Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Bulletin of the American Mathematical Society
Bulletin of the American Mathematical Society
ISSN 1088-9485(online) ISSN 0273-0979(print)

 

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
Published electronically: October 17, 2006
MathSciNet review: 2265010
Full-text PDF

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.


References [Enhancements On Off] (What's this?)

  • [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, http://dx.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 (56 #11084)
  • [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 0654337 (58 #31657)
  • [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 (99a:92033), http://dx.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, http://dx.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 (99k:92041)
  • [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 (2003m:92070), http://dx.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 (2000h:60086), http://dx.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 (2002c:92034), http://dx.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 (86a:92002)
  • [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 (89h:92052), http://dx.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 (90e:92068), http://dx.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, http://dx.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 (2003e:60223), http://dx.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 (89m:92045), http://dx.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 (2003m:92080), http://dx.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 (2001a:91064)
  • [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.

Similar Articles

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: http://dx.doi.org/10.1090/S0273-0979-06-01148-7
PII: S 0273-0979(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.