Homeland Security What Can Mathematics Do

1. 1 Homeland Security What Can Mathematics Do?

2. 2 Dealing with terrorism requires detailed planning of preventive measures and responses. Both require precise reasoning and extensive analysis.

3. 3 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.

4. 4 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.

5. 5 What Can Math Models Do For Us?

6. 6 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.

7. 7 OUTLINE Examples of Homeland Security Research at Rutgers that Use Mathematics Examples of Research Projects I am Involved in One Example in Detail

8. 8 OUTLINE Examples of Homeland Security Research at Rutgers that Use Mathematics Examples of Research Projects I am Involved in One Example in Detail

9. 9 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

10. 10 TRANSPORTATION AND BORDER SECURITY Statistical analysis of flight/aircraft inspections The Math: Statistics Port-of-entry inspection algorithms The Math: Statistics + �combinatorial optimization�

11. 11 TRANSPORTATION AND BORDER SECURITY Vessel tracking for homeland defense The Math: geometry + calculus

12. 12 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

13. 13 COMMUNICATION SECURITY Exploiting analogies between computer viruses and biological viruses The Math: Differential equations, dynamical systems

14. 14 COMMUNICATION SECURITY Information privacy: Identity theft Privacy of health care data The Math: Number theory (cryptography), Statistics

15. 15 FOOD AND WATER SUPPLY SECURITY Using economic weapons to protect against agroterrorism The Math: �Game Theory� Optimization

16. 16 SURVEILLANCE/DETECTION Detecting a bioterrorist attack using �syndromic surveillance� The Math: Statistics, Data Mining, Discrete Math

17. 17 SURVEILLANCE/DETECTION Weapons detection and identification (dirty bombs, plastic explosives) The Math: Linear algebra, Statistics, �Data Mining� (computer science)

18. 18 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

19. 19 RESPONDING TO AN ATTACK Exposure/Toxicology Modeling dose received Rapid risk and exposure characterization The Math: Differential Equations, Probability

20. 20 RESPONDING TO AN ATTACK Simulating evacuation of complex transportation facilities The Math: Computer simulation

21. 21 RESPONDING TO AN ATTACK Emergency Communications Rapid networking at emergency locations Rapid �telecollaboration� The Math: discrete math, network analysis

22. 22 OUTLINE Examples of Homeland Security Research at Rutgers that Use Mathematics Examples of Research Projects I am Involved in One Example in Detail

23. 23 The Bioterrorism Sensor Location Problem

24. 24 Early warning is critical This is a crucial factor underlying government�s plans to place networks of sensors/detectors to warn of a bioterrorist attack

25. 25 Two Fundamental Problems Sensor Location Problem (SLP): Choose an appropriate mix of sensors decide where to locate them for best protection and early warning

26. 26 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?

27. 27 The Sensor Location Problem: Algorithmic Tools

28. 28 Algorithmic Approaches I : Greedy Algorithms

29. 29 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.

30. 30 Algorithmic Approaches II : Variants of Classic Facility Location Theory Methods

31. 31 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.

32. 32 Location Theory

33. 33

34. 34 Algorithmic Approaches II : Variants of Classic Location Theory Methods: Complications We don�t 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.

35. 35 The Pattern Interpretation Problem

36. 36 The Pattern Interpretation Problem It will be up to the Decision Maker to decide how to respond to an alarm from the sensor network.

37. 37 Approaching the PIP: Minimizing False Alarms

38. 38 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.

39. 39 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.

40. 40 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.

41. 41 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) > T. Alternative decision rule: alarm if two sensors reach 90% of threshold, three reach 75% of threshold, etc. Alarm if: f(x) > T for some x, or if f(x1) > .9T and f(x2) > .9T for some x1,x2, or if f(x1) > .75T and f(x2) > .75T and f(x3) > .75T for some x1,x2,x3.

42. 42

43. 43

44. 44

45. 45

46. 46 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 0�s and 1�s; count 1 if word wi appears in the message, 0 otherwise.

47. 47 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.

48. 48 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.

49. 49 �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

50. 50 �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

51. 51 �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

52. 52 �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.

53. 53 �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�)

54. 54

55. 55 OUTLINE Examples of Homeland Security Research at Rutgers that Use Mathematics Examples of Research Projects I am Involved in One Example in Detail

56. 56 Mathematics and Bioterrorism: Graph-theoretical Models of Spread and Control of Disease

57. 57 Mathematics and Bioterrorism: Graph-theoretical Models of Spread and Control of Disease

58. 58 Great concern about the deliberate introduction of diseases by bioterrorists has led to new challenges for mathematical scientists.

59. 59 I got involved right after September 11 and the anthrax attacks.

60. 60 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.

61. 61 Models of the Spread and Control of Disease through Social Networks

62. 62 The Basic Model

63. 63 Example of a Social Network

64. 64 More About States

65. 65 The State Diagram for a Smallpox Model

66. 66

67. 67 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

68. 68 Moving From State to State

69. 69 Threshold Processes

70. 70 Threshold Processes II

71. 71 Basic 2-Threshold Process

72. 72

73. 73

74. 74 Irreversible 2-Threshold Process

75. 75

76. 76

77. 77 Complications to Add to Model

78. 78 Periodicity

79. 79 Periodicity II

80. 80 Periodicity III

81. 81 Periodicity IV

82. 82 Conversion Sets

83. 83 1-Conversion Sets

84. 84 1-Conversion Sets

85. 85 Irreversible 1-Conversion Sets

86. 86 Conversion Sets for Odd Cycles

87. 87 Conversion Sets for Odd Cycles

88. 88

89. 89

90. 90 Conversion Sets for Odd Cycles

91. 91

92. 92

93. 93 Irreversible Conversion Sets for Odd Cycles

94. 94 Vaccination Strategies

95. 95 Vaccination Strategies

96. 96 Vaccination Strategies

97. 97 Vaccination Strategy I: Worst Case (Adversary Infects Two)Two Strategies for Adversary

98. 98 The �alternation� between your choice of a defensive strategy and your adversary�s 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.

99. 99 Vaccination Strategy I Adversary Strategy Ia

100. 100 Vaccination Strategy I Adversary Strategy Ib

101. 101 Vaccination Strategy II: Worst Case (Adversary Infects Two)Two Strategies for Adversary

102. 102 Vaccination Strategy II Adversary Strategy IIa

103. 103 Vaccination Strategy II Adversary Strategy IIb

104. 104 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.

105. 105 k-Conversion Sets

106. 106 k-Conversion Sets II

107. 107 NP-Completeness

108. 108 k-Conversion Sets in Regular Graphs

109. 109 k-Conversion Sets in Regular Graphs II

110. 110 k-Conversion Sets in Grids

111. 111 Toroidal Grids

112. 112

113. 113 4-Conversion Sets in Toroidal Grids

114. 114

115. 115 4-Conversion Sets for Rectangular Grids

116. 116 4-Conversion Sets for Rectangular Grids

117. 117 4-Conversion Sets for Rectangular Grids

118. 118 More Realistic Models

119. 119 Alternative Models to Explore

120. 120 More Realistic Models

121. 121 Alternative Model to Explore

122. 122 The Case d = 2, p = 1/2

123. 123 The Case d = 2, p = 1/2

124. 124 The Case d = 2, p = 1/2

125. 125

126. 126 The Case d = 2, p = 1/2

127. 127 The Case d = 2, p = 1/2

128. 128 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.

129. 129 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.

130. 130

131. 131 What about a deliberate release of smallpox?

132. 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. 133 Why shouldn�t these approaches work in the defense against bioterrorism?

134. 134