P R E S E N T A T I O N

B Y

A A R O N M O S H E R

Progressive Temporal Models for a Zombie Apocalypse

Topic Overview

Imagine that you are the CDC faced with the possibility of an impending epidemic

What do you want to know? How fast will the disease spread?

How many people will it affect?

What is the best response?

What you need: A mathematical description of the disease, the population, and

the general environment.

Tools to analyze the results

Abstract

Define basic assumptions and conditions What is a Zombie? What can be assumed about the population? What can be assumed about the location?

Propose a series of progressive ODE models by gradually loosening conditions

Basic SI model to cumulative SIMBER model

Test and compare models with numerical and empirical results

Zombie tag trial Numerical methods Linearization

Present and interpret results If time, discuss a PDE model using chemo-

taxis/diffusion to model decision factors

D E F I N I T I O N O F A Z O M B I E

P O P U L A T I O N A S S U M P T I O N S

S P A T I A L C O N D I T I O N S

Base assumptions and conditions

Concerning Zombies

The zombie myth originated as part of the Haitian voodoo legend

Evil sorcerers were thought to possess the power to revive the recently dead as mindless servants

Became a popular part of American film culture Night of the Living Dead (1968)

Dawn of the Dead (1978), remake in 2004

28 Days Later (2002)

Traditionally thought to be the result of curses/magic. Modern films often depict zombies as the result of a biological agent (terrestrial or otherwise)

For the purposes of modeling, these are equivalent

Definition

In the following models, we define a zombie as any infected or cursed subject that exhibits the following qualities:

A loss of higher order reasoning functions

An amplified hunger/survival response

Is capable of spreading the infection/curse via direct contact (i.e. a bite)

Remains mobile, and is capable of spreading infection indefinitely unless removed by an external source

Note: By the above definition, zombies will actively work to spread infection.

This differs greatly from other diseases

Other Conditions

Like the above films, the model will focus on a single isolated population within a relatively short time span

Ignore global mass transit, and assume that there is zero flux in or out of the defined area

Assume that natural births/deaths have a negligible impact on the overall population (i.e. total population is constant)

Assume that the population is uniformly mixed Every individual has an equal chance of encountering any other

individual

Some conditions may be loosened in later models

G E N E R A L D E R I V A T I O N S

B A S E SI

C U M U L A T I V E SIMBER

Ordinary Differential Models

Model Foundations

We propose that the rate of new infections is proportional to the rate of contact between subjects who can catch the disease (susceptibles) and subjects that can transmit the disease (infectives)

Propose that the rate of contact, is proportional to the probability of contact between susceptibles and infectives

Rate of infection = c * P( contact ) for some constant c

Suppose we have a constant, isolated, uniformly mixed population P that has N subjects.

We partition P into distinct classes P1,P2,P3,…,Pn each with N1,N2,N3,…,Nn subjects.

(continued)

Because P is uniformly mixed, we know that the probability that any individual will encounter a subject from class Pi is

Thus, the probability that a subject from class Pi will contact a subject from class Pj is

1N

N i

2)1(

N

NN ji

Assumptions of SI model

We consider a population of N subjects that is split into two classes S is the number of people who can become zombies (susceptibles)

I is the number of zombies (infectives)

We know that if I = 0, there will be new infections. Similarly, if S = 0 there cannot be any new infections

Assume that zombies are immortal and indestructible

Assume there is no delay, or incubation time, until zombies become infective

Graphically we have

Infectives ISusceptibles

SBitten by zombies

Base SI Model

b is a positive, real valued constant

The system can be solved by a separation of variables

SI Model:

Analytic Solution:

SIN

b

dt

dI

SIN

b

dt

dS

ISN

2

2

)1(

)1(

t

t

Ce

NNtS

Ce

NtI

1)(

1)(

2

0

)1(

N

bN

I

INC

o

Discussion of SI model

Population is conserved (dN/dt = 0 )

Notice that the constant b has units of subjects/second

Can be though of as a conditional rate

Has many „nice‟ properties Has an analytic solution

Represents a worst case scenario that always results in epidemic

Introduction of B

Suppose it is found that after a susceptible is bitten, it takes some average time T before they become fully infective.

