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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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After Pearl Harbor: Mathematics and mathematicians played a vitally important role in the US World War II effort.

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Critical War-Effort Contributions Included:

•Code breaking.

•Creation of the mathematics-based field of Operations Research:

logistics

optimal scheduling

inventory

strategic planning

Enigma machine

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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

• IV. Dealing with Climate Change

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I. Vaccination Strategies for Control of a Highly Infectious Disease

Naturally occurring

Smallpox

Deliberately released by“bioterrorists”?

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The Model: Moving From State to State

Diseases spread through social networksSocial Network = GraphNodes = PeopleEdges = contact

SI modelOnce in infected state, stay there.

Times are discrete: t = 0, 1, 2, …

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 toKah Loon Ng

DIMACSfor 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 = 1

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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

Firefighting on Trees:

78912 11

324161512 6

12 1131111 3 1

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22

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Firefighting on Trees

Greedy Optimal

= 7 = 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 inspections.

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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 Guardboat in a tour of theharbor 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 presenceor 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 ora0 R a2 R1

•Container classified in category 1 iff it hasa1 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 Taxicabsand/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 VehiclesIssues 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/alarms

52A simulation of taxicab locations

at morning rush hour

Manhattan, New York City

...........

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.. .

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..

.. .

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..

+GPS tracking

deviceNuclear sensor

device

dirty bomb?

Nuclear Detectionusing Taxicabs

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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

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1500 2000 2500 3000 3500 4000

Number of Sensors

Po

we

r

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

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

1.02

1500 2000 2500 3000 3500 4000

Number of Sensors

Po

wer

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 Chicago514 heat-related deaths3300 excess emergency admissions

–2003 heat wave in Europe35,000 deaths

–Food spoilage on Antarctica

expeditionsNot 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 WarmingAreas 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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• 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, onset time, and severity of heat events)

Optimal Locations for Shelters in Extreme Heat Events

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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

a

e

d c

b

f

1

1

1

1

1

1

1’s represent distances along edges

Nodes are placesfor users/evacuees or facilities

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Location Theory

a

e

d c

b

f

1

1

1

1

1

1

1’s represent distances along edges

d(x,y) = length of shortest routefrom x to ySo, d(a,c) = 2.

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a

e

d c

b

f

1

1

1

1

1

1 u1

u3

u2

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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a

e

d c

b

f

1

1

1

1

1

1 u1

u3

u2

x=a: d(x,ui)=1+1+2=4x=b: d(x,ui)=2+0+1=3

x=c: d(x,ui)=3+1+0=4x=d: d(x,ui)=2+2+1=5

x=e: d(x,ui)=1+3+2=6x=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?