Networks and Spatial Economics

, Volume 16, Issue 2, pp 687–721 | Cite as

Finding Outbreak Trees in Networks with Limited Information

  • David ReyEmail author
  • Lauren Gardner
  • S. Travis Waller


Real-time control of infectious disease outbreaks represents one of the greatest epidemiological challenges currently faced. In this paper we address the problem of identifying contagion patterns responsible for the spread of a disease in a network, which can be applied in real-time to evaluate an ongoing outbreak. We focus on the scenario where limited information, i.e. infection reports which may or may not include the actual source, is available during an ongoing outbreak and we seek the most likely infection tree that spans at least a set of known infected nodes. This problem can be represented using a maximum likelihood constrained Steiner tree model where the objective is to find a spanning tree with an assignment of integer nodes weights. We propose a novel formulation and solution method based on a two-step heuristic which (1) reduces the initial graph using a polynomial time algorithm designed to find feasible infection paths and (2) solves an exact mixed integer linear programming reformulation of the maximum likelihood model on the resulting subgraph. The proposed methodology can be applied to outbreaks which may evolve from multiple sources. Simulated contagion episodes are used to evaluate the performance of our solution method. Our results show that the approach is computationally efficient and is able to reconstruct a significant proportion of the outbreak, even in the context of low levels of information availability.


Contagion patterns Social contact networks Network optimization Integer programming Shortest path 


