Homeland Security What Can Mathematics Do?

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Homeland Security What Can Mathematics Do?
Fred Roberts Chair, Rutgers University Homeland
Security Research Initiative Director, DIMACS
Center
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• Dealing with terrorism requires detailed planning
  of preventive measures and responses.
• Both require precise reasoning and extensive
  analysis.

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• Experimentation or field trials are often
  prohibitively expensive or unethical and do not
  always lead to fundamental understanding.
• Therefore, mathematical modeling becomes an
  important experimental and analytical tool.

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• Mathematical models have become important tools
  in preparing plans for defense against terrorist
  attacks, especially when combined with powerful,
  modern computer methods for analyzing and/or
  simulating the models.

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What Can Math Models Do For Us?
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What Can Math Models Do For Us?

• Sharpen our understanding of fundamental
  processes
• Compare alternative policies and interventions
• Help make decisions.
• Prepare responses to terrorist attacks.
• Provide a guide for training exercises and
  scenario development.
• Guide risk assessment.
• Predict future trends.

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OUTLINE

• Examples of Homeland Security Research at Rutgers
  that Use Mathematics
• Examples of Research Projects I am Involved in
• One Example in Detail

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OUTLINE

• Examples of Homeland Security Research at Rutgers
  that Use Mathematics
• Examples of Research Projects I am Involved in
• One Example in Detail

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TRANSPORTATION AND BORDER SECURITY

• Pattern recognition for machine-assisted baggage
  searches
• The Math Linear algebra Pattern defined as
  a vector
• Border security decision support software
• The Math Computer models

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TRANSPORTATION AND BORDER SECURITY

• Statistical analysis of flight/aircraft
  inspections
• The Math Statistics
• Port-of-entry inspection algorithms
• The Math Statistics combinatorial
  optimization

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TRANSPORTATION AND BORDER SECURITY

• Vessel tracking for homeland defense
• The Math
• geometry calculus

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COMMUNICATION SECURITY

• Resource-efficient security protocols for
  providing data confidentiality and authentication
  in cellular, ad hoc, and wireless local area
  networks
• The Math
• Network Analysis
• Number theory Cryptography

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COMMUNICATION SECURITY

• Exploiting analogies between computer viruses and
  biological viruses
• The Math Differential equations, dynamical
  systems

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COMMUNICATION SECURITY

• Information privacy
• Identity theft
• Privacy of health care data
• The Math
• Number theory (cryptography),
• Statistics

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FOOD AND WATER SUPPLY SECURITY

• Using economic weapons to protect against
  agroterrorism
• The Math
• Game Theory
• Optimization

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SURVEILLANCE/DETECTION

• Detecting a bioterrorist attack using syndromic
  surveillance
• The Math
• Statistics, Data Mining, Discrete Math

Anthrax bacillus
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SURVEILLANCE/DETECTION

• Weapons detection and identification (dirty
  bombs, plastic explosives)
• The Math
• Linear algebra,
• Statistics,
• Data Mining (computer science)

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SURVEILLANCE/DETECTION

• Biometrics
• Face, gait, voice, iris recognition
• Non-verbal behavior detection (lying or telling
  the truth?) (applications to interrogation)
• The Math
• Optimization, linear algebra, statistics

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RESPONDING TO AN ATTACK

• Exposure/Toxicology
• Modeling dose received
• Rapid risk and exposure characterization
• The Math
• Differential Equations, Probability

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RESPONDING TO AN ATTACK

• Simulating evacuation of complex transportation
  facilities
• The Math
• Computer simulation

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RESPONDING TO AN ATTACK

• Emergency Communications
• Rapid networking at emergency locations
• Rapid telecollaboration
• The Math
• discrete math, network analysis

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OUTLINE

• Examples of Homeland Security Research at Rutgers
  that Use Mathematics
• Examples of Research Projects I am Involved in
• One Example in Detail

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The Bioterrorism Sensor Location Problem
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• Early warning is critical
• This is a crucial factor underlying governments
  plans to place networks of sensors/detectors to
  warn of a bioterrorist attack

The BASIS System
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Two Fundamental Problems

• Sensor Location Problem (SLP)
• Choose an appropriate mix of sensors
• decide where to locate them for best protection
  and early warning

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Two Fundamental Problems

• Pattern Interpretation Problem (PIP) When
  sensors set off an alarm, help public health
  decision makers decide
• Has an attack taken place?
• What additional monitoring is needed?
• What was its extent and location?
• What is an appropriate response?

