Contact network epidemiology: Bond percolation applied to infectious disease prediction and control
Author:
Lauren Ancel Meyers
Journal:
Bull. Amer. Math. Soc. 44 (2007), 63-86
MSC (2000):
Primary 92D30, 92C60, 92B05, 60K35, 82B43
DOI:
https://doi.org/10.1090/S0273-0979-06-01148-7
Published electronically:
October 17, 2006
MathSciNet review:
2265010
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: Mathematics has long been an important tool in infectious disease epidemiology. I will provide a brief overview of compartmental models, the dominant framework for modeling disease transmission, and then contact network epidemiology, a more powerful approach that applies bond percolation on random graphs to model the spread of infectious disease through heterogeneous populations. I will derive important epidemiological quantities using this approach and provide examples of its application to issues of public health.
- [Abb52] H. Abbey, An examination of the reed-frost theory of epidemics, Human Biology 24 (1952), no. 3, 201-233.
- [AM91] R. M. Anderson and R. M. May, Infectious Diseases of Humans, Dynamics and Control, Oxford University Press, Oxford, 1991.
- [AO04] L. A. N. Amaral and J. M. Ottino, Complex networks - augmenting the framework for the study of complex systems, Eur. Phys. J. B. 38 (2004), 147-162.
- [BA99] Albert-László Barabási and Réka Albert, Emergence of scaling in random networks, Science 286 (1999), no. 5439, 509–512. MR 2091634, https://doi.org/10.1126/science.286.5439.509
- [Bai75] Norman T. J. Bailey, The mathematical theory of infectious diseases and its applications, 2nd ed., Hafner Press [Macmillan Publishing Co., Inc.] New York, 1975. MR 0452809
- [BB04] D. Bernoulli and S. Blower, An attempt at a new analysis of the mortality caused by smallpox and of the advantages of inoculation to prevent it, Reviews in Medical Virology 14 (2004), no. 5, 275-288.
- [Bec77] Niels Becker, Estimation for discrete time branching processes with application to epidemics, Biometrics 33 (1977), no. 3, 515–522 (English, with French summary). MR 654337, https://doi.org/10.2307/2529366
- [Bei03] Designated hospitals in Beijing meeting SARS treatment demands, People's Daily Online, 2003.
- [BMST97] Frank Ball, Denis Mollison, and Gianpaolo Scalia-Tomba, Epidemics with two levels of mixing, Ann. Appl. Probab. 7 (1997), no. 1, 46–89. MR 1428749, https://doi.org/10.1214/aoap/1034625252
- [BMT03] C. M. Booth, L. M. Matukas, G. A. Tomlinson, A. R. Rachlis, D. B. Rose, H. A. Dwosh, S. L. Walmsley, T. Mazzulli, M. Avendano, P. Derkach, I. E. Ephtimios, I. Kitai, B. D. Mederski, S. B. Shadowitz, W. L. Gold, L. A. Hawryluck, E. Rea, J. S. Chenkin, D. W. Cescon, S. M. Poutanen, and A. S. Detsky, Clinical features and short-term outcomes of 144 patients with SARS in the greater Toronto area, JAMA (2003), 289.21.JOC30885.
- [BPMss] S. Bansal, B. Pourbohloul, and L. A. Meyers, A comparative analysis of influenza vaccination programs, PLoS Medicine 3 (2006), e387.
- [BRO02] Tom Britton and Philip D. O’Neill, Bayesian inference for stochastic epidemics in populations with random social structure, Scand. J. Statist. 29 (2002), no. 3, 375–390. MR 1925565, https://doi.org/10.1111/1467-9469.00296
- [CDC03] CDC, Efficiency of quarantine during an epidemic of severe acute respiratory syndrome-Beijing, China, 2003, Morbidity and Mortality Weekly Report 52 (2003), no. 43, 1037-1040.
- [CDC04] CDC informational bulletin, Public health guidance for community-level preparedness and response to severe acute respiratory syndrome (SARS) version 2, supplement D: Community containment measures, including non-hospital isolation and quarantine, 2004. http://www.cdc.gov/ncidod/sars/guidance/D/pdf/lessons.pdf
- [CHECC03] G. Chowell, J. M. Hyman, S. Eubank, and C. Castillo-Chavez, Scaling laws for the movement of people between locations in a large city, Physical Review E 68 (2003), 066102.
- [CNN03a] CNN.com, Berkeley turns away students from SARS-hit regions, 2003.
- [CNN03b] CNN.com, SARS closing Beijing schools, 2003.