  1. AJ D A R (2007) Beast: Bayesian evolutionary analysis by sampling trees. BMC Evol Biol 7:214CrossRefGoogle Scholar
  2. Anderson R, May R (1991) Infectious diseases of humans: dynamics and control. Oxford University PressGoogle Scholar
  3. Balthrop J, Forrest S, Newman M, Williamson M (2004) Email networks and the spread of computer viruses. Science 304(5670):527–529CrossRefGoogle Scholar
  4. Barabási A L, Albert R (1999) Emergence of scaling in random networks. Science 286:509–512CrossRefGoogle Scholar
  5. Broeck W V, Gioannini C, Gonċalves B, Quaggiotto M, Colizza V, Vespignani A (2011) The gleamviz computational tool, a publicly available software to explore realistic epidemic spreading scenarios at the global scale. BMC BMC Infect Dis 11(1):37CrossRefGoogle Scholar
  6. Clauset A, Shalizi C, Newman M (2009) Power-law distributions in empirical data. SIAM Rev 51(4):661–703. doi: 10.1137/070710111 CrossRefGoogle Scholar
  7. Coleman J, Menzel H, Katz E (1966) Medical innovations: a diffusion study. Bobbs Merrill, New YorkGoogle Scholar
  8. Cummings D, Burke D, Epstein J M, Singa R, Chakravarty S (2002) Toward a containment strategy for smallpox bioterror: an individual-based computational approach. Brookings Institute PressGoogle Scholar
  9. D B V C, B G H H, JJ R A V (2009) The modelling of global epidemics: stochastic dynamics and predictability. Proc Natl Acad Sci USA 106:21484–21489CrossRefGoogle Scholar
  10. DT H, M CT, DJ S, L M, JK F, J W, MEJ W (2003) The construction and analysis of epidemic trees with reference to the 2001 uk foot-and-mouth outbreak. Proc R Soc B 270:121–127CrossRefGoogle Scholar
  11. Dunham J (2005) An agent-based spatially explicit epidemiological model in mason. J Artif Societies and Social Simulation 9(1):3Google Scholar
  12. Erath A, Löchl M, Axhausen K W (2009) Graph-theoretical analysis of the swiss road and railway networks over time. Netw Spat Econ 9(3):379–400CrossRefGoogle Scholar
  13. Eubank S, Guclu H, Kumar V, Marathe M, Srinivasan A, Toroczkai Z, Wang N (2004) Modeling disease outbreaks in realistic urban social networks. Nature 429:180–184CrossRefGoogle Scholar
  14. Fajardo D, Gardner L (2013) Inferring contagion patterns in social contact networks with limited infection data. networks and spatial economicsGoogle Scholar
  15. Ferguson N, Cummings D, Fraser C, Cajka J, Cooley P, Burke D (2006) Strategies for mitigating an influenza pandemic. Nature 442:448–452CrossRefGoogle Scholar
  16. Gardner L M, Fajardo D, Waller S T (2012) Inferring infection-spreading links in an air traffic network. Transp Res Rec: J Transp Res Board 2300(1):13–21. doi: 10.3141/2300-02 CrossRefGoogle Scholar
  17. Gardner L M, Fajardo D, Travis W S (2014) Inferring contagion patterns in social contact networks using a maximum likelihood approach. ASCE, natural hazards reviewGoogle Scholar
  18. Garey M, Johnson D (1977) The rectilinear Steiner tree problem is NP-complete. SIAM J Appl Math 32(4):826–834. doi: 10.1137/0132071 CrossRefGoogle Scholar
  19. Gastner M T, Newman M E (2006) The spatial structure of networks. Eur Phys J B-Condens Matter Complex Syst 49(2):247–252CrossRefGoogle Scholar
  20. Gonzales M, Hidalgo C, Barabási A L (2008) Understanding individual human mobility patterns. Nature 453:479–482Google Scholar
  21. Gouveia L, Magnanti T L (2003) Network flow models for designing diameter-constrained minimum-spanning and steiner trees. Networks 41(3):159–173. doi: 10.1002/net.10069 CrossRefGoogle Scholar
  22. Gouveia L, Simonetti L, Uchoa E (2011) Modeling hop-constrained and diameter-constrained minimum spanning tree problems as steiner tree problems over layered graphs. Math Program 128(1–2):123–148. doi: 10.1007/s10107-009-0297-2 CrossRefGoogle Scholar
  23. Graham R L, Hell P (1985) On the history of the minimum spanning tree problem. Ann Hist Comput 7(1):43–57. doi: 10.1109/MAHC.1985.10011 CrossRefGoogle Scholar
  24. Hagberg A A, Schult D A, Swart P J (2008) Exploring network structure, dynamics, and function using networkX. In: Proceedings of the 7th python in science conference (SciPy2008), Pasadena, pp 11–15Google Scholar
  25. Hasan S, Ukkusuri S (2011) A contagion model for understanding the propagation of hurricane warning information. Transp Res B 45:1590–1605CrossRefGoogle Scholar
  26. Hoogendoorn S P, Bovy P H (2005) Pedestrian travel behavior modeling. Netw Spat Econ 5(2):193–216CrossRefGoogle Scholar
  27. Hwang F K, Richards D S (1992) Steiner tree problems. Networks 22 (1):55–89. doi: 10.1002/net.3230220105 CrossRefGoogle Scholar
  28. Illenberger J, Nagel K, Flötteröd G (2013) The role of spatial interaction in social networks. Netw Spat Econ 13(3):255–282CrossRefGoogle Scholar
  29. Jombart T, Eggo RM, Dodd P, Balloux F (2009) Spatiotemporal dynamics in the early stages of the 2009 a/h1n1 influenza pandemic. PLoS currents influenzaGoogle Scholar
  30. Kinney R, Crucitti P, Albert R, Latora V (2005) Modeling cascading failures in the north american power grid. Eur Phys J B 46(1):101–107CrossRefGoogle Scholar
  31. Lam W H, Huang H J (2003) Combined activity/travel choice models: time-dependent and dynamic versions. Netw Spat Econ 3(3):323–347CrossRefGoogle Scholar
  32. Liberti L, Cafieri S, Tarissan F (2009) Reformulations in mathematical programming : a computational approach. In: Foundations of computational intelligence volume 3 - global optimization. SpringerGoogle Scholar
  33. Luo W, Tay W P, Leng M (2013) Identifying infection sources and regions in large networks. IEEE Trans Sigs Process 61(11):2850–2865CrossRefGoogle Scholar
  34. Murray J (2002) Mathematical biology, 3rd edn. SpringerGoogle Scholar
  35. Newman M, Forrest S, Balthrop J (2002) Email networks and the spread of computer viruses. Phys Rev E 66(3)Google Scholar
  36. P L, M S, A R (2009) Reconstructing the initial global spread of a human influenza pandemic: a bayesian spatial-temporal model for the global spread of h1n1pdm. PLoS currents influenzaGoogle Scholar
  37. Ramadurai G, Ukkusuri S (2010) Dynamic user equilibrium model for combined activity-travel choices using activity-travel supernetwork representation. Netw Spat Econ 10(2):273–292CrossRefGoogle Scholar
  38. Roche B, Drake J, Rohani P (2011) An agent-based model to study the epidemiological and evolutionary dynamics of influenza viruses. BMC Bioinforma 12 (1):87CrossRefGoogle Scholar
  39. Roorda M J, Carrasco J A, Miller E J (2009) An integrated model of vehicle transactions, activity scheduling and mode choice. Transp Res B Methodol 43(2):217–229CrossRefGoogle Scholar
  40. Rosenwein M B, Wong R T (1995) A constrained steiner tree problem. European journal of operational researchGoogle Scholar
  41. Rosseel M (1968) Comments on a paper by romesh saigal: a constrained shortest route problem. Oper Res 16(6):1232–1234CrossRefGoogle Scholar
  42. Sachtjen M, Carreras B, Lynch V (2000) Disturbances in a power transmission system. Phys Rev E 61(5):4877–4882CrossRefGoogle Scholar
  43. Saigal R (1968) A constrained shortest route problem. Oper Res 16(1):205–209CrossRefGoogle Scholar
  44. Santos M, Drummond L M, Uchoa E (2010) A distributed dual ascent algorithm for the hop-constrained steiner tree problem. Oper Res Lett 38(1):57–62. doi: 10.1016/j.orl.2009.09.008 CrossRefGoogle Scholar
  45. Schintler L A, Kulkarni R, Gorman S, Stough R (2007) Using raster-based gis and graph theory to analyze complex networks. Netw Spat Econ 7(4):301–313CrossRefGoogle Scholar
  46. Sornette D (2003) Why stock markets crash: critical events in complex financial systems. Princeton University PressGoogle Scholar
  47. V C A B, M B A V (2006) The modelling of global epidemics: Stochastic dynamics and predictability. Bull Math Biol 68:1893–1921CrossRefGoogle Scholar
  48. Voss S (1999) The steiner tree problem with hop constraints. Annals of operations researchGoogle Scholar
  49. Wallace R, HoDac H, Lathrop R, Fitch W (2007) A statistical phylogeography of influenza a h5n1. Proc Natl Acad Sci USA 104(11):4473–4478CrossRefGoogle Scholar
  50. Wesolowski A, Buckee C, Bengtsson L, Wetter E, Lu X, Tatem A (2014) Commentary: containing the ebola outbreak–the potential and challenge of mobile network data. PLOS currents outbreaksGoogle Scholar
  51. Yen J Y (1971) Finding the k shortest loopless paths in a network. Management scienceGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.School of Civil and Environmental EngineeringUNSW AustraliaSydneyAustralia
  2. 2.School of Civil and Environmental EngineeringUNSW Australia and NICTASydneyAustralia

Personalised recommendations