
Figure 2: α = 0.6 , β = 0.2
18
Figure 3: α = 0.33 , β = 0.33
Figure 4: α = 0.28 , β = 0.36
19
Figure 5: α = 0.28 , β = 0.36
Figure 6: α = 0.26 , β = 0.37
20
Figure 7: α = 0.26 , β = 0.37
Figure 8: α = 0.20 , β = 0.40
21
Figure 9: α = 0.20 , β = 0.40
5.2
Graphical Analysis
In Figure 1, the infection rate is at one hundred percent, meaning that for every contact between the human population and the zombie population, the zombie will infect the
human. Consequently, this means the zombie eradication rate will be zero, and humans
are defenseless against zombies. This leads to the rapid depletion of the human population
where almost all of the human population turns into the zombie population, over the time
span of about two years. The zombie population exceeds the initial human population as
the number of human births before the eradication of humans would add to the zombie
population, and the removed population would remain near zero. All populations are in
equilibrium when the human population is depleted. This agrees with the qualitative analysis.
In Figure 2, the infection rate is at sixty percent and the zombie eradication rate is at
twenty percent, signifying that the zombies have the upper hand upon contact. Therefore,
the human population is again diminished over the time period of about four years. The
zombie population approaches about 5 billion, and the removed population approaches
about 2.5 billion. All populations are in equilibrium when the human population is depleted, this again agrees with the qualitative analysis.
Figure 3 represents equal chances of all possibilities at contact, zombie infects human,
human kills zombie, or human escapes. Again the human population is depleted over the
time period of about 40 years. Additionally, the zombie population is sustained at about
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500 million, and the removed population just exceeds 8 billion. Moreover, once the human
population is exhausted, the populations are in equilibrium.
In Figure 4 the infection rate is twenty-eight percent, and the zombie eradication rate
is thirty-six percent. One would think that the humans have the upper-hand and would
be able to sustain their population; however, since half of the naturally deceased humans
come back as zombies, the human population will still be eliminated over the period of
about 2000 years. The zombie population is sustained at about 50 million and the removed population just exceeds 20 billion over this time frame. Similarly, when the human
population dies out, the system is at equilibrium.
Figure 5 is a reference to Figure 4 to show that the zombie population is positive.
In Figure 6 the infection rate is twenty-six percent and the zombie eradication rate is
thirty-seven percent. Finally, both the zombie population and the human population are
sustained. There is no longer a possibility for an equilibrium of all of the populations,
as the removed population will always increase since the natural reproductive rate of the
humans is positive. The human population approaches 700 million and the zombie population approaches 40 million.
Figure 7 is a reference to Figure 6 to show that the zombie population is positive.
In Figure 8 the infection rate is twenty percent and the zombie eradication rate is forty
percent. Clearly, humanity finally has the upper hand, the human population is sustained
and approaches just less than 5 billion. The zombie population can never be fully eradicated, as naturally deceased humans will always turn into zombies. The zombie population
is maintained at just over 30 million.
Figure 9 is a reference to Figure 8 to show that the zombie population is positive.
The precipice of the relationship between ”α” and ”β” that distinguishes when the human population either dies out or is sustained was determined in the qualitative analysis.
In general,
β > (
d + 2r
)α
2r
β > (
2951
)α
2162
23
And the specific condition used for these graphs was,
1
β = (1 − α)
2
Substituting and solving for ”α” and ”β”,
α <
β >
1081
≈ 0.2681
4032
2951
≈ 0.3659
8064
This is agreed with by graphs, as Figure 4 had a infection rate of 28% and a zombie
eradication rate of 36% and the human population died out in this scenario. In this case,
the value for ”α” was slightly too high, and the value for ”β” was slightly too low to sustain
the human population.
Furthermore, Figure 6 had an infection rate of 26% and a zombie eradication rate of
37% and the human population was sustained. Since ”α” was less than 26.81% and ”β”
was greater than 36.59% the human population was able to be maintained.
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6
Conclusion
Unfortunately, in most scenarios the model demonstrates that the upcoming zombie
apocalypse will lead to the cataclysmic annihilation of the human race. The qualitative
analysis and graphical solution produces many well-founded conclusions, and can potentially be modified to fit similar scenarios. These are some of the main conclusions drawn
from this model.
Firstly, the initial population of one zombie is enough to decimate the entirety of the
global population of 7.174 billion humans. This is true so long as the ratio between ”α”
and ”β” is,
β>
2951
2162
Secondly, it has been proven that the zombie population can never be eradicated. This
is based in the assumption that all humans embody a dormant version of the incurable
virus that activates upon death, transforming them into the zombie population. Therefore
a percentage of all naturally deceased humans will add to the zombie population.
Thirdly, the populations are only in equilibrium when the human population is zero,
leaving the zombie and removed population to be anything.
Lastly, it was proved that it is possible to have a sustained human and zombie population. When the zombie eradication rate is approximately 1.365 times larger than the
infection rate, the human population is sustainable. However, all three populations cannot
be in equilibrium with a human population that is greater than zero. This is due to the
fact that humans continually reproduce with a positive net reproductive rate. Even if the
human and zombie population are in equilibrium, the removed population will continually
increase.
This model could be potentially modified with different initial conditions or a different
ratio for how many naturally deceased humans transfer into both the zombie and removed
population. Furthermore, the model could be elaborated to include other populations such
as quarantined populations or a cured population.
All in all, there remains a glimmer of hope for the human race, in the face of flesheating adversity, so long as they examine this model and understand the mathematical
possibilities of the zombie infested future.
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7
Bibliography
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[2] Guillermo Abramson. ”Mathematical Modeling of the Spread of Infectious Diseases.”
University of New Mexico (2001). Web. 24 Mar. 2015.
[3] Lucas C. Pulley. ”Analyzing Predator-Prey Models Using Systems of Ordinary Differential Equations.” Southern Illinois University. Honors Thesis (2011). Paper 344.
Web. 24 Mar. 2015.
[4] Bailey Steinworth, and Xing Zhang. ”Zombification of the Planet” (2010). Web. 24
Mar. 2015.
[5] Denzel Alexander, and Alles Rebel. ”Predator Prey Models Using Differential Equations in Latex” (2012). Web. 24 Mar. 2015.
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