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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 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. smallpox
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? 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 Anthrax
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 simulation 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: arrangements designs codes patterns schedules assignments
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 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 (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.
