Homeland Security:
What can Mathematics
Do?
Examples from Work at
CCICADA
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Founded 2009 as a DHS University Center
of Excellence
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Why CCICADA? Methods of mathematics and
computer science have become important tools in
preparing plans for defense against terrorist attacks
or natural disasters, especially when combined with
powerful, modern computer methods for analysis
and simulation.
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Are you Serious?? What Can
Mathematics do For Us?
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5
.
After Pearl Harbor: Mathematics and
mathematicians played a vitally important role in
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the US World War II effort.
Critical War-Effort Contributions Included:
•Code breaking.
Enigma machine
•Creation of the mathematics-based field of Operations
Research:
logistics
optimal scheduling
inventory
strategic planning
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But: Terrorism is Different.
Can Math and Computer Science
Really Help?
5+2=?
1, 2, 3, …
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I’ll Illustrate with Math and
Computer Science Projects I’m
Involved in.
There are Many Others
• I. Vaccination Strategies for Control of a
Highly Infectious Disease
• II. Inspecting Containers at Ports for
Weapons of Mass Destruction
• III. Putting Nuclear Detectors in Taxicabs or
Police Cars
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• IV. Dealing with Climate Change
I. Vaccination Strategies for Control
of a Highly Infectious Disease
Smallpox
Naturally occurring
Deliberately released by
“bioterrorists”?
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The Model: Moving From
State to State
Diseases spread through
social networks
Social Network = Graph
Nodes = People
Edges = contact
t=0,1,2, …
SI model
Once in infected state, stay there.
Times are discrete: t = 0, 1, 2, …
= infected
= susceptible
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Disease Process
Highly Infectious Disease: You change your state
from to at time t+1 if at least one of your
neighbors have state at time t. You never leave
state .
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Vaccination Strategies
Let’s say you have a limited amount of vaccine
available each time period, say v doses.
Whom should you vaccinate?
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Vaccination Strategies
More precisely: What vaccination strategy minimizes
number of people ultimately infected if a disease
breaks out with one infection?
Sometimes called the firefighter problem:
alternate fire spread and firefighter placement.
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Some Results on the Firefighter
Problem
Thanks to
Kah Loon Ng
DIMACS
for some of the following slides,
slightly modified by me
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Three doses of vaccine per time period (v = 3)
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v=3
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v=3
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v=3
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v=3
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v=3
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v=3
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v=3
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Some questions that can be asked (but
not necessarily answered!)
• Can the fire be contained?
• How many time steps are required before fire is
contained?
• How many firefighters per time step are necessary?
• What fraction of all nodes will be saved (burnt)?
• Does where the fire breaks out matter?
• Fire starting at more than 1 node?
• What about other types of social networks?
• How do we construct graphs to minimize damage?
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Containing Fires in d-dimensional Grids
Fire starts at only one node:
d = 2: Impossible to contain the fire with 1
firefighter per time step
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Containing Fires in d-dimensional Grids
d = 2: Two firefighters per time step needed to contain the
fire.
8 time steps
18 burnt
nodes
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Firefighting on Trees
Epidemic starts at the root. Number doses of vaccine: v = 127
Firefighting on Trees
Greedy algorithm:
For each node x, define
weight (x) = number descendants of x + 1
Algorithm: At each time step, place
firefighter at node that has not been saved
such that weight (x) is maximized.
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Firefighting on Trees
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Firefighting on Trees:
12
8
9
2
6
1
1
3
1 1
7
5
1 3
1
6
11
1
4
1
2 1
2
3
1
1
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Firefighting on Trees
Greedy
=7
Optimal
=9
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Firefighting on Trees
Theorem (Hartnell and Li, 2000): For any tree
with one fire starting at the root and one
firefighter to be deployed per time step, the
greedy algorithm always saves more than ½ of
the nodes that any algorithm saves.
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II. Algorithms for Port of Entry
Inspection for WMDs
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Port of Entry Inspection Algorithms
•Goal: Find ways to intercept illicit
nuclear materials and weapons
destined for the U.S. via the
maritime transportation system
•Currently inspecting only small
% of containers arriving at ports
•Even inspecting 8% of containers in Port of
NY/NJ might bring international trade to a halt
•So we need faster and more efficient ways to do
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inspections.
