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Illness-death model: statistical perspective and differential equations

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Abstract

The aim of this work is to relate the theory of stochastic processes with the differential equations associated with multistate (compartment) models. We show that the Kolmogorov Forward Differential Equations can be used to derive a relation between the prevalence and the transition rates in the illness-death model. Then, we prove mathematical well-definedness and epidemiological meaningfulness of the prevalence of the disease. As an application, we derive the incidence of diabetes from a series of cross-sections.

Keywords

Fix-Neyman competing risks model Illness-death model Multistate models Non-parametric estimation of transition rates Incidence Prevalence Markov processes Kolmogorov Differential Equations 
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Notes

Acknowledgements

This paper uses data from SHARE Waves 1, 2, 4 and 5. The SHARE data collection has been primarily funded by the European Commission through FP5 (QLK6-CT-2001-00360), FP6 (SHARE-I3: RII-CT-2006-062193, COMPARE: CIT5-CT-2005-028857, and FP7 (SHARE-PREP: Nr. 211909, SHARE-LEAP: Nr. 227822, SHARE M4: Nr. 261982). Additional funding from the German Ministry of Education and Research, the U.S. National Institute on Aging (U01_AG09740-13S2, P01_AG005842, P01_AG08291, P30_AG12815, R21_AG025169, Y1-AG-4553-01, IAG_BSR06-11, OGHA_04-064) and from various national funding sources is gratefully acknowledged (see www.share-project.org).

