This document is the final project for an ODEs class analyzing a zombie outbreak mathematically. It summarizes several mathematical models of zombie infection spread and human-zombie interactions. The basic HZR model tracks humans, zombies, and removed individuals over time. Analysis shows stable fixed points when either humans or zombies go extinct. An expanded HIZR model includes infected individuals and finds the zombie apocalypse equilibrium is stable. The document also analyzes a predator-prey inspired HZ model and a non-dimensionalized version.

1. MAT 119B: ODEs
FINAL PROJECT
Work reproduced by:
Anthony Khuu; Blane Downing-Jennings
Jason Goss; Ke Li
Connor Duthie; Chieu Vuong
Instructor: T. LEWIS
abstract
“How did we fare the zombie apocalypse?” Will there be anyone around to ask this question? In our project,
we take an analytical mosey through how zombies stand to threaten every aspect of human existence. With most
work recreated from, referred to, and inspired by Robert Smith?’s publication *(W.Z.A.!) “When Zombies Attack!:
Mathematical Modelling of an Outbreak of Zombie Infection,” we plan to come to the realistic conclusions of the
severity of any zombie outbreak, or perhaps “apocalypse”.
Introduction
Do we know about zombies? They’ve already invaded our culture, right? We know how they move and spread
and that their existence, were they to spring up, immediately stands in opposition to our own. There are hundreds
of movies centered around them. Why? All zombies are boringly similar and all of the movies about them are the
basically same. So then why do we have so many of these movies and why are they here to stay? Perhaps zombies
are metaphors for things we aren’t quite prepared for or don’t take serious, despite earnestly looming at the thought
of their(its) existence. Zombies are things we aren’t prepared for because they are a threat so unfamiliar from what
we associate as actual threats, and they are threats. Zombies are things that we can’t control once they exist, they
are the things that can’t be contained by the time we have organized the effort and ability to contain them. It is
with this thought in mind that we here talk about zombies.
The first goal of this paper is to recreate and extend the models of Smith?’s W.Z.A.!, with the added goals
of analysing and relating other scientific models to the study. The scientific motivation of the paper (background,
context, etc).
1 HZR Basic Model
Let us begin with Smith?’s basic model (with notation changed from W.Z.A.!), where we consider three
classes, Human (H), Zombie (Z), and Removed (R). Humans reproduce and are brought into the system at a birth
rate of π, while the rate of natural, non-zombie related, deaths are represented by parameter δ. Humans can become
zombies through transmission rate β, when losing an encounter with a zombie. Each human also has the ability of
defeating a zombie during an encounter by destroying its brain (parameter α), sending the zombie into a removed
class. The removed class consists of individuals who have died through human-zombie altercations or by natural
causes. Humans in the removed class can become reanimated and become a zombie (parameter ζ).
1

2. 1 HZR BASIC MODEL
The chance that a human encounters a zombie in the system is H/N, with N being the total population. We
have an average member of the population making contact to transmit the fatal zombie-transmitted “bacteria” with
βN others per unit of time. Therefore, the number of zombie being created through zombie-related attacks per unit
of time per zombie is βN(H/N)Z = βHZ.
Similarly, each human has the possibility of defeating a zombie during an altercation at a rate of αN per unit
of time, with the probability Z/N of making contact with a zombie. The number of zombies destroyed through these
altercations is (αN)(Z/N)H = αHZ This very basic model is given by
H = π − βHZ − δH
Z = βHZ − αHZ + ζR
R = αHZ − ζR.
By choosing a short timescale where birth and natural deaths are negligible, we can set π = 0 and δ = 0.
Then by setting H = Z = R = 0, we have
βHZ = 0
βHZ − αHZ + ζR = 0
αHZ − ζR = 0.
Immediately from the H = 0 equation, we have either H∗
= 0 or Z∗
= 0 as our fixed points. With H∗
= 0,
we have the zombie apocalypse equilibrium (H∗
, Z∗
, R∗
) = (0, N, 0), where no humans are left alive and only zombies
roam the world. When Z∗ = 0, we a zombie-free equilibrium with (H∗
, Z∗
, R∗
) = (N, 0, 0). The Jacobian of this
system is
J =
−βZ −βH 0
βZ − αZ βH − αH ζ
αZ αH −ζ
The Jacobian at the zombie apocalypse equilibrium is
J(0, N, 0) =
−βN 0 0
βN − αN 0 ζ
αN 0 −ζ
We have det(J - λI) = -λ(-βN - λ)(-ζ - λ). This gives us λ1,2,3 = 0, -βN, -ζ.
The Jacobian at the zombie-free equilibrium is
J(N, 0, 0) =
0 −βN 0
0 βN − αN ζ
0 αN −ζ
We have det(J - λI) = -λ(λ2
+ [ζ - (β - α)N]λ - βζN).
When solving for eigenvalues, we have λ1,2,3 = 0,
−(ζ − (β − α)N) ± ((ζ − (β − α)N)2 + 4βζN
2
.
Since 4βζN > 0, we always have a positive value in the square root, which implies that it may be evaluated for all
parameter values.
2

