When Zombies attack, or
Mathematical model of doomsday scenario.
Vira Babenko
University of Utah
October 30, 2014
Vira Babenko
When Zombies attack, or Mathematical model of doomsday sc
Background
Definition
A zombie is a reanimated human corpse that feeds on living flesh
Usually brought about through an outbreak or epidemic.
Vira Babenko
When Zombies attack, or Mathematical model of doomsday sc
History
Vodou (Afro-Carribean spiritual belief system): a dead person can
be revived by a bokor, or sorcerer. Zombies remain under the
control of the bokor since they have no will of their own.
There are several possible etymologies of the word zombie:
jumbie, which comes from the Carribean term for ghost.
zonbi, used in the Louisiana Creole or the Haitian Creole.
Vira Babenko
When Zombies attack, or Mathematical model of doomsday sc
Zombification
Davis traveled to Haiti in 1982 and, as a result of his investigations,
claimed that a living person can be turned into a zombie by two
special powders (neurotoxin and drugs) being introduced into the
blood stream. One powder contains an extremely powerful
neurotoxin that temporarily paralyzes the human nervous system
and it creates a state of hibernation. The main organs, such as the
heart and lungs, and all of the bodily functions, operate at minimal
levels during this state of hibernation. What turns these human
beings into zombies is the lack of oxygen to the brain. As a result
of this, they suffer from brain damage.
Davis, Wade, 1988 Passage of Darkness - The Ethnobiology of the
Haitian Zombie, Simon and Schuster pp. 14, 60-62.
Vira Babenko
When Zombies attack, or Mathematical model of doomsday sc
Current state of affairs
George Romero’s 1968 film “Night of the living Dead”
...
The Walking Dead series
CDC has emergency preparedness plans for the Zombie
Apocalypse.
Vira Babenko
When Zombies attack, or Mathematical model of doomsday sc
Basic SIR Model (no zombies)
Consider three basic classes (compartments):
Susceptible (S)
Infected (I)
Recovered (R)
Vira Babenko
When Zombies attack, or Mathematical model of doomsday sc
Basic Model
Consider three basic classes:
Susceptible (S)
Zombie (Z) (classical pop-culture zombie: slow moving,
cannibalistic and undead). Zombies zombify humans via
mass-action interaction.
Removed (R)
Vira Babenko
When Zombies attack, or Mathematical model of doomsday sc
Constructing the Model
Interactions between groups:
Susceptibles can become deceased through natural causes, i.e.,
non-zombie-related death (parameter δ).
The removed class consists of individuals who have died, either
through attack or natural causes.
Humans in the removed class can resurrect and become a zombie
(parameter ζ).
Susceptibles can become zombies through transmission via an
encounter with a zombie (transmission parameter β).
Zombies move to the removed class upon being “defeated”. This
can be done by removing the head or destroying the brain of the
zombie (parameter α).
the birth rate is a constant Π.
Vira Babenko
When Zombies attack, or Mathematical model of doomsday sc
Mathematical Model
Figure: The basic model
S 0 = Π − βSZ − δS
Z 0 = βSZ + ζR − αSZ
R 0 = δS + αSZ − ζR
Average member of the population makes contact sufficient to
transmit infection with βN others per unit time (N-total
population without infection. The probability that a random
contact by a zombie is made with a susceptible is S/N =⇒:
(βN)(S/N)Z = βSZ
Vira Babenko
When Zombies attack, or Mathematical model of doomsday sc
Analysis of critical points (equilibria).
The ODEs satisfy S 0 + Z 0 + R 0 = Π and hence S + Z + R → ∞ as
t → ∞, if Π 6= 0. Clearly S 9 ∞, so this results in a doomsday
scenario: an outbreak of zombies will lead to the collapse of
civilization, as large numbers of people are either zombified or
dead.
If we assume that the outbreak happens over a short timescale,
then we can ignore birth and background death rates. We set
Π = δ = 0:
−βSZ
βSZ + ζR − αSZ
αSZ − ζR
= 0
= 0
= 0
Two equilibria: (S, Z , R) = (N, 0, 0) (all humans; disease-free) and
(S, Z , R) = (0, N, 0) (all zombies; doomsday).
Vira Babenko
When Zombies attack, or Mathematical model of doomsday sc
The Jacobian matrix of the system
dS
dt
dZ
dt
dR
dt
is
= −βSZ
= βSZ + ζR − αSZ
= αSZ − ζR

