Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

A frailty model for intervention effectiveness against disease transmission when implemented with unobservable heterogeneity

. Infectious Diseases Prevention and Control Branch, Public Health Agency of Canada, Ottawa, Ontario, Canada, K1A 0K9

For an intervention against the spread of communicable diseases, the idealized situation is when individuals fully comply with the intervention and the exposure to the infectious agent is comparable across all individuals. Some level of non-compliance is likely where the intervention is widely implemented. The focus is on a more accurate view of its effects population-wide. A frailty model is applied. Qualitative analysis, in mathematical terms, reveals how large variability in compliance renders the intervention less effective. This finding sharpens our vague, intuitive and empirical notions. An effective reproduction number in the presence of frailty is defined and is shown to be invariant with respect to the time-scale of disease progression. This makes the results in this paper valid for a wide spectrum of acute and chronic infectious diseases. Quantitative analysis by comparing numerical results shows that they are also robust with respect to assumptions on disease progression structure and distributions, such as with or without the latent period and the assumed distributions of latent and infectious periods.

  Figure/Table
  Supplementary
  Article Metrics

Keywords Frailty model; heterogeneity; infectious diseases control; intervention; reproduction number

Citation: Ping Yan. A frailty model for intervention effectiveness against disease transmission when implemented with unobservable heterogeneity. Mathematical Biosciences and Engineering, 2018, 15(1): 275-298. doi: 10.3934/mbe.2018012

References

  • [1] R. Anderson,R. May, null, Infectious Diseases of Humans, Dynamics and Control, Oxford University Press, 1991.
  • [2] O. Diekmann, J. A. P. Heesterbeek and T. Britton, Mathematical Tools for Understanding Infectious Disease Dynamics, Princeton Series in Theoretical and Computational Biology, Princeton University Press, 2013.
  • [3] K. Dietz, Some problems in the theory of infectious diseases transmission and control, in Epidemic Models: their Structure and Relation to Data (ed. Denis Mollison), Cambridge University Press, (1995), 3-16.
  • [4] M. Greenwood,G. Yule, An inquiry into the nature of frequency distributions representative of multiple happenings with particular reference to occurrence of multiple attacks of diseases or of repeated accidents, Journal of the Royal Statistical Society, 83 (1920): 255-279.
  • [5] S. Goldstein, Operational representation of Whittaker's confluent hypergeometric function and Weber's parabolic cylinder function, Proc. London Math. Soc., 2 (1932): 103-125.
  • [6] P. Hougaard, Life table methods for heterogenous populations: Distributions describing the heterogeneity, Biometrika, 71 (1984): 75-83.
  • [7] P. Hougaard, Frailty models for survival data, Lifetime Data Analysis, 1 (1995): 255-273.
  • [8] E. K. Lenzi,E. P. Borges,R. S. Mendes, A q-generalization of Laplace transforms, Journal of Physics A: Mathematical and General, 32 (1999): 8551-8561.
  • [9] K. S. Lomax, Business failures, another example of the analysis of failure data, Journal of the American Statistics Association, 49 (1954): 847-852.
  • [10] J. Ma,D. Earn, Generality of the final size formula for an epidemic of a newly invading infectious disease, Bulletin of Mathematical Biology, 68 (2006): 679-702.
  • [11] A. W. Marshall and I. Olkin, Life Distributions, Structure of Nonparametric, Semiparametric and Parametric Families, Springer, 2007.
  • [12] S. R. Naik, The q-Laplace transforms and applications, Chapter 7 of Pathway Distributions, Autoregressive Processes and Their Applications, PhD Thesis, Mahatima Gandhi University, India, (2008).
  • [13] A. Olivieri, Heterogeneity in survival models, applications to pensions and life annuities, Belgian Actuarial Bulletin, 6 (2006): 23-39.
  • [14] S. Picoli,R. S. Mendes,L. C. Malacarne,R. P. B. Santos, q-distributions in complex systems: A brief review, Brazilian Journal of Physics, 39 (2009): 468-474.
  • [15] S. Ross, Stochastic Processes, Second Edition, Wiley and Sons Inc, 1996.
  • [16] R. K. Saxena, A study of the generalized Stieltjes transform, Lecturer in Mathematics, M.B. College, Udaipur, 25 (1959): 340-355.
  • [17] J. F. Steffensen, Deux problèms du calcul des probabilités, Ann. Inst. H. Poincaré, 3 (1933): 319-344.
  • [18] J. W. Vaupel,K. G. Manton,E. Stallard, The impact of heterogeneity in individual frailty on the dynamics of mortality, Demography, 16 (1979): 439-354.
  • [19] H. W. Watson,F. Galton, On the probability of extinction of families, J. Anthropol. Inst. Great Britain and Ireland, 4 (1874): 138-144.
  • [20] P. Yan,Z. Feng, Variability order of the latent and the infectious periods in a deterministic SEIR epidemic model and evaluation of control effectiveness, Mathematical Biosciences, 224 (2010): 43-52.
  • [21] O. Yürekli, A theorem on the generalized Stieltjes transform and its applications, Journal of Mathematical Analysis and Applications, 168 (1992): 63-71.

 

This article has been cited by

  • 1. Gerardo Chowell, Amna Tariq, James M. Hyman, A novel sub-epidemic modeling framework for short-term forecasting epidemic waves, BMC Medicine, 2019, 17, 1, 10.1186/s12916-019-1406-6
  • 2. Ping Yan, Gerardo Chowell, , Quantitative Methods for Investigating Infectious Disease Outbreaks, 2019, Chapter 6, 183, 10.1007/978-3-030-21923-9_6
  • 3. Ping Yan, Gerardo Chowell, , Quantitative Methods for Investigating Infectious Disease Outbreaks, 2019, Chapter 9, 317, 10.1007/978-3-030-21923-9_9

Reader Comments

your name: *   your email: *  

© 2018 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved