1. Mathematical Epidemiology
(And Zombies!)
Meagan Durbin
4/19/2013
Math 5001

2. Durbin, 1
Looking on the internet for a concrete definition of epidemiology will result in severalexplanations that
seem to conflict one another. Epidemiology can be interpreted and explained in severaldifferent ways,
but Merriam-Webster’s Medical Dictionary1
yields the following definition:
Unfortunately, the dictionary definition is rather muddled in and of itself. Given that epidemiology stems
from the word ‘epidemic’ (an outbreak of an infectious disease), one can clearly see that epidemiology is
the essentially the study of the hows and whys of infectious diseases in a population. This includes how
the disease spreads,the causes and effects of a given disease for a specified population, as well as the
transmission patterns of a specific pathogen. Epidemiologists (scientists in the field of epidemiology) help
with study design, the collection and analysis of data,and the interpretation and dissemination of results
of those studies.
Although manipulating and analyzing data sets to understand the effects of a disease is easily seen as a
direct mathematical approach, mathematics is gaining a foothold in the heart of epidemiology in the form
of math modeling. Sophisticated models can show how people in a specific population interact and spread
disease, then show the spread of the disease through that population to other populations. To do this,
mathematical epidemiologists or epidemiological mathematicians borrow methods of modeling from
chemistry, physics, ecology, biology, and other sciences.
While, at first, a mathematical model of a disease sounds like a difficult task for a novelty, but models are
extremely useful. Since equations are not the ‘real world’ epidemiologists can study the spread of various
diseases without involving living things. Altering equations is far less dangerous than manipulating a
pathogen’s genomes. For example, scientists in Holland were studying H5N4 (the bird flu), and
ultimately created a strain that is easily transmissible between mammals.2
This is rather unfortunate,
because the H5N4 is a virulent strain of the flu with a high mortality rate,and before this, only infected
humans who were in direct contact with an infected bird. Along with being relatively safe,mathematical
models can be built with limited understanding of the pathogen and then built upon as more information
is gathered, making it a very useful tool that can be updated in real-time.
Like many branches of science, epidemiological models have applications to seemingly unrelated fields.
Computer science is the most obvious connection. Epidemiological models can be applied to a computer
virus moving its way through a network, as computer viruses are transmitted through a network in a
similar way to influenza transmitted between humans. An unseen connection is epidemiology in pop
culture through zombies. Traditionally, through films like Night of the Living Dead,a zombie is a
1
"Epidemiology." Merriam-Webster. Merriam-Webster, n.d. Web. 01 Apr. 2013.
2
Landau, Elizabeth. "Bird Flu Research Resumes." CNN. Cable News Network, 23 Jan. 2013. Web.
ep·i·de·mi·ol·o·gy noun -jē (Medical Dictionary)
1. a branch of medical science that deals with the incidence, distribution, and control of
disease in a population
2. the sum of the factors controlling the presence or absence of a disease or pathogen

3. Durbin, 2
reanimated human corpse whose main directive is to feed on human brain tissue.3
However, 28 Days
Later is a film in the horror genre that depicts a very different type of zombie. In the film, the ‘zombies’
are actually humans that are infected with a ‘rage virus’ which is simply a mutated rabies virus that
escaped from a research laboratory.4
Unlike vampires and werewolves,it seems like pop culture cannot let go of the idea of zombies. Part of
this reason could be that disease is the last remaining predator of mankind and dead people are a very
serious reminder (and traditionally a very serious cause) of disease. Part of this reason could be that a
zombie plague seems to be the most plausible out of all of the other mythical creatures.
As previously stated,the ‘zombie virus’ in 28 Days Later is a mutated rabies virus. While rabies DNA is
fairly stable in a natural setting, the virus could theoretically be modified by extremist groups and
unleashed in biological warfare. Other than that,there are naturally occurring parasites and prions in the
animal world that are transmitted in ways similar to the way pop culture shows zombie infection to be
transmitted. The scary part is that these specific parasites and prions induce very traditional zombie-esque
symptoms.
