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, systemof ordinary differential equations, and is considered a disease model or an interactingspecies model. Using qualitative analysis and graphical solutions, the equilibrium of thehuman population, zombie population, and the removed populations are determinedand classified. Furthermore, the effectiveness of the elimination of the entire globalhuman population due to zombie intervention is found using the most recent censusdata, 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, andhumanity needs to prepare, if we plan on surviving. Clearly, the first step to combatingthe living dead is to create a mathematical model to determine the effects on the humanpopulation. First the nature of zombies must be understood.

Zombies are resurrected human corpses, that have only the desire to devour humanflesh. The inevitable, initial cause for the creation of zombies is yet to be determined, butwill most likely be due to a virus, exposure to radiation, or genetic modification. In thisconcept, I will be elucidating the highly contagious viral model. The transmission of theinfection, via bites or scratches, will lead to the conversion of the human population intothe zombie population. Also, for unknown causes, all uninfected humans also encompass adormant version of the virus that activates upon death, which will supplement the zombiepopulation. Additionally, zombies do not decay over time, nor do they reproduce or dienaturally, due to the virus, but can be terminated by intensive brain damage (eg. gunshot to the head), beheading, or disintegration. Lastly, it is assumed that zombies willnot entirely devour a human, meaning that all infected humans will shift to the effectivezombie 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 andspace, 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 threeoutcomes at contact between humans and zombies: zombie infects the human, the humankills 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 beclassified as a mix of an interacting species and a disease model. The following will describewhat constitutes each population.

Firstly, humans reproduce and die naturally, according to the logistic model, and theycan 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, basichuman nature has taught us that humans will fight back in attempts of survival and willkill a percentage of the zombie population. Furthermore, a percentage of the naturally de-ceased humans will come back as zombies. Since, humans are rational beings, we can chooseto eliminate ourselves before turning, and this will be considered a part of the naturallydeceased human population. For the purposes of this model it is assumed that half of thenaturally 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 ofthe human population that chose to eliminate themselves, which again for the purposes ofthis 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 andare, 1.870%, 0.789%, 1.081%, 10 billion, and 7.174 billion, respectively. Variable parameterswill 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 thezombie 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 addto the effective zombie population.

11. Humans have a dormant version of the virus that will cause them to become infectedafter 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 dienaturally. The human population increases proportionally to the size of the population.This constant of proportionality is known as the birth rate, ”b”. Additionally, the humanpopulation decreases proportionally to the size of the population. This constant of propor-tionality is known as the death rate, ”d”.

H ′ = bH − dH

This can be rewritten as,H ′ = (b− d)H

Where the net natural reproductive rate is,

r = b− d

So,H ′ = rH

Furthermore, it is assumed that there is limited resources and space on earth, whichmeans there cannot be unlimited exponential growth. There is a maximum capacity calledthe 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 ofthe population, let this constant of proportionality be ”d”.

Thus the equation becomes,

H ′ = (b− CH)H − (d+ aH)H

This can be rewritten as,

H ′ = (b− d)H − (a+ c)H2

The carrying capacity, ”K”, is set to,

K =b− d

a+ c=

r

a+ c

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Thus,

a+ c =r

K

So the human population differential equation can be written as,

H ′ = rH − (r

K)H2

For the purposes of this model, the equation is most useful if the overall birth rate isseparated from the overall death rate because the overall death rate will be used in boththe zombie and removed differential equations.

Therefore the equation is best written as,

H ′ = bH − dH − (r

K)H2

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 throughthe term − r

KH2.

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 netnatural reproductive rate ”r” is 1.081%.

Therefore,

a+ c =r

K=

0.01081

10, 000, 000, 000= 1.081 × 10−12

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 deathrate, so the equation is,

H ′ = bH − (d+ (r

K)H)H

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 totalnatural deaths,

−(d+ (r

K)H)H

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. Zombieswill 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 ofproportionality is the infection rate ”α”.

Thus the differential equation for human population is finalized as,

H ′ = bH − (d+ (r

K)H)H − αHZ

3.2 Zombie Population

As the human population decreases proportionally to the contact between zombiesand humans, where the constant of proportionality is the infection rate ”α”, the zombiepopulation increases by the same amount.

So,Z ′ = 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 ′ = αHZ − βHZ

Since humans have a dormant version of the virus that activates upon death, half ofthe humans that die naturally resurrect and add to the zombie population.

Therefore, the zombie differential equation is finalized as,

Z ′ = αHZ − βHZ +1

2(d+ (

r

K)H)H)

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3.3 Removed Population

The zombies that are killed by humans turn into the removed population. They remainpermanently deceased.

So,R′ = βHZ

Again, humans have a dormant version of the virus that will cause them to becomeinfected after death, but they can choose to eliminate themselves before turning into azombie, where half of the naturally deceased human population will have the ability to doso. This means half of the humans that die naturally will shift into the removed population.

