Advertisement

A Model for Coupled Outbreaks Contained by Behavior Change

  • John M. Drake
  • Andrew W. Park
Chapter

Abstract

Large epidemics such as the recent Ebola crisis in West Africa occur when local efforts to contain outbreaks fail to overcome the probabilistic onward transmission to new locations. As a result, there may be large differences in total epidemic size from similar initial conditions. This work seeks to determine the extent to which the effects of behavior changes and metapopulation coupling on epidemic size can be characterized. While mathematical models have been developed to study local containment by social distancing, intervention and other behavior changes, their connection to larger-scale transmission is relatively underdeveloped. We make use of the assumption that behavior changes limit local transmission before susceptible depletion to develop a time-varying birth-death process capturing the dynamic decrease of the transmission rate associated with behavior changes. We derive an expression for the mean outbreak size of this model and show that the distribution of outbreak sizes is approximately geometric. This allows a probabilistic extension whereby infected individuals may initiate new outbreaks. From this model we characterize the overall epidemic size as a function of the behavior change rate and the probability that an infected individual starts a new outbreak. We find good agreement between the analytical results and stochastic simulations leading to novel findings including critical learning rates that demarcate large and small epidemic sizes.

Keywords

Ebola Epidemic model Behavior change Transmission rate Birth-death process Metapopulation model 

Notes

Acknowledgments

Research reported here was supported by the National Institute Of General Medical Sciences of the National Institutes of Health under Award Number U01GM110744 and the National Science Foundation under Rapid Award Number 1515194. The content is solely the responsibility of the authors and does not necessarily reflect the official views of the National Institutes of Health or the National Science Foundation.

References

  1. 1.
    Achlioptas, D., D’Souza, R.M., Spencer, J.: Explosive percolation in random networks. Science 323(5920), 1453–5 (2009)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Ball, F., Britton, T., House, T., Isham, V., Mollison, D., Pellis, L., Scalia Tomba, G.: Seven challenges for metapopulation models of epidemics, including households models. Epidemics 10, 63–67 (2015)Google Scholar
  3. 3.
    Brauer, F.: A simple model for behaviour change in epidemics. BMC Public Health 11 Suppl 1, S3 (2011)Google Scholar
  4. 4.
    Drake, J.M., Bakach, I., Just, M.R., ORegan, S.M., Gambhir, M., Fung, I.C.H.: Transmission models of historical Ebola outbreaks. Emerging Infect. Dis. 21, 1447–1450 (2015)Google Scholar
  5. 5.
    Drake, J.M., Chew, S.K., Ma, S.: Societal learning in epidemics: intervention effectiveness during the 2003 SARS outbreak in Singapore. PLOS One 1, e20 (2006)CrossRefGoogle Scholar
  6. 6.
    Drake, J.M., Kaul, R.B., Alexander, L.W., Regan, S.M.O., Kramer, M., Pulliam, J.T., Ferrari, M.J., Park, A.W.: Ebola cases and health system demand in Liberia. PLOS Biol. 13(1), e1002,056 (2015)Google Scholar
  7. 7.
    Guo, D., Li, K.C., Peters, T.R., Snively, B.M., Poehling, K.A., Zhou, X.: Multi-scale modeling for the transmission of influenza and the evaluation of interventions toward it. Sci. Rep. 5, 8980 (2015)CrossRefGoogle Scholar
  8. 8.
    Johnson, N.L., Kotz, S., Kemp, A.W.: Univariate Discrete Distributions. Wiley, New York (1992)Google Scholar
  9. 9.
    Keeling, M.: The implications of network structure for epidemic dynamics. Theor. Popul. Biol. 67(1), 1–8 (2005)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Kendall, D.G.: On the generalized “birth-and-death” process. Ann. Math. Stat. 19(1), 1–15 (1948)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Lai, P.C., Wong, C.M., Hedley, A.J., Lo, S.V., Leung, P.Y., Kong, J., Leung, G.M.: Understanding the spatial clustering of severe acute respiratory syndrome (SARS) in Hong Kong. Environ. Health Perspect. 112(15), 1550–1556 (2004)CrossRefGoogle Scholar
  12. 12.
    Lofgren, E.T., Halloran, M.E., Rivers, C.M., Drake, J.M., Porco, T.C., Lewis, B., Yang, W., Vespignani, A., Shaman, J., Eisenberg, J.N.S., Eisenberg, M.C., Marathe, M., Scarpino, S.V., Alexander, K.A., Meza, R., Ferrari, M.J., Hyman, J.M., Meyers, L.A., Eubank, S.: Mathematical models: a key tool for outbreak response. Proc. Natl. Acad. Sci. USA 111(51), 18,095–18,096 (2014)Google Scholar
  13. 13.
    Ma, J., Earn, D.J.D.: Generality of the final size formula for an epidemic of a newly invading infectious disease. Bull. Math. Biol. 68(3), 679–702 (2006)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Poletto, C., Pelat, C., Levy-Bruhl, D., Yazdanpanah, Y., Boelle, P.Y., Colizza, V.: Assessment of the Middle East respiratory syndrome coronavirus (MERS-CoV) epidemic in the Middle East and risk of international spread using a novel maximum likelihood analysis approach. Eurosurveillance 19(23), 20824 (2014)Google Scholar
  15. 15.
    Ruan, S., Wang, W.: Dynamical behavior of an epidemic model with a nonlinear incidence rate. J. Differ. Equ. 188(1), 135–163 (2003)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    UN: World Population Prospects: The 2015 Revision, Key Findings and Advance Tables. Technical Report, United Nations, Department of Economic and Social Affairs, Population Division (2015)Google Scholar
  17. 17.
    WHO Ebola Response Team: Ebola Virus Disease in West Africa - The First 9 Months of the Epidemic and Forward Projections. New Engl. J. Med. 371(16), 1481–1495 (2014)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Odum School of EcologyUniversity of GeorgiaAthensUSA

Personalised recommendations