- [DdJM98] O. Diekmann, M. C. M. De Jong, and J. A. J. Metz, A deterministic epidemic model taking account of repeated contacts between the same individuals, J. Appl. Probab. 35 (1998), no. 2, 448–462. MR 1641833, https://doi.org/10.1239/jap/1032192860
- [DGL03] C. A. Donnelly, A. C. Ghani, G. M. Leung, A. J. Hedley, C. Fraser, S. Riley, L. J. Abu-Raddad, L-M. Ho, T-Q. Thach, P. Chau, K-P. Chan, T-H. Lam, L-Y. Tse, T. Tsang, S-H. Liu, J. H. B. Kong, E. M. C. Lau, N. M. Ferguson, and R. M. Anderson, Epidemiological determinants of spread of causal agent of severe acute respiratory syndrome in Hong Kong, The Lancet (2003), 1.
- [DH02] Klaus Dietz and J. A. P. Heesterbeek, Daniel Bernoulli’s epidemiological model revisited, Math. Biosci. 180 (2002), 1–21. John A. Jacquez memorial volume. MR 1950745, https://doi.org/10.1016/S0025-5564(02)00122-0
- [Dur99] Rick Durrett, Stochastic spatial models, SIAM Rev. 41 (1999), no. 4, 677–718. MR 1722998, https://doi.org/10.1137/S0036144599354707
- [EGK04] S. Eubank, H. Guclu, V. S. A. Kumar, M. V. Marathe, A. Srinivasan, Z. Toroczkai, and N. Wang, Modelling disease outbreaks in realistic urban social networks, Nature 429 (2004), 180-184.
- [FBMBss] M. Ferrari, S. Bansal, L. A. Meyers, and O. N. Bjornstad, Network frailty and the geometry of herd immunity, Proceedings of the Royal Society (London) B (in press).
- [FG00] N. M. Ferguson and G. P. Garnett, More realistic models of sexually transmitted disease transmission dynamics: sexual partnership networks, pair models, and moment closure, Sex. Transm. Dis. 27 (2000), no. 10, 600.
- [FHCF82] J. P. Fox, C. E. Hall, M. K. Cooney, and H. M. Foy, Influenza virus infections in Seattle families, 1975-1979, American Journal of Epidmiology 116 (1982), 212-227.
- [FKG03] C. P. Farrington, M. N. Kanaan, and N. J. Gay, Branching process models for surveillance of infectious diseases controlled by mass vaccination, Biostatistics 4 (2003), no. 2, 279-295.
- [Gle96] W. P. Glezen, Emerging infections: pandemic influenza, Epidemiology Reviews 18 (1996), no. 1, 64-76.
- [Gra83] P. Grassberger, Critical behavior of the general epidemic process and dynamical percolation, Mathematical Biosciences 63 (1983), no. 2, 157-172.
- [Ham06] W. H. Hamer, Epidemic disease in England--the evidence of variability and persistency of type, The Lancet i (1906), 733-739.
- [Het00] Herbert W. Hethcote, The mathematics of infectious diseases, SIAM Rev. 42 (2000), no. 4, 599–653. MR 1814049, https://doi.org/10.1137/S0036144500371907
- [HY84] Herbert W. Hethcote and James A. Yorke, Gonorrhea transmission dynamics and control, Lecture Notes in Biomathematics, vol. 56, Springer-Verlag, Berlin, 1984. With a foreword by Paul J. Wiesner and Willard Cates, Jr. MR 766910
- [JM78] L. C. Jennings and J. A. R. Miles, A study of acute respiratory disease in the community of Port Chalmers, Journal of Hygiene 81 (1978), 67-75.
- [KG99] A. Kleczkowski and B. T. Grenfell, Mean field-type equations for spread of epidemics: the `small world' model, Physica A 274 (1999), no. 1-2, 1-385.
- [KM27] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proceedings of the Royal Society (London) A 115 (1927), 700-721.
- [KRM97] M. J. Keeling, D. A. Rand, and A. J. Morris, Correlation models for childhood diseases, Proceedings of the Royal Society (London) B 264 (1997), 1149-1156.
- [KWM03] M. J. Keeling, M. E. Woolhouse, R. M. May, G. Davies, and B. T. Grenfell, Modelling vaccination strategies against foot-and-mouth disease, Nature 421 (2003), no. 6919, 136-42.
- [LCC03] M. Lipsitch, T. Cohen, B. Cooper, J. M. Robins, S. Ma, L. James, G. Gopalakrishna, S. K. Chew, C. C. Tan, M. H. Samore, D. Fisman, and M. Murray, Transmission dynamics and control of severe acute respiratory syndrome, Science (2003), 1086616.