3. 2 HIZR MODEL
2 HIZR Model
Smith? further expands the HZR model to include a class of infected individuals, which now represent the
humans that have lost an encounter with another zombie.
H = π − βHZ − δH
I = βHZ − ρI − δI
Z = ρI − αHZ + ζR
R = αHZ − ζR + δI + δH
The Jacobian1
of this system is given by
J(S, I, Z, R) =
−βZ 0 −βH 0
βZ −ρ βH 0
−αZ ρ −αH ζ
αZ 0 αH −ζ
For our zombie-free equilibrium with (S∗
, I∗
, Z∗
, R∗
) = (N, 0, 0, 0),
J(N, 0, 0, 0) =
0 0 −βN 0
0 −ρ βN 0
0 ρ −αN ζ
0 0 αN −ζ
This gives us 0 = −λ[-λ3
- (ρ + ζ + αN)λ2
- (ραN + ρζ − ρβN)λ + ρζβN].
We will always have a positive eigenvalue since ρζβN > 0, implying that the zombie-free equilibrium is unsta-
ble.
At our zombie apocalypse equilibrium with (S∗
, I∗
, Z∗
, R∗
) = (0, 0, N, 0), we have
J(0, 0, N, 0) =
−βN 0 0 0
βN −ρ 0 0
−αN ρ 0 ζ
αN 0 0 −ζ
The eigenvalues are λ = 0, -βN, -ρ, -ζ. Since these are all non positive eigenvalues, the zombie apocalypse equilibrium
is stable.
1With chosen time-scale in which π = δ = 0, to find our equilibrium points.
3

4. 4 PREDATOR/PREY
3 Our Basic Model
We have created our own basic model in which we consider only two classes, Human and Zombie. This
model closely resembles The Walking Dead zombie lore, as every deceased human with their brain intact becomes
reanimated into a zombie, regardless of any previous interaction with a zombie. We also changed the birthrate π to
be dependent on H, since Robert Smith didn’t take into account the number of births to vary for very small and
very large human populations. We also changed the system so that defeated zombies are completely removed from
the system, since a zombie can no longer reanimate again after having its brain destroyed.
H = πH − δH − βHZ
Z = βHZ − αHZ
Again, we choose a short time-scale where birth and natural death are negligible, and we set π = δ = 0, allowing us
to set up the equilibrium equations
H = βHZ = 0
Z = βHZ − αHZ = 0
This gives us our usual fixed points when H* = 0 and Z* = 0.
The Jacobian of our basic HZ model is
J =
−βZ −βH
(β − α)Z (β − α)H
The Jacobian at the zombie-free equilibrium
J(N, 0) =
0 −βN
0 (β − α)N
Giving us λ = 0, (β - α)N. We have a change in stability of Z* = 0 when β = α. Which shows that a zombie-free
equilibrium could be stable α > β, which denotes humans being able to kill zombies at a higher rate than zombies
are killing humans. This works because in this, our most basic model, δ = 0 and so it is simply a battle only of who
kills the other off first. It is a much more dire model when human death means zombie birth!
The Jacobian at the zombie apocalypse equilibrium
J(0, N) =
−βN 0
(β − α)N 0
Giving us λ = 0, -βN Since β > 0 at all times, the zombie apocalypse equilibrium will always be stable. DANG!
4 Predator/Prey
This model is a change in the SZR model from the paper by Robert Smith? in that we are jettisoning one
variable, the R, because we are assuming that once the zombies or humans are killed, they are removed from the
system entirely and cannot be resurrected to become new zombies.
4