Port of Entry Inspection Algorithms
•My work on port of entry inspection has gotten
me and my students to some remarkable places.
Me on a Coast Guard
boat in a tour of the
harbor in Philadelphia
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Sequential Decision Making Problem
•Containers arriving to be classified into categories.
•Simple case: 0 = “okay”, 1 = “suspicious”
•Inspection scheme: specifies which inspections are
to be made based on previous observations. You don’t
have to do every inspection on every container.
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Sequential Decision Making Problem
•Containers have attributes:
–Does ship’s manifest set off an “alarm”?
Yes = 1, No = 0
–What is the neutron or Gamma emission
count? Is it above threshold?
Yes = 1, No = 0
–Does a radiograph image come up positive?
Yes = 1, No = 0
–Does an induced fission test come up positive?
Yes = 1, No = 0
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Sequential Decision Making Problem
•Then: Container corresponds to a binary string (bit
string) like 011001
•This container has a “Yes” on the second, third, and
sixth attributes.
•So: Container classification takes a bit string and
decides if the container is “suspicious” (call it 1) or
“okay” (call it 0).
•A decision rule F takes a bit string and decides if it
corresponds to a suspicious or okay container.
011001
F(011001)
If attributes 2, 3, and 6 are present, assign container to
category F(011001).
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Sequential Decision Making Problem
•Given a container, test its attributes until know
enough to calculate whether it is suspicious or
okay.
•An inspection scheme tells us in which order to
test the attributes to minimize cost.
•Even this simplified problem is hard
computationally.
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Binary Decision Tree Approach
•Tests measure presence
or absence of attributes: so 0 or 1
•Classification is 1 or 0
•Binary Decision Tree:
–Nodes are tests a0, a1, etc. or
categories 1 or 0
–Two arrows (“arcs”) exit from
each test node, labeled left and
right.
–Take the right arc when test says
the attribute is present (1), left arc
otherwise
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Binary Decision Tree Approach
•Reach category 1 from the
root only through the path
a0 to a1 to 1.
•Container is classified in
category 1 iff it has both
attributes a0 and a1 .
•Corresponding Decision
Rule
• F(11) = 1, F(10) = F(01)
= F(00) = 0.
Figure 1
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Binary Decision Tree Approach
•Reach category 1 from the
root only through the path a1
to a0 to 1.
•Container is classified in
category 1 iff it has both
attributes a0 and a1 .
•Corresponding Decision Rule:
• F(11) = 1, F(10) = F(01) = F(00)
= 0.
•Note: Different tree, same
Decision Rule
Figure 2
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Binary Decision Tree Approach
•Reach category 1 from
the root by:
a0 L to a1 R a2 R 1 or
a0 R a2 R1
•Container classified in
category 1 iff it has
a1 and a2 and not a0 or
a0 and a2 and possibly a1.
•Corresponding Decision Rule:
• F(111) = F(101) = F(011) = 1,
F(abc) = 0 otherwise.
Figure 3
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Binary Decision Tree Approach
•This binary decision
tree corresponds to the
same Decision rule
F(111) = F(101) =
F(011) = 1, F(abc) = 0
otherwise.
However, it has one less
test node ai. So, it is
more efficient if all tests
are equally costly and
equally likely.
Figure 4
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Binary Decision Tree Approach
•So we have seen that a given Decision Rule may
correspond to different binary decision trees.
•How do we find a binary decision tree
corresponding to a Decision Rule?
•How do we find a least cost one?
Port of Long Beach
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Binary Decision Tree Approach
•For small n = number of attributes, can try to find
least cost binary decision tree by trying all
possible binary decision trees corresponding to the
Decision Rule F.
•Even for n = 4, not practical. (n = 4 at Port of
Long Beach-Los Angeles)
•Methods developed at CCICADA work for n up
to 20.
Port of Long Beach
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III. Nuclear Detection using Taxicabs
and/or Police Cars
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Nuclear Detection Using Vehicles
• Distribute GPS tracking and nuclear detection
devices to taxicabs or police cars in a metropolitan
area.
– Feasibility: New technologies are making devices
portable, powerful, and cheaper.
