Skip to main content
SpringerLink
Log in

Mechanistic Models with Spatial Structures and Reactive Behavior Change

  • Chapter
  • First Online:
Quantitative Methods for Investigating Infectious Disease Outbreaks

Part of the book series: Texts in Applied Mathematics ((TAM,volume 70))

  • 4707 Accesses

Abstract

As we have emphasized in Chaps. 4 and 5, simple homogeneous models of transmission or growth dynamics often yield an early exponential epidemic growth phase even when the population is stratified into different groups (e.g., age, gender, regions). However, recent work has highlighted the presence of early sub-exponential growth patterns in case incidence from empirical outbreak data. This suggests that integrating detailed and often unobserved heterogeneity into simple mechanistic models could open the door to a new and exciting research area to better understand the role of heterogeneity on key transmission parameters, epidemic size, stochastic extinction, the effects of interventions, and disease forecasts.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 79.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Abdoulaye, B., Moussa, S., Daye, K., Boubakar, B. S., Cor, S. S., Idrissa, T., et al. (2014). Experience on the management of the first imported Ebola virus disease case in Senegal. The Pan African Medical Journal, 22(Suppl. 1), 6.

    Google Scholar 

  • Albert, R., & Barabasi, A. L. (2002). Statistical mechanics of complex networks. Reviews of Modern Physics, 74(1), 47–97.

    Article  MathSciNet  Google Scholar 

  • Althaus, C. L. (2014). Estimating the reproduction number of Ebola virus (EBOV) during the 2014 outbreak in West Africa. arXiv preprint. arXiv:1408.3505.

    Google Scholar 

  • Anderson, R. M., & May, R. M. (1991) Infectious diseases of humans, dynamics and control. Oxford: Oxford University Press.

    Google Scholar 

  • Apolloni, A., Poletto, C., Ramasco, J. J., Jensen P., & Colizza, V. (2014). Metapopulation epidemic models with heterogeneous mixing and travel behaviour. Theoretical Biology and Medical Modelling, 11, 3.

    Article  Google Scholar 

  • Appoloni, A., Poletto, C., & Colizza, V. (2013). Age-specific contacts and travel patterns in the spatial spread of 2009 H1N1 influenza pandemic. BMC Infectious Diseases, 13, 176.

    Article  Google Scholar 

  • Arino, J., Davis, J. R., Hartley, D., Jordan, R., Miller, J. M., & van den Driessche, P. (2005). A multi-species epidemic model with spatial dynamics. Mathematical Medicine and Biology, 22(2), 129–142.

    Article  Google Scholar 

  • Arino, J., Jordan, R., & van den Driessche, P. (2007). Quarantine in a multi-species epidemic model with spatial dynamics. Mathematical Biosciences, 206(1), 46–60.

    Article  MathSciNet  Google Scholar 

  • Bacaër, N., & Ait Dads el, H. (2011). Genealogy with seasonality, the basic reproduction number, and the influenza pandemic. Journal of Mathematical Biology, 62(5), 741–762.

    Google Scholar 

  • Ball, F. G., Britton, T., House, T., Isham, V., Mollison, D., Pellis, L., et al. (2015). Seven challenges for metapopulation models of epidemics, including households models. Epidemics, 10, 63–67.

    Article  Google Scholar 

  • Ball, F. G., Sirl, D., & Trapman, P. (2009). Threshold behaviour and final outcome of an epidemic on a random network with household structure. Advances in Applied Probability, 41, 765–796.

    Article  MathSciNet  Google Scholar 

  • Bansal, S., Grenfell, B. T., & Meyers, L. A. (2007). When individual behaviour matters: Homogeneous and network models in epidemiology. Journal of the Royal Society Interface, 4(16), 879–891.

    Article  Google Scholar 

  • Barabási, A. L., & Albert, R. (1999). Emergence of scaling in random network. Science, 286, 509–512.

    Article  MathSciNet  Google Scholar 

  • Belik, V., Geisel, T., & Brockmann, D. (2011). Natural human mobility patterns and spatial spread of infectious diseases. Physical Review X, 1, 011001.

    Article  Google Scholar 

  • Blythe, S. P., & Castillo-Chavez, C. (1989). Like-with-like preference and sexual mixing models. Mathematical Biosciences, 96, 221–238.

    Article  Google Scholar 

  • Breakwell, L., Gerber, A. R., Greiner, A. L., Hastings, D. L., Mirkovic, K., Paczkowski, M. M., et al. (2016). Early identification and prevention of the spread of Ebola in high-risk African countries. MMWR Supplements, 65(3), 21–27.

    Article  Google Scholar 

  • Capaldi, A., Behrend, S., Berman, B., Smith, J., Wright, J., & Lloyd, A. L. (2012). Parameter estimation and uncertainty quantification for an epidemic model. Mathematical Biosciences & Engineering, 9, 553–576.