5. 4 PREDATOR/PREY
The model arises from the assumptions that humans grow at a rate proportional to the present rate of hu-
mans, and that they die off at a rate proportional to the interaction rate between humans and zombies, which is
modeled in the vein of Michaelis-Menton kinetics as simply the product of the H and Z variables. We assume that
Zombies grow at a rate proportional to the interaction between zombies and humans, and that they deteriorate at a
rate proportional to the amount of zombie present in the system. The system is as follows:
H’ = αH - βHZ
Z’ = δ HZ - γZ
To analyze these equations, we first nondimensionalize them, which rids us of one parameter, and we then plot null-
clines (which are horizontal and vertical lines) and examine their intersections to find fixed points.
Our first non-dimensionalized equations are derived from the Lotka-Volterra equations, which just include the H and
Z classes. In these equations, we again find two fixed points, with one of them being (0,0). The fixed point at (0,0)
is a saddle, and the non-trivial fixed point is a non-linear center, which we can see because of the conserved quantity
K that we will derive.
H’ = (π - δ)H - βHZ
Z’ = (β - α)HZ - rZ
Here, r is the deterioration rate of the zombies.
Let H(0) = Ho
Let Z(0) = Zo
Let T =
1
r
H’ = (π − δ)TH - βTSZZo
Z’ = (β − α)THZHo - rTZ
=⇒ S’ =
π − δ
r
H -
βZo
r
HZ = aH - bHZ
=⇒ Z’ =
β − α
r
HZ - Z = cHZ - Z
where a =
π − δ
r
, b =
βZo
r
, and c =
β − α
r
The Jacobian of our non-dimensionalized system is
J =
α − bZ −bH
cZ cH − 1
Nullclines of this system are H’: H = 0, Z =
a
b
Z’: Z = 0, H =
1
c
To find the stability of the system, we look at our
equilibrium points (0,0) and (
a
b
,
1
c
)
J(0, 0) =
α 0
0 −1
λ = α, -1 The point (0,0) is a saddle when α > 0, and it is stable when α < 0
J(
1
c
,
a
b
) =


0 −
b
cac
b
0


5

6. 4 PREDATOR/PREY
Evaluating for the eigenvalues gives us λ2
+ α = 0
=⇒ λ = ± i
√
α
This implies that we have a conserved quantity, K = ae−bZ
x e−cx
So, in applying this back to our model there are two such fixed points, at (H, Z) = (0, 0) and (H, Z) =
(α/β, γ/δ). The fixed point at (0,0) is a saddle point, and the other fixed point is a center, which can be seen by
examining the respective Jacobians. The center here is actually a nonlinear center, so there are closed orbits for all
sufficiently close-by initial conditions. The conserved quantity is
K = Zα
∗ e−Z∗β
∗ Hγ
∗ e−H∗δ
The interpretation of the solutions to this equation are that the populations of humans and zombies oscillate nonlin-
early around the non-zero fixed point, and these solutions correspond to perpetual and cyclic war between humans
and zombies.
For our second set of non-dimensionalized equations, we assume a conservation of individuals, so we can throw out
R as a class. Then the equations reduce to just H and Z classes, and we get dynamics similar to the full SZR model
in Robert Smith’s paper.
We use N = H = Z = K
H’ = πT - βTZoHZ - TδH
Z’ = βTHoHZ +
HoT
ZRoR
− aHoTHZ
R’ =
cT
Ro
H +
aHoZoT
Ro
HZ − HoTR
So we want some of πT = 1, bTZo = 1, cT = 1, bTHo = 1,
cT
Ro
= 1,
aHoZoT
Ro
= 1, HoT = 1,
HoTRo
Z
= 1
After setting (H*, Z*, R*) = (N, 0, 0), (0, N, 0), and (0, 0, N) as our fixed points, we get =⇒ πT = 1, bTN = 1, dT
= 1,
cT
N
= 1, aNT = 1, Ho T = 1 When using T =
1
bN
as our timescale, we get our set of equations as
H’ =
π
Nb
- HZ -
c
bN
H
Z’ = HZ +
c
BN
R -
a
b
HZ
R’ =
c
bN
H +
a
b
HZ -
Ho
bN
R
6