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The Sensor Location Problem Algorithmic Tools
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Algorithmic Approaches I Greedy Algorithms
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Greedy Algorithms

• Find the most important location first and locate
  a sensor there.
• Find second-most important location.
• Etc.
• Builds on earlier work at Institute for Defense
  Analyses (Grotte, Platt)
• Steepest ascent approach.
• No guarantee of optimality.
• In practice, gets pretty close to optimal
  solution.

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Algorithmic Approaches II Variants of Classic
Facility Location Theory Methods
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Location Theory

• Where to locate facilities to best serve users
• Often deal with a network with vertices, edges,
  and distances along edges
• Users u1, u2, , un located at vertices
• One approach locate the facility at vertex x
  chosen so that
• is minimized.

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Location Theory
1s represent distances along edges
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u1
u2
u3
xa ?d(x,ui)1124 xb ?d(x,ui)2013 xc ?
d(x,ui)3104 xd ?d(x,ui)2215 xe ?d(x,ui
)1326 xf ?d(x,ui)0235 xb is optimal
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Algorithmic Approaches II Variants of Classic
Location Theory Methods Complications

• We dont have a network with vertices and edges
  we have points in a city
• Sensors can only be at certain locations (size,
  weight, power source, hiding place)
• We need to place more than one sensor
• Instead of users, we have places where
  potential attacks take place.
• Potential attacks take place with certain
  probabilities.
• Wind, buildings, mountains, etc. add
  complications.

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The Pattern Interpretation Problem
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The Pattern Interpretation Problem

• It will be up to the Decision Maker to decide how
  to respond to an alarm from the sensor network.

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Approaching the PIP Minimizing False Alarms
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Approaching the PIP Minimizing False Alarms

• One approach Redundancy. Require two or more
  sensors to make a detection before an alarm is
  considered confirmed
• Require same sensor to register two alarms
  Portal Shield requires two positives for the same
  agent during a specific time period.

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Approaching the PIP Minimizing False Alarms

• Redundancy II Place two or more sensors at or
  near the same location. Require two proximate
  sensors to give off an alarm before we consider
  it confirmed.
• Redundancy drawbacks cost, delay in confirming
  an alarm.

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Approaching the PIP Using Decision Rules

• Existing sensors come with a sensitivity level
  specified and sound an alarm when the number of
  particles collected is sufficiently high above
  threshold.

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Approaching the PIP Using Decision Rules

• Let f(x) number of particles collected at
  sensor x in the past 24 hours. Sound an alarm if
  f(x) gt T.
• Alternative decision rule alarm if two sensors
  reach 90 of threshold, three reach 75 of
  threshold, etc.
• Alarm if
• f(x) gt T for some x,
• or if f(x1) gt .9T and f(x2) gt .9T for some
  x1,x2,
• or if f(x1) gt .75T and f(x2) gt .75T and
  f(x3) gt .75T for some x1,x2,x3.

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Monitoring Message Streams Algorithmic Methods
for Automatic Processing of Messages
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Objective
Monitor huge communication streams, in
particular, streams of textualized communication,
to automatically detect pattern changes and
"significant" events
Motivation monitoring email traffic, news,
communiques, faxes, voice intercepts (with speech
recogntion)
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Technical Approaches

• Given stream of text in any language.
• Decide whether "events" are present in the flow
  of messages.
• Event new topic or topic with unusual level of
  activity.
• Initial Problem Retrospective or Supervised
  Event Identification Classification into
  pre-existing classes. Given example messages on
  events/topics of interest, algorithm detects
  instances in the stream.

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• More Complex Problem Prospective Detection or
  Unsupervised Filtering
• Classes change - new classes or change meaning
• A difficult problem in statistics
• Recent new C.S. approaches
• Semi-supervised Learning
• Algorithm suggests a possible new event/topic
• Human analyst labels it determines its
  significance

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The Approach Bag of Words

• List all the words of interest that may arise in
  the messages being studied w1, w2,,wn
• Bag of words vector b has k as the ith entry if
  word wi appears k times in the message.
• Sometimes, use bag of bits Vector of 0s and
  1s count 1 if word wi appears in the message, 0
  otherwise.

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The Approach Bag of Words

• Key idea how close are two such vectors?
• Known messages have been classified into
  different groups group 1, group 2,
• A message comes in. Which group should we put it
  in? Or is it new?
• You look at the bag of words vector associated
  with the incoming message and see if fits
  closely to typical vectors associated with a
  given group.

