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Graph-theoretical Models of the Spread and Control of Disease Fred Roberts, DIMACS

Graph-theoretical Models of the Spread and Control of Disease Fred Roberts, DIMACS. smallpox. Models of the Spread and Control of Disease through Social Networks. Diseases are spread through social networks.

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Graph-theoretical Models of the Spread and Control of Disease Fred Roberts, DIMACS

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  1. Graph-theoretical Models of the Spread and Control of DiseaseFred Roberts, DIMACS smallpox

  2. Models of the Spread and Control of Disease through Social Networks • Diseases are spread through social networks. • “Contact tracing” is an important part of any strategy to combat outbreaks of infectious diseases, whether naturally occurring or resulting from bioterrorist attacks.

  3. 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)

  4. 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.

  5. 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 has been considered by the CDC, Dept. of Homeland Security, Dept. of Health and Human Services, etc.

  6. 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

  7. Aside: Related Work • The model to be presented arose from the study of opinion formulation in groups. • Similar models are used in distributed computing. • The models we will study are sometimes called threshold networks.

  8. The Model: Moving From State to State Let si(t) give the state of vertex i at time t. Simplified Model: Two states 0 and 1. Times are discrete: t = 0, 1, 2, …

  9. First Try: Majority Processes Basic Majority Process: You change your state at time t+1 if a majority of your neighbors have the opposite state at time t. (No change in case of “ties”) Useful in models of spread of opinion. Disease interpretation? Cure if majority of your neighbors are uninfected. Does this make sense?

  10. Majority Processes II Irreversible Majority Process: You change your state from 0 to 1 at time t+1 if a majority of your neighbors have state 1 at time t. You never leave state 1. (No change in case of “ties”) Disease interpretation? Infected if sufficiently many of your neighbors are infected.

  11. Basic Majority Process

  12. Irreversible Majority Process

  13. Aside: Distributed Computing Majority processes are studied in distributed computing. Goal: Eliminate damage caused by failed processors (vertices) or at least to restrict their influence. Do this by maintaining replicated copies of crucial data and, when a fault occurs, letting a processor change “state” if a majority of its neighbors are in a different state. Other applications of similar ideas in distributed computing: distributed database management, quorum systems, fault local mending.

  14. Second Try: 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?

  15. 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.

  16. Basic 2-Threshold Process

  17. Irreversible 2-Threshold Process

  18. 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.

  19. 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.

  20. 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.

  21. 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?

  22. 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))

  23. 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.

  24. 1-Conversion Sets k = 1. What are the conversion sets in a basic 1-threshold process?

  25. 1-Conversion Sets k = 1. The only conversion set in a basic 1-threshold process is the set of all vertices. For, if any two adjacent vertices have 0 and 1 in the initial state, then they keep switching between 0 and 1 forever. What are the irreversible 1-conversion sets?

  26. Irreversible 1-Conversion Sets k = 1. Every single vertex x is an irreversible 1-conversion set if the graph is connected. We make it 1 and eventually all vertices become 1 by following paths from x.

  27. Conversion Sets for Odd Cycles C2p+1 2-threshold process. What is a conversion set?

  28. Conversion Sets for Odd Cycles C2p+1. 2-threshold process. Place p+1 1’s in “alternating” positions.

  29. Conversion Sets for Odd Cycles We have to be careful where we put the initial 1’s. p+1 1’s do not suffice if they are next to each other.

  30. Irreversible Conversion Sets for Odd Cycles What if we want an irreversible conversion set under an irreversible 2-threshold process? Same set of p+1 vertices is an irreversible conversion set. Moreover, everyone gets infected in one step.

  31. Vaccination Strategies Mathematical models are very helpful in comparing alternative vaccination strategies. The problem is especially interesting if we think of protecting against deliberate infection by a bioterrorist.

  32. Vaccination Strategies If you didn’t know whom a bioterrorist might infect, what people would you vaccinate to be sure that a disease doesn’t spread very much? (Vaccinated vertices stay at state 0 regardless of the state of their neighbors.) Try odd cycles again. Consider an irreversible 2-threshold process. Suppose your adversary has enough supply to infect two individuals. Strategy 1: “Mass vaccination”: make everyone 0 and immune in initial state.

  33. Vaccination Strategies In C5, mass vaccination means vaccinate 5 vertices. This obviously works. In practice, vaccination is only effective with a certain probability, so results could be different. Can we do better than mass vaccination? What does better mean? If vaccine has no cost and is unlimited and has no side effects, of course we use mass vaccination.

  34. Vaccination Strategies What if vaccine is in limited supply? Suppose we only have enough vaccine to vaccinate 2 vertices. Consider two different vaccination strategies: Vaccination Strategy I Vaccination Strategy II

  35. Vaccination Strategy I: Worst Case (Adversary Infects Two)Two Strategies for Adversary Adversary Strategy Ia Adversary Strategy Ib

  36. 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.

  37. Vaccination Strategy I Adversary Strategy Ia

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