Most mathematical models for the spread of disease use differential equations based on uniform mixing assumptions1 or ad hoc models for the contact process2,3,4. Here we explore the use of dynamic bipartite graphs to model the physical contact patterns that result from movements of individuals between specific locations. The graphs are generated by large-scale individual-based urban traffic simulations built on actual census, land-use and population-mobility data. We find that the contact network among people is a strongly connected small-world-like5 graph with a well-defined scale for the degree distribution. However, the locations graph is scale-free6, which allows highly efficient outbreak detection by placing sensors in the hubs of the locations network. Within this large-scale simulation framework, we then analyse the relative merits of several proposed mitigation strategies for smallpox spread. Our results suggest that outbreaks can be contained by a strategy of targeted vaccination combined with early detection without resorting to mass vaccination of a population.
Access optionsAccess options
Subscribe to Journal
Get full journal access for 1 year
only $3.90 per issue
All prices are NET prices.
VAT will be added later in the checkout.
Rent or Buy article
Get time limited or full article access on ReadCube.
All prices are NET prices.
Kaplan, E., Craft, D. & Wein, L. Emergency response to a smallpox attack: the case for mass vaccination. Proc. Natl Acad. Sci. USA 99, 10935–10940 (2002)
Halloran, M., Longini, I. M. Jr, Nizam, A. & Yang, Y. Containing bioterrorist smallpox. Science 298, 1428–1432 (2002)
Kretzschmar, M. & Morris, M. Measures of concurrency in networks and the spread of infectious disease. Math. Biosci. 133, 165–195 (1996)
Keeling, M. The effects of local spatial structure on epidemiological invasions. Proc. R. Soc. Lond. B 266, 859–867 (1999)
Watts, D. & Strogatz, S. Collective dynamics of small-world networks. Nature 393, 440–442 (1998)
Albert, R. & Barabási, A.-L. Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97 (2002)
Zwingle, E. Megacities. Natl Geogr. Mag. 202, 70–99 (2002)
Neff, J. M., Lane, J. M., Fulginiti, V. A. & Henderson, D. A. Contact vaccinia: transmission of vaccinia from smallpox vaccination. J. Am. Med. Assoc. 288, 1901–1905 (2002)
Ferguson, N. et al. Planning for smallpox outbreaks. Nature 425, 681–685 (2003)
Eubank, S. in Proc. ACM Symp. Appl. Comput. (eds Maniatty, W. & Szymanski, B.) 139–145 (ACM Press, New York, 2002)
Barrett, C. L. et al. TRANSIMS: Transportation Analysis Simulation System (Technical Report LA-UR-00–1725, Los Alamos National Laboratory, 2000)
Chowell, G., Hyman, J. M., Eubank, S. & Castillo-Chavez, C. Scaling laws for the movement of people between locations in a large city. Phys. Rev. E 68, 066102 (2003)
Dorogovtsev, S. N., Goltsev, A. V. & Mendes, J. F. F. Pseudo-fractal scale-free web. Phys. Rev. E 65, 066122 (2002)
Szabó, G., Alava, M. & Kertész, J. Structural transitions in scale-free networks. Phys. Rev. E 67, 056102 (2003)
Jin, E., Girvan, M. & Newman, M. Structure of growing networks. Phys. Rev. E 64, 046132 (2001)
Ravasz, E., Somera, A., Mongru, D., Oltvai, Z. & Barabási, A.-L. Hierarchical organization of modularity in metabolic networks. Science 297, 1551–1555 (2002)
Albert, R., Jeong, H. & Barabási, A.-L. Error and attack tolerance of complex networks. Nature 406, 378–382 (2000)
Newman, M. Assortative mixing in networks. Phys. Rev. Lett. 89, 208701 (2002)
Pastor-Satorras, R. & Vespignani, A. Immunization of complex networks. Phys. Rev. E 65, 036104 (2002)
Lloyd, A. & May, R. How viruses spread among computers and people. Science 292, 1316–1317 (2001)
Callaway, C., Newman, M., Strogatz, S. & Watts, D. Network robustness and fragility: percolation on random graphs. Phys. Rev. Lett. 85, 5468–5471 (2000)
Cohen, R., Erez, K., ben Avraham, D. & Havlin, S. Breakdown of the internet under intentional attack. Phys. Rev. Lett. 86, 3682–3685 (2001)
Eubank, S., Anil Kumar, V., Marathe, M. V., Srinivasan, A. & Wang, N. in Proc. ACM-SIAM Symp. Discrete Algorithms (ed Munro, I.) 711–720 (SIAM Press, Philadelphia, 2004)
Dall, J. & Christensen, M. Random geometric graphs. Phys. Rev. E 66, 016121 (2002)
Fenner, F., Henderson, D., Arita, I., Jezek, Z. & Ladnyi, I. Smallpox and its Eradication (World Health Organization, Geneva, 1988)
Eichner, M. & Dietz, K. Transmission potential of smallpox: estimates based on detailed data from an outbreak. Am. J. Epidemiol. 158, 110–117 (2003)
Keeling, M. & Grenfell, B. T. Individual-based perspectives on R0 . J. Theor. Biol. 203, 51–61 (2000)
Newman, M., Strogatz, S. & Watts, D. Properties of highly clustered networks. Phys. Rev. E 68, 026121 (2003)
We thank G. Korniss, G. Istrate and the Fogarty International Center at the National Institutes of Health for useful discussions, and acknowledge the work of all the members of the TRANSIMS and EpiSims team. The EpiSims project is funded by the National Infrastructure Simulation and Analysis Program (NISAC) at the Department of Homeland Security. The TRANSIMS project was funded by the Department of Transportation. H.G. was supported in part by the National Science Foundation (Division of Materials Research) and Z.T. by the Department of Energy. We thank the anonymous referees for their helpful suggestions.
The authors declare that they have no competing financial interests.
Contains related URLs, references, Figures, and extensive model details. (HTM 83 kb)
About this article
Chaos, Solitons & Fractals (2020)
Scientific Reports (2019)
Soft Computing (2019)
Validation of modified radio-frequency identification tag firmware, using an equine population case study
PLOS ONE (2019)
Future Internet (2019)