Challenges for Discrete Mathematics and Theoretical Computer Science in Defense Against Bioterrorism

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Challenges for Discrete Mathematicsand
Theoretical Computer Sciencein Defense Against
Bioterrorism
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• Great concern about the deliberate introduction
  of diseases by bioterrorists has led to new
  challenges for mathematical scientists.
• 
  smallpox

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• Dealing with bioterrorism requires detailed
  planning of preventive measures and responses.
• Both require precise reasoning and extensive
  analysis.
• Understanding infectious systems requires being
  able to reason about highly complex biological
  systems, with hundreds of demographic and
  epidemiological variables.
• Intuition alone is insufficient to fully
  understand the dynamics of such systems.

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• 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.

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• Mathematical models have become important tools
  in analyzing the spread and control of infectious
  diseases and plans for defense against
  bioterrorist attacks, especially when combined
  with powerful, modern computer methods for
  analyzing and/or simulating the models.

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What Can Math Models Do For Us?
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What Can Math Models Do For Us?

• Sharpen our understanding of fundamental
  processes
• Compare alternative policies and interventions
• Help make decisions.
• Prepare responses to bioterrorist attacks.
• Provide a guide for training exercises and
  scenario development.
• Guide risk assessment.
• Predict future trends.

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• What are the challenges for mathematical
  scientists in the defense against disease?
• This question led DIMACS, the Center for Discrete
  Mathematics and Theoretical Computer Science, to
  launch a special focus on this topic.
• Post-September 11 events soon led to an emphasis
  on bioterrorism.

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DIMACS Special Focus on Computational and
Mathematical Epidemiology 2002-2005
Anthrax
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Methods of Math. and Comp. Epi.

• Math. models of infectious diseases go back to
  Daniel Bernoullis mathematical analysis of
  smallpox in 1760.

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• Hundreds of math. models since have
• highlighted concepts like core population in
  STDs

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• Made explicit concepts such as herd immunity for
  vaccination policies

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• Led to insights about drug resistance, rate of
  spread of infection, epidemic trends, effects of
  different kinds of treatments.

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• The size and overwhelming complexity of modern
  epidemiological problems -- and in particular the
  defense against bioterrorism -- calls for new
  approaches and tools.

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The Methods of Mathematical and Computational
Epidemiology

• Statistical Methods
• long history in epidemiology
• changing due to large data sets involved
• Dynamical Systems
• model host-pathogen systems, disease spread
• difference and differential equations
• little systematic use of todays powerful
  computational methods

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The Methods of Mathematical and Computational
Epidemiology

• Probabilistic Methods
• stochastic processes, random walks, percolation,
  Markov chain Monte Carlo methods
• simulation
• need to bring in more powerful computational
  tools

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Discrete Math. and Theoretical Computer Science

• Many fields of science, in particular molecular
  biology, have made extensive use of DM broadly
  defined.

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Discrete Math. and Theoretical Computer Science
Contd

• Especially useful have been those tools that make
  use of the algorithms, models, and concepts of
  TCS.
• These tools remain largely unused and unknown in
  epidemiology and even mathematical epidemiology.

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DM and TCS Continued

• These tools are made especially relevant to
  epidemiology because of
• Geographic Information Systems

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DM and TCS Continued

• Availability of large and disparate computerized
  databases on subjects relating to disease and the
  relevance of modern methods of data mining.

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DM and TCS Continued

• The increasing importance of an evolutionary
  point of view in epidemiology and the relevance
  of DM/TCS methods of phylogenetic tree
  reconstruction.

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Challenges for Discrete Math and Theoretical
Computer Science in Bioterrorism Defense
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What are DM and TCS?

• DM deals with
• arrangements
• designs
• codes
• patterns
• schedules
• assignments

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TCS deals with the theory of computer algorithms.

• During the first 30-40 years of the computer age,
  TCS, aided by powerful mathematical methods,
  especially DM, probability, and logic, had a
  direct impact on technology, by developing
  models, data structures, algorithms, and lower
  bounds that are now at the core of computing.