Because of these naturally occurring pathogens and/or fear of biological warfare,many groups are
fervently preparing for the zombie apocalypse.5
Other groups are mocking preparedness attempts,
essentially stating that the human race is doomed in the event of a zombie infection. This poses a unique
opportunity for mathematicians and epidemiologists to study the outbreak of a zombie virus.
Since zombies are (hopefully) not realand thus nothing about them can be quantitatively measured,an
easy way to study the zombie apocalypse would be to study the zombie viruses themselves. An effective
tool in mathematical modeling is the S-I-R model, first introduced by Kermack and McKendrick in 1927
as a way to study an “infectious disease that confers an immunity.”6
A zombie outbreak is theoretically an
infectious disease,and a destroyed zombie is technically immune to becoming re-infected, thus the basic
S-I-R model in this paper can be adapted to show a zombie outbreak’s effects on a human population.
To do this, a population is divided into three groups. In the original model, these groups are:Susceptible,
Infected,and Recovered. In the adapted model, these groups are renamed Humans, Zombies, and
Defeated,respectively. These groups interact in a way that is very similar to substances undergoing a
chemical reaction. And their interactions can be written in a way similar to chemical equations.7
3
Night of the Living Dead. Dir. George A. Romero. By John Russo. Perf. Duane Jones, Judith O'Dea, Marilyn Eastman, and Karl
Hardman. Continental Distributing, Inc., 1968. DVD.
4
28 Days Later. Dir. Juan Carlos Fresnadillo. By Juan Carlos Fresnadillo. Fox Atomic, 2007. DVD.
5
"The Zombie ApocalypseIs Going To Be AWESOME!" Zombie Prep Network. N.p., 13 Sept. 2012.
6
Kermack, W. O., and A. G. McKendrick. "A Contribution to theMathematical Theory of Epidemics." RoyalSociety
Publishing, 1 Aug. 1927. Web.
7
Sibert, Gwen. "ACT Chemistry Notes: Types of Equations." Types of Equations. Virginia Tech Department of Chemistry, n.d.
Web.

4. S = Susceptible, Humans
I = Infected, Zombies
R = Recovered,“defeated” zombies
H + Z → 2Z, Human interaction with a
zombie produces two zombies.
Z → D, The zombie is eventually defeated
and becomes inanimate.
Because these interactions mirror chemical equations, they are bound by the same set of laws, and the
same techniques are used to turn them into analyzable equations. The Law of Mass Action states that the
rate of a chemical reaction is directly proportional to the concentrations of the reactants.8
Michaelis-
Menten Kinetics allows a parameter to be assigned to these rates.9
Thus, the interactions transform into
the following system of differential equations with initial conditions H(0) = H0, Z(0) = Z0, D(0) = D0, α
denoting the rate at which zombies are destroyed, and β denoting the infection rate of humans
Before analyzing the system of equations, it is best to nondimensionalize, or scale them to remove any
attached dimensions in either the variables H, Z, and D,or parameters H0,Z0, D0, α, and β. To do this,
groupings of parameters and one variable are multiplied together and defined as a new ‘dimensionless’
variable. This new variable now has the same numerical value no matter what system is used to measure
it.10
The dimensionless variables then replace the original, unscaled equations and then the equations are
simplified. The scales used to nondimensionalize the above equations as well as the dimensionless
equations are below, with scaled initial conditions 𝑢(0) = 1, 𝑣(0) = 𝜖, 𝑤(0) = 0, and 𝜖 =
𝐼0
𝑆0
.
8
"Law of Mass Action." Science World. Wolfram Research, n.d. Web.
9
Cornish-Bowden, Athel. "Fundamentals of EnzymeKinetics." Portland Press, n.d. Web.
10
Lin, C. C., and Lee A. Segel. "Chapter 6." Mathematics Applied to Deterministic Problems in the Natural Sciences.
Philadelphia, PA: Soc. for Industrial and Applied Mathematics, 1988. N. pag. Print.
