Finite Dimensional State Representation of Linear and Nonlinear Delay Systems

  • Odo Diekmann
  • Mats Gyllenberg
  • J. A. J. Metz


We consider the question of when delay systems, which are intrinsically infinite dimensional, can be represented by finite dimensional systems. Specifically, we give conditions for when all the information about the solutions of the delay system can be obtained from the solutions of a finite system of ordinary differential equations. For linear autonomous systems and linear systems with time-dependent input we give necessary and sufficient conditions and in the nonlinear case we give sufficient conditions. Most of our results for linear renewal and delay differential equations are known in different guises. The novelty lies in the approach which is tailored for applications to models of physiologically structured populations. Our results on linear systems with input and nonlinear systems are new.


Linear chain trick Delay-differential equation Renewal equation Markov chain Physiologically structured populations Epidemic models 

Mathematics Subject Classification

34K17 93C23 92D25 



We thank Michael Mackey for turning our attention to the works [31] and [16]. Hans Metz’ work benefitted from the support from the “Chair Modélisation Mathématique et Biodiversité of Veolia Environnement-École Polytechnique-Muséum National d’Histoire Naturelle-Fondation X”.


  1. 1.
    Asmussen, S.: Applied Probability and Queues. Wiley, New York (1987)MATHGoogle Scholar
  2. 2.
    Barbour, A.D., Reinert, G.: Approximating the epidemic curve. Electron. J. Probab. 18, 1–30 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Breda, D., Diekmann, O., Gyllenberg, M., Scarabel, F., Vermiglio, R.: Pseudospectral discretization of nonlinear delay equations: new prospects for numerical bifurcation analysis. SIAM J. Appl. Dyn. Syst. 15, 1–23 (2016)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Calsina, À., Saldaña, J.: A model of physiologically structured population dynamics with a nonlinear individual growth rate. J. Math. Biol. 33, 335–364 (1995)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    de Roos, A.M.: Numerical methods for structured population models: the Escalator Boxcar train. Numer. Methods Partial Differ. Equ. 4, 173–195 (1988)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    de Roos, A.M., Persson, L.: Population and Community Ecology of Ontogenetic Development. Princeton University Press, Princeton (2013)Google Scholar
  7. 7.
    Diekmann, O., Gyllenberg, M.: Equations with infinite delay: blending the abstract and the concrete. J. Differ. Equ. 252, 819–851 (2012)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Diekmann, O., Metz, H.: Exploring linear chain trickery for physiologically structured populations. In: Bij het afscheid van Prof. Dr. H.A. Lauwerier, TW in Beeld, CWI, Amsterdam, pp. 73–84 (1988)Google Scholar
  9. 9.
    Diekmann, O., Getto, Ph, Gyllenberg, M.: Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars. SIAM J. Math. Anal. 39, 1023–1069 (2007)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Diekmann, O., Gyllenberg, M., Metz, J.A.J.: Finite dimensional state representation of physiologically structured population models (in preparation)Google Scholar
  11. 11.
    Diekmann, O., Gyllenberg, M., Metz, J.A.J.: Steady-state analysis of structured population models. Theor. Popul. Biol. 63, 309–338 (2003)CrossRefMATHGoogle Scholar
  12. 12.
    Diekmann, O., Heesterbeek, J.A.P., Roberts, M.G.: The construction of next-generation matrices for compartmental epidemic models. J. R. Soc. Interface 7, 873–885 (2010)CrossRefGoogle Scholar
  13. 13.
    Diekmann, O., van Gils, S.A., Verduyn Lunel, S.M., Walther, H.-O.: Delay Equations. Functional-, Complex-, and Nonlinear Analysis. Springer, New York (1995)MATHGoogle Scholar
  14. 14.