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The Approach Bag of Words

• Your performance can improve over time.
• You learn how to classify better.
• Typically you do this automatically and try to
  program a machine to learn from past data.

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Bag of Words Example

• Words
• w1 bomb, w2 attack, w3 strike
• w4 train, w5 plane, w6 subway
• w7 New York, w8 Los Angeles, w9 Madrid, w10
  Tokyo, w11 London
• w12 January, w13 March

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Bag of Words

• Message 1
• Strike Madrid trains on March 1.
• Strike Tokyo subway on March 2.
• Strike New York trains on March 11.
• Bag of words b1 (0,0,3,2,0,1,1,0,1,1,0,0,3)
• w1 bomb, w2 attack, w3 strike
• w4 train, w5 plane, w6 subway
• w7 New York, w8 Los Angeles, w9 Madrid, w10
  Tokyo, w11 London
• w12 January, w13 March

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Bag of Words

• Message 2
• Bomb Madrid trains on March 1.
• Attack Tokyo subway on March 2.
• Strike New York trains on March 11.
• Bag of words b2 (1,1,1,2,0,1,1,0,1,1,0,0,3)
• w1 bomb, w2 attack, w3 strike
• w4 train, w5 plane, w6 subway
• w7 New York, w8 Los Angeles, w9 Madrid, w10
  Tokyo, w11 London
• w12 January, w13 March

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Bag of Words

• Note that b1 and b2 are close
• b1 (0,0,3,2,0,1,1,0,1,1,0,0,3)
• b2 (1,1,1,2,0,1,1,0,1,1,0,0,3)
• Close could be measured using distance d(b1,b2)
  number of places where b1,b2 differ (Hamming
  distance between vectors).
• Here d(b1,b2) 3
• The messages are similar could belong to the
  same class of message.

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Bag of Words

• Message 3
• Go on on strike against Madrid trains on March 1.
• Go on strike against Tokyo subway on March 2.
• Go on strike against New York trains on March 11.
• Bag of words b3 same as b1.
• BUT message 3 is quite different from message
  1.
• Shows trickiness of problem. Maybe missing some
  key words like go or maybe we should use pairs
  of words like on strike (bigrams)

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Streaming Data

• We often have just one shot at the data as it
  comes streaming by because there is so much of
  it. This calls for powerful new algorithms.

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OUTLINE

• Examples of Homeland Security Research at Rutgers
  that Use Mathematics
• Examples of Research Projects I am Involved in
• One Example in Detail

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Mathematics and Bioterrorism Graph-theoretical
Models of Spread and Control of Disease
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Mathematics and Bioterrorism Graph-theoretical
Models of Spread and Control of Disease
Warning Next Few Slides Contain Graphic Material
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• Great concern about the deliberate introduction
  of diseases by bioterrorists has led to new
  challenges for mathematical scientists.
• 
  

smallpox
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• I got involved right after September 11 and the
  anthrax attacks.
• 
  

anthrax
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• Bioterrorism issues are typical of many homeland
  security issues.
• The rest of this talk will emphasize
  bioterrorism, but many of the messages apply to
  homeland security in general.

Waiting on line to get smallpox vaccine during
New York City smallpox epidemic 1947
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Models of the Spread and Control of Disease
through Social Networks

• Diseases are spread through social networks.
• This is especially relevant to sexually
  transmitted diseases such as AIDS.
• Contact tracing is an important part of any
  strategy to combat outbreaks of diseases such as
  smallpox, whether naturally occurring or
  resulting from bioterrorist attacks.

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The Basic Model
Social Network Graph Vertices People Edges
contact State of a Vertex simplest model 1
if infected, 0 if not infected (SI Model) More
complex models SI, SEI, SEIR, etc. S
susceptible, E exposed, I infected, R
recovered (or removed)
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Example of a Social Network
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More About States
Once you are infected, can you be cured? If you
are cured, do you become immune or can you
re-enter the infected state? We can build a
directed graph reflecting the possible ways to
move from state to state in the model.
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The State Diagram for a Smallpox Model
The following diagram is from a Kaplan-Craft-Wein
(2002) model for comparing alternative responses
to a smallpox attack. This has been considered by
the Centers for Disease Control (CDC) and Office
of Emergency Preparedness in Dept. of Health and
Human Services.
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The Stages