−βZ
βZ − αZ
αZ
Vira Babenko
−βS
βS − αS
αS

0
ζ 
−ζ
When Zombies attack, or Mathematical model of doomsday sc
The Jacobian at the disease-free equilibrium is


0
−βS
0
J(N, 0, 0) = 0 βS − αS ζ 
0
αS
−ζ
And the characteristic equation becomes
0 = det(J − λI ) = −λ λ2 + (ζ − (β − α)N)λ − βζN
and the eigenvalues are
λ = 0,
−(ζ − (β − α)N) ±
p
(ζ − (β − α)N)2 + 4βζN
2
This will have a root for λ that is positive, and therefore the
zombie-free equilibrium is not stable.
Vira Babenko
When Zombies attack, or Mathematical model of doomsday sc
The Jacobian at the “doomsday” equilibrium

−β Z̄
0

J(0, Z̄ , 0) = β Z̄ − αZ̄ 0
αZ̄
0
is

0
ζ 
−ζ
And the characteristic equation becomes
0 = det(J − λI ) = −λ(−β Z̄ − λ)(−ζ − λ)
and the eigenvalues are
λ = 0,
λ = −β Z̄ ,
λ = −ζ.
Since all eigenvalues of the doomsday equilibrium are negative or
zero, it is asymptotically stable. It follows that, in a short
outbreak, zombies will likely infect everyone.
Vira Babenko
When Zombies attack, or Mathematical model of doomsday sc
Figure: Basic model outbreak scenario
(here α = 0.005, β = 0.0095, ζ = 0.0001, δ = 0.0001). Susceptibles are
quickly eradicated and zombies take over, infecting everyone.
Vira Babenko
When Zombies attack, or Mathematical model of doomsday sc
Model with Latent Infection
Additional assumptions:
1
Susceptibles first move to an infected class once infected and
remain there for some period of time.
2
Infected individuals can still die a “natural” death before
becoming a zombie; otherwise, they become a zombie.
Vira Babenko
When Zombies attack, or Mathematical model of doomsday sc
Model with Latent Infection
Figure: Model with Latent Infection
S0
I0
Z0
R0
=
=
=
=
Π − βSZ − δS
βSZ − ρI − δI
ρI + ζR − αSZ
δS + δI + αSZ − ζR
Vira Babenko
When Zombies attack, or Mathematical model of doomsday sc
Analysis of equilibria
If Π 6= 0, then infection overwhelms the population.
So we assume short time scale: Π = δ = 0.
In this case critical points are:
Z =0
(S, I , Z , R) = (N, 0, 0, 0)
S =0
(S, I , Z , R) = (0, 0, N, 0)
Coexistence is again not possible!
Vira Babenko
When Zombies attack, or Mathematical model of doomsday sc
Which equilibrium is stable?
The Jacobian matrix of the system
S0
I0
Z0
R0
is

=
=
=
=
−βZ
 βZ

−αZ
αZ
Π − βSZ − δS
βSZ − ρI − δI
ρI + ζR − αSZ
δS + δI + αSZ − ζR
0 −βS
−ρ βS
ρ −αS
0
αS
Vira Babenko

0
0

ζ 
−ζ
When Zombies attack, or Mathematical model of doomsday sc
Disease-free case (N, 0, 0, 0)
Consider


−λ
0
−βN
0
 0 −ρ − λ
βN
0 

det(J(N, 0, 0, 0) − λI ) = det 
 0
ρ
−αN − λ
ζ 
0
0
αN
−ζ − λ
= −λ(−λ3 − (ρ + ζ + αN)λ2 − (ραN + ρζ − ρβN)λ + ρζβN)
has positive root!!! Hence, the disease-free equilibrium is again
unstable.
Vira Babenko
When Zombies attack, or Mathematical model of doomsday sc
Doomsday case (0, 0, N, 0)
Consider


