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

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

2. Founded 2009 as a DHS University Center of Excellence

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

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

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

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

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

11. t=0,1,2, … 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, … = infected = susceptible

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

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

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

17. v = 3

18. v = 3

19. v = 3

20. v = 3

21. v = 3

22. v = 3

23. v = 3

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

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

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. 26 Firefighting on Trees: 22 12 8 9 7 11 2 6 1 5 1 6 1 4 2 3 3 1 3 1 1 1 1 1 1 2 1 Firefighting on Trees

30. = 7 = 9 Firefighting on Trees Greedy Optimal

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. II. Algorithms for Port of Entry Inspection for WMDs

33. 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. Port of Entry Inspection Algorithms

34. My work on port of entry inspection has gotten me and my students to some remarkable places. Port of Entry Inspection Algorithms Me on a Coast Guard boat in a tour of the harbor in Philadelphia

36. Containersarriving to be classified intocategories. • 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. Sequential Decision Making Problem

37. 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 Sequential Decision Making Problem

38. Then: Container corresponds to abinary 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 ruleF takes a bit string and decides if it corresponds to a suspicious or okay container. Sequential Decision Making Problem 011001 F(011001) If attributes 2, 3, and 6 are present, assign container to category F(011001).

39. 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. Sequential Decision Making Problem

40. Testsmeasure 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 Binary Decision Tree Approach

41. 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. Binary Decision Tree Approach Figure 1

42. 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 Binary Decision Tree Approach Figure 2

43. 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. Binary Decision Tree Approach Figure 3

44. 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. Binary Decision Tree Approach Figure 4

45. 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? Binary Decision Tree Approach Port of Long Beach

46. 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. Binary Decision Tree Approach Port of Long Beach

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

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. 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. 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.