• Row 1 Untraced and in various stages of
  susceptibility or infectiousness.
• Row 2 Traced and in various stages of the queue
  for vaccination.
• Row 3 Unsuccessfully vaccinated and in various
  stages of infectiousness.
• Row 4 Successfully vaccinated dead

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Moving From State to State
Let si(t) give the state of vertex i at time
t. Two states 0 and 1. Times are discrete t
0, 1, 2,
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Threshold Processes
Basic k-Threshold Process You change your state
at time t1 if at least k of your neighbors
have the opposite state at time t. Disease
interpretation? Cure if sufficiently many of your
neighbors are uninfected. Does this make sense?
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Threshold Processes II
Irreversible k-Threshold Process You change
your state from 0 to 1 at time t1 if at
least k of your neighbors have state 1 at
time t. You never leave state 1. Disease
interpretation? Infected if sufficiently many of
your neighbors are infected. Special Case k
1 Infected if any of your neighbors is
infected.
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Basic 2-Threshold Process
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Irreversible 2-Threshold Process
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Complications to Add to Model

• k 1, but you only get infected with a certain
  probability.
• You are automatically cured after you are in the
  infected state for d time periods.
• You become immune from infection (cant re-enter
  state 1) once you enter and leave state 1.
• A public health authority has the ability to
  vaccinate a certain number of vertices, making
  them immune from infection.

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Periodicity
State vector s(t) (s1(t), s2(t), ,
sn(t)). First example, s(1) s(3) s(5)
, s(0) s(2) s(4) s(6) Second example
s(1) s(2) s(3) ... In all of these
processes, because there is a finite set of
vertices, for any initial state vector s(0),
the state vector will eventually become periodic,
i.e., for some P and T, s(tP) s(t) for
all t gt T. The smallest such P is called the
period.
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Periodicity II
First example the period is 2. Second example
the period is 1. Both basic and irreversible
threshold processes are special cases of
symmetric synchronous neural networks. Theorem
(Goles and Olivos, Poljak and Sura) For
symmetric, synchronous neural networks, the
period is either 1 or 2.
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Periodicity III
When period is 1, we call the ultimate state
vector a fixed point. When the fixed point is
the vector s(t) (1,1,,1) or (0,0,,0), we
talk about a final common state. One problem of
interest Given a graph, what subsets S of the
vertices can force one of our processes to a
final common state with entries equal to the
state shared by all the vertices in S in the
initial state?
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Periodicity IV
Interpretation Given a graph, what subsets S
of the vertices should we plant a disease with so
that ultimately everyone will get it? (s(t) ?
(1,1,,1)) Economic interpretation What set of
people do we place a new product with to
guarantee saturation of the product in the
population? Interpretation Given a graph, what
subsets S of the vertices should we vaccinate
to guarantee that ultimately everyone will end up
without the disease? (s(t) ? 0,0,,0))
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Conversion Sets
Conversion set Subset S of the vertices that
can force a k-threshold process to a final common
state with entries equal to the state shared by
all the vertices in S in the initial state. (In
other words, if all vertices of S start in same
state x 1 or 0, then the process goes to a
state where all vertices are in state
x.) Irreversible k-conversion set if
irreversible process.
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1-Conversion Sets
k 1. What are the conversion sets in a basic
1-threshold process?
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1-Conversion Sets
k 1. The only conversion set in a basic
1-threshold process is the set of all vertices.
For, if any two adjacent vertices have 0 and 1 in
the initial state, then they keep switching
between 0 and 1 forever. What are the
irreversible 1-conversion sets?
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Irreversible 1-Conversion Sets
k 1. Every single vertex x is an
irreversible 1-conversion set if the graph is
connected. We make it 1 and eventually all
vertices become 1 by following paths from x.
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Conversion Sets for Odd Cycles
C2p1 2-threshold process. What is a conversion
set?
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Conversion Sets for Odd Cycles
C2p1. 2-threshold process. Place p1 1s in
alternating positions.
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Conversion Sets for Odd Cycles
We have to be careful where we put the initial
1s. p1 1s do not suffice if they are next to
each other.
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Irreversible Conversion Sets for Odd Cycles
What if we want an irreversible conversion set
under an irreversible 2-threshold process? Same
set of p1 vertices is an irreversible
conversion set. Moreover, everyone gets infected
in one step.
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Vaccination Strategies
If you didnt know whom a bioterrorist might
infect, what people would you vaccinate to be
sure that a disease doesnt spread very much?
(Vaccinated vertices stay at state 0 regardless
of the state of their neighbors.) Try odd cycles
again. Consider an irreversible 2-threshold
process. Suppose your adversary has enough supply
to infect two individuals. Strategy 1 Mass
vaccination make everyone 0 and immune in
initial state.
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Vaccination Strategies
In C5, mass vaccination means vaccinate 5
vertices. This obviously works. In practice,
vaccination is only effective with a certain
probability, so results could be different. Can
we do better than mass vaccination? What does
better mean? If vaccine has no cost and is
unlimited and has no side effects, of course we
use mass vaccination.
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Vaccination Strategies
What if vaccine is in limited supply? Suppose we
only have enough vaccine to vaccinate 2
vertices. Consider two different vaccination
strategies
Vaccination Strategy I
Vaccination Strategy II
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Vaccination Strategy I Worst Case (Adversary
Infects Two)Two Strategies for Adversary
Adversary Strategy Ia
Adversary Strategy Ib
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The alternation between your choice of a
defensive strategy and your adversarys choice of
an offensive strategy suggests we consider the
problem from the point of view of game
theory.The Food and Drug Administration is
studying the use of game-theoretic models in the
defense against bioterrorism.
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Vaccination Strategy I Adversary Strategy Ia
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Vaccination Strategy I Adversary Strategy Ib
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Vaccination Strategy II Worst Case (Adversary
Infects Two)Two Strategies for Adversary
Adversary Strategy IIa
Adversary Strategy IIb
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Vaccination Strategy II Adversary Strategy IIa
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Vaccination Strategy II Adversary Strategy IIb
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Conclusions about Strategies I and II

