Zombie Outbreak Model

William Burke, Sean Brown, George Lopez, Teri Bennett

December 15, 2015

Abstract

In this paper we look at the potential growth of a model population inthe event of a zombie outbreak. In this epidemic model, the zombie virusis carried by the living dead and transmitted to susceptible individuals viadirect attack. We introduce an exposed class comprised of members whohave been attacked and become sick, but have not yet died and come backto life as a zombie. Individuals in this exposed class have the ability tobe treated and be brought back to the class of susceptible members of thepopulation. We also include in our model the ability for susceptible indi-viduals to attack and kill zombies, rendering them inanimate and foreverunable to infect another susceptible member of the model population.

1 Introduction

The entire population contained within the model can be categorized intothree classes of individuals: S for susceptible, E for exposed, and Z for infected.The members of class S are ordinary citizens not infected by the zombie virus.The members of class E have been exposed to the virus by direct attack by amember of class Z. There is a latency period in which the exposed individual isnot a zombie yet. Members of class E are unable to infect members of S. Themembers of class Z are fully undead zombies, capable of infecting members ofclass S.

In this model for a zombie outbreak, members of class E have been infectedwith the zombie virus, but are able to be treated for this viral zombie infectionwith a drug and brought back to the class S of susceptible individuals. Therate of treatment success is denoted by ρ. The rate at which members of classE are converted to undead infected zombies is given by the constant γ. Themembers of classes S and E have a natural death rate µ. Since the membersof class Z are already dead they do not have a natural death rate, but insteadhave a natural rate of decay which can also be described by µ. The only waythat a member of class Z, a zombie, can be killed is if a susceptible individualcuts off its head.

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Thus, for the model we have:λ: Birth rate/recruitment rateρ: Rate of treatment successγ: Conversion rateµ: Natural death rate for S, E, and natural rate of decay for Z.

The incidence rate β (the rate at which members of S are attacked by mem-bers of Z and become exposed is given by,p1: Probability of infection per-attackc1: Per-capita contact rate between Z and S.

β = p1c1

The killing rate α (the rate at which members of S attack and destroymembers of Z is given by,p2: Probability of chopping off zombie’s headc2: Per-capita contact rate between S and Z.

α = p2c2

Then, the incidence rate is given by β SZN , and the killing rate is given by

αSZN , where N = S +E +Z is the total number of individuals contained in themodel population.

2 Zombie Outbreak Model

The compartmental model is then,

Figure 1: Compartmental Model for Zombie Outbreak

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The model is given by the series of differential equations,dSdt = λ− β SZN + ρE − µSdEdt = β SZN − E(ρ+ µ+ γ)dZdt = γE − µZ − αSZN

where N = S + E + Z is the total number of individuals contained in themodel population.

We wish to know under what conditions will the zombie virus be contained,and what conditions are necessary for there to be a total zombie outbreakamongst the population contained within the model.

3 Model Analysis

3.1 Disease Free Equilibrium

When the model is disease free there are no exposed or infected individuals.Every member in the model is a member of the susceptible class S. Setting Eand Z both equal to zero gives us the disease free equilibrium (DFE),

DFE =

(λ

µ, 0, 0

).

We used Maple code (see Appendix) to assist us in our calculations of theJacobian matrix evaluated at the DFE, which is given by,

JDFE =

−µ ρ −β0 −(γ + µ+ ρ) β0 γ −(µ+ α)

.Subsequently, we were able to calculate the eigenvalues for the above matrix,

λ1 = −µ

λ2 =1

2

(−α− γ − 2µ− ρ−

√α2 − 2αγ + 4βγ + γ2 − 2αρ+ 2γρ+ ρ2

)λ3 =

1

2

(−α− γ − 2µ− ρ+

√α2 − 2αγ + 4βγ + γ2 − 2αρ+ 2γρ+ ρ2

).