Therefore, the removed population is finalized as,

R′ = βHZ +1

2(d+ (

r

K)H)H)

3.4 Overall System of Differential Equations

The overall system of differential equations is,

H ′ = bH − (d+ (r

K)H)H − αHZ

Z ′ = αHZ − βHZ +1

2(d+ (

r

K)H)H)

R′ = βHZ +1

2(d+ (

r

K)H)H)

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4 Qualitative Analysis

4.1 Analyzing Equilibrium for All Populations

This analysis will determine the equilibrium for all three of the populations. Equilib-rium is when the rate of change of each population is zero. By setting each differentialequation to zero, the nullcline equations are resolved. Subsequently, the equilibrium pointsare 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 intothe 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

H ′ = bH − dH − (r

K)H2 − αHZ

H ′ = rH − (r

K)H2 − αHZ

0 = H(r − (r

K)H − αZ)

Z-nullclines

Z ′ = αHZ − βHZ + (d

2)H + (

r

2K)H2

0 = H((α− β)Z + (d

2) + (

r

2K)H)

R-nullclines

R′ = βHZ + (d

2)H + (

r

2K)H2

0 = H(βZ + (d

2) + (

r

2K)H)

1. H-nullcline: H = 0

2. H-nullcline: r − ( rK )H − αZ = 0

3. Z-nullcline: H = 0

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4. Z-nullcline: (α− β)Z + (d2) + ( r2K )H = 0

5. R-nullcline: H = 0

6. R-nullcline: βZ + (d2) + ( r2K )H = 0

4.1.2 Equilibrium Points

After solving each combination of the systems of nullclines, the only valid equilibriumpoint was found to be,

(H,Z,R) = (0, Z,R)

Some of the other equilibrium points were just subsets of the above equilibrium with specificvalues of Z, but they do not need to be included as the above equilibrium contains thesepoints. 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 thepopulations must be greater or equal to zero. Also, one equilibrium point was found to beundefined, and cannot be used.

This equilibrium point indicates that all three populations will be at equilibrium when-ever the human population is depleted (equal to zero) for any potential zombie populationand removed population.

4.1.3 Jacobian

The jacobian allows the nonlinear system equations to be linearized, which makes itpossible to classify the type of equilibrium at the equilibrium point, by determining theeigenvalues.

To determine the jacobian, the partial derivatives for each differential equation areplaced into a matrix as such,

f(H,Z,R) = rH − (r

K)H2 − αHZ

g(H,Z,R) = αHZ − βHZ + (d

2)H + (

r

2K)H2

h(H,Z,R) = βHZ + (d

2)H + (

r

2K)H2

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J(H,Z,R) =

∂f∂H

∂f∂Z

∂f∂R

∂g∂H

∂g∂Z

∂g∂R

∂h∂H

∂h∂Z

∂h∂R

.

J(H,Z,R) =

r − (2rK )H − αZ −αH 0

αZ − βZ + d2 + ( r

K )H αH − βH 0

βZ + d2 + ( r

K )H βH 0

.

Specifically, at the equilibrium point (0,Z,R) the jacobian becomes,

J(0, Z,R) =

r − αZ 0 0

(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 thejacobian has multiple columns of zeros, it is clear that the determinant will be equal tozero. In this scenario, the eigenvalues can be determined to classify the equilibrium.

J(0, Z,R) − λI =

(r − αZ) − λ 0 0

(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 equilibriumpoint (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 atequilibrium is when the human population is zero. This raises the question if it is possible tohave 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 ignoringthe 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

H ′ = bH − dH − (r

K)H2 − αHZ

H ′ = rH − (r

K)H2 − αHZ

0 = H(r − (r

K)H − αZ)

Z-nullclines

Z ′ = αHZ − βHZ + (d

2)H + (

r

2K)H2

0 = H((α− β)Z + (d

2) + (

r

2K)H)

1. H-nullcline: H = 0

2. H-nullcline: r − ( rK )H − αZ = 0

3. Z-nullcline: H = 0

4. Z-nullcline: (α− β)Z + (d2) + ( r2K )H = 0

4.2.2 Equilibrium Points

After solving all of the combinations of systems of nullclines two valid equilibrium pointswere found.

(H,Z) = (0, Z)

(H,Z) = (2K(β − sα

2β − α),

b

2β − α)

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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 zombiepopulations are sustained, and that the human population is not equal to zero we cansolve the inequality,

H = 2K(β − sα

2β − α) > 0

for the ratio between ”α” and ”β” that will yield a human population that is greater thanzero.

2K(β − sα) > 0

2Kβ − 2Ksα > 0

2Kβ > 2Ksα

β > sα

Remember,

s =d+ 2r

2r=

2951

2162

So,

β > (d+ 2r

2r)α

β > (2951

2162)α

4.2.3 Classification

Using mathematical software, the classifications for the equilibrium points for the hu-man and zombie populations were determined.

