Mathematics and Bioterrorism: Graph-theoretical Models of Spread and Control of Disease
Great concern about the deliberate introduction of diseases by bioterrorists has led to new challenges for mathematical scientists. smallpox
Waiting on line to get smallpox vaccine during New York City smallpox epidemic Bioterrorism issues are typical of many homeland security issues. This talk will emphasize bioterrorism, but many of the “messages” apply to homeland security in general.
Outline 1. The role of mathematical sciences in the fight against bioterrorism. 2. Methods of computational and mathematical epidemiology 2a. Other areas of mathematical sciences 2b. Discrete math and theoretical CS 3. Graph-theoretical models of spread and control of disease
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;
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.
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 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 Availability of large and disparate computerized databases on subjects relating to disease and the relevance of modern methods of data mining: Issues involve detection surveillance (monitoring) streaming data analysis clustering visualization of data
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.
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. Heavy use of DM in phylogenetic tree reconstruction Might help in identification of source of an infectious agent
Models of the Spread and Control of Disease through Social Networks • Diseases are spread through social networks. • This is especially relevant to sexually transmitted diseases such as AIDS. • “Contact tracing” is an important part of any strategy to combat outbreaks of diseases such as smallpox, whether naturally occurring or resulting from bioterrorist attacks.
The Basic Model Social Network = Graph Vertices = People Edges = contact State of a Vertex: simplest model: 1 if infected, 0 if not infected (SI Model) More complex models: SI, SEI, SEIR, etc. S = susceptible, E = exposed, I = infected, R = recovered (or removed)
More About States Once you are infected, can you be cured? If you are cured, do you become immune or can you re-enter the infected state? We can build a digraph reflecting the possible ways to move from state to state in the model.
The State Diagram for a Smallpox Model The following diagram is from a Kaplan-Craft-Wein (2002) model for comparing alternative responses to a smallpox attack. This is being considered by the CDC and Office of Homeland Security.
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
Moving From State to State Let si(t) give the state of vertex i at time t. Two states 0 and 1. Times are discrete: t = 0, 1, 2, …
Threshold Processes Basic k-Threshold Process: You change your state at time t+1 if at least k of your neighbors have the opposite state at time t. Disease interpretation? Cure if sufficiently many of your neighbors are uninfected. Does this make sense?
Threshold Processes II Irreversible k-Threshold Process: You change your state from 0 to 1 at time t+1 if at least k of your neighbors have state 1 at time t. You never leave state 1. Disease interpretation? Infected if sufficiently many of your neighbors are infected. Special Case k = 1: Infected if any of your neighbors is infected.
Complications to Add to Model • k = 1, but you only get infected with a certain probability. • You are automatically cured after you are in the infected state for d time periods. • You become immune from infection (can’t re-enter state 1) once you enter and leave state 1. • A public health authority has the ability to “vaccinate” a certain number of vertices, making them immune from infection.
Periodicity State vector: s(t) = (s1(t), s2(t), …, sn(t)). First example, s(1) = s(3) = s(5) = …, s(0) = s(2) = s(4) = s(6) = … Second example: s(1) = s(2) = s(3) = ... In all of these processes, because there is a finite set of vertices, for any initial state vector s(0), the state vector will eventually become periodic, i.e., for some P and T, s(t+P) = s(t) for all t > T. The smallest such P is called the period.
Periodicity II First example: the period is 2. Second example: the period is 1. Both basic and irreversible threshold processes are special cases of symmetric synchronous neural networks. Theorem (Goles and Olivos, Poljak and Sura): For symmetric, synchronous neural networks, the period is either 1 or 2.
Periodicity III When period is 1, we call the ultimate state vector a fixed point. When the fixed point is the vector s(t) = (1,1,…,1) or (0,0,…,0), we talk about a final common state. One problem of interest: Given a graph, what subsets S of the vertices can force one of our processes to a final common state with entries equal to the state shared by all the vertices in S in the initial state?
Periodicity IV Interpretation: Given a graph, what subsets S of the vertices should we plant a disease with so that ultimately everyone will get it? (s(t) (1,1,…,1)) Economic interpretation: What set of people do we place a new product with to guarantee “saturation” of the product in the population? Interpretation: Given a graph, what subsets S of the vertices should we vaccinate to guarantee that ultimately everyone will end up without the disease? (s(t) 0,0,…,0))
Conversion Sets Conversion set: Subset S of the vertices that can force a k-threshold process to a final common state with entries equal to the state shared by all the vertices in S in the initial state. (In other words, if all vertices of S start in same state x = 1 or 0, then the process goes to a state where all vertices are in state x.) Irreversible k-conversionset if irreversible process.
1-Conversion Sets k = 1. What are the conversion sets in a basic 1-threshold process?