    Article  MathSciNet  Google Scholar 

  • Chowell, G., Hengartner, N. W., Castillo-Chavez, C., Fenimore, P. W., & Hyman, J. M. (2004). The basic reproductive number of Ebola and the effects of public health measures: The cases of Congo and Uganda. Journal of Theoretical Biology, 229(1), 119–126.

    Article  MathSciNet  Google Scholar 

  • Chowell, G., Hincapie-Palacio, D., Ospina, J., Pell, B., Tariq, A., Dahal, S., et al. (2016). Using phenomenological models to characterize transmissibility and forecast patterns and final burden of Zika epidemics. PLOS Currents Outbreaks, 8. https://doi.org/10.1371/currents.outbreaks.f14b2217c902f453d9320a43a35b9583

  • Chowell, G., Rivas, A. L., Hengartner, N. W., Hyman, J. M., & Castillo-Chavez, C. (2006). The role of spatial mixing in the spread of foot-and-mouth disease. Preventive Veterinary Medicine, 73(4), 297–314.

    Article  Google Scholar 

  • Chowell, G., & Rothenberg, R. (2018). Spatial infectious disease epidemiology: On the cusp. BMC Medicine, 16(1), 192.

    Article  Google Scholar 

  • Chowell, G., Sattenspiel, L., Bansal, S., & Viboud, C. (2016). Mathematical models to characterize early epidemic growth: A review. Physics of Life Reviews, 18, 66–97.

    Article  Google Scholar 

  • Chowell, G., Viboud, C., Hyman, J. M., & Simonsen, L. (2015). The Western Africa ebola virus disease epidemic exhibits both global exponential and local polynomial growth rates. PLoS Currents, 7. https://doi.org/10.1371/currents.outbreaks.8b55f4bad99ac5c5db3663e916803261

  • Chowell, G., Viboud, C., Simonsen, L., Merler, S., & Vespignani, A. (2017). Perspectives on model forecasts of the 2014–2015 Ebola epidemic in West Africa: Lessons and the way forward. BMC Medicine, 15, 42.

    Article  Google Scholar 

  • Chowell, G., Viboud, C., Simonsen, L., & Moghadas, S. (2016). Characterizing the reproduction number of epidemics with early sub-exponential growth dynamics. Journal of the Royal Society Interface, 13(123). https://doi.org/10.1098/rsif.2016.0659

    Article  Google Scholar 

  • Danon, L., Ford, A. P., House, T., Jewell, C. P., Keeling, M. J., Roberts, G. O., et al. (2011). Networks and the epidemiology of infectious disease. Interdisciplinary Perspectives on Infectious Diseases, 2011, 284909.

    Article  Google Scholar 

  • Diekmann, O., & Heesterbeek, J. A. P. (2000). Mathematical epidemiology of infectious diseases: Model building, analysis and interpretation. Mathematical and computational biology (Vol. 5). Chichester: Wiley.

    MATH  Google Scholar 

  • Eames, K. T., & Keeling, M. J. (2003). Contact tracing and disease control. Proceedings of the Royal Society of London B: Biological Sciences, 270(1533), 2565–2571.

    Article  Google Scholar 

  • Eichner, M., Dowell, S. F., & Firese, N. (2011). Incubation period of ebola hemorrhagic virus subtype zaire. Osong Public Health and Research Perspectives, 2(1), 3–7.

    Article  Google Scholar 

  • Fasina, F., Shittu, A., Lazarus, D., Tomori, O., Simonsen, L., Viboud, C., et al. (2014). Transmission dynamics and control of Ebola virus disease outbreak in Nigeria, July to September 2014. Eurosurveillance, 19(40), 20920.

    Google Scholar 

  • Fenichel, E. P., Castillo-Chavez, C., Ceddia, M. G., Chowell, G., Parra, P. A. G., Hickling, G. J., et al. (2011). Adaptive human behavior in epidemiological models. Proceedings of the National Academy of Sciences, 108(15), 6306–6311.

    Article  Google Scholar 

  • Fraser, C. (2007). Estimating individual and household reproduction numbers in an emerging epidemic. PLoS One, 2(8), e758.

    Article  Google Scholar 

  • Goldstein, E., Paur, K., Fraser, C., Kenah, E., Wallinga, J., & Lipsitch, M. (2009). Reproductive numbers, epidemic spread and control in a community of households. Mathematical Biosciences, 221, 11–25.

    Article  MathSciNet  Google Scholar 

  • Hadeler, K. P., & Castillo-Chavez, C. (1995). A core group model for disease transmission. Mathematical Biosciences, 128, 41–55.

    Article  Google Scholar 

  • Halloran, M. E., Longini, I. M., Nizam, A., & Yang, Y. (2002). Containing bioterrorist smallpox. Science, 298(5597), 1428–1432.