7. 4.1 Predator/Prey with Logistic Prey Growth 4 PREDATOR/PREY
Using T =
1
Ho
, we get
H’ =
π
H
-
bN
Ho
HZ -
c
Ho
H
Z’ =
bN
Ho
HZ + R -
aN
Ho
HZ
R’ =
c
Ho
H +
aN
Ho
HZ - R
=⇒
H’ = πTH - cTH - bZoTHZ =⇒ H’ = (π - c)TH - bNTHZ
Z’ = bHoTRZ - aHoTHZ =⇒ Z’ = (b - a)NTHZ
We can choose T to be either
1
π − c
or
1
bN
or
1
(b − a)N
This gives us three pairs of non-dimensionalized equations
(H’, Z’) = (H -
bN
π − c
HZ,
b − a
π − c
NHZ), (
π − c
bN
H - HZ,
b − a
b
HZ), (
π − c
(b − a)N
H -
b
b − a
HZ, HZ)
When examining the first dimensionalized equation, we see that we have (H*, Z*) = (0, 0) and (0,
π − c
bN
) as our
equilibrium points.
To analyze these equilibrium points we use the Jacobian of the first non-dimensionalized system
J =



1 −
bN
π − c
Z
bN
π − c
H
b − a
π − c
NZ
b − a
π − c
NH



At (0,0), we get a trivial Jacobian
J(0, 0) =
1 0
0 0
with eigenvalues λ = 0, 1, implying the (0, 0) equilibrium point is unstable. At our (0,
π − c
bN
) fixed point,
J(0,
π − c
bN
) =
0 0
b
a
b
which gives us λ = 0, b. This fixed point is stable when b < 0.
4.1 Predator/Prey with Logistic Prey Growth
We consider the previous model, but with humans having a logistic growth rate, so that the population levels out at
some carrying capacity. The equations are as follows
We can analyze the fixed points of this system similarly to above, but the dynamics of the fixed point changes from
a nonlinear center to a stable spiral. This means that the population oscillations will die down and converge to the
fixed point where there is a kind of coexistence of humans and zombies, where the humans are regenerating fast
enough to sustain a stable zombie population.
7

8. 4.2 Competitive Predator/Prey 4 PREDATOR/PREY
4.2 Competitive Predator/Prey
This is another extension of the predator prey interactions, but where both species have a logistic growth rate, and
are competing for resources.
8

9. 4.2 Competitive Predator/Prey 4 PREDATOR/PREY
These equations are linear with negative slope, and intersect as in the diagrams. There are 4 fixed points in
the equations, (0,0), one with positive zombie population and zero humans, one with zero zombie populations and
positive humans, and one with positive populations of both. The stability of the last 3 fixed points differs based on
the slopes of the nullclines. If the humans null cline is steeper than the zombie nullcline, the last fixed point is stable,
so every trajectory with nonzero initial conditions converges to the fixed point with both populations being positive.
However, if the zombie nullcline is steeper than the human nullcline, the last fixed point is unstable, while the two
fixed points where one of the populations dies out are stable, with the unstable manifold goint from the origin to the
unstable fixed point with positive populaion. So, trajectories starting from the right of the unstable manifold converge
to the fixed point where all the zombies die off, and trajectories starting from the left of the unstable manifold result
in all human dying off. Now, these equations do not consider the case where zombies have a deterioration factor, so
these fixed points are stable.
9

10. 6 OUR HUMAN AGGRESSION MODEL
5 Budworm Model
We include this model as motivation for the next model.
To further study predator-prey models, we analysed a Spruce Budworm population model. The basic budworm
population equation, without predation, is given by the basic logistic model
N’ = rN(1-
N
K
)
with r an appropriate birth rate of budworms.
Here is the same model, but now with a predation rate defined as a function p(N) =
βPN2
No2 + N2
.
N’ = rN(1-
N
K
) -
βPN2
No2 + N2
Where N represents the budworm population (the prey), while P is the bird population (the predators). β is
the efficiency of the predators, while r is the growth rate of the budworms. K is the carrying capacity, which limits
the potential growth of the budworms due to foilage density and other territorial resources. Naturally the presence
of a predator is bad news for the budworm, as the consumption of the prey increases, then the budworm’s population
will be reduced. However, the rate at which predation happens will reach a level of saturation. This idea is of critical
importance for our analysis, and it is the typical behaviour of this function with which we focus.
In order to adapt this model to a human vs. zombies model, we set zombies as the prey, with humans this
time as the predators of the walking undead, so we have
Z’ = βHZ(1 -
Z
K
) - (HZ)
γZ2
Ho + Z2
In this system, the carrying capaity (parameter K) is now dependent on the amount of resources (food, water,
shelter) available for the humans to live off of. (It is here that we note that this carrying capacity only hinders the
growth of zombies for any positive value of K, no matter how large... i.e. Z/K > 0 ⇒ (1 − Z/K) < 1. Here it
serves us to omit such restrictions on zombie growth based on the conclusions the the rate of zombie “birth” is only
dependant on the direct interaction of humans and zombies times an “aggression/turn” rate based on average zombie
behaviour.) Also, zombies only grow until there are no more humans and then the system halts.
(H = 0 ⇒ all flow stops).
6 Our Human Aggression Model
During our work on the simplified model, and after having discussed what could most realistically be added
to it, we thought to the rate at which humans killed zombies, or perhaps an “aggression” value for human-zombie
interaction. Where it made sense that the “aggression” of mind-less zombies could be accurately represented by a
constant, could humans’? It made sense that we should view what would happen if the response to the presence of
zombies were to be slow at first and then approach some sort of maximum aggression toward the zombie pestilence.
Our thoughts turned the Budworm model described above and the function we here rename α(Z).
10