𝐻 + 𝑍
𝛽
→ 2𝑍
𝑍
𝛼
→ 𝐷
𝐻′( 𝑡) = −𝛽𝐻
𝑍′( 𝑡) = −𝛽𝐻𝑍 + 2𝛽𝐻𝑍 − 𝛼𝑍 = (𝛽𝐻 − 𝛼)𝑍
𝐷′( 𝑡) = 𝛼𝑍
S = S0u
I = I0v
R = R0w
t =
𝜏
𝛽𝑠0
𝛽𝑆0
2
𝑢′ = −𝛽𝑆0
2
𝑢𝑣
𝛽𝑆0
2
𝑣′ = (𝛽𝑆0 𝑢 − 𝛼)𝑆0 𝑣
𝛽𝑆0
2
𝑤′ = 𝛼𝑆0
2
𝑣
𝑢′ = −𝑢𝑣
𝑣′ = ( 𝑢 − 𝑟) 𝑣
𝑤′ = 𝑟𝑣
𝑟 =
𝛼
𝛽𝑆0

5. Durbin, 1
The dimensionless equations u’, v’, and w’ denote the rate of change in the human, zombie, and destroyed
populations, respectively. Nondimensionalization almost always results in dimensionless groupings that
are assigned a new notation. In this model, r denotes the ratio of zombie removal to zombie creation rate.
More interestingly, 1/𝑟 signifies the initial zombie ‘birth’ rate. 0 < 𝑟 < 1 indicates an outbreak of the
zombie infection, while 𝑟 > 1 suggests the number of zombies quickly tend to zero.
To analyze what happens to each of these populations, the solution(s) to the system is(are) found. These
solutions are called steady states,because they symbolize are when the groups at an equilibrium. Luckily
(for humans), the only solutions to this system occur when there are no zombies left (v=0). Since this
system is closed (𝑢′ + 𝑣′ + 𝑤′ = 0) and the initial population is denoted by 𝑢(0) + 𝑣(0) + 𝑤(0) = 1 +
𝜖, the population size of humans at time τ is 𝑢( 𝜏) = 1 + 𝜖 − 𝑣( 𝜏) − 𝑤( 𝜏).
To find the sizes of the human and destroyed populations as the number of zombies tend to zero, the ratio
u’/w’ is analyzed and solved by the method of separable ordinary differential equations.
𝑢′
𝑤′
= −
𝑢𝑣
𝑟𝑣
⇒
𝑑𝑢
𝑑𝑤
= −
𝑢
𝑟
⇒ ∫
1
𝑢
𝑑𝑢 = ∫−
1
𝑟
𝑑𝑤 ⇒ ln 𝑢 = −
𝑤
𝑟
⇒ 𝑢 = 𝑒−
𝑤
𝑟
To further analyze u and w, the equations 𝑢( 𝜏) = 1 + 𝜖 − 𝑤( 𝜏) and 𝑢 = 𝑒−
𝑤
𝑟 are plotted on a w-u axis,
with varying values of r.
The orange dots in Figure 1 indicate the values of u and w as the zombie population collapses. When r is
larger than one, there are many more humans left than when r is less than one, which fits with the original
Figure 1

6. Durbin, 2
assumptions about 1/r indicating the zombie ‘birth’ rate. A conclusion from this model can be made, that
is, as long as zombies are destroyed faster than they are created,the human population will survive.
The drawbacks to this model stem from the fact that it is very limited. This model does not account for
human population growth or anything else that may affect the human population during an outbreak of
any type. Also, the model is riddled with hidden assumptions. For example, since the population is
constant, the assumption was made that the infection happens over a very quick time scale. No
consideration was given to how the zombies were being defeated. Aiding humans in their fight to destroy
zombies is much easier than aiding in zombie decomposition. The hidden assumption is that it does not
matter how zombies are destroyed, as long as they are being destroyed.
Hidden assumptions can cripple a model and inadvertently turn out unreliable or biased results. For
example, Phillip Munz’s article “When Zombies Attack!" deals with a zombie outbreak similar to the one
above, except he adds in the possibility of reanimation.11
Unfortunately, every instance of a zombie
outbreak in his models results in the ‘doomsday scenario’ (zombies destroying the human population) to
be the only stable outcome. However,a hidden assumption is buried in his interactions:
“Zombies move to the removed class upon being
‘defeated’. This can be done by removing the head or
destroying the brain of the zombie.”