– Some police departments are already
experimenting with nuclear detectors.
• Taxicabs are a good example because their
movements are subject to considerable uncertainty –
confusing the “bad guys” as to where we are
searching.
• Send out signals if the vehicles are getting close to
nuclear sources.
• Analyze the information (both locations and nuclear
signals) to detect potential location of a source.
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Nuclear Detection Using Vehicles
•
•
•
•
Issues of Concern in our Project:
Our discussions with law enforcement suggest
reluctance to depend on the private sector (e.g.,
taxicab drivers) in surveillance
However, are there enough police cars to get
sufficient “coverage” in a region?
How many vehicles are needed for sufficient
coverage?
How does the answer depend upon:
– Routes vehicles take?
– Range of the detectors?
– False positive and false negative rates of
detectors?
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Detectors in Vehicles – Model
Components
• In our early work, we did not have a
specific model of vehicle movement.
• We assumed that vehicles are
randomly moved to new locations in
the region being monitored each time
period.
• If there are many vehicles with
sufficiently random movements, this is
a reasonable first approximation.
• It is probably ok for taxicabs, less so
for police cars.
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Vehicles – Clustering of Events
• Definition of Clusters:
– Unusually large number of events/patterns
clumping within a small region of time, space or
location in a sequence
– A cluster of alarms suggests there is a source
• Use statistical methods developed at
CCICADA to see if there is a cluster:
• Statistical methods we use are called Scan
Statistics
– Scan entire study area and seek to locate region(s)
with unusually high likelihood of events/alarms51
Nuclear Detection
using
Taxicabs
Manhattan, New York City
.
.
.
.. . .
.
.
........
.
.
.
.
.
.
.
+
GPS tracking
device
Nuclear sensor
device
. .
.
dirty
bomb?
A simulation of taxicab locations
at morning rush hour
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Number of Vehicles Needed
• The required number of vehicles in the surveillance
network can be determined by statistical power analysis
– The larger # of vehicles, the higher power of detection
• An illustrative example:
– A surveillance network covers area 4000 ft by 10000 ft
 Roughly equal to the area of the roads and
sidewalks of Mid/Downtown Manhattan
– N vehicles are randomly moving around in the area
 Fix key parameters
– Effective range of a working detector
– False positive & false negative rates for detectors
– The ranges and rates we used are not realistic, but we
wanted to test general methods, & not be tied to today’s
technology
– A fixed nuclear source randomly placed in the area
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Number of Vehicles Needed
•
•
•
•
•
First Model
Effective range of detector: 150 ft.
False positive rate 2%
False negative rate 5%
Varied number of vehicles (= number of sensors)
and ran at least 50 computer simulations for each
number of vehicles.
For each, measure the power = P(D=1/S=1) =
probability of detection of a source.
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Number of Vehicles (Sensors) Needed
• Sensor range=150 feet, false positive=2%, false negative=5%.
Detection Power
0.8
0.7
0.6
Power
0.5
0.4
0.3
0.2
0.1
0
1500
2000
2500
3000
3500
4000
Number of Sensors
Conclusion: Need 4000 vehicles to even get 75% power.
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Number of Vehicles Needed
• NYPD has 3000+ vehicles in 76 precincts in
5 boroughs. Perhaps 500 to 750 are in streets
of Mid/Downtown Manhattan at one time.
• Preliminary conclusion: The number of
police cars in Manhattan would not be
sufficient to even give 30% power.
• So, if we want to use vehicles, we need
a larger fleet, as in taxicabs.
Modified Model
• What if we have a better detector, say with
an effective range of 250 ft.?
• Don’t change assumptions about false
positive and false negative rates.
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Number of Vehicles (Sensors) Needed
•Sensor range=250 feet, false positive=2%, false negative=5%.
Detection Power
1.02
1
0.98
Power
0.96
0.94
0.92
0.9
0.88
0.86
1500
2000
2500
3000
3500
4000
Num ber of Sensors
Conclusion: 2000 vehicles already give 93% power.
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Number of Vehicles Needed
• There are not enough police cars to
accomplish this kind of coverage.
• Taxicabs could do it.