In order for the DFE to be stable, the real part of all eigenvalues of theJDFE must be negative. λ1 and λ2 can be seen to be less than zero, but inorder for the DFE to be stable, λ3 must also be less than zero. Setting λ3 < 0gives us,

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1

2

(−α− γ − 2µ− ρ+

√α2 − 2αγ + 4βγ + γ2 − 2αρ+ 2γρ+ ρ2

)< 0√

α2 − 2αγ + 4βγ + γ2 − 2αρ+ 2γρ+ ρ2 < −α− γ − 2µ− ρα2 − 2αγ + 4βγ + γ2 − 2αρ+ 2γρ+ ρ2 < (−α− γ − 2µ− ρ)2

2αγ + 4αµ+ 2αρ+ 4γµ+ 2γρ+ 4µ2 + 4ρµ > −2αγ + 4βγ − 2αρ+ 2γρ

4αγ − 4βγ + 4αρ+ 4αµ+ 4γµ+ 4ρµ+ 4µ2 > 0

α(γ + µ) + ρ(α+ µ) + µ(γ + µ) > βγ.

Thus, in order for the real part of all eigenvalues of the Jacobian matrix forthe DFE to be negative,

βγ

α(γ + µ) + ρ(α+ µ) + µ(γ + µ)< 1.

So our R0, or basic reproduction number, can be given by,

R0 =βγ

α(γ + µ) + ρ(α+ µ) + µ(γ + µ)

which satisfies the criteria that R0 must be less than one for the disease freeequilibrium to be stable.

3.2 Endemic Equilibrium

For the endemic equilibrium (EE), we set all three differential equations forthe populations of classes S, E, and Z at time t equal to zero and solved foreach of the variables (see Appendix). We get the EE to be,

EE = (S∗, E∗, Z∗)

where S∗, E∗, and Z∗ are given by,

S∗ =λ(αγ2 + 2αγµ+ 2αγρ+ αµ2 + 2αµρ+ αρ2 − βγ2 − 2βγµβγρ− βµ2 − βµρ)/

(αβγ2 + 2αβγµ+ αβγρ+ αβµ2 + αβµρ+ αγ2µ+ 2αγµ2 + 2αγµρ+ αµ3

+ 2αµ2ρ+ αµρ2 − β2γ2 − β2γµ)

E∗ =βλ(αγ + αµ+ αρ− βγ + γµ+ µ2 + µρ)/(αβγ2 + 2αβγµ+ αβγρ+ αβµ2

+ αβµρ+ αγ2µ+ 2αγµ2 + 2αγµρ+ αµ3 + 2αµ2ρ+ αµρ2 − β2γ2 − β2γµ)

Z∗ =− λ(αγ + αµ+ αρ− βγ)(αγ + αµ+ αρ− βγ + γµ+ µ2 + µρ)/((αβγ2

+ 2αβγµ+ αβγρ+ αβµ2 + αβµρ+ αγ2µ+ 2αγµ2 + 2αγµρ+ αµ3 + 2αµ2ρ

+ αµρ2 − β2γ2 − β2γµ)µ).

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Using these values we can then find the Jacobian matrix evaluated at theendemic equilibrium,

JEE =

−αβγ+αβµ+αβρ+αγµ+αµ

2+αµρ−β2γ+βµραγ+αµ+αρ−βγ−βµ ρ βµ(ρ+µ+γ)

αγ+αµ+αρ−βγβ(αγ+αµ+αρ−βγ+γµ+µ2+µρ)

αγ+αµ+αρ−βγ−βµ −(γ + µ+ ρ) − βµ(γ+µ+ρ)αγ+αµ+αρ−βγ

−α(αγ+αµ+αρ−βγ−γµ+µ2+µρ)

αγ+αµ+αρ−βγ−βµ γ βγµαγ+αµ+αρ−βγ

.The criteria for stability of an equilibrium point of a model is all real parts

of the eigenvalues of the Jacobian matrix evaluated at that equilibrium musthave negative real parts. An equivalent statement is to say that the trace of thesame Jacobian matrix must be negative and the determinant must be positive.

The trace of the JEE , again found using Maple (see Appendix), can be seento be negative. The determinant is given by,

|JEE | = −β(αγ + αµ+ αρ− βγ + γµ+ µ2 + µρ)µ(µ+ γ)

αγ + αµ+ αρ− βγ − βµ.

