Risk factors in the illness-death model: Simulation study and the partial differential equation about incidence and prevalence

Autoři: Annika Hoyer aff001;  Sophie Kaufmann aff001;  Ralph Brinks aff001
Působiště autorů: Institute for Biometrics and Epidemiology, German Diabetes Center, Leibniz Center for Diabetes Research at Heinrich Heine University Düsseldorf, Düsseldorf, Germany aff001;  Hiller Research Unit for Rheumatology, Heinrich Heine University Düsseldorf, Düsseldorf, Germany aff002
Vyšlo v časopise: PLoS ONE 14(12)
Kategorie: Research Article
doi: 10.1371/journal.pone.0226554


Recently, we developed a partial differential equation (PDE) that relates the age-specific prevalence of a chronic disease with the age-specific incidence and mortality rates in the illness-death model (IDM). With a view to planning population-wide interventions, the question arises how prevalence can be calculated if the distribution of a risk-factor in the population shifts. To study the impact of such possible interventions, it is important to deal with the resulting changes of risk-factors that affect the rates in the IDM. The aim of this work is to show how the PDE can be used to study such effects on the age-specific prevalence of a chronic disease, to demonstrate its applicability and to compare the results to a discrete event simulation (DES), a frequently used simulation technique. This is done for the first time based on the PDE which only needs data on population-wide epidemiological indices and is related to the von Foerster equation. In a simulation study, we analyse the effect of a hypothetical intervention against type 2 diabetes. We compare the age-specific prevalence obtained from a DES with the results predicted from modifying the rates in the PDE. The DES is based on 10000 subjects and estimates the effect of changes in the distributions of risk-factors. With respect to the PDE, the change of the distribution of risk factors is synthesized to an effective rate that can be used directly in the PDE. Both methods, DES and effective rate method (ERM) are capable of predicting the impact of the hypothetical intervention. The age-specific prevalences resulting from the DES and the ERM are consistent. Although DES is common in simulating effects of hypothetical interventions, the ERM is a suitable alternative. ERM fits well into the analytical theory of the IDM and the related PDE and comes with less computational effort.

Klíčová slova:

Cohort studies – Death rates – Epidemiology – Medical risk factors – Partial differential equations – Simulation and modeling


1. Brinks R, Landwehr S. Age-and time-dependent model of the prevalence of non-communicable diseases and application to dementia in Germany. Theoretical population biology. 2014;92:62–68. doi: 10.1016/j.tpb.2013.11.006 24333220

2. Brinks R, Landwehr S. A new relation between prevalence and incidence of a chronic disease. Mathematical Medicine and Biology. 2015;32:425–435. doi: 10.1093/imammb/dqu024 25576933

3. Brinks R, Hoyer A, Landwehr S. Surveillance of the Incidence of Non-Communicable Diseases (NCDs) with Sparse Resources: A Simulation Study Using Data from a National Diabetes Registry, Denmark, 1995-2004. PloS one. 2016;11:e0152046. doi: 10.1371/journal.pone.0152046 27023438

4. Cox DR. Regression models and life tables (with discussion). Journal of the Royal Statistical Society. 1972;34:187–220.

5. Kalbfleisch JD, Prentice RL. The Statistical Analysis of Failure Time Data. John Wiley & Sons; 2011.

6. Bland M. An introduction to medical statistics. Oxford University Press (UK); 2015.

7. Jeon CY, Lokken RP, Hu FB, van Dam RM. Physical activity of moderate intensity and risk of type 2 diabetes: a systematic review. Diabetes Care. 2007;30:744–752. doi: 10.2337/dc06-1842 17327354

8. Brinks R, Hoyer A, Kuss O, Rathmann W. Projected Effect of Increased Active Travel in German Urban Regions on the Risk of Type 2 Diabetes. PLoS one. 2015;10:e0122145. doi: 10.1371/journal.pone.0122145 25849819

9. Law AM. Simulation Modeling & Analysis. Fourth Edition. McGraw-Hill, New York. 2007.

10. Narayan KMV, Boyle JP, Thompson TJ, Gregg EW, Williamson DF. Effect of BMI on Lifetime Risk for Diabetes in the U.S. Diabetes Care. 2007;30:1562–1566. doi: 10.2337/dc06-2544 17372155

11. Berentzen TL, Jakobsen MU, Halkjaer J, Tjonneland A, Sorensen TI, Overvad K. Changes in waist circumference and the incidence of diabetes in middle-aged men and women. PloS one. 2011;6:e23104. doi: 10.1371/journal.pone.0023104 21829698

12. Von Foerster H. Some remarks on changing populations. In: The Kinetics of Cellular Proliferation, Stohlman F. Jr., editor. Greene and Stratton, New York.