    Diekmann, O., Gyllenberg, M., Metz, J.A.J., Nakaoka, S., de Roos, A.M.: Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example. J. Math. Biol. 61, 277–318 (2010)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Diekmann, O., Gyllenberg, M., Huang, H., Kirkilionis, M., Metz, J.A.J., Thieme, H.R.: On the formulation and analysis of general deterministic structured population models. II. Nonlinear theory. J. Math. Biol. 43, 157–189 (2001)MATHGoogle Scholar
  16. 16.
    Fargue, D.: Réductibilité des systèmes héréditaires à des systèmes dynamiques (régis par des équations différentielles ou aux dérivées partielles). C. R. Acad. Sci. Paris Sér. B 277, 471–473 (1973)Google Scholar
  17. 17.
    Farina, L., Rinaldi, S.: Positive Linear Systems. Theory and Applications. Wiley, New York (2000)CrossRefMATHGoogle Scholar
  18. 18.
    Gripenberg, G., Londen, S.-O., Staffans, O.: Volterra Integral and Functional Equations. Cambridge University Press, Cambridge (1990)CrossRefMATHGoogle Scholar
  19. 19.
    Gurtin, M.E., MacCamy, R.C.: Non-linear age-dependent population dynamics. Arch. Ration. Mech. Anal. 54, 281–300 (1974)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Gyllenberg, M.: Nonlinear age-dependent population dynamics in continuously propagated bacterial cultures. Math. Biosci. 62, 45–74 (1982)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Gyllenberg, M.: Stability of a nonlinear age-dependent population model containing a control variable. SIAM J. Appl. Math. 43, 1418–1438 (1983)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Gyllenberg, M.: Mathematical aspects of physiologically structured populations: the contributions of JAJ Metz. J. Biol. Dyn. 1, 3–44 (2007)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Kermack, W.O., McKendrick, A.G.: A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. Ser. A 115, 700–721 (1927)CrossRefMATHGoogle Scholar
  24. 24.
    Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations: Analysis and Numerical Solution. European Mathematical Society, Zürich (2006)CrossRefMATHGoogle Scholar
  25. 25.
    Leung, K.Y., Diekmann, O.: Dangerous connections: on binding site models of infectious disease dynamics. J. Math. Biol. 74, 619–671 (2017)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    MacDonald, M.: Time Lags in Biological Models. Lecture Notes in Biomathematics, vol. 27. Springer, Berlin (1978)Google Scholar
  27. 27.
    MacDonald, M.: Biological Delay Systems. Cambridge University Press, Cambridge (1989)MATHGoogle Scholar
  28. 28.
    Metz, J.A.J., Diekmann, O.: Exact finite dimensional representations of models for physiologically structured populations. I. The abstract foundations of linear chain trickery. In: Goldstein, J.A., Kappel, F., Schappacher, W. (eds.) Differential Equations with Applications in Biology, Physics and Engineering. Lecture Notes in Pure & Applied Mathematics, vol. 133, pp. 269–289. Marcel Dekker, New York (1991)Google Scholar
  29. 29.
    Miller, J.C.: A note on a paper by Erik Volz: SIR dynamics in random networks. J. Math. Biol. 62, 349–358 (2011)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Miller, J.C., Volz, E.M.: Incorporating disease and population structure into models of SIR disease in contact networks. In: PLoS ONE [E], vol 8, no. 8, Public Library of Science, San Francisco, pp. 1–14 (2013)Google Scholar
  31. 31.
    Vogel, T.: Théorie des Systèmes Évolutifs. Gauthier-Villars, Paris (1965)MATHGoogle Scholar
  32. 32.
    Zadeh, L.A., Desoer, C.A.: Linear System Theory: The State Space Approach. McGraw-Hill, New York (1963)MATHGoogle Scholar
  33. 33.
    Zadeh, L.A., Polak, E.: System Theory. McGraw-Hill, New York (1969)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • Odo Diekmann
    • 1
  • Mats Gyllenberg
    • 2
  • J. A. J. Metz
    • 3
    • 4
    • 5
  1. 1.Department of MathematicsUniversity of UtrechtUtrechtThe Netherlands
  2. 2.Department of Mathematics and StatisticsUniversity of HelsinkiHelsinkiFinland
  3. 3.Mathematical Institute and Institute of BiologyLeiden UniversityLeidenThe Netherlands
  4. 4.Evolution and Ecology ProgramInternational Institute for Applied Systems AnalysisLaxenburgAustria
  5. 5.Naturalis Biodiversity CenterLeidenThe Netherlands

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