“New zombies can only come from two sources:
 The resurrected from the newly deceased
(removed group).
 Susceptibles who have ‘lost’ an encounter
with a zombie.”
Munz’s assumption is that deceased humans and defeated zombies belong to the same group. This
assumption leads to humans fighting an infinite number of zombies, since new zombies are being
resurrected from a group that contains defeated zombies. This is more than likely the reason why the only
stable scenario is the zombies defeating humans. Of course the doomsday scenario is steady when
humans, in effect,cannot defeat zombies. For a more realistic model to hold, “Defeated Zombie” and
“Potential Reanimation” need to be separate groups.
Recognizing the shortcomings of the previous two models allows us to create an improved third model;
one that distinguishes between passive and active destruction of zombies and that includes human growth
but keeps human deaths that are not due to a zombie attack separate from human deaths by zombie. Also,
this third model should list its assumptions so as to combat any hidden assumptions.
11
Munz, Phillip. When Zombies Attack! Infectious Disease Modelling Research Progress. Nova Science Publishers, Inc., 2009.
Web.

7. Durbin, 3
This model uses the definition of a zombie as “a relentlessly aggressive reanimated human corpse driven
by a biological infection12
”. Since a zombie by this definition is essentially a walking human corpse, we
will assume that zombies will decompose to a point where they are incapable of infecting humans. We
also assume that other than the passive rotting, the only way to incapacitate a zombie is for a human to
destroy it.
Conversely, the only way for a human to become a zombie is to be bitten by a zombie. There are no
outside means of infection, and the conversion from human to zombie happens very quickly (there is no
latency period). Also, in an altercation between a human and a zombie, it is possible for the human to be
killed but not converted (the human is essentially eaten by the zombie). In the complete absence of
zombies, the human population behaves logistically. We assume that the birth and death rate are
equivalent (for simplicity), thus the growth rate is equal to 1.
This model deals with the same three groups: Humans (H), Zombies (Z), and Destroyed (D). An
interaction between a human and zombie produces three possible outcomes; the human is converted
resulting in two zombies, the human destroys the zombie, or the zombie destroys the human. Also, a
zombie can be passively destroyed by decomposition. The symbolic interactions and the corresponding
equations (from the Law of Mass Action) are as follows:
Where α is the conversion rate of humans to zombies, β is the rate at which humans destroy zombies, γ is
the rate at which zombies destroy but not convert humans, η is the passive destruction rate of zombies, K
is the carrying capacity for humans, and r is the growth rate of humans. Since three of the reactions are
outcomes of the interactions between a human and a zombie, α, β, and γ are the proportions of ρ, the
interaction rate between humans and zombies, and 𝛼 + 𝛽 + 𝛾 = 𝜌. This model takes ρ=1, so that every
human will interact with one zombie. Also, since D is not present in H’ and Z’, we do not need to include
D’ in our analysis.
12
Matt Mogk, Everything You Ever Wanted to Know About Zombies, (New York: Gallery, 2011), 6.
𝐻 + 𝑍 → 2𝑍
𝐻 + 𝑍 → 𝐻 + 𝐷
𝐻 + 𝑍 → 𝐷 + 𝑍
𝑍 → 𝐷
𝑑𝐻
𝑑𝑡
= 𝐻′ = 𝐻 ( 𝑟 (1 −
𝐻
𝑘
)− ( 𝛼 + 𝛾) 𝑍)
𝑑𝑍
𝑑𝑡
= 𝑍′ = 𝑍(( 𝛼 − 𝛽) 𝐻 − 𝜂)
𝑑𝐷
𝑑𝑡
= 𝐷′ = 𝑍(( 𝛽 + 𝛾) 𝐻 + 𝜂)

8. Durbin, 4
The nondimensionalized equations and scales for H’ and Z’ are:
𝑢′ = 𝑢( 𝑟̂(1 − 𝑢) − (1 + 𝛾) 𝑣)
𝑣′ = 𝑣 ((1 − 𝛽̂) 𝑢 − 𝜂̂)
𝑇 =
𝜏
𝛼𝑁
𝐾 = 𝑁
𝐻 = 𝑢𝑁
𝑍 = 𝑣𝑁
Where N is the initial human population.
