# 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).

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