• If you can only vaccinate two individuals
• Vaccination Strategy II never leads to more than
  two infected individuals, while Vaccination
  Strategy I sometimes leads to three infected
  individuals (depending upon strategy used by
  adversary).
• Thus, Vaccination Strategy II is better.

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k-Conversion Sets
k-conversion sets are complex. Consider the
graph K4 x K2.
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k-Conversion Sets II
Exercise (a). The vertices a, b, c, d, e form a
2-conversion set. (b). However, the vertices
a,b,c,d,e,f do not. Interpretation Immunizing
one more person can be worse! (Planting a
disease with one more person can be worse if you
want to infect everyone.) Note the same does
not hold true for irreversible k-conversion
sets.
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NP-Completeness
Problem Given a positive integer d and a graph
G, does G have a k-conversion set of size at
most d? Theorem (Dreyer 2000) This problem is
NP-complete for fixed k gt 2. (NP-complete
probably implies we will never have an efficient
computer algorithm for solving the
problem.) (Whether or not it is NP-complete for
k 2 remains open.) Same conclusions for
irreversible k-conversion set.
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k-Conversion Sets in Regular Graphs
G is r-regular if every vertex has degree
r. Set of vertices is independent if there are no
edges. Theorem (Dreyer 2000) Let G (V,E)
be a connected r-regular graph and D be a set
of vertices. (a). D is an irreversible
r-conversion set iff V-D is an independent
set. (b). D is an r-conversion set iff V-D
is an independent set and D is not an
independent set.
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k-Conversion Sets in Regular Graphs II
Corollary (Dreyer 2000) (a). The size of the
smallest irreversible 2- conversion set in Cn
is ceilingn/2. (b). The size of the smallest
2-conversion set in Cn is ceiling(n1)/2. ce
ilingx smallest integer at least as big as
x. This result agrees with our observation.
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k-Conversion Sets in Grids
Let G(m,n) be the rectangular grid graph with
m rows and n columns.
G(3,4)
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Toroidal Grids
The toroidal grid T(m,n) is obtained from the
rectangular grid G(m,n) by adding edges from
the first vertex in each row to the last and from
the first vertex in each column to the
last. Toroidal grids are easier to deal with
than rectangular grids because they form regular
graphs Every vertex has degree 4. Thus, we can
make use of the results about regular graphs.
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T(3,4)
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4-Conversion Sets in Toroidal Grids
Theorem (Dreyer 2000) In a toroidal grid T(m,n)
(a). The size of the smallest 4-conversion set
is maxn(ceilingm/2), m(ceilingn/2) m or n
odd mn/2 1 m, n even (b). The size of
the smallest irreversible 4-conversion set is as
above when m or n is odd, and it is mn/2
when m and n are even.

