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Challenges for Discrete Mathematics and Theoretical Computer Science in Defense Against Bioterrorism. Great concern about the deliberate introduction of diseases by bioterrorists has led to new challenges for mathematical scientists. smallpox.

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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

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


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.

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.

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.

What Can Math Models Do For Us?

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.

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.

DIMACS Special Focus on Computational and Mathematical Epidemiology 2002-2005


Methods of Math. and Comp. Epi.

Math. models of infectious diseases go back to Daniel Bernoulli’s mathematical analysis of smallpox in 1760.

Hundreds of math. models since have:

highlighted concepts like core population in STD’s;

  • Made explicit concepts such as herd immunity for vaccination policies;

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

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

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 today’s powerful computational methods

The Methods of Mathematical and Computational Epidemiology

Probabilistic Methods

stochastic processes, random walks, percolation, Markov chain Monte Carlo methods


need to bring in more powerful computational tools

Discrete Math. and Theoretical Computer Science

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

Discrete Math. and Theoretical Computer Science Cont’d

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.

DM and TCS Continued

These tools are made especially relevant to epidemiology because of:

Geographic Information Systems

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.

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.

Challenges for Discrete Math and Theoretical Computer Science in Bioterrorism Defense

What are DM and TCS?

DM deals with:







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.

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.

So How are DM/TCS Relevant to the Fight Against Bioterrorism?

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

AT&T 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

  • 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?

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.

1c. Visualization

Large data sets are sometimes best understood by visualizing them.

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.

1d. Data Cleaning

Disease detection problem: Very “dirty” data:

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

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.

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?

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

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.

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 t+1 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.

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.

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?

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

2b. Distributed Computing:

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.

2c. Spread of Opinion

2c. Spread of Opinion

Of relevance to bioterrorism.

Dynamic models of how opinions spread through social networks.

Your opinion changes at time t+1 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.

3. Evolution

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.

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.

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.

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.

4. Decision Making/Policy Analysis

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

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

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)

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.

4c. Game Theory

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.

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

5. Operations Research (continued)

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

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.

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.

Would DM/TCS help with a deliberate outbreak of Anthrax?

What about a deliberate release of smallpox?

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

Why shouldn’t these approaches work in the defense against bioterrorism?

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