We split the population N into 3 classes: Susceptibles S

Infectives I

Bitten B

Let v = 1/T, then an average of vB exposed subjects will become infective per unit time

Graphically

Bitten B Infectives ISusceptibles

SBecome infective

after time TBitten by Zombies

SIB Model

b, v are positive real valued constants

vBdt

dI

vBSIN

b

dt

dB

SIN

b

dt

dS

BISN

2

2

)1(

)1(

Introduction of M and R

Next, we allow zombies to be removed/quarantined Perhaps due to a trained responder class M

Susceptibles are able to fight back

Like infection, we propose that the rate of removal will be proportional to the probability of contact

M may be more resistant and/or efficient than S

Susceptibles S

Removed R

Bitten B Infectives I

Medical M

SIMBR Model

bs, bm are the infection constants for the susceptibles and responders respectively

rmi is the removal constant of I by M. Similarly for rmb, rsi, rsb

*Note: With a certain selection of constants, this model can be reduced to the SIR model discussed in [3]

MIN

rSI

N

rMB

N

rSB

N

r

dt

dR

MIN

rSI

N

rvB

dt

dI

vBMBN

rSB

N

rMI

N

bSI

N

b

dt

dB

MIN

b

dt

dM

SIN

b

dt

dS

misimbsb

misi

mbsbms

m

s

2222

22

2222

2

2

)1()1()1()1(

)1()1(

)1()1()1()1(

)1(

)1(

RBMISN

Introduction of E

Logic would dictate that, given the option, people would attempt to evacuate or find shelter

Assume that such shelter can only be provided by trained responders

The rate of evacuation will be proportional to the rate of contact between S and M

Susceptibles S

Removed R

Bitten B Infectives I

Medical M

Evacuated E

SIMBER Model

e is the conditional rate of evacuation by class M

MIN

rSI

N

rMB

N

rSB

N

r

dt

dR

SMN

e

dt

dE

MIN

rSI

N

rvB

dt

dI

vBMBN

rSB

N

rMI

N

bSI

N

b

dt

dB

MIN

b

dt

dM

SMN

eSI

N

b

dt

dS

misimbsb

misi

mbsbms

m

s

2222

2

22

2222

2

22

)1()1()1()1(

)1(

)1()1(

)1()1()1()1(

)1(

)1()1(

Loosening Conditions

What if the population isn‟t constant or isolated? Notice that evacuated or removed subjects no longer influence the population

What if there is a group of K responders outside the normal population that can be deployed in the case of an emergency

We define a variable N‟ as the current number of active subjects

We propose that, up to the maximum capacity K, responders will be deployed at a rate proportional to the severity of infection

e.g. the number of infected cases I

BMISN '

IK

Md

dt

dM

1

Cumulative SIMBER Model

Adjusts for inactive portions of the population, and allows for a deployment of trained personnel

MIN

rSI

N

rMB

N

rSB

N

r

dt

dR

SMN

e

dt

dE

MIN

rSI

N

rvB

dt

dI

vBMBN

rSB

N

rMI

N

bSI

N

b

dt

dB

MIN

bI

K

Md

dt

dM

SMN

eSI

N

b

dt

dS

misimbsb

misi

mbsbms

m

s

2222

2

22

2222

2

22

)1'()1'()1'()1'(

)1'(

)1'()1'(

)1()1'()1'()1'(

)1'(1

)1'()1'(

M O D E L V A L I D A T I O N

L I N E A R I Z A T I O N A N D B A S I C R E P R O D U C T I O N N U M B E R

D E V E L O P M E N T O F C O N T R O L P R O C E D U R E S

Analysis of Ordinary Models

Model Verification

To test the validity of the model, we would like to compare results to real world data

Due to the nature of the disease, there is no readily available statistical data for comparison

Simple SI/SIB model can be approximated by a modified version of tag

Trial of 9 subjects, with one initial „it‟

Every person tagged becomes an „it‟ and remains infected until the end of trial

Subjects exhibited an incubation time on the order of 1 second

Experimental SI Trial 1

For Trial 1, b was found to be ~.428, and v was assumed to be ~1

0

1

2

3

4

5

6

7

8

9

10

0 10 20 30 40 50 60 70

time (s)

Infected persons I(t)

Trial

I Analytic

I Numeric SIB

Experimental SI Trial 2

For Trail 2, b was found to be ~.220, v~1

0

1

2

3

4

5

6

7

8

9

10

0 20 40 60 80 100 120 140

time (s)

Infected Persons I(t)

Trial 2

Analytic

SIB Numeric

Experimental SI Trail 3

For Trial 3, be was found to be ~.785, v~1 0

1

2

3

4

5

6

7

8

9

10

0 5 10 15 20 25 30 35

Infected Persons I(t)

Trial 3

Analytic

Numeric SIB

Experimental SI Trial 4

For Trial 4, b was found to be ~.410, v~1

0

1

2

3

4

5

6

7

8

9

10

0 10 20 30 40 50 60 70 80

Infected Persons I(t)

Trial 4

Analytic SI

Numeric SIB

Discussion of Experimental Trials

Possible Sources of Error: Small population, so infection over time is a discrete function

Measurement error

Wide variation of movement speed and fitness levels

The effect of incubation time is more pronounced within smaller time intervals

Significant for P-value of .001 based on an F test results greater than 80.