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Part of the Proof Recall that D is an
irreversible 4-conversion set in a 4-regular
graph iff V-D is independent. V-D
independent means that every edge u,v in G
has u or v in D. In particular, the ith row
must contain at least ceilingn/2 vertices in D
and the ith column at least ceilingm/2 vertices
in D (alternating starting with the end vertex of
the row or column). We must cover all rows and
all columns, and so need at least
maxn(ceilingm/2), m(ceilingn/2) vertices
in an irreversible 4-conversion set.
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4-Conversion Sets for Rectangular Grids
More complicated methods give Theorem (Dreyer
2000) The size of the smallest 4-conversion set
and smallest irreversible 4-conversion set in a
grid graph G(m,n) is 2m 2n - 4
floor(m-2)(n-2)/2
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4-Conversion Sets for Rectangular Grids
Consider G(3,3) 2m 2n - 4
floor(m-2)(n-2)/2 8. What is a smallest
4-conversion set and why 8?
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4-Conversion Sets for Rectangular Grids
Consider G(3,3) 2m 2n - 4
floor(m-2)(n-2)/2 8. What is a smallest
4-conversion set and why 8? All boundary
vertices have degree lt 4 and so must be included
in any 4-conversion set. They give a conversion
set.
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More Realistic Models

• Many oversimplifications. For instance
• What if you stay infected only a certain number
  of days?
• What if you are not necessarily infective for the
  first few days you are sick?
• What if your threshold k for changes from 0 to 1
  changes depending upon how long you have been
  uninfected?

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Alternative Models to Explore
Consider an irreversible process in which you
stay in the infected state (state 1) for d time
periods after entering it and then go back to the
uninfected state (state 0). Consider a
k-threshold process in which we vaccinate a
person in state 0 once k-1 neighbors are infected
(in state 1). Etc. -- let your imagination roam
free ...
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More Realistic Models

• Our models are deterministic. How do
  probabilities enter?
• What if you only get infected with a certain
  probability if you meet an infected person?
• What if vaccines only work with a certain
  probability?
• What if the amount of time you remain infective
  exhibits a probability distribution?

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Alternative Model to Explore
Consider an irreversible 1-threshold process in
which you stay infected for d time periods and
then enter the uninfected state. Assume that you
get infected with probability p if at least one
of your neighbors is infected. What is the
probability that an epidemic will end with no one
infected?
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The Case d 2, p 1/2
Consider the following initial state
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The Case d 2, p 1/2
With probability 1/2, vertex a does not get
infected at time 1. Similarly for vertex
b. Thus, with probability 1/4, we stay in the
same states at time 1.
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The Case d 2, p 1/2
Suppose vertices are still in same states at time
1 as they were at time 0. With probability 1/2,
vertex a does not get infected at time 2.
Similarly for vertex b. Also after time 1,
vertices c and d have been infected for two
time periods and thus enter the uninfected
state. Thus, with probability 1/4, we get to the
following state at time 2
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The Case d 2, p 1/2
Thus, with probability 1/4 x 1/4 1/16, we
enter this state with no one infected at time
2. However, we might enter this state at a later
time. It is not hard to show (using the theory
of finite Markov chains) that we will end in
state (0,0,0,0). (This is the only absorbing
state in an absorbing Markov chain.). Thus with
probability 1 we will eventually kill the disease
off entirely.
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The Case d 2, p 1/2
Is this realistic? What might we do to modify
the model to make it more realistic?
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How do we Analyze this or More Complex Models for
Graphs?

• Computer simulation is an important tool.
• Example At the Johns Hopkins University and the
  Brookings Institution, Donald Burke and Joshua
  Epstein have developed a simple model for a
  region with two towns totalling 800 people. It
  involves a few more probabilistic assumptions
  than ours. They use single simulations as a
  learning device. They also run large numbers of
  simulations and look at averages of outcomes.

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How do we Analyze this or More Complex Models for
Graphs?

• Burke and Epstein are using the model to do what
  if experiments
• What if we adopt a particular vaccination
  strategy?
• What happens if we try different plans for
  quarantining infectious individuals?
• There is much more analysis of a similar nature
  that can be done with graph-theoretical models.

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Would Graph Theory help with a deliberate
outbreak of Anthrax?
131

• What about a deliberate release of smallpox?

132

• Similar approaches, using mathematical models
  based on mathematical methods, have proven useful
  in many other fields, to
• make policy
• plan operations
• analyze risk
• compare interventions
• identify the cause of observed events

133

• Why shouldnt these approaches work in the
  defense against bioterrorism?

134
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