For the equilibrium point,

(H,Z) = (0, Z)

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It is determined to be a degenerate nodal sink.

For the equilibrium point,

(H,Z) = (2K(β − sα

2β − α),

b

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 beplotted 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 infectshuman, human kills zombie, and human escapes), the percent of the sum of the outcomesmust 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 remainingpercent chance that the zombie escapes, that is not included as a parameter, as this wouldjust 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 zombieis equal to the chance that a human escapes, as to have comparative numerical values forthe parameters ”α” and ”β”. The differential equations do not call for this, and any valuefor ”α” and ”β” can be used so long as the sum does not exceed 100%.

However, for the following graphs it is assumed that,

β =1

2(1 − α)

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

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Figure 3: α = 0.33 , β = 0.33

Figure 4: α = 0.28 , β = 0.36

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Figure 5: α = 0.28 , β = 0.36

Figure 6: α = 0.26 , β = 0.37

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Figure 7: α = 0.26 , β = 0.37

Figure 8: α = 0.20 , β = 0.40

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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 con-tact between the human population and the zombie population, the zombie will infect thehuman. Consequently, this means the zombie eradication rate will be zero, and humansare defenseless against zombies. This leads to the rapid depletion of the human populationwhere almost all of the human population turns into the zombie population, over the timespan of about two years. The zombie population exceeds the initial human population asthe number of human births before the eradication of humans would add to the zombiepopulation, and the removed population would remain near zero. All populations are inequilibrium when the human population is depleted. This agrees with the qualitative anal-ysis.

In Figure 2, the infection rate is at sixty percent and the zombie eradication rate is attwenty 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. Thezombie population approaches about 5 billion, and the removed population approachesabout 2.5 billion. All populations are in equilibrium when the human population is de-pleted, 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 thetime 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 humanpopulation is exhausted, the populations are in equilibrium.

In Figure 4 the infection rate is twenty-eight percent, and the zombie eradication rateis thirty-six percent. One would think that the humans have the upper-hand and wouldbe able to sustain their population; however, since half of the naturally deceased humanscome back as zombies, the human population will still be eliminated over the period ofabout 2000 years. The zombie population is sustained at about 50 million and the re-moved population just exceeds 20 billion over this time frame. Similarly, when the humanpopulation 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 isthirty-seven percent. Finally, both the zombie population and the human population aresustained. 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 thehumans is positive. The human population approaches 700 million and the zombie popu-lation 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 fortypercent. Clearly, humanity finally has the upper hand, the human population is sustainedand approaches just less than 5 billion. The zombie population can never be fully eradi-cated, as naturally deceased humans will always turn into zombies. The zombie populationis 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 hu-man population either dies out or is sustained was determined in the qualitative analysis.

In general,

β > (d+ 2r

2r)α

β > (2951

2162)α

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And the specific condition used for these graphs was,

β =1

2(1 − α)

Substituting and solving for ”α” and ”β”,

α <1081

4032≈ 0.2681

β >2951

8064≈ 0.3659

This is agreed with by graphs, as Figure 4 had a infection rate of 28% and a zombieeradication 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 sustainthe human population.

Furthermore, Figure 6 had an infection rate of 26% and a zombie eradication rate of37% 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 zombieapocalypse will lead to the cataclysmic annihilation of the human race. The qualitativeanalysis and graphical solution produces many well-founded conclusions, and can poten-tially be modified to fit similar scenarios. These are some of the main conclusions drawnfrom this model.

Firstly, the initial population of one zombie is enough to decimate the entirety of theglobal 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. Thisis based in the assumption that all humans embody a dormant version of the incurablevirus that activates upon death, transforming them into the zombie population. Thereforea 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 pop-ulation. When the zombie eradication rate is approximately 1.365 times larger than theinfection rate, the human population is sustainable. However, all three populations cannotbe in equilibrium with a human population that is greater than zero. This is due to thefact that humans continually reproduce with a positive net reproductive rate. Even if thehuman and zombie population are in equilibrium, the removed population will continuallyincrease.

This model could be potentially modified with different initial conditions or a differentratio for how many naturally deceased humans transfer into both the zombie and removedpopulation. Furthermore, the model could be elaborated to include other populations suchas quarantined populations or a cured population.

All in all, there remains a glimmer of hope for the human race, in the face of flesh-eating adversity, so long as they examine this model and understand the mathematicalpossibilities of the zombie infested future.

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7 Bibliography

References

[1] Philip Munz, Ioan Hudea, Joe Imad, Robert J. Smith?. ”When Zombies Attack!:Mathematical Modeling of a Zombie Infection.” University of Ottawa. Nova SciencePublishers, Inc. (2009). Web. 24 Mar. 2015.

[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 Dif-ferential 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. 24Mar. 2015.

[5] Denzel Alexander, and Alles Rebel. ”Predator Prey Models Using Differential Equa-tions in Latex” (2012). Web. 24 Mar. 2015.

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