11. 6 OUR HUMAN AGGRESSION MODEL
H’ = πH - βHZ - δH
Z’ = βHZ + δH - [α(Z)]HZ
Here we see the graphs that will look familiar and are perhaps the most realistic, without near zero human aggression.
We see that for any initial size of human population the fixed point is human annihilation, i.e. the fixed point where
all humans are dead and there remains a fixed population of zombies.
11

12. 6 OUR HUMAN AGGRESSION MODEL
Here we see a change that reflects a much less dramatic natural death rate and in return a much higher relative input
of humans (birthrate is a much higher proportion to natural death), there is human growth for an interval but once
again eventually zombies will exist alone.
Here we amp up human aggression in a major way, along with our much higher birthrate than natural death rate.
Not only that, but with our adjusted α such that we start with an inflated amount of aggression (much higher
than zero) we easily destroy the zombies and breed excessively in celebration. This model can be thought of (in a
context out of zombies) as a humanity that procreates excessively, dies selectively, is hyper efficient at organizing
and preparedness and is highly capable of overcoming persistent problems.) This is the super-human vs. a relatively
docile threat.
12

13. 7 BIEBER FEV ER MODEL
These are the same parameters of the super-human, however the rate of natural death is now 110 percent the
birthrate. Zombie anarchy ensues, even with the much more aggressive/efficient humans we have humans losing to a
fixed amount of zombies.
7 Bieber Fever Model
Robert Smith? models Bieber Fever’s epidemic as a varied SIR model, similar to his work for the Zombie
Infection models. In the Bieber Fever model, we consider three classes: Susceptible (S), Bieber-infected (B), and
Removed (R). Unlike the Zombie Infection models, the Removed class in the Beiber Fever system contain individuals
who are immune to infection. These Removed individuals represent people that have lost interest in Justin Bieber,
but their immunity to Bieber Fever can disappear under special conditions.
This model introduces a parameter M, which represents the total amount of media that includes a certain
proportions ( ) of positive (P) and negative (N) media coverage. This is given by the equation
M = P + (1 − )N.
Media events can come in the form of song releases, merchandise, or even guest appearances on television.
The proportion ( ) is included to represent how much positive media is shown in reference to the negative media that
is present.
Parameter Symbol InitialValues Units
Susceptible S 1500 people
Bieber-infected B 3 people
Recovered R 0 people
Recruitment rate π 10 people month−1
Transmission rate β 8.3 x 10−4
people−1
month−
1
Maturation rate µ 1/144 month−1
Boredom rate b 1/24, 2 month−1
Positive media rate P 2 month−1
Negative media rate N 1 month−1
Positive media proportion 0.75 n/a
13