- [LCH03] Y. S. Leo, M. Chen, B. H. Heng, C. C. Lee, N. Paton, B. Ang, P. Choo, S. W. Lim, A. E. Ling, M. L. Ling, B. K. Tay, P. A. Tambyah, Y. T. Lim, G. Gopalakrishna, S. Ma, L. James, P. L. Ooi, S. Lim, K. T. Goh, Sk. K. Chew, and C. C. Tan, Severe acute respiratory syndrome - Singapore, 2003, Morbidity and Mortality Weekly Report 52 (2003), no. 18, 405.
- [LEA01] F. Liljeros, C. R. Edling, L. A. N. Amaral, H. E. Stanley, and Y. Aberg, The web of human sexual contacts, Nature 411 (2001), 907-908.
- [LEA03] F. Liljeros, C. R. Edling, and L. A. N. Amaral, Sexual networks: implications for the transmission of sexually transmitted diseases, Microbes (2003).
- [LH05] I. M. Longini and M. E. Halloran, Strategy for distribution of influenza vaccine to high-risk groups and children, American Journal of Epidemiology 161 (2005), 303-306.
- [LHNY04] I. M. Longini, M. E. Halloran, A. Nizam, and Y. Yang, Containing pandemic influenza with antiviral agents, American Journal of Epidemiology 159 (2004), 623-633.
- [LKMF82] I. M. Longini, J. S. Koopman, A. S. Monto, and J. P. Fox, Estimating household and community transmission parameters of influenza, American Journal of Epidemiology 115 (1982), 736-751.
- [LM01] A. L. Lloyd and R. M. May, Epidemiology. How viruses spread among computers and people, Science 292 (2001), no. 5520, 1316.
- [Lon88] Ira M. Longini Jr., A mathematical model for predicting the geographic spread of new infectious agents, Math. Biosci. 90 (1988), no. 1-2, 367–383. Nonlinearity in biology and medicine (Los Alamos, NM, 1987). MR 958149, https://doi.org/10.1016/0025-5564(88)90075-2
- [LP89] Claude Lefèvre and Philippe Picard, On the formulation of discrete-time epidemic models, Math. Biosci. 95 (1989), no. 1, 27–35. MR 1001289, https://doi.org/10.1016/0025-5564(89)90049-7
- [MKL85] A. S. Monto, J. S. Koopman, and I. M. Longini, The Tecumseh study of illness. XIII. Influenza infection and disease, 1976-1981, American Journal of Epidemiology 121 (1985), 811-822.
- [MNMS03] L. A. Meyers, M. E. J. Newman, M. Martin, and S. Schrag, Applying network theory to epidemics: Control measures for mycoplasma pneumoniae outbreaks, Emerging Infectious Diseases 9 (2003), no. 2, 204.
- [MNP06] L. A. Meyers, M. E. J. Newman, and B. Pourbohloul, Predicting epidemics on directed contact networks, Journal of Theoretical Biology 240 (2006), 400-418.
- [Mor95] M. Morris, Data driven network models for the spread of disease, Epidemic Models: Their Structure and Relation to Data (D. Mollison, ed.), Cambridge University Press, Cambridge, 1995, pp. 302-322.
- [MPN05] Lauren Ancel Meyers, Babak Pourbohloul, M. E. J. Newman, Danuta M. Skowronski, and Robert C. Brunham, Network theory and SARS: predicting outbreak diversity, J. Theoret. Biol. 232 (2005), no. 1, 71–81. MR 2106112, https://doi.org/10.1016/j.jtbi.2004.07.026
- [New02] M. E. J. Newman, Spread of epidemic disease on networks, Phys. Rev. E (3) 66 (2002), no. 1, 016128, 11. MR 1919737, https://doi.org/10.1103/PhysRevE.66.016128
- [New05] -, Threshold effects for two pathogens spreading on a network, Physical Review Letters 95 (2005).
- [Org03] World Health Organization, Severe acute respiratory syndrome (SARS), 2003.
- [PMS05] B. Pourbohloul, L. A. Meyers, D. M. Skowronski, M. Krajden, D. M. Patrick, and R. C. Brunham, Modeling control strategies of respiratory pathogens, Emerg. Infect. Dis. 11 (2005), no. 8, 1249-1256.
- [PSV01] R. Pastor-Satorras and A. Vespignani, Epidemic spreading in scale-free networks, Phys. Rev. Lett. 86 (2001), no. 14, 3200.
- [PVJ98] N. Paneth and P. Vinten-Johansen, A rivalry of foulness: Official and unofficial investigations of the London cholera epidemic of 1854, American Journal of Public Health 88 (1998), no. 10, 1545-1553.
