# 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## 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

- Andersen PK, Borgan Ø, Gill RD, Keiding N (1992) Statistical models based on counting processes. Springer, BerlinzbMATHGoogle Scholar
- 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
- 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
- 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 92:62–68CrossRefzbMATHGoogle Scholar
- Brinks R, Landwehr S (2015a) Change rates and prevalence of a dichotomous variable: simulations and applications. PLoS ONE 10(3):e0118955CrossRefGoogle Scholar
- 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 zbMATHGoogle Scholar
- 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
- 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
- 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
- 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):76–90CrossRefzbMATHGoogle Scholar
- 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
- Chubb MC, Jacobsen KH (2010) Mathematical modeling and the epidemiological research process. Eur J Epidemiol 25(1):13–19CrossRefGoogle Scholar
- Day NE, Breslow NE (1980) The analysis of case–control studies. International Agency for Research on Cancer, LyonGoogle Scholar
- DuChateau P, Zachmann D (2012) Applied partial differential equations. Dover books on mathematics. Dover Publications, MineolaGoogle Scholar
- 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
- Federal Statistical Office of Germany: Lifetables for Germany 1896–2009 (2011). https://www.destatis.de/
- Fisz M (1963) Probability theory and mathematical statistics. Wiley, New YorkzbMATHGoogle Scholar
- Fix E, Neyman J (1951) A simple stochastic model of recovery, relapse, death and loss of patients. Hum Biol 23:205–241Google Scholar
- Kalbfleisch J, Prentice R (2002) The statistical analysis of failure time data, 2nd edn. Wiley, New YorkCrossRefzbMATHGoogle Scholar
- Kamke E (1983) Differentialgleichungen Lösungsmethoden und Lösungen. Teubner, StuttgartzbMATHGoogle Scholar
- Keiding N (1991) Age-specific incidence and prevalence: a statistical perspective. J R Stat Soc A 154:371–412MathSciNetCrossRefzbMATHGoogle Scholar
- Keiding N (2006) Event history analysis and the cross-section. Stat Med 25(14):2343–2364MathSciNetCrossRefGoogle Scholar
- Kermack WO, McKendrick AG (1927) A contribution to the mathematical theory of epidemics. Proc R Soc Lond A Math Phys Eng Sci 115:700–721CrossRefzbMATHGoogle Scholar
- 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
- 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
- Muench H (1934) Derivation of rates from summation data by the catalytic curve. J Am Stat Assoc 29(185):25–38CrossRefGoogle Scholar
- 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
- 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
- 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.
- 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
- Reid WT (1972) Riccati differential equations. Elsevier, AmsterdamzbMATHGoogle Scholar
- Szklo M, Nieto J (2014) Epidemiology. Jones & Bartlett Learning, BurlingtonGoogle Scholar
- 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
- 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
- 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
- Walter W (1998) Ordinary differential equations. Graduate texts in mathematics. Springer, New YorkCrossRefGoogle Scholar
- Yang G (2013) Neyman, Markov processes and survival analysis. Lifetime Data Anal 19(3):393–411MathSciNetCrossRefzbMATHGoogle Scholar
- Zachmanoglou E, Thoe D (1986) Introduction to partial differential equations with applications. Dover books on mathematics. Dover Publications, MineolaGoogle Scholar