References

  1. Andersen PK, Borgan Ø, Gill RD, Keiding N (1992) Statistical models based on counting processes. Springer, BerlinMATHGoogle Scholar
  2. Börsch-Supan A, Hank K, Jürges H (2005) A new comprehensive and international view on ageing: introducing the ‘survey of health, ageing and retirement in Europe’. Eur J Ageing 2(4):245–253.  https://doi.org/10.1007/s10433-005-0014-9 CrossRefGoogle Scholar
  3. Börsch-Supan A, Brandt M, Hunkler C, Kneip T, Korbmacher J, Malter F, Schaan B, Stuck S, Zuber S (2013) Data resource profile: the survey of health, ageing and retirement in Europe (SHARE). Int J Epidemiol 42:992–1001CrossRefGoogle Scholar
  4. Brinks R, Landwehr S (2014) Age-and time-dependent model of the prevalence of non-communicable diseases and application to dementia in Germany. Theor Popul Biol 92:62–68CrossRefMATHGoogle Scholar
  5. Brinks R, Landwehr S (2015a) Change rates and prevalence of a dichotomous variable: simulations and applications. PLoS ONE 10(3):e0118955CrossRefGoogle Scholar
  6. Brinks R, Landwehr S (2015b) A new relation between prevalence and incidence of a chronic disease. Math Med Biol 32(4):425–435.  https://doi.org/10.1093/imammb/dqu024 MATHGoogle Scholar
  7. Brinks R, Landwehr S, Icks A, Koch M, Giani G (2013) Deriving age-specific incidence from prevalence with an ordinary differential equation. Stat Med 32(12):2070–2078MathSciNetCrossRefGoogle Scholar
  8. Brinks R, Bardenheier BH, Hoyer A, Lin J, Landwehr S, Gregg EW (2015a) Development and demonstration of a state model for the estimation of incidence of partly undetected chronic diseases. BMC Med Res Methodol 15(1):98.  https://doi.org/10.1186/s12874-015-0094-y CrossRefGoogle Scholar
  9. Brinks R, Hoyer A, Kuss O, Rathmann W (2015b) Projected effect of increased active travel in German urban regions on the risk of type 2 diabetes. PLoS ONE.  https://doi.org/10.1371/journal.pone.0122145 Google Scholar
  10. Brunet RC, Struchiner CJ (1999) A non-parametric method for the reconstruction of age-and time-dependent incidence from the prevalence data of irreversible diseases with differential mortality. Theor Popul Biol 56(1):76–90CrossRefMATHGoogle Scholar
  11. Carstenson B, Kristensen JK, Ottosen P, Borch-Johnsen K (2008) The Danish National Diabetes Register: trends in incidence, prevalence and mortality. Diabetologia 51(12):2187–2196CrossRefGoogle Scholar
  12. Chubb MC, Jacobsen KH (2010) Mathematical modeling and the epidemiological research process. Eur J Epidemiol 25(1):13–19CrossRefGoogle Scholar
  13. Day NE, Breslow NE (1980) The analysis of case–control studies. International Agency for Research on Cancer, LyonGoogle Scholar
  14. DuChateau P, Zachmann D (2012) Applied partial differential equations. Dover books on mathematics. Dover Publications, MineolaGoogle Scholar
  15. Egeberg A, Kristensen LE (2017) Impact of age and sex on the incidence and prevalence of psoriatic arthritis. Ann Rheum Dis.  https://doi.org/10.1136/annrheumdis-2017-211980 Google Scholar
  16. Federal Statistical Office of Germany: Lifetables for Germany 1896–2009 (2011). https://www.destatis.de/
  17. Fisz M (1963) Probability theory and mathematical statistics. Wiley, New YorkMATHGoogle Scholar
  18. Fix E, Neyman J (1951) A simple stochastic model of recovery, relapse, death and loss of patients. Hum Biol 23:205–241Google Scholar
  19. Kalbfleisch J, Prentice R (2002) The statistical analysis of failure time data, 2nd edn. Wiley, New YorkCrossRefMATHGoogle Scholar
  20. Kamke E (1983) Differentialgleichungen Lösungsmethoden und Lösungen. Teubner, StuttgartMATHGoogle Scholar
  21. Keiding N (1991) Age-specific incidence and prevalence: a statistical perspective. J R Stat Soc A 154:371–412MathSciNetCrossRefMATHGoogle Scholar
  22. Keiding N (2006) Event history analysis and the cross-section. Stat Med 25(14):2343–2364MathSciNetCrossRefGoogle Scholar
  23. Kermack WO, McKendrick AG (1927) A contribution to the mathematical theory of epidemics. Proc R Soc Lond A Math Phys Eng Sci 115:700–721CrossRefMATHGoogle Scholar
  24. Lozano R, Naghavi M, Foreman K, Lim S, Shibuya K, Aboyans V, Abraham J, Adair T, Aggarwal R, Ahn SY (2013) Global and regional mortality from 235 causes of death for 20 age groups in 1990 and 2010: a systematic analysis for the global burden of disease study 2010. Lancet 380(9859):2095–2128CrossRefGoogle Scholar
  25. Mahiane GS, Ouifki R, Brand H, Delva W, Welte A (2012) A general hiv incidence inference scheme based on likelihood of individual level data and a population renewal equation. PLoS ONE 7(9):1–11.  https://doi.org/10.1371/journal.pone.0044377 CrossRefGoogle Scholar
  26. Muench H (1934) Derivation of rates from summation data by the catalytic curve. J Am Stat Assoc 29(185):25–38CrossRefGoogle Scholar
  27. Murray CJ, Lopez AD (1996) Global and regional descriptive epidemiology of disability: incidence, prevalence, health expectancies and years lived with disability. Glob Burd Dis 1:201–246Google Scholar
  28. Ng M, Fleming T, Robinson M (2014) Global, regional, and national prevalence of overweight and obesity in children and adults during 1980–2013: a systematic analysis for the global burden of disease study 2013. Lancet 384(9945):766–781CrossRefGoogle Scholar
  29. Perera G, Pedersen L, Ansel D, Alexander M, Arrighi HM, Avillach P, Foskett N, Gini R, Gordon MF, Gungabissoon U et al. (2017) Dementia prevalence and incidence in a federation of European electronic health record databases: the European medical informatics framework resource. Alzheimers Dement.  https://doi.org/10.1016/j.jalz.2017.06.2270.
  30. Rait G, Walters K, Bottomley C, Petersen I, Iliffe S, Nazareth I (2010) Survival of people with clinical diagnosis of dementia in primary care: cohort study. BMJ 341:c3584CrossRefGoogle Scholar
  31. Reid WT (1972) Riccati differential equations. Elsevier, AmsterdamMATHGoogle Scholar
  32. Szklo M, Nieto J (2014) Epidemiology. Jones & Bartlett Learning, BurlingtonGoogle Scholar
  33. Tamayo T, Brinks R, Hoyer A, Ku O, Rathmann W (2016) Prvalenz und Inzidenz von Diabetes mellitus in Deutschland. Dtsch Arzteblatt Int 113(11):177–182.  https://doi.org/10.3238/arztebl.2016.0177 Google Scholar
  34. Termorshuizen F, Dorigo-Zetsma J, De Melker H, van den Hof S, Conyn-van Spaendonck M (2000) The prevalence of antibodies to hepatitis a virus and its determinants in the Netherlands: a population-based survey. Epidemiol Infect 124(03):459–466CrossRefGoogle Scholar
  35. Vos T, Barber RM, Bell B, Bertozzi-Villa A, Biryukov S, Bolliger I, Charlson F, Davis A, Degenhardt L, Dicker D (2015) Global, regional, and national incidence, prevalence, and years lived with disability for 301 acute and chronic diseases and injuries in 188 countries, 1990–2013: a systematic analysis for the global burden of disease study 2013. Lancet 386(9995):743CrossRefGoogle Scholar
  36. Walter W (1998) Ordinary differential equations. Graduate texts in mathematics. Springer, New YorkCrossRefGoogle Scholar
  37. Yang G (2013) Neyman, Markov processes and survival analysis. Lifetime Data Anal 19(3):393–411MathSciNetCrossRefMATHGoogle Scholar
  38. Zachmanoglou E, Thoe D (1986) Introduction to partial differential equations with applications. Dover books on mathematics. Dover Publications, MineolaGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Hiller Research Unit for RheumatologyUniversity Hospital DuesseldorfDuesseldorfGermany
  2. 2.Institute for Biometry and EpidemiologyGerman Diabetes CenterDuesseldorfGermany

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