13. Chubb MC, Jacobsen KH. Mathematical modeling and the epidemiological research process. European Journal of Epidemiology. 2010;25:13–19. doi: 10.1007/s10654-009-9397-9 19859816

14. Keiding N. Age-specific incidence and prevalence: a statistical perspective. Journal of the Royal Statistical Society. Series A (Statistics in Society). 1991;371–412. doi: 10.2307/2983150

15. Robert Koch-Institut and Destatis. Distribution of the population to groups in terms of body mass index in percent. Classification: years, Germany, age, sex, body mass index. 2003. Available at: http://www.gbe-bund.de/gbe10/trecherche.prc_them_rech?tk=5800&tk2=6000&p_uid=gast&p_aid=10171277&p_sprache=D&cnt_ut=11&ut=6150 (Last accessed May 15, 2019).

16. Lindström J, Louheranta A, Mannelin M, Rastas M, Salminen V, Eriksson J, Uusitupa M, Tuomilehto J, Finnish Diabetes Prevention Study Group. The Finnish Diabetes Prevention Study (DPS): Lifestyle intervention and 3-year results on diet and physical activity. Diabetes Care. 2003;26:3230–3236.

17. Tamayo T, Brinks R, Hoyer A, Kuss O, Rathmann W. The Prevalence and Incidence of Diabetes in Germany. Deutsches Arzteblatt International. 2016;133:177–182.

18. Statistisches Bundesamt. Sterbetafel 2012/2014—Methoden- und Ergebnisbericht zur laufenden Berechnung von Periodensterbetafeln für Deutschland und die Bundesländer. Available at: https://www.destatis.de/DE/Themen/Gesellschaft-Umwelt/Bevoelkerung/Sterbefaelle-Lebenserwartung/_inhalt.html#sprg233418 (Last accessed May 15, 2019).

19. Tobias DK, Pan A, Jackson CL, O’Reilly EJ, Ding EL, Willett WC, Manson JE, Hu FB. Body-mass index and mortality among adults with incident type 2 diabetes. New England Journal of Medicine. 2014;370:233–244. doi: 10.1056/NEJMoa1304501 24428469

20. Saydah SH, Loria CM, Eberhardt MS, Brancati FL. Subclinical states of glucose intolerance and risk of death in the U.S. Diabetes Care. 2001;24:447–453. doi: 10.2337/diacare.24.3.447 11289466

21. Brinks R, Landwehr S, Fischer-Betz R, Schneider M, Giani G. Lexis Diagram and Illness-Death Model: Simulating Populations in Chronic Disease Epidemiology. PLoS one. 2014;9:e106043. doi: 10.1371/journal.pone.0106043 25215502

22. Brinks R, Landwehr S, Icks A, Koch M, Giani G. Deriving age-specific incidence from prevalence with an ordinary differential equation. Statistics in Medicine. 2013;32:2070–2078. doi: 10.1002/sim.5651 23034867

23. Dahlquist G, Björk A. Numerical Methods. Prentice-Hall Inc. Englewood Cliffs, New Jersey. 1964.

24. Brennan A, Chick SE, Davies R. A taxonomy of model structures for economic evaluation of health technologies. Health Economics. 2006;15:1295–1310. doi: 10.1002/hec.1148 16941543

25. Brinks R, Landwehr S. Change rates and prevalence of a dichotomous variable: simulations and applications. PloS one. 2015;10:e0118955. doi: 10.1371/journal.pone.0118955 25749133

Článek vyšel v časopise


2019 Číslo 12