14. 7 BIEBER FEV ER MODEL
The equations for the Bieber Fever model are
S’ = π - βSB - PS + (1 - )NB - µS + PR - (1 - )NS
B’ = βSB + PS - (1 - )NB - bB - µR
R’ = bB + (1 - )NS - PR - µR
People are introduced into the Susceptible class at a recruitment rate of π. The members of the Susceptible
class can become infected with Bieber Fever in one of two ways: direct contact with an infected individual (at a
transmission rate β) or infected through enough positive media exposure ( P).
Once an individual becomes infected with Bieber Fever and becomes a Bieber Zombie, he or she can become
cured of this infection in a variety of ways. Because the target demographic for Bieber’s propaganda are kids and
teenagers, the most effective cure for Bieber Fever is for an infected individual to grow up beyond the age of 17 years-
old (at a maturation rate µ) and be removed form the system entirely. In fact, individuals in both the Susceptible
and Removed class can leave the system through this method as well.
An infected individual can also grow tired and bored of Justin Bieber (at a rate b), entering the Removed class
where they grant themselves a temporary immunity from Bieber Fever. When Bieber fans are affected by a negative
media (at a proportion of (1 − )N), they will then revert back into a Susceptible case. Furthermore, a Susceptible
individual that is affected by this proportion of negative media will enter the Removed class, also granting themselves
a temporary immunity from Justin Bieber.
Once in the Removed class, individuals can not be directly infected, but they can re-enter the Susceptible
state through positive media (at a proportion of P). Members of the Removed class are the hardest to infect, as
they can only become infected after they re-enter the Susceptible class through this method.
The initial value of the Susceptible class in our graphs were set at 1,500, which could represent a local high
school since it is close to the number of students in an average high school in California (currently there are 1,089
high school students for every high school). Since the target demographic of Justin Bieber’s propaganda is from the
5-17 year age range, it is fitting to choose a time frame of 12 years (
1
µ
months) for the model. The two choices for
the rate of boredom (b) are:
b =
1
24
, which will take an individual two years to become bored of Justin Bieber, and
b = 2, where a person will get bored fairly quickly in just two weeks.
The positive media proportion ( ) was given a value of 0.75, since propaganda and advertisements generally
favor in promoting Justin Bieber rather than hurting his image. Since Justin Bieber releases and is featured in, on
average, 15 singles every year, and has around 9 appearances a year in films and television shows, it was fitting to
set the positive media rate P = 2 times a month as an initial value. The negative media rate is set at N = 1 to
represent celebrity gossip sites such as TMZ that might occasionally spread bad press on Justin Bieber to boost its
viewership.
The Justin Bieber model shares many similarities to the HZR Zombie models, but with the key difference
that a disease-free equilibrium does not exist. Again, we use T = S + B + R to represent the total population, and
denote T’ = π - µN as the change of total population in this system.
14

15. 8 MEDIA ABSENSE MODEL
Solving for T’ = 0, we have T* =
π
µ
as our total population equilibrium. Since the initial value of π = 10
people
month
,
the population equilibrium in our analysis and parameters is T* = 1440 people.
8 Media Absense Model
One of the models presented in Robert Smith’s work is an absense of media model, which sets P = N = 0,
not allowing positive nor negative media to effect the system. The equations are now given as
S’ = π - βSB - µS
B’ = βSB - bB - µR
R’ = bB - µR
This model has the disease-free equilibrium (
π
µ
, 0, 0), and the endemic equilibrium (
r
β
,
br
µ(b + µ)
,
r
b + µ
), where
r = π −
µ(b + µ)
β
.
The endemic equilibrium only exists when r > 0 =⇒ R0 =
πβ
µ(b + µ)
Solving for the Jacobian gives us
J =


−βB − µ βS 0
βB βS − b − µ 0
0 b −µ

 .
Solving for eigenvalues yields =⇒ λ2
+ (βB + µ + βS + b + µ)λ + βbB + βµB - βµS + µb + µ2
= 0.
At the disease-free equilibrium, the constant term of the characteristic equation c satisfies c = -βπ + µ(b +
µ). This implies that c < 0 when R0 > 1. Since the coefficient of λ is always positive, it follows that: 1the disease-free
equilibrium is unstable ⇐⇒ R0 > 1.
At the endemic equilibrium, c = βπ - µ(b + µ). We can see that c > 0 when R0 > 1, and the endemic equilibrium
is stable whenever it exists.
15

16. 8 MEDIA ABSENSE MODEL
16

17. 9 COUNTINUOUS POSITIV E MEDIA MODEL
9 Countinuous Positive Media Model
The more interesting Bieber Fever model we will look at deals with a scenario in which the media will only
be in favor of Justin Bieber. To accomplish this, it is sufficient to set = 1, which gives us the new model
S’ = π - βSB - PS + µS + PR
B’ = βSB + PS - bB - µR
R’ = bB - PR - µR
When solving for possible fixed points, if P > 0 then there is no disease-free equilibrium. This implies that
Bieber Fever will always spread without any negative media reducing its growth. The fixed points of the system
are
R∗
=
b
P + µ
B∗
, S∗
=
(b + µ)B∗
βB∗ + P
and g(B∗
) =
Pbβ
P + µ
B∗2
+ [πβ - (β + P + µ)(b + µ) +
P2
b
P + µ
]B∗
+ Pπ = 0
Take limP →∞ =⇒
g(B∗
)
P
= -µB* + π = 0.
=⇒ B∗
=
π
µ
= T
This implies that the absence of negative Justin Bieber media coverage will allow Bieber Fever to continue
infecting individuals until the Bieber class consists of the entire population.
17