Computation of R0

“The basic reproduction number…R0 is defined as the number of secondary cases generated by a primary infectious case…in an entirely susceptible population” [5]

Clearly, if R0>1 then an infection will spread, and if R0<1 an infection will die out

R0 can be computed from a linear approximation of a system about some equilibrium point

Linearization of SIMBR

The first step is to find an equilibrium point such that all derivatives are zero

For SIMBR: Let X=[ S, I, M, B, R ]T, then X*=[ S0, 0, M0, 0, 0 ]T

Now, consider only the compartments where the infection is in active progression (B and I)

Define Fj as the rate of appearance of new infection is compartment j

Define Vj as the rate of transfer in/out of compartment j by all other means

• Vj+ is the rate at which individuals enter

• Vj- is the rate at which individuals leave

Then, for any compartment j

jjjjj

jVFVVF

dt

dX

)(

(continued)

Clearly then, for the SIMBR model we have

Now we use the Jacobian of F and V about X*

vB

N

MIbSIb

XFms

2)1()(

2

2

)1(

)1()(

N

MIrSIr

vBN

MBrSBr

XVmisi

mbsb

0

)1(0

)( 2

00*

v

N

MbSb

XJms

F

2

00

2

00

*

)1(0

0)1(

)(

N

MrSr

vN

MrSr

XJmisi

mbsb

V

Theorem

From [5] we have that R0=ρ(JFJV-1), where ρ(A)

denotes the maximum eigenvalue of a matrix A

Doing the computations we find .

It can then be shown the system is locally asymptotically stable if R0<1, and locally asymptotically unstable if R0>1.

))1()((

)()1(2

0000

00

2

0

NvMrSrMrSr

MbSbNvR

mbsbmisi

ms

R0= .75 < 1 R0 = 1.25 > 1

Model Stability

Reminder of Important Figures

SI Model:

SIMBER Model:

R0:

t

t

Ce

NNtS

Ce

NtI

1)(

1)(

))1()((

)()1(2

0000

00

2

0

NvMrSrMrSr

MbSbNvR

mbsbmisi

ms

MIN

rSI

N

rMB

N

rSB

N

r

dt

dR

SMN

e

dt

dE

MIN

rSI

N

rvB

dt

dI

vBMBN

rSB

N

rMI

N

bSI

N

b

dt

dB

MIN

bI

K

Md

dt

dM

SMN

eSI

N

b

dt

dS

misimbsb

misi

mbsbms

m

s

2222

2

22

2222

2

22

)1'()1'()1'()1'(

)1'(

)1'()1'(

)1()1'()1'()1'(

)1'(1

)1'()1'(

SIN

b

dt

dI

SIN

b

dt

dS

2

2

)1(

)1(

Development of Control Procedures

Remember that the SI model was said to have „nice‟ properties Existence of an analytic solution

Notice that for a given set of constants and initial conditions, dI/dt|(SI) ≥ dI/dt|(SIMBER).

Thus a solution to the SIMBER model will be bounded above by the analytic solution to the SI model

From here, one can predict the magnitude of response necessary to contain infection

(1) Statistically determine model constants (2) Use the Analytic SI solution to predict the severity of an infection at a

specified response time (3) Use the expression for R0 to solve for the number of

responders, M, necessary to contain a zombie infection

Pause for Questions

Concluding Remarks

Results Developed several models to fit a wide range of assumptions Analyzed models through linearization to compute the basic reproduction number

R0. Developed a series of Partial differential models for further study

Other applications of Zombie Models With a careful selection of constants, can be used to model the spread of

gossip/misinformation

Topics of further study: Solution and/or analysis of Partial Differential models Greater exploration of Numerical methods

• Stability• Efficiency• Non regular boundary conditions

Questions?

Acknoledgements

Dr. Jessica Sklar: Capstone Professor

Dr. Bryan Dorner: References to the heat equation

Dr. Mei Zhu: Capstone go-to advisor

Bibliography

[1] Burden, Richard L, Numerical Analysis, 8th edition, pgs 687-736, Thomson Brooks/Cole (2005)

[2] Kreyszig, Erwin, Advanced Engineering Mathematics, 7th edition, pgs 1075-1082, John Wiley & Sons, Inc, New York, (1993)

[3] Murray, J.D, Mathematical Biology 2nd edition, pgs 236-249, 610-617, 650-655, Springer (1989)

[4] Weinberger, Hans F, A First Course in Partial Differential Equations, 1st

edition, pgs 58-60, Blaisdell Publishing, New York (1965)

[5] Y. Dumont, F. Chiroleu, C. Domerg, "On a temporal model for the Chikun-gunya disease: Modeling theroy, and numerics", Mathematical Bio-sciences, 213 (2008): 80-91