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. 2012 Jun 20;67(3):483–507. doi: 10.1007/s00285-012-0558-1

Studying the recovery procedure for the time-dependent transmission rate(s) in epidemic models

Anna Mummert 1,✉
PMCID: PMC7080094  PMID: 22714651

Abstract

Determining the time-dependent transmission function that exactly reproduces disease incidence data can yield useful information about disease outbreaks, including a range potential values for the recovery rate of the disease and could offer a method to test the “school year” hypothesis (seasonality) for disease transmission. Recently two procedures have been developed to recover the time-dependent transmission function, β(t), for classical disease models given the disease incidence data. We first review the β(t) recovery procedures and give the resulting formulas, using both methods, for the susceptible-infected-recovered (SIR) and susceptible-exposed-infected-recovered (SEIR) models. We present a modification of one procedure, which is then shown to be identical to the other. Second, we explore several technical issues that appear when implementing the procedure for the SIR model; these are important when generating the time-dependent transmission function for real-world disease data. Third, we extend the recovery method to heterogeneous populations modeled with a certain SIR-type model with multiple time-dependent transmission functions. Finally, we apply the β(t) recovery procedure to data from the 2002–2003 influenza season and for the six seasons from 2002–2003 through 2007–2008, for both one population class and for two age classes. We discuss the consequences of the technical conditions of the procedure applied to the influenza data. We show that the method is robust in the heterogeneous cases, producing comparable results under two different hypotheses. We perform a frequency analysis, which shows a dominant 1-year period for the multi-year influenza transmission function(s).

Keywords: Epidemic models, Time-dependent transmission rate, Recovery procedure, Heterogeneous population disease model, Seasonality, Separability

References

  1. Bailey N. The mathematical theory of infectious diseases and its applications. London: Charles Griffin and Company; 1975. [Google Scholar]
  2. Biggerstaff M, Balluz L. Self-reported influenza-like illness during the 2009 H1N1 influenza pandemic—United States, September 2009–March 2010. Morbid Mortal Weekly Rep. 2011;60:37–41. [PubMed] [Google Scholar]
  3. Brauer F, van den Driessche P, Wu J, editors. Mathematical epidemiology. Berlin: Springer; 2008. [Google Scholar]
  4. Capistrán M, Moreles M, Lara B. Parameter estimation of some epidemic models. The case of recurrent epidemics caused by respiratory syncytial virus. Bull Math Biol. 2009;71:1890–1901. doi: 10.1007/s11538-009-9429-3. [DOI] [PubMed] [Google Scholar]
  5. Carrat F, Vergu E, Ferguson N, Lemaitre M, Cauchemez S, Leach S, Valleron A. Time lines of infection and disease in human influenza: a review of volunteer challenge studies. Am J Epidemiol. 2008;167:775–785. doi: 10.1093/aje/kwm375. [DOI] [PubMed] [Google Scholar]
  6. Grassly N, Fraser C. Seasonal infectious disease epidemiology. Proc Roy Soc Lond B: Biol Sci. 2006;273:2541–2550. doi: 10.1098/rspb.2006.3604. [DOI] [PMC free article] [PubMed] [Google Scholar]
  7. Hadeler K. Parameter identification in epidemic models. Math Biosci. 2011;229:185–189. doi: 10.1016/j.mbs.2010.12.004. [DOI] [PubMed] [Google Scholar]
  8. Hadeler K (2012) Parameter estimation in epidemic models: simplified formulas. Can Appl Math Q (2012, to appear)
  9. Hethcote H. Modeling heterogeneous mixing in infectious disease dynamics. In: Isham V, Medley G, editors. Models for infectious human diseases: their structure and relation to data. Cambridge: Cambridge University Press; 1996. pp. 215–238. [Google Scholar]
  10. Keeling M, Rohani P. Modeling infectious diseases. Princeton: Princeton University Press; 2008. [Google Scholar]
  11. Kermack W, McKendrick A. A contribution to the mathematical theory of epidemics. Proc Roy Soc Lond A. 1927;115:700–721. doi: 10.1098/rspa.1927.0118. [DOI] [Google Scholar]
  12. Lagarias J, Reeds J, Wright M, Wright P. Convergence properties of the Nelder-Mead simplex method in low dimensions. SIAM J Optim. 1998;9(1):112–147. doi: 10.1137/S1052623496303470. [DOI] [Google Scholar]
  13. Mkhatshwa T, Mummert A. Modeling super-spreading events for infectious diseases: case study SARS. IAENG Int J Appl Math. 2011;41:82–88. [Google Scholar]
  14. Murray J. Mathematical biology 1: An introduction. Berlin: Springer; 2002. [Google Scholar]
  15. Pollicott M, Wang H, Weiss H. Extracting the time-dependent transmission rate from infection data via solution of an inverse ODE problem. J Biol Dyn. 2012;6:509–523. doi: 10.1080/17513758.2011.645510. [DOI] [PubMed] [Google Scholar]
  16. Ponciano J, Capistrán M. First principles modeling of nonlinear incidence rates in seasonal epidemics. PLoS Comput Biol. 2011;7:e1001 079. doi: 10.1371/journal.pcbi.1001079. [DOI] [PMC free article] [PubMed] [Google Scholar]

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