−βN − λ
0
0
0
 βN
−ρ − λ 0
0 

det(J(0, 0, N, 0) − λI ) = det 
 −αN
ρ
−λ
ζ 
αZ
0
0 −ζ − λ
= (−βN − λ)(−ρ − λ)(−λ)(−ζ − λ).
All eigenvalues are nonpositive. Hence, the doomsday equilibrium
is asymptotically stable.
Vira Babenko
When Zombies attack, or Mathematical model of doomsday sc
Figure: An outbreak with latent infection. Zombies still take over, but it
takes approximately twice as long.
Vira Babenko
When Zombies attack, or Mathematical model of doomsday sc
Model with Quarantine
Additional assumptions to consider:
1
The quarantined area only contains members of the infected or
zombie populations (entering at rates κ and σ, respectively).
2
There is a chance some members will try to escape, but any
that tried to would be killed before finding their “freedom”
(parameter γ).
3
These killed individuals enter the removed class and may later
become reanimated as “free” zombies.
Vira Babenko
When Zombies attack, or Mathematical model of doomsday sc
Model with Quarantine
S0
I0
Z0
R0
Q0
=
=
=
=
=
Π − βSZ − δS
βSZ − ρI − δI − κI
ρI + ζR − αSZ − σZ
δS + δI + αSZ − ζR + γQ
κI + σZ − γQ
Vira Babenko
When Zombies attack, or Mathematical model of doomsday sc
Figure: An outbreak with quarantine. The effect of quarantine is to
slightly delay the time of eradication of humans..
Vira Babenko
When Zombies attack, or Mathematical model of doomsday sc
Model with Treatment
Additional assumptions to consider:
1
Since we have treatment, we no longer need the quarantine.
2
The cure will allow zombies to return to their original human
form regardless of how they became zombies in the first place.
3
Any cured zombies become susceptible again; the cure does
not provide immunity.
Vira Babenko
When Zombies attack, or Mathematical model of doomsday sc
Model with Treatment
S0
I0
Z0
R0
=
=
=
=
Π − βSZ − δS + cZ
βSZ − ρI − δI
ρI + ζR − αSZ − cZ
δS + δI + αSZ − ζR
Vira Babenko
When Zombies attack, or Mathematical model of doomsday sc
Figure: An outbreak with treatment. Humans are not eradicated, but
only exist in low numbers.
Vira Babenko
When Zombies attack, or Mathematical model of doomsday sc
Impulsive Eradication Model
S0
Z0
R0
∆Z
=
=
=
=
Π − βSZ − δS, t 6= tn
βSZ + ζR − αSZ , t 6= tn
δS + αSZ − ζR, t 6= tn
−knZ , t = tn ,
where k ∈ (0, 1] is the kill ratio and n denotes the number of
attacks required until kn > 1
Vira Babenko
When Zombies attack, or Mathematical model of doomsday sc
Vira Babenko
When Zombies attack, or Mathematical model of doomsday sc
Discussion
An outbreak of zombies infecting humans is likely to be disastrous,
unless extremely aggressive tactics are employed against the
undead. While aggressive quarantine may eradicate the infection,
this is unlikely to happen in practice. A cure would only result in
some humans surviving the outbreak, although they will still
coexist with zombies. Only sufficiently frequent attacks, with
increasing force, will result in eradication, assuming the available
resources can be mustered in time.
Furthermore, these results assumed that the timescale of the
outbreak was short, so that the natural birth and death rates could
be ignored. If the timescale of the outbreak increases, then the
result is the doomsday scenario: an outbreak of zombies will result
in the collapse of civilization, with every human infected, or dead.
This is because human births and deaths will provide the undead
with a limitless supply of new bodies to infect, resurrect and
convert. Thus, if zombies arrive, we must act quickly and
decisively to eradicate them before they eradicate us.
Vira Babenko
When Zombies attack, or Mathematical model of doomsday sc
References
“WHEN ZOMBIES ATTACK!: MATHEMATICAL
MODELING OF AN OUTBREAK OF ZOMBIE INFECTION”
by Munz, Hudea and Smith in Infectious Disease Modeling
Research Progress, Editors: J.M. Tchuenche and C. Chiyaka,
2009, pp. 133–150.
Vira Babenko
When Zombies attack, or Mathematical model of doomsday sc