• There are other problems with our
model as it relates to police cars:
– Police cars tend to remain in their own
region/precinct.
– Police cars don’t move around as
randomly or as frequently as taxicabs
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Hybrid Model: Police Cars +
Taxicabs
• Keeping detectors with effective range of 250 ft.,
false positive and false negative rates of 2% and
5%, respectively.
• Use 500 police cars split into 25 in each of 20
regions.
• In addition, use 2000 taxicabs ranging through
the whole region.
• Now get 98% power.
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Detectors in Cell Phones
• Similar ideas for placing sensors in cell phones have
been proposed and tested by the Radiation Laboratory
at Purdue University and at Lawrence Livermore.
• At a meeting with the NYC Police Department, where
we presented our taxicab and police car work, we were
encouraged to explore applying our methods to the cell
phone idea.
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IV: Dealing with Climate Change
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Climate and Health
Concerns about global warming.
Resulting impact on health
–Of people
–Of animals
–Of plants
–Of ecosystems
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Climate and Health
•Some early warning signs:
–1995 extreme heat event in Chicago
514 heat-related deaths
3300 excess emergency admissions
–2003 heat wave in Europe
35,000 deaths
–Food spoilage on Antarctica
expeditions
Not cold enough to store
food in the ice
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Climate and Health
•Some early warning signs:
–Malaria in the African Highlands
–Dengue epidemics
–Floods, hurricanes
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Extreme Events due to Global Warming
•We anticipate an increase in number and severity of
extreme events due to global warming.
•More heat waves.
•More floods, hurricanes.
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Extreme Events due to Global Warming
Areas of Emphasis At CCICADA
•Evacuations during extreme heat events
•Rolling power blackouts during extreme heat events
•Emergency vehicle rerouting after floods
•Note: similar emphasis on “heat events” at the Centers for
Disease Control and Prevention (CDC)
•We work with the CDC and our students have interned there.
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Evacuations during Extreme Heat
Events
One response to such events: evacuation of most
vulnerable individuals to climate controlled
environments.
Mathematical challenges:
Where to locate the evacuation centers?
Whom to send where?
Goals include minimizing travel time, keeping facilities to
their maximum capacity
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Optimal Locations for Shelters in
Extreme Heat Events
• Work based in Newark NJ
• Data includes locations of potential shelters, travel
distance from each city block to potential shelters, and
population size and demographic distribution on each
city block.
• Determined “at risk” age groups and their likely levels
of healthcare needed to avoid serious problems
• Computing optimal routing plans for at-risk
population to minimize adverse health outcomes and
travel time
• Using techniques of probabilistic mixed integer
programming and aspects of location theory constrained
by shelter capacity (based on predictions of duration,68
onset time, and severity of heat events)
Location Theory
• Old problem in Operations research: Where to
locate facilities (fire houses, garbage dumps,
evacuation centers, etc.) to best serve “users”
• Often deal with a network with nodes, edges, and
distances along edges
• Users (evacuees) u1, u2, …, un are located at nodes
• One approach: locate the facility at node x chosen
so that sum of distances to users is minimized.
n
• Minimize: d(x,ui) where d(x,ui) is distance x to ui
i=1
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Location Theory
f
1
a
1
1
e
b
1
1
d
1
Nodes are places
for users/evacuees or facilities
c
1’s represent
distances along
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edges
Location Theory
f
1
a
1
1
e
b
1
1
d
1
d(x,y) = length of shortest route
from x to y
So, d(a,c) = 2.
c
1’s represent
distances along
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edges
1
f
1
a
1
u1
e
b
u2
1
1
d
1
c
u3
Given evacuees at u1, u2, u3, where do we place a
facility to minimize the sum of distances to the
people being evacuated?
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1
f
1
a
1
u1
e
b
u2
1
1
d
1
c
x=a: d(x,ui)=1+1+2=4
u3
x=b: d(x,ui)=2+0+1=3
x=c: d(x,ui)=3+1+0=4
x=d: d(x,ui)=2+2+1=5
x=e: d(x,ui)=1+3+2=6
x=f: d(x,ui)=0+2+3=5
x=b is optimal
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Mathematics, computer science, and homeland
security: What can you do to make the world a
safer place?
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