    Article  Google Scholar 

  • Hethcote, H. W. (2000). The mathematics of infectious diseases. SIAM Review, 42(4), 599–653.

    Article  MathSciNet  Google Scholar 

  • Jacquez, J. A. (1996). Compartmental analysis in biology and medicine. Dexter, MI: Michigan Thompson-Shore Inc.

    MATH  Google Scholar 

  • Keeling, M. J., & Eames, K. T. (2005). Networks and epidemic models. Journal of the Royal Society Interface, 2(4), 295–307.

    Article  Google Scholar 

  • Keeling, M. J., & Rohani, P. (2008). Modeling infectious diseases in humans and animals. Princeton, NJ: Princeton University Press.

    Book  Google Scholar 

  • Kenah, E., Chao, D. L., Matrajt, L., Halloran, M. E., & Longini, I. M. Jr. (2011). The global transmission and control of influenza. PLoS One, 6(5), e19515.

    Article  Google Scholar 

  • Kiskowski, M. (2014). Three-scale network model for the early growth dynamics of 2014 West Africa Ebola epidemic. PLOS Currents Outbreaks. https://doi.org/10.1371/currents.outbreaks.b4690859d91684da963dc40e00f3da81

  • Kiskowski, M., & Chowell, G. (2015). Modeling household and community transmission of Ebola virus disease: Epidemic growth, spatial dynamics and insights for epidemic control. Virulence, 7(2), 63–73.

    Google Scholar 

  • Longini, I. M. Jr., & Koopman, J. S. (1982). Household and community transmission parameters from final distributions of infections in households. Biometrics, 38, 115–126.

    Article  Google Scholar 

  • Longini, I. M., Jr., Halloran, M. E., Nizam, A., Yang, Y., Xu, S., Burke, D. S., Cummings, D. A., & Epstein, J. M. (2007). Containing a large bioterrorist smallpox attack: A computer simulation approach. International Journal of Infectious Diseases, 11(2), 98–108.

    Article  Google Scholar 

  • Marguta, R., & Parisi, A. (2015). Impact of human mobility on the periodicities and mechanisms underlying measles dynamics. Journal of the Royal Society Interface, 12(104), 20141317.

    Article  Google Scholar 

  • Martín, A. C., Derrough, T., Honomou, P., Kolie, N., Diallo, B., Konê, M., et al. (2016). Social and cultural factors behind community resistance during an Ebola outbreak in a village of the Guinean Forest region, February 2015: A field experience. International Health, 8, 227–229.

    Article  Google Scholar 

  • Merler, S., Ajelli, M., Fumanelli, L., Gomes, M. F., Piontti, A. P., Rossi, L., et al. (2015). Spatiotemporal spread of the 2014 outbreak of Ebola virus disease in Liberia and the effectiveness of non-pharmaceutical interventions: A computational modelling analysis. The Lancet Infectious Diseases, 15(2), 204–211.

    Article  Google Scholar 

  • Murray, G. D., & Cliff, A. D. (1977). A stochastic model for measles epidemics in a multi-region setting. Transactions of the Institute of British Geographers, 2, 158–174.

    Article  Google Scholar 

  • Nishiura, H., & Chowell, G. (2014). Feedback from modelling to surveillance of Ebola virus disease. Eurosurveillance, 19(37), pii=20908.

    Google Scholar 

  • Olu, O. O., Lamunu, M., Nanyunja, M., Dafae, F., Samba, T., Sempiira, N., et al. (2016). Contact tracing during an outbreak of Ebola virus disease in the western area districts of Sierra Leone: Lessons for future Ebola outbreak response. Frontiers in Public Health, 4, 130.

    Google Scholar 

  • Pandey, A., Atkins, K. E., Medlock, J., Wenzel, N., Townsend, J. P., Childs, J. E., et al. (2014). Strategies for containing Ebola in West Africa. Science, 346(6212), 991–995.

    Article  Google Scholar 

  • Pellis, L., Ball, F., Bansal, S., Eames, K., House, T., Isham, V., et al. (2015). Eight challenges for network epidemic models. Epidemics, 10, 58–62.

    Article  Google Scholar 

  • Pellis, L., Ball, F., & Trapman, P. (2012). Reproduction numbers for epidemic models with households and other social structures. I. Definition and calculation of R0. Mathematical Biosciences, 235, 85–97.

    Article  MathSciNet  Google Scholar 

  • Pellis, L., Ferguson, N. M., & Fraser, C. (2009). Threshold parameters for a model of epidemic spread among households and workplaces. Journal of the Royal Society Interface, 6, 979–987.

    Article  Google Scholar 

  • Riley, S. (2007). Large-scale spatial-transmission models of infectious disease. Science, 316(5829), 1298–1301.