- [RFD03] S. Riley, C. Fraser, C. A. Donnelly, A. C. Ghani, L. J. Abu-Raddad, A. J. Hedley, G. M. Leung, L-M. Ho, T-H. Lam, T. Q. Thach, P. Chau, K-P. Chan, S-V. Lo, P-Y. Leung, T. Tsang, W. Ho, K-H. Lee, E. M. C. Lau, N. M. Ferguson, and R. M. Anderson, Transmission dynamics of the etiological agent of SARS in Hong Kong: Impact of public health interventions, Science (2003), 1086478.
- [Rot01] R. B. Rothenberg, How a net works: implications of network structure for the persistence and control of sexually transmitted diseases and HIV, Sexually Transmitted Diseases 28 (2001), 63-68.
- [RSF01] T. A. Reichart, N. Sugaya, D. S. Fedson, W. P. Glezen, L. Simonsen, and M. Tashiro, The Japanese experience with vaccinating school-children against influenza, New England Journal of Medicine 344 (2001), 889-896.
- [RST97] R. B. Rothenberg, C. Sterk, K. Toomey, J. Potterat, D. Johnson, M. Schrader, and S. Hatch, Using social network and ethnographic tools to evaluate syphilis transmission, Sexually Transmitted Diseases 25 (1997), no. 3, 154-160.
- [Sma02] Smallpox: A great and terrible scourge, 2002, National Library of Medicine (National Institutes of Health) website: http://www.nlm.nih.gov/exhibition/smallpox/sp_variolation.html
- [SS88] Lisa Sattenspiel and Carl P. Simon, The spread and persistence of infectious diseases in structured populations, Math. Biosci. 90 (1988), no. 1-2, 341–366. Nonlinearity in biology and medicine (Los Alamos, NM, 1987). MR 958148, https://doi.org/10.1016/0025-5564(88)90074-0
- [SWS02] L. M. Sander, C. P. Warren, I. M. Sokolov, C. Simon, and J. Koopman, Percolation on heterogeneous networks as a model for epidemics, Math. Biosci. 180 (2002), 293–305. John A. Jacquez memorial volume. MR 1950759, https://doi.org/10.1016/S0025-5564(02)00117-7
- [TPGC82] L. H. Taber, A. Paredes, W. P. Glezen, and R. B. Couch, Infection with influenza A/Victoria virus in Houston families, 1976, Journal of Hygiene 86 (1982), 303-313.
- [VdPvVdV98] C. P. B. Van der Ploeg, C. van Vliet, S. J. de Vlas, J. O. Ndinya-Achola, L. Fransen, G. J. van Oortmarssen, and J. D. F. Habbema, A microsimulation model for decision support in STD control, Interfaces 28 (1998), 84-100.
- [Volss] E. Volz, SIR dynamics in structured populations with heterogeneous connectivity, Journal of Mathematical Biology (in press).
- [Wat99] Duncan J. Watts, Small worlds, Princeton Studies in Complexity, Princeton University Press, Princeton, NJ, 1999. The dynamics of networks between order and randomness. MR 1716136
- [WEH05] D. Weycker, J. Edelsberg, M. E. Halloran, I. M. Longini, A. Nizam, V. Ciuryla, and G. Oster, Population-wide benefits of routine vaccination of children against influenza, Vaccine 23 (2005), no. 10, 1284-1293.
- [XHE04] R-H. Xu, J-F. He, M. R. Evans, G-W. Peng, H. E. Field, D-W. Yu, C-K. Lee, H-M. Luo, W-S. Lin, P. Lin, L-H. Li, W-J. Liang, J-Y. Lin, and A. Schnur, Epidemiologic clues to SARS origin in China, Emerg. Infect. Dis. 10 (2004), no. 6.
Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 92D30, 92C60, 92B05, 60K35, 82B43
Retrieve articles in all journals with MSC (2000): 92D30, 92C60, 92B05, 60K35, 82B43
Additional Information
Lauren Ancel Meyers
Affiliation:
Section of Integrative Biology, and Institute for Cellular and Molecular Biology, The University of Texas at Austin, Austin, Texas 78712
Email:
laurenmeyers@mail.utexas.edu
DOI:
https://doi.org/10.1090/S0273-0979-06-01148-7
Received by editor(s):
July 23, 2006
Published electronically:
October 17, 2006
Additional Notes:
This article is based on a lecture presented January 14, 2006, at the AMS Special Session on Current Events, Joint Mathematics Meetings, San Antonio, TX
Article copyright:
© Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.