The hats are defined as:
𝑟̂ =
1
𝛼𝑁
; The conversion rate of the human population, or the “birth rate” of zombies
𝛾̂ =
𝛾
𝛼
; The ratio of human death to human conversion
𝜂̂ =
𝜂
𝛼𝑁
; The ratio of passive destruction of zombies to the conversion rate of human population, or the
passive “death rate” of zombies
𝛽̂ =
𝛽
𝛼
; The ratio of active destruction of zombies to the conversion rate of humans, or the active “death
rate” of zombies
Note that if 𝛽̂ ≥ 1, v’ will always be negative, and the zombie population will collapse. For this reason,
we only consider 0 < 𝛽̂ < 1, or 𝛼 > 𝛽 (which means zombies are created at a rate faster than humans can
kill them).
By algebraic manipulation, three steady states were found (denoted (u*,v*)):
( 𝑢∗, 𝑣∗) = (0,0)
( 𝑢∗, 𝑣∗) = (1,0)
( 𝑢∗, 𝑣∗) = (
−𝜂̂
𝛽̂ − 1
,
𝑟̂( 𝛽̂ + 𝜂̂ − 1)
( 𝛽̂ − 1)( 𝛾̂ + 1)
)
Unfortunately, the extra variables and parameters resulted in a system that is not as easily solved as in the
first example. Because of this, we turn to phase plane analysis, which involves a linearization matrix (also
known as a Jacobian matrix)13
. Instead of going through the tedious process of finding the eigenvalues of
our system, we will only look at the trace and determinant of this Jacobian matrix. When the trace of the
Jacobian is negative and the determinant is positive, the resulting steady state is stable; meaning a slight
perturbation to the values is predictable and will result in the system returning to those values.14
13
"Jacobian Matrix." Mathworld. Wolfram Research, n.d. Web.
14
"Phase Plane Analysis." N.p., n.d. Web. <http://link.springer.com/content/pdf/bbm%3A978-0-387-22437-4%2F1>.

9. Durbin, 1
The Jacobian matrix A used in the linear stability analysis of these steady states is as follows:
𝐴(𝑢∗, 𝑣∗) = [
𝑟̂(1 − 2𝑢∗) − (1 + 𝛾̂) 𝑣∗ (1 + 𝛾̂) 𝑢∗
(1 − 𝛽̂) 𝑣∗ (1 − 𝛽̂) 𝑢∗ − 𝜂̂
]
The trace and determinant of this Jacobian matrix are:
𝑇( 𝑢∗, 𝑣∗) = (1 − 2𝑟̂ − 𝛽̂) 𝑢∗ − (1 + 𝛾̂) 𝑣∗ + 𝑟̂ − 𝜂̂
𝐷( 𝑢∗, 𝑣∗) = 𝜂̂(1 + 𝛾̂) 𝑣∗ + 𝑟̂(2𝑢∗ − 1)( 𝑢∗( 𝛽̂ − 1) + 𝜂̂)
The trivial steady state ( 𝑢∗, 𝑣∗) = (0,0) corresponds to the absence of humans and zombies.
𝑇𝑟𝑎𝑐𝑒[ 𝐴(0,0)] = 𝑇(0,0) = 𝑟̂ − 𝜂̂ =
1−𝜂
𝑁𝛼
is always positive.
𝐷𝑒𝑡[ 𝐴(0,0)] = 𝐷(0,0) = −𝑟̂ 𝜂̂ = −
𝜂
( 𝑁𝛼)2
is always negative.
This steady state is always unstable.
The steady state ( 𝑢∗, 𝑣∗) = (1,0) corresponds to the absence of zombies and the presence of humans.