    Article  Google Scholar 

  • Rushton, S. P., & Mautner, A. (1955). The deterministic model of a simple epidemic for more than one community. Biometrika, 42, 126–132.

    Article  MathSciNet  Google Scholar 

  • Sattenspiel, L. (2009). The geographic spread of infectious diseases: Models and applications. Princeton, NJ: Princeton University Press.

    Book  Google Scholar 

  • Sattenspiel, L., & Dietz, K. (1995). A structured epidemic model incorporating geographic mobility among regions. Mathematical Biosciences, 128(1–2), 71–91.

    Article  Google Scholar 

  • Shrivastava, S. R., Shrivastava, P. S., & Ramasamy, J. (2014). Utility of contact tracing in reducing the magnitude of Ebola disease. Germs, 4(4), 97.

    Article  Google Scholar 

  • Simini, P., González, M. C., Maritan, A., & Barabási, A. L. (2012). A universal model for mobility and migration patterns. Nature, 484, 96–100.

    Article  Google Scholar 

  • Szendroi, B., & Csányi, G. (2004). Polynomial epidemics and clustering in contact networks. Proceedings of the Royal Society of London. Series B: Biological Sciences, 271(Suppl. 5), S364–S366.

    Google Scholar 

  • The World Health Organization. (2014). WHO declares end of Ebola outbreak in Nigeria. World Health Organization Media Statement. Retrieved October 20.

    Google Scholar 

  • The World Health Organization Emergency Response Team. (2014). Ebola virus disease in West Africa - The first 9 months of the epidemic and forward projections. New England Journal of Medicine, 371(16), 1481–1495.

    Article  Google Scholar 

  • Tizzoni, M., Bajardi, P., Poletto, C., Ramasco, J. J., Balcan, D., Goncalves, B., et al. (2012). Real-time numerical forecast of global epidemic spreading: Case study of 2009 A/H1N1pdm. BMC Medicine, 10, 165.

    Article  Google Scholar 

  • Viboud, C., Bjornstad, O. N., Smith, D. L., Simonsen, L., Miller, M. A., & Grenfell, B. T. (2006). Synchrony, waves, and spatial hierarchies in the spread of influenza. Science, 312(5772), 447–451.

    Article  Google Scholar 

  • Viboud, C., Simonsen, L., Chowell, G. (2016). A generalized-growth model to characterize the early ascending phase of infectious disease outbreaks. Epidemics 15, 27–37.

    Article  Google Scholar 

  • Vincenot, C. E., & Moriya, K. (2011). Impact of the topology of metapopulations on the resurgence of epidemics rendered by a new multiscale hybrid modeling approach. Ecological Informatics, 6, 177–186.

    Article  Google Scholar 

  • Watts, D. J., & Strogatz, S. H. (1998). Collective dynamics of ’small-world’ networks. Nature, 393(6684), 440.

    Article  Google Scholar 

  • Weinberger, D. M., Krause, T. G., Molbak, K., Cliff, A., Briem, H., Viboud, C., et al. (2012). Influenza epidemics in Iceland over 9 decades: Changes in timing and synchrony with the United States and Europe. American Journal of Epidemiology, 176(7), 649–655.

    Article  Google Scholar 

  • Wilson, E. B., & Worcester, J. (1945). The spread of an epidemic. Proceedings of the National Academy of Sciences of the United States of America, 31, 327–333.

    Google Scholar 

  • Xia, Y. C., Bjørnstad, O. N., & Grenfell, B. T. (2004). Measles metapopulation dynamics: A gravity model for epidemiological coupling and dynamics. The American Naturalist, 164, 267–281.

    Article  Google Scholar 

  • Xiao, Y., Zhou, Y., & Tang, S. (2011). Modelling disease spread in dispersal networks at two levels. Mathematical Medicine and Biology: A Journal of the IMA, 28, 227–244.

    Article  MathSciNet  Google Scholar 

  • Yan, P. (2018). A frailty model for intervention effectiveness against disease transmission when implemented with unobservable heterogeneity. Mathematical Biosciences & Engineering, 15(1), 275–298.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Infectious Diseases Prevention and Control Branch, Public Health Agency of Canada, Ottawa, ON, Canada

    Ping Yan

  2. Department of Statistics and Actuarial Science, Faculty of Mathematics, University of Waterloo, Waterloo, ON, Canada

    Ping Yan

  3. School of Public Health, Georgia State University, Atlanta, GA, USA

    Gerardo Chowell

Authors
  1. Ping Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gerardo Chowell
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Crown

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Yan, P., Chowell, G. (2019). Mechanistic Models with Spatial Structures and Reactive Behavior Change. In: Quantitative Methods for Investigating Infectious Disease Outbreaks. Texts in Applied Mathematics, vol 70. Springer, Cham. https://doi.org/10.1007/978-3-030-21923-9_9

Download citation

Publish with us

Policies and ethics

Search

Navigation