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DM and TCS have found extensive use in many areas
of science and public policy, for example in
Molecular Biology. These tools, which seem
especially relevant to problems of epidemiology,
are not well known to those working on public
health problems.
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So How are DM/TCS Relevant to the Fight Against
Bioterrorism?
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1. Detection/Surveillance

• 1a. Streaming Data Analysis
• When you only have one shot at the data
• Widely used to detect trends and sound alarms in
  applications in telecommunications and finance
• ATT uses this to detect fraudulent use of credit
  cards or impending billing defaults
• Columbia has developed methods for detecting
  fraudulent behavior in financial systems
• Uses algorithms based in TCS
• Needs modification to apply to disease detection

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• Research Issues
• Modify methods of data collection, transmission,
  processing, and visualization
• Explore use of decision trees, vector-space
  methods, Bayesian and neural nets
• How are the results of monitoring systems best
  reported and visualized?
• To what extent can they incur fast and safe
  automated responses?
• How are relevant queries best expressed, giving
  the user sufficient power while implicitly
  restraining him/her from incurring unwanted
  computational overhead?

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1b. Cluster Analysis

• Used to extract patterns from complex data
• Application of traditional clustering algorithms
  hindered by extreme heterogeneity of the data
• Newer clustering methods based on TCS for
  clustering heterogeneous data need to be modified
  for infectious disease and bioterrorist
  applications.

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1c. Visualization

• Large data sets are sometimes best understood by
  visualizing them.

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1c. Visualization (continued)

• Sheer data sizes require new visualization
  regimes, which require suitable external memory
  data structures to reorganize tabular data to
  facilitate access, usage, and analysis.
• Visualization algorithms become harder when data
  arises from various sources and each source
  contains only partial information.

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1d. Data Cleaning

• Disease detection problem Very dirty data

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1d. Data Cleaning (continued)

• Very dirty data due to
• manual entry
• lack of uniform standards for content and formats
• data duplication
• measurement errors
• TCS-based methods of data cleaning
• duplicate removal
• merge purge
• automated detection

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1e. Dealing with Natural Language Reports

• Devise effective methods for translating natural
  language input into formats suitable for
  analysis.
• Develop computationally efficient methods to
  provide automated responses consisting of
  follow-up questions.
• Develop semi-automatic systems to generate
  queries based on dynamically changing data.

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1f. Cryptography and Security

• Devise effective methods for protecting privacy
  of individuals about whom data is provided to
  biosurveillance teams -- data from emergency
  dept. visits, doctor visits, prescriptions
• Develop ways to share information between
  databases of intelligence agencies while
  protecting privacy?

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1f. Cryptography and Security (continued)

• Specifically How can we make a simultaneous
  query to two datasets without compromising
  information in those data sets? (E.g., is
  individual xx included in both sets?)
• Issues include
• insuring accuracy and reliability of responses
• authentication of queries
• policies for access control and authorization

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2. Social Networks

• Diseases are often spread through social contact.
• Contact information is often key in controlling
  an epidemic, man-made or otherwise.
• There is a long history of the use of DM tools in
  the study of social networks Social networks as
  graphs.

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2a. Spread of Disease through a Network

• Dynamically changing networks discrete times.
• Nodes (individuals) are infected or non-infected
  (simplest model).
• An individual becomes infected at time t1 if
  sufficiently many of its neighbors are infected
  at time t. (Threshold model)
• Analogy saturation models in economics.
• Analogy spread of opinions through social
  networks.

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Complications and Variants

• Infection only with a certain probability.
• Individuals have degrees of immunity and
  infection takes place only if sufficiently many
  neighbors are infected and degree of immunity is
  sufficiently low.
• Add recovered category.
• Add levels of infection.
• Markov models.
• Dynamic models on graphs related to neural nets.

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Research Issues

• What sets of vertices have the property that
  their infection guarantees the spread of the
  disease to x of the vertices?
• What vertices need to be vaccinated to make
  sure a disease does not spread to more than x of
  the vertices?
• How do the answers depend upon network structure?
• How do they depend upon choice of threshold?

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These Types of Questions Have Been Studied in
Other Contexts Using DM/TCS

• 2b. Distributed Computing

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• 2b. Distributed Computing (continued)
• Eliminating damage by failed processors -- when a
  fault occurs, let a processor change state if a
  majority of neighbors are in a different state or
  if number is above threshold.
• Distributed database management.
• Quorum systems.
• Fault-local mending.

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2c. Spread of Opinion
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2c. Spread of Opinion

• Of relevance to bioterrorism.
• Dynamic models of how opinions spread through
  social networks.
• Your opinion changes at time t1 if the number of
  neighboring vertices with the opposite opinion at
  time t exceeds threshold.
• Widely studied.
• Relevant variants confidence in your opinion (
  immunity) probabilistic change of opinion.

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3. Evolution
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3. Evolution (continued)

• Models of evolution might shed light on new
  strains of infectious agents used by
  bioterrorists.
• New methods of phylogenetic tree reconstruction
  owe a significant amount to modern methods of
  DM/TCS.
• Phylogenetic analysis might help in
  identification of the source of an infectious
  agent.

