Modeling Disease Transmission Across Social Networks

1. Modeling Disease Transmission Across Social Networks DIMACS seminar February 7, 2005 Stephen Eubank Virginia Bioinformatics Institute Virginia Tech eubank@vt.edu Simulation Science Laboratory

2. Variations on a Theme I. Estimating a Social Network II. Varieties of Social Networks III. Characterizing Networks for Epidemiology Simulation Science Laboratory

3. Translation • Compute structural properties of very large graphs • Which ones? • Are local properties enough? • Structural properties should be robust • How? need efficient algorithms • Generate constrained random graphs • for experiment • Chung-Lu, Reed-Molloy, MCMC • for analysis • preserve independence as much as possible Simulation Science Laboratory

4. 2 N ~2 If not uniform mixing, what? Network model ODE model Homogenous Isotropic ? alternativenetworks . . .

5. Do Local Constraints Fix Global Properties? • N vertices  ~ 2N2graphs(non-identical vertices  few symmetries) • E edges  ~ N2E graphs • Degree distribution  ?? graphs • Clustering coefficient  ?? graphs • What additional constraints  ?? graphs equivalent w.r.t. epidemics? Simulation Science Laboratory

6. Estimating a social network • Synthetic population • Survey (diary) based activity templates • Iterative solution to a large game • Assigning locations for activities (depends on travel times) • Planning routes • Estimating travel times (depends on activity locations) Simulation Science Laboratory

7. Example Synthetic Household Simulation Science Laboratory

8. Example Route Plans first person in household second person in household

9. Estimating Travel Times by Microsimulation intersection with multipleturn buffers (not internallydivided into grid cells) single-cell vehicle multiple-cell vehicle 7.5 meter  1 lane cellularautomaton grid cells

10. Typical Family’s Day Work Lunch Work Carpool Carpool Shopping Home Home Car Car Daycare Bus School Bus time Simulation Science Laboratory

11. Others Use the Same Locations time Simulation Science Laboratory

12. Time Slice of a Social Network Simulation Science Laboratory

13. Activities Adapt to Situation Home Home Simulation Science Laboratory

14. Example: Smallpox Response Efficacy # deaths per initial infected by day 100

15. Part II: Varieties of Social Networks • Definition of vertex • People • Concepts (location, role in society, group) • Definition of edge • Effective contact • Proximity • Weights • Edges: Interaction strength / probability of transmission • Vertices: “importance” • Time dependence • Directionality Simulation Science Laboratory

16. A Social Network: multipartite labeled graph People (8.8 million) • Vertex attributes: • age • household size • gender • income • … Simulation Science Laboratory

17. A Social Network: bipartite labeled graph Locations (1 million) • Vertex attributes: • (x,y,z) • land use • … Simulation Science Laboratory

18. A Social Network: bipartite labeled graph • Edge attributes: • activity type: shop, work, school • (start time 1, end time 1) • probability of transmitting Simulation Science Laboratory

19. A Social Network: projection onto people Simulation Science Laboratory

20. A Social Network: projection onto people [t1,t2] [t2,t3] [t3,t4] [t4,t5] Simulation Science Laboratory

21. A Social Network: projection over time Simulation Science Laboratory

22. Dendrogram: actual path disease takes Simulation Science Laboratory

23. A Social Network: bipartite labeled graph Simulation Science Laboratory

24. A Social Network: projection onto locations Simulation Science Laboratory

25. A Social Network: projection onto locations t2 t3 t4 Simulation Science Laboratory

26. A Social Network: projection over time Simulation Science Laboratory

27. Disease Dynamics & Scenario Determine Relevant Projections • People projection: edge if people co-located • communicable disease + vaccination/isolation • Location projection: directed edge if travel between locations • contamination, quarantine • Time dependence: almost periodic • Important time scales set by disease dynamics: • Infectious period • Duration of contact for transmission Simulation Science Laboratory

28. Example: Person-person graph

29. Person-person graph (~ dendrogram with ptransmission = 1)

30. Dendrogram with ptransmission << 1

31. Geographic spread

32. Characterizing EpiSims Networks • Degree distributions • Pointwise clustering: ratio of # triangles to # possible • Assortative mixing by degree, age, … • Shortest path length distribution • Expansion Simulation Science Laboratory

33. Degree Distribution, location-location

34. Degree Distribution, people-people

35. Sensitivity to parameters

36. Sensitivity to parameters

37. Assortative Mixing in EpiSims Graphs • Static people - people projection is assortative • by degree (~0.25) • but not as strongly by age, income, household size, … This is • Like other social networks • Unlike • technological networks, • Erdos-Renyi random graphs • Barabasi-Albert networks Simulation Science Laboratory

38. Removing high degree people useless

39. Removing high degree locations better

40. Clustering coefficient vs degree Simulation Science Laboratory

41. Characterizing Networks for Epidemiology • Question: how to change a network to reduce [casualties]? • Constraints: • Don’t know ahead of time where outbreak begins • Minimize impact on other social functions of network • Don’t know true network, only estimated one • Incorporate dependence on pathogen properties • Optimization: • Propose edge/vertex removal based on measurable (local) properties • Quickly estimate effect of new structure • How does propagation depend on structure? Simulation Science Laboratory

42. Suggested Metric Nk(i) = Number of distinct people connected to person i by a (shortest) path of length k • “k-betweenness”, “pointwise k-expansion” • Important k values are related to ratio of incubation to response times • Shortest path vs any path: depends on probability of transmission • Given N1(i), ..., Nk(i), can construct analog for non-shortest path of length k • Assumes static graph, but expect graph to change • Simple cases incorporate intuitively important properties • For k=1, N1(i) = d(i) • For k=2, includes degree distribution, clustering, assortativity by degree Simulation Science Laboratory

43. Comparison to “usual suspects” • Harder to measure in real networks • Difficult to work with analytically • Perturbative expansions (say, around tree-like structure) are lacking a small parameter to expand in • Describes how clustering should be combined with degree • Degree alone determines neither vulnerability nor criticality • Betweenness is global, sensitive to small changes • Usual statistics don’t incorporate time scales naturally Simulation Science Laboratory

44. Degree alone determines neither vulnerability nor criticality Same degree distribution Different assortative mixing by degree Introduce index case uniformly at random, what color (degree) is vulnerable?Top graph: degree 1, 80% of the timeBottom graph: degree 4, 80% of the time Critical vertex Simulation Science Laboratory

45. Use depends on how disease is introduced • Introduction uniformly distributed,consider distribution over all people: mean, variance, … • Introduction concentrated on specific part of graph,consider distribution over k-neighborhood • Introduction by malicious agent, consider worst case or tail Simulation Science Laboratory

46. Conclusion Progress on many fronts, but plenty more to be done: • Estimating large social networks • Building efficient, scalable simulations • Understanding structure of social networks • Determining how structure affects disease spread Simulation Science Laboratory