18. 9 COUNTINUOUS POSITIV E MEDIA MODEL
(In the second graph, where did everyone go? Everyone is recovered thanks to a really high boredom with Bieber.)
Here, along with their captions, are the graphs from the Bieber Fever paper that are of particular interest to us.
Look familiar? This results, when boredom and aggression/efficiency are appropriately paralleled, say very similar
things. With constant media coverage (or resilient/aggressive pestilence) the highest we can hope for, while not being
super-human, is basic mutual destruction, but realistically we will be wiped out.
Conclusion
The real world result gained from this paper should be that the human race should be over prepared for as many
possibilities as are possible, even/especially zombies. The papers we studied during this project were well thought
out and presented. Though there are areas in them where satisfactory answers weren’t given to the questions being
raised. Smith?’s papers were a good starting off point for other concerns about zombie lore that others could be
concerned with. There is nothing that we would have done differently had there been no paper before it about the
subject.
18

19. 10 HIZRQ MODEL
other models reviewed but not within the main paper
10 HIZRQ Model
Robert Smith’s quarantine model introduces an area available only to infected and zombie individuals, entering
at rates of κ (for infected) and σ (for zombies). Individuals that are quarantined will leave at a rate of γ, representing
members killed while attempting to flee from the quarantine zone.
The model equations are
H = π − βHZ − δH
I = βHZ − ρI − δI − κI
Z = ρI − αHZ + ζR − σZ
R = αHZ − ζR + δI + δH + γQ
Q = κI + σZ − γQ
Again, for our own basic HZ model, we wanted the birthrate to be dependent on the number of humans,
and for defeated zombies to leave the system forever. By applying our modifications to Robert Smith?’s quarantine
model, our own quarantine model equations then become
flowchart
H = πH − βHZ − δH
I = βHZ − ρI − δI − κI
Z = ρI − αHZ + ζR − σZ
R = γQ − ζR + δI + δH
Q = κI + σZ − γQ
Again, as we look at our model with a short time-scale and set π = δ = 0, we have (H*, I*, Z*, R*, Q*) = (N, 0, 0,
0, 0), (0, 0, Z, R, Q) as our fixed points.
The Jacobian of our quarantine model is
J =
−βZ 0 −βH 0 0
βZ −(ρ + κ) βH 0 0
−αZ ρ −(αH + σ) ζ 0
αZ 0 αH −ζ γ
0 κ σ 0 −γ
Evaluated at our zombie apocalypse equlibirium
19

20. 11 TREATMENT MODEL
J(0, 0, Z, R, Q) =
−βZ 0 0 0 0
βZ −(ρ + κ) 0 0 0
−αZ ρ −σ ζ 0
αZ 0 0 −ζ γ
0 κ σ 0 −γ
This gives us λ1,−,5 = -βZ, -(ρ + κ), -σ, -ζ, -γ. Since all parameters are positive, the zombie apocalypse equilibrium
is stable at all times.
The Jacobian at our zombie-free fixed point
J(N, 0, 0, 0, 0) =
0 0 −βN 0 0
0 −(ρ + κ) βN 0 0
0 ρ −(αN + σ) ζ 0
0 0 αN −ζ γ
0 κ σ 0 −γ
By solving for λ we have λ = 0, -(ρ + κ), -(αN + σ), -ζ, -γ. This implies that the zombie-free equilibrium is stable as
well, as infected individuals and zombies are quarantined before they can cause enough harm to the system.
11 Treatment Model
In W.Z.A.!, Smith? gives us a Treatment model in which we are able to find a cure for a zombie disease. The
cure would allow a member of the zombie class to return to the Human class, but it does not provide that individual
with immunity from becoming infected again. Zombies that were reanimated from the Removed class are able to be
cured and return back to the living Human class as well. This model also does not include a Quarantine class since
there is treatment.
The equations for this model are given by
H = π − βHZ − δH + cZ
I = βHZ − ρI − δI
Z = ρI + ζR − αHZ − cZ
R = δH + δI + αHZ − ζR
Due to the cZ in the H equation and the −cZ in the Z equation, we are able to get a new equilibrium point aside
from our usual zombie-free and zombie apocalypse scenarios.
As in our previous evaluations, we choose a time frame in which π = δ = 0 when finding our equilibrium points as
we set H = Z = I = R = 0.
-βHZ + cZ = 0
βHZ - ρI = 0
ρI + ζR - αHZ - cZ = 0
αHZ - ζR = 0
20