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3a. Some Relevant Tools of DM/TCS

• Information-theoretic bounds on tree
  reconstruction methods.
• Optimal tree refinement methods.
• Disk-covering methods.
• Maximum parsimony heuristics.
• Nearest-neighbor-joining methods.
• Hybrid methods.
• Methods for finding consensus phylogenies.

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3b. New Challenges for DM/TCS

• Tailoring phylogenetic methods to describe the
  idiosyncracies of viral evolution -- going beyond
  a binary tree with a small number of
  contemporaneous species appearing as leaves.
• Dealing with trees of thousands of vertices, many
  of high degree.
• Making use of data about species at internal
  vertices (e.g., when data comes from serial
  sampling of patients).
• Network representations of evolutionary history -
  if recombination has taken place.

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3b. New Challenges for DM/TCS Continued

• Modeling viral evolution by a collection of trees
  -- to recognize the quasispecies nature of
  viruses.
• Devising fast methods to average the quantities
  of interest over all likely trees.

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4. Decision Making/Policy Analysis
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4. Decision Making/Policy Analysis (continued)

• DM/TCS have a close historical connection with
  mathematical modeling for decision making and
  policy making.
• Mathematical models can help us
• understand fundamental processes
• compare alternative policies and interventions
• provide a guide for scenario development
• guide risk assessment
• aid forensic analysis
• predict future trends

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4a. Consensus

• DM/TCS fundamental to theory of group decision
  making/consensus
• Based on fundamental ideas in theory of voting
  and social choice
• Key problem combine expert judgments (e.g.,
  rankings of alternatives) to make policy

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4a. Consensus Continued

• Prior application to biology (Bioconsensus)
• Find common pattern in library of molecular
  sequences
• Find consensus phylogeny given alternative
  phylogenies
• Developing algorithmic view in consensus theory
  fast algorithms for finding the consensus policy
• Special challenge re bioterrorism/epidemiology
  instead of many decision makers and few
  candidates, could be few decision makers and
  many candidates (lots of different parameters to
  modify)

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4b. Decision Science

• Formalizing utilities and costs/benefits.
• Formalizing uncertainty and risk.
• DM/TCS aid in formalizing optimization problems
  and solving them maximizing utility, minimizing
  pain,
• Bringing in DM-based theory of meaningful
  statements and meaningful statistics.
• Some of these ideas virtually unknown in public
  health applications.
• Challenges are primarily to apply existing tools
  to new applications.

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4c. Game Theory
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4c. Game Theory (continued)

• History of use in military decision making
• Relevant to conflicts bioterrorism
• DM/TCS especially relevant to multi-person games
• Of use in allocating scarce resources to
  different players or different components of a
  comprehensive policy.
• New algorithmic point of view in game theory
  finding efficient procedures for computing the
  winner or the appropriate resource allocation.

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5. Operations Research

• O.R. a traditional tool in defense.
• Many applications in planning defense against
  attacks by bioterrorists.
• Methods of Discrete Optimization/Queueing
  relevant to
• size of stockpiles of vaccines
• allocation of medications
• analysis of bottlenecks in treatment facilities

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5. Operations Research (continued)

• Challenges are not primarily development of new
  methods, but modification of existing O.R.
  methods to apply to new contexts.

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6. Some Additional Relevant DM/TCS Topics

• 6a. Order-Theoretic Concepts
• Relevance of partial orders and lattices.
• The exposure set (set of all subjects whose
  exposure levels exceed some threshold) is a
  common construction in dimension theory of
  partial orders.
• Point lattices may be useful for visualizing the
  relationships of contigency tables to effect
  measures and cut-off choices.

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6b. Combinatorial Group Testing

• Natural or human-induced epidemics might require
  us to test samples from large populations at
  once.
• Combinatorial group testing arose from need for
  mathematical methods to test millions of WWII
  draftees for syphilis.
• Identify all positive cases in large population
  by
• dividing items into subsets
• testing if subset has at least one positive item
• iterating by dividing into smaller groups.

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Would DM/TCS help with a deliberate outbreak of
Anthrax?
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• What about a deliberate release of smallpox?

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• Similar approaches, using mathematical models
  based in DM/TCS, have proven useful in many other
  fields, to
• make policy
• plan operations
• analyze risk
• compare interventions
• identify the cause of observed events

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• Why shouldnt these approaches work in the
  defense against bioterrorism?