1. 1 Homeland Security: What can Mathematics Do? Examples from Work at CCICADA

2. 2 Founded 2009 as a DHS University Center of Excellence

3. 3 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.

4. 4 Are you Serious?? What Can Mathematics do For Us?

5. 5

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

7. 7 Critical War-Effort Contributions Included: Code breaking. Creation of the mathematics-based field of Operations Research: logistics optimal scheduling inventory strategic planning Enigma machine

8. 8 But: Terrorism is Different. Can Math and Computer Science Really Help? 5 + 2 = ? 1, 2, 3, …

9. 9 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

10. 10 I. Vaccination Strategies for Control of a Highly Infectious Disease Naturally occurring Smallpox Deliberately released by “bioterrorists”?

11. 11 The Model: Moving From State to State Diseases spread through social networks Social Network = Graph Nodes = People Edges = contact SI model Once in infected state, stay there. Times are discrete: t = 0, 1, 2, … t=0,1,2, … = infected = susceptible

12. 12 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.

13. 13 Vaccination Strategies Let’s say you have a limited amount of vaccine available each time period, say v doses. Whom should you vaccinate?

14. 14 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.

15. 15 Some Results on the Firefighter Problem Thanks to Kah Loon Ng DIMACS for some of the following slides, slightly modified by me

16. 16 Three doses of vaccine per time period (v = 3)

17. 17 v = 3

18. 18 v = 3

19. 19 v = 3

20. 20 v = 3

21. 21 v = 3

22. 22 v = 3

23. 23 v = 3

24. 24 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?

25. 25 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

26. 26 Containing Fires in d-dimensional Grids d = 2: Two firefighters per time step needed to contain the fire. 8 time steps 18 burnt nodes

27. 27 Firefighting on Trees Epidemic starts at the root. Number doses of vaccine: v = 1

28. 28 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.

29. 29 Firefighting on Trees Firefighting on Trees: 7 8 9 12 11 32416151 2 6 1211 3 111131 26 22

30. 30 Firefighting on Trees GreedyOptimal = 7 = 9

31. 31 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.

32. 32 II. Algorithms for Port of Entry Inspection for WMDs

33. 33 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.

34. 34 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

35. 35

36. 36 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.

37. 37 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

38. 38 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. 011001F(011001) If attributes 2, 3, and 6 are present, assign container to category F(011001).

39. 39 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.

40. 40 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 a 0, a 1, 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

41. 41 Binary Decision Tree Approach Reach category 1 from the root only through the path a 0 to a 1 to 1. Container is classified in category 1 iff it has both attributes a 0 and a 1. Corresponding Decision Rule F(11) = 1, F(10) = F(01) = F(00) = 0. Figure 1

42. 42 Binary Decision Tree Approach Reach category 1 from the root only through the path a 1 to a 0 to 1. Container is classified in category 1 iff it has both attributes a 0 and a 1. Corresponding Decision Rule: F(11) = 1, F(10) = F(01) = F(00) = 0. Note: Different tree, same Decision Rule Figure 2

43. 43 Binary Decision Tree Approach Reach category 1 from the root by: a 0 L to a 1 R a 2 R 1 or a 0 R a 2 R1 Container classified in category 1 iff it has a 1 and a 2 and not a 0 or a 0 and a 2 and possibly a 1. Corresponding Decision Rule: F(111) = F(101) = F(011) = 1, F(abc) = 0 otherwise. Figure 3

44. 44 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 a i. So, it is more efficient if all tests are equally costly and equally likely. Figure 4

45. 45 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

46. 46 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

47. 47 III. Nuclear Detection using Taxicabs and/or Police Cars

48. 48 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.

49. 49 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 ?

50. 50 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.

51. 51 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

52. 52 A simulation of taxicab locations at morning rush hour Manhattan, New York City................................ + GPS tracking device Nuclear sensor device dirty bomb? Nuclear Detection using Taxicabs

53. 53 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

54. 54 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.

55. 55 Number of Vehicles (Sensors) Needed Sensor range=150 feet, false positive=2%, false negative=5%. Conclusion: Need 4000 vehicles to even get 75% power.

56. 56 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.

57. 57 Number of Vehicles (Sensors) Needed Sensor range=250 feet, false positive=2%, false negative=5%. Conclusion: 2000 vehicles already give 93% power.

58. 58 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

59. 59 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.

60. 60 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.

61. 61 IV: Dealing with Climate Change

62. 62 Climate and Health Concerns about global warming. Resulting impact on health –Of people –Of animals –Of plants –Of ecosystems

63. 63 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

64. 64 Climate and Health Some early warning signs: –Malaria in the African Highlands –Dengue epidemics –Floods, hurricanes

65. 65 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.

66. 66 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.

67. 67 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

68. 68 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

69. 69 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) u 1, u 2, …, u n 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,u i ) where d(x,u i ) is distance x to u i i=1

70. 70 Location Theory a e d c b f 1 1 1 1 1 1 1’s represent distances along edges Nodes are places for users/evacuees or facilities

71. 71 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 route from x to y So, d(a,c) = 2.

72. 72 a e d c b f 1 1 1 1 1 1 u1u1 u3u3 u2u2 Given evacuees at u 1, u 2, u 3, where do we place a facility to minimize the sum of distances to the people being evacuated?

73. 73 a e d c b f 1 1 1 1 1 1 u1u1 u3u3 u2u2 x=a:  d(x,u i )=1+1+2=4x=b:  d(x,u i )=2+0+1=3 x=c:  d(x,u i )=3+1+0=4x=d:  d(x,u i )=2+2+1=5 x=e:  d(x,u i )=1+3+2=6x=f:  d(x,u i )=0+2+3=5 x=b is optimal

74. 74 Mathematics, computer science, and homeland security: What can you do to make the world a safer place?