21. 13 COUNTINUOUS NEGATIV E MEDIA MODEL
The new equilibrium and Jacobians are
(S*, I*, Z*, R*) = (
c
β
,
c
ρ
Z*, Z*,
αc
ζβ
Z*)
J =
βZ 0 −βH + c 0
βZ −ρ βH 0
−αZ ρ −(αH + c) ζ
αZ 0 αH −ζ
Since H, Z, I, and R are all non-zero values, this equilibrium point implies co-existence among the Humans and
Zombies.
To test the stability of the co-existence equilibrium, we compute the Jacobian at this point
J(
c
β
,
c
ρ
Z∗
, Z∗
,
αc
ζβ
Z∗
) =
βZ∗
0 −βH + c 0
βZ∗
−ρ βH 0
−αZ∗
ρ −(
αc
β
+ c) ζ
αZ∗
0
αc
β
−ζ
Evaluating for the eigenvalues gives us -(βZ - λ)(-λ[λ2
+ ( ρ +
αc
β
+ c + ζ)λ +
ζαc
β
+
ραc
β
+ ρζ + cζ]). Immediately
we can see that λ = 0, β Z.
For the quadratic λ2
+ ( ρ +
αc
β
+ c + ζ)λ +
ζαc
β
+
ραc
β
+ ρζ + cζ, we see that all coefficients are positive. Thus,
the next two values of λ are positive as well and this tells us that the co-existence equilibrium is unstable.
12 Impulse Model
Our final analysis of Smith?’s W.Z.A.! models will conclude with the impulsive eradication scenario, where we
attempt to control the spread of the zombie infestation with strategic large-scale attacks. These attacks could be in
the form of large infantry hit-and-run sweeps, artillery strikes, or even the release of a biological counter-attack. We
again use Smith?’s basic HZR model, but we add the impulsive criterion to get
H = π − βHZ − δH, t = tn
Z = βHZ + ζR − αHZ, t = tn
R = δH + αHZ − ζ, t = tnR
H = −knZ, t = tn
where k ∈ (0, 1] is the kill ratio and n denotes the number of attacks required until kn > 1.
13 Countinuous Negative Media Model
The final Bieber Fever model will look at the opposite scenario of our previous model. We will create a
scenario where the airwaves are filled with nothing but slander and negative media attention, possibly due to the
recent London photographer and Anne Frank museum incidents in the past few months. In this scenario we will set
the positive media proportion = 0
The equations for this model are
21

22. 13 COUNTINUOUS NEGATIV E MEDIA MODEL
S’ = π - βSB + NB - µS - (1 - )NS
B’ = βSB - NB - bB - µR
R’ = bB + NS - µR
In this model, a disease-free equilibrium exists at (S*, B*, R*) = (
π
µ + N
, 0,
πN
µ(µ + N)
) and R0, neg =
πβ
(µ + N)(b + µ + N)
.
When we set N very large, we will conversely have R0, neg become very small and the disease-free equilibrium becomes
stable.
As we take limN → ∞ =⇒ (S*, B*, R*) = (0, 0,
π
µ
). This equilibrium point implies that with only negative media
available for Justin Bieber, the public will eventually become bored of Justin Bieber and immune to Bieber Fever’s
infectious capabilities.
As we can verify from our human vs. zombies equation, our fixed points are (H*, Z*) = (0, Z) and (K, 0),
which are our zombie apocalypse and zombie-free equilibriums. The second equilibrium point suggests that without
zombies in the system, humans will populate the world with as many offspring that is sustainable.
To find the stability of the zombie apocalypse and zombie-free scenarios, we are able to utilize the second
derivative of H
22

23. 13 COUNTINUOUS NEGATIV E MEDIA MODEL
H” = π(1 -
2H
K
) -
2H(βZ + 1)
(Ho2 + H2)2
H”(0, Z) = π, which indicates that the zombie apocalypse scenario is unstable as humans will continue to grow
For our zombie-free scenario, H”(K, 0) = -π -
2K
(Ho2 + K2)2
< 0 since our parameters are always posi-
tive.
This predator prey model gives us a switch in stability from our usual zombie-free and zombie apocalypse
equilibrium.
23