𝑇(1,0) = 1 − 𝑟̂ − 𝛽̂ − 𝜂̂ = −
𝜂−𝑁𝜇+1
𝑁𝛼
is negative for 𝜂 > 𝑁𝜇 − 1
𝐷(1,0) = 𝑟̂( 𝛽̂ + 𝜂̂ − 1) =
𝜂−𝑁𝜇
( 𝑁𝛼)2
is positive for 𝜂 > 𝑁𝜇
This steady state is stable for 𝜂 > 𝑁𝜇 (> 𝑁𝜇 − 1), with 𝜇 = 𝛼 − 𝛽.
Recall, we are only considering 𝛼 > 𝛽, due to our assumption that otherwise the zombie population will
collapse. If we consider 𝛼 < 𝛽, or 𝜇 < 0 , our hypothesis is confirmed, resulting in T(1,0) that is always
negative and D(1,0) that is always positive, and a stable state for all values of u and v.
The steady state ( 𝑢∗, 𝑣∗) = (
−𝜂̂
𝛽̂−1
,
𝑟̂( 𝛽̂+𝜂̂−1)
( 𝛽̂−1)( 𝛾̂+1)
) = (𝑢 𝜖,𝑣𝜖) corresponds to the presence of humans and
zombies.
𝑇( 𝑢 𝜖,𝑣𝜖) =
𝑟̂ 𝜂̂
𝛽̂−1
= −
𝜂
𝑁2 𝛼𝜇
is always negative
𝐷( 𝑢 𝜖,𝑣𝜖) =
𝑟̂ 𝜂̂( 𝛽̂+𝜂̂−1)
𝛽̂−1
=
𝜂( 𝑁𝜇−𝜂)
𝑁𝜇( 𝑁𝛼)2
is positive for 𝜂 < 𝑁𝜇
This steady state is stable for 𝜂 < 𝑁𝜇, with 𝜇 = 𝛼 − 𝛽
Again, if we consider 𝜇 < 0, we find 𝑇( 𝑢 𝜖,𝑣𝜖) to always be negative and 𝐷( 𝑢 𝜖,𝑣𝜖) to be always positive,
making this an unstable state for all of u and v.

10. Durbin, 2
If we graph the values of 𝜂 and 𝜇, we can
easily see the values that make each steady
state stable:
The blue shaded region in Figure 2
indicates the values of η and μ that
ensure (u*, v*)=(1,0) is stable.
The red shaded region in Figure 2
indicates the values of η and μ that
ensure (u*, v*)=(uε,vε) is stable.
From these results, we can conclude several things:
1. If humans destroy the zombies faster than the zombies can convert humans into more zombies,
zombies will be eliminated fairly quickly. (𝜇 < 0)
2. Otherwise, if the passive destruction rate of zombies (rotting, walking off cliffs, etc.) is larger
than the net increase of zombies, humans also come out triumphantly (𝑁𝜇 < 𝜂). Note, this
situation is feasible only when 0 < 𝜇 ≪ 1 and 0 ≪ 𝜂 < 1 (The net increase of zombies needs to
be very, very small, and the passive destruction rate of zombies needs to be fairly large).
3. When zombies create more zombies faster than they die off by any means (𝜂 < 𝑁𝜇),, this creates
an endemic situation, and this endemic situation is avoidable by controlling the infection rate of
humans and/or the destruction rates of zombies.
Although this is a model of a fictional disease,it gives realinsight on the research methods of
mathematical epidemiology. Just as the CDC uses a zombie outbreak as an analogy to other emergency
situations15
, epidemiological models of zombie outbreaks mirror very real communicable diseases. And,
while many people scoff at those preparing for a zombie pandemic, the people preparing for a zombie
pandemic are well prepared for anything nature can throw at them. Now, they have another tool to add to
their disaster kits.
“Most people don't believe something can happen until it already has. That's not stupidity or weakness,
that's just human nature.” 16
15
"Preparedness 101: Zombie Apocalypse." Centers for Disease Control and Prevention. N.p., n.d. Web. 11 Jan. 2013.
16
Brooks, Max. World War Z: An Oral History of the Zombie War. New York: Crown, 2006. Print.
Figure 1