In order for the EE to be stable, the determinant must be positive. Setting|JEE | > 0 gives us,

−β(α(γ + µ) + ρ(α+ µ) + µ(γ + µ)− βγ)µ(µ+ γ)

αγ + αµ+ αρ− βγ − βµ> 0

α(γ + µ) + ρ(α+ µ) + µ(γ + µ)− βγαγ + αµ+ αρ− βγ − βµ

< 0

α(γ + µ) + ρ(α+ µ) + µ(γ + µ)− βγ < 0

α(γ + µ) + ρ(α+ µ) + µ(γ + µ) < βγ.

And thus,βγ

α(γ + µ) + ρ(α+ µ) + µ(γ + µ)> 1

which satisfies the requirement that R0 must be greater than one for the EE tobe stable. Thus both the DFE and the EE are stable equilibria.

4 Parameter Values and Numerical Simulation

For the simulation of our model, we used the population size of New YorkCity in 2014 as 8491080 [2]. We set our parameter values at,

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β = 0.0095

α = 0.005

µ = 0.0001

ρ = 0.0001

λ = 849

which we found using a previous paper on the topic [1].

For the following simulations, the parameter value that was changed wasγ, the conversion rate from class E to class Z, as well as the intial number ofmembers in each class. The Matlab code that we used for this simulation isshown below.

function Final_Zombie_Project

global lambda beta rho mu gamma alpha N;

N = 8491080;

lambda=849; %Birth rate

beta=.0095; %Infection probability of susceptible when confronting by zombie

gamma=.001; %Conversion rate

mu=.0001; %Natural per-capita death rate

rho=.0001; %Treatment rate

alpha=.005; %Death rate of zombie when confronted by susceptible

tend = 100000;

u0 = [8491080; 0; 0;];

[tsol, usol] = ode45(@rhs, [0, tend], u0);

Ssol = usol(:,1); Esol = usol(:,2); Zsol = usol(:,3);

plot(tsol, Ssol, ’b’); hold on;

plot(tsol, Esol, ’g’);

plot(tsol, Zsol, ’r’);

labels = [0 1 2 3 4 5 6 7 8 9 10];

set(gca, ’XTickLabel’, labels);

hold off;

title(’Zombie Outbreak Model’)

xlabel(’Time (Years)’)

ylabel(’Population’)

end

function udot = rhs(t, u)

global lambda beta rho mu gamma alpha N;

S = u(1); E=u(2); Z=u(3);

Sdot = lambda - beta*S*Z/N + rho*E - mu*S;

Edot = beta*S*Z/N - rho*E - mu*E - gamma*E;

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Zdot = gamma*E - mu*Z - alpha*Z*S/N;

udot = [Sdot; Edot; Zdot;];

end

4.1 Disease Free Equilibrium

Figure 2: Disease Free Population

In the scenario above, there is no disease which means that there are nohumans that were exposed or infected to the zombie virus. As the graph shows,there are no zombies. There are only people who are susceptible to becoming azombie, but zombies do not exist. With no infection to infect susceptibles, thesusceptible population stabilizes and remains constant as time goes on becausethe birth rate is equal to the per-capita death rate. This means that there arethe same number of people being born as there are people who died that year.Therefore, the population remains constant at the initial population.

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Figure 3: R0 < 1, Population half S, half Z

This scenario shows what happens when half the population of NYC arezombies and the other half are susceptible individuals who are not yet infected,with zero people exposed. When half the population are zombies, there is animmediate rise in the exposed population because there is a large number ofzombies attacking susceptible people. Since the zombies are attacking people,there is a decline of people who are not exposed or infected. They lose a majorityof their population but a small proportion of the population is able to livelong enough to avoid being exposed. As people are being exposed, there arealso exposed people becoming zombies and increasing the zombie population.However, each group is experiencing a significant decline in their populationexcept susceptibles, resulting in the exposed and zombie populations dying outin roughly five years. The zombie population dies out earlier in roughly two yearsbecause susceptibles grow in number and kill more zombies than exposed canbe converted into zombies. Therefore, the disease is eradicated after five yearsand the susceptible population grows until it stabilizes at the total population.

