Abstract
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.
Similar content being viewed by others
References
AJ D A R (2007) Beast: Bayesian evolutionary analysis by sampling trees. BMC Evol Biol 7:214
Anderson R, May R (1991) Infectious diseases of humans: dynamics and control. Oxford University Press
Balthrop J, Forrest S, Newman M, Williamson M (2004) Email networks and the spread of computer viruses. Science 304(5670):527–529
Barabási A L, Albert R (1999) Emergence of scaling in random networks. Science 286:509–512
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):37
Clauset A, Shalizi C, Newman M (2009) Power-law distributions in empirical data. SIAM Rev 51(4):661–703. doi:10.1137/070710111
Coleman J, Menzel H, Katz E (1966) Medical innovations: a diffusion study. Bobbs Merrill, New York
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 Press
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 106:21484–21489
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 270:121–127
Dunham J (2005) An agent-based spatially explicit epidemiological model in mason. J Artif Societies and Social Simulation 9(1):3
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):379–400
Eubank S, Guclu H, Kumar V, Marathe M, Srinivasan A, Toroczkai Z, Wang N (2004) Modeling disease outbreaks in realistic urban social networks. Nature 429:180–184
Fajardo D, Gardner L (2013) Inferring contagion patterns in social contact networks with limited infection data. networks and spatial economics
Ferguson N, Cummings D, Fraser C, Cajka J, Cooley P, Burke D (2006) Strategies for mitigating an influenza pandemic. Nature 442:448–452
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
Gardner L M, Fajardo D, Travis W S (2014) Inferring contagion patterns in social contact networks using a maximum likelihood approach. ASCE, natural hazards review
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
Gastner M T, Newman M E (2006) The spatial structure of networks. Eur Phys J B-Condens Matter Complex Syst 49(2):247–252
Gonzales M, Hidalgo C, Barabási A L (2008) Understanding individual human mobility patterns. Nature 453:479–482
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
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
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
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 11–15
Hasan S, Ukkusuri S (2011) A contagion model for understanding the propagation of hurricane warning information. Transp Res B 45:1590–1605
Hoogendoorn S P, Bovy P H (2005) Pedestrian travel behavior modeling. Netw Spat Econ 5(2):193–216
Hwang F K, Richards D S (1992) Steiner tree problems. Networks 22 (1):55–89. doi:10.1002/net.3230220105
Illenberger J, Nagel K, Flötteröd G (2013) The role of spatial interaction in social networks. Netw Spat Econ 13(3):255–282
Jombart T, Eggo RM, Dodd P, Balloux F (2009) Spatiotemporal dynamics in the early stages of the 2009 a/h1n1 influenza pandemic. PLoS currents influenza
Kinney R, Crucitti P, Albert R, Latora V (2005) Modeling cascading failures in the north american power grid. Eur Phys J B 46(1):101–107
Lam W H, Huang H J (2003) Combined activity/travel choice models: time-dependent and dynamic versions. Netw Spat Econ 3(3):323–347
Liberti L, Cafieri S, Tarissan F (2009) Reformulations in mathematical programming : a computational approach. In: Foundations of computational intelligence volume 3 - global optimization. Springer
Luo W, Tay W P, Leng M (2013) Identifying infection sources and regions in large networks. IEEE Trans Sigs Process 61(11):2850–2865
Murray J (2002) Mathematical biology, 3rd edn. Springer
Newman M, Forrest S, Balthrop J (2002) Email networks and the spread of computer viruses. Phys Rev E 66(3)
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 influenza
Ramadurai G, Ukkusuri S (2010) Dynamic user equilibrium model for combined activity-travel choices using activity-travel supernetwork representation. Netw Spat Econ 10(2):273–292
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):87
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):217–229
Rosenwein M B, Wong R T (1995) A constrained steiner tree problem. European journal of operational research
Rosseel M (1968) Comments on a paper by romesh saigal: a constrained shortest route problem. Oper Res 16(6):1232–1234
Sachtjen M, Carreras B, Lynch V (2000) Disturbances in a power transmission system. Phys Rev E 61(5):4877–4882
Saigal R (1968) A constrained shortest route problem. Oper Res 16(1):205–209
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
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):301–313
Sornette D (2003) Why stock markets crash: critical events in complex financial systems. Princeton University Press
V C A B, M B A V (2006) The modelling of global epidemics: Stochastic dynamics and predictability. Bull Math Biol 68:1893–1921
Voss S (1999) The steiner tree problem with hop constraints. Annals of operations research
Wallace R, HoDac H, Lathrop R, Fitch W (2007) A statistical phylogeography of influenza a h5n1. Proc Natl Acad Sci USA 104(11):4473–4478
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 outbreaks
Yen J Y (1971) Finding the k shortest loopless paths in a network. Management science
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rey, D., Gardner, L. & Waller, S.T. Finding Outbreak Trees in Networks with Limited Information. Netw Spat Econ 16, 687–721 (2016). https://doi.org/10.1007/s11067-015-9294-6
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11067-015-9294-6