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Figure 4: R0 < 1, Population equal for S, E, and Z

The scenario above shows what happens when all populations are initiallyequal in size. This graph behaves similarly to the graph in the scenario wherehalf the population are zombies and the other half are susceptible with zeroexposed initially. The susceptibles decrease drastically in a matter of days andthe exposed population grows. With the growth of the exposed class, a portion ofthis class become zombies. However, people are able to treat the exposed and killzombies at a larger rate than the rate it takes for an exposed person to becomea zombie. After a few years, the zombies die out and the exposed populationvanishes as the susceptible class grows to make up the total population.

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4.2 Endemic Equilibrium

Figure 5: R0 > 1, Population half S, half Z

In this scenario, half the population are zombies and the rest are all suscep-tibles. This is similar to the scenario prior except that the endemic equilibriumis stable in this case. This means that the scenario is an epidemic in whichthe zombie population grows to be the largest population. Here, one can seethat susceptibles drastically decline. This is due to the amount of people be-ing attacked by zombies and becoming exposed. There is a larger portion ofexposed people becoming zombies in this scenario resulting in the growing pop-ulation of zombies. However, the zombie population declines because there areless people to infect and less people exposed. This results in each populationstabilizing at a specific population size. This is because each population checksthe other population. Zombies infect people, people become exposed, exposedpeople are either treated or become zombies and susceptibles kill zombies. Thisrelationship results in a coexistence between the populations.

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Figure 6: R0 > 1, Population equal for S, E, and Z

In the above scenario, all populations are equal to each other initially. This issimilar to the scenario prior except that the endemic equilibrium is stable in thiscase. Meaning, that the scenario is an epidemic in which the zombie populationgrows to be the largest population. Here, susceptibles drastically decline as theexposed population grows in size. Since zombies are attacking people, there isa large rate of people becoming exposed. There is a larger portion of exposedpeople becoming zombies in this scenario resulting in the growing populationof zombies. However, like in the scenario prior, the zombie population declinesbecause there are less people to infect and less people exposed resulting in eachpopulation stabilizing at a specific population size. Each population adds toand takes away from the other populations creating the stabilization. Therefore,there is a coexistence between the populations.

5 Conclusion

From the results above, it can be seen that the spread of the zombie outbreakmodel we’ve applied to our population is dependent on many factors. Since theeventual outbreak, or containment, of the zombie virus is dependent on R0,which is expressed as,

R0 =βγ

α(γ + µ) + ρ(α+ µ) + µ(γ + µ)

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we can see that whether or not the infection is contained is dependent onβ, γ, α, ρ, and µ. In our Matlab code, the parameter that we changed togive use an R0 < 1 and R0 > 1 was γ. This is the rate at which members ofclass E transition to class Z and are able to infect the susceptible individualsin the population. The lower the value of γ, the more time there is to treatan exposed individual and bring them back to class S, and vice-versa. This isbecause the average amount of time an exposed individual spends in class Ecan be described by 1/γ. R0 can also be raised to a value greater than one byincreasing the rate of contact β between susceptible individuals and zombies.

If the rate of contact β, and the conversion rate γ are high, there are stillways to contain the virus and keep R0 < 1. The rate at which susceptibleindividuals are able to kill the zombies α can be increased. The treatmentsuccess rate ρ can also clearly be increased. Perhaps a good way to contain thevirus would be to come up with a higher quality, or more effective drug to treatthe virus. The last way to contain the zombie outbreak would be to increase µ,the natural death and decay rates for all individuals in the model, but this wouldnot be desirable as it would necessarily mean that the population of susceptibleindividuals, as well as those exposed and infected, would die more quickly.

In all, our model has been able to successfully describe the conditions nec-essary for a zombie virus endemic, as well as its containment, in a model popu-lation where there is a latency period in which individuals can be treated, butnot cured.

References

[1] Joe Imad Robert J. Smith Philip Munz, Ioan Hudea. When zombies attack!:Mathematical modelling of an outbreak of zombie infection. Living on Earth,October 2009.

[2] NYC Planning. Current population estimates.http://www.nyc.gov/html/dcp/html/census/popcur.shtml, 2014.

6 Apendix

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