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Equivalences between age structured models and state dependent distributed delay differential equations

1 Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montréal, H3A 0B9, Canada
2 Département de mathématiques et de statistique, Université de Montréal, 2920 chemin de la Tour, Montréal, H3T 1J4, Canada
3 Department of Physiology, McGill University, 3655 Promenade Sir-William-Osler, Montréal, H3G 1Y6, Canada

Special Issues: Recent Advances in Mathematical Population Dynamics

We use the McKendrick equation with variable ageing rate and randomly distributed mat-uration time to derive a state dependent distributed delay differential equation. We show that the resulting delay differential equation preserves non-negativity of initial conditions and we characterise local stability of equilibria. By specifying the distribution of maturation age, we recover state depen-dent discrete, uniform and gamma distributed delay differential equations. We show how to reduce the uniform case to a system of state dependent discrete delay equations and the gamma distributed case to a system of ordinary differential equations. To illustrate the benefits of these reductions, we convert previously published transit compartment models into equivalent distributed delay differential equations.
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Keywords delay differential equations; age structured populations; transit compartment models; linear chain technique; state dependent delays

Citation: Tyler Cassidy, Morgan Craig, Antony R. Humphries. Equivalences between age structured models and state dependent distributed delay differential equations. Mathematical Biosciences and Engineering, 2019, 16(5): 5419-5450. doi: 10.3934/mbe.2019270

References

  • 1. A. G. McKendrick, Applications of mathematics to medical problems, Proc. Edinburgh Math. Soc., 44 (1925), 98–130.
  • 2. E. Trucco, Mathematical models for cellular systems. The Von Foerster equation. Part II, Bull. Math. Biophys., 27 (1965), 449–471.
  • 3. J. A. J Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, vol. 68 of Lecture Notes in Biomathematics. Berlin, Heidelberg: Springer, 3 ed., 1986.
  • 4. H. L. Smith, Reduction of structured population models to threshold-type delay equations and functional differential equations: a case study, Math. Biosci., 113 (1993), 1–23.
  • 5. M. Craig, A. R. Humphries and M. C. Mackey, A mathematical model of granulopoiesis incor-porating the negative feedback dynamics and kinetics of G-CSF/neutrophil binding and internal-ization, Bull. Math. Biol., 78 (2016), 2304–2357.
  • 6. T. Cassidy and A. R. Humphries, A mathematical model of viral oncology as an immuno-oncology instigator, Math. Med. Biol., Online First (2019), dqz008. Available from: https://doi.org/10.1093/imammb/dqz008.
  • 7. A. Otto and G. Radons, Transformations from variable delays to constant delays with applica-tions in engineering and biology, in Time Delay Syst. (T. Insperger, T. Ersal, and G. Orosz, eds.), vol. 7 of Advances in Delays and Dynamics, 169–183, Cham: Springer International Publishing, 2017.
  • 8. S. Bernard, Moving the boundaries of granulopoiesis modelling, Bull. Math. Biol., 78 (2016), 2358–2363.
  • 9. J. M. Mahaffy, J. Bélair and M. C. Mackey, Hematopoietic model with moving boundary condi-tion and state dependent delay: application in erythropoiesis, J. Theor. Biol., 190 (1998), 135–146.
  • 10. D. R. Cox, Regression models and life-tables, J. R. Stat. Soc., 34 (1972), 187–220.
  • 11. E. L. Kaplan and P. Meier, Nonparametric estimation from incomplete observations, J. Am. Stat. Assoc., 53 (1958), 457.
  • 12. D. Câmara de Souza, M. Craig, T. Cassidy, et al., Transit and lifespan in neutrophil production: implications for drug intervention, J. Pharmacokinet. Pharmacodyn., 45 (2018),59–77.
  • 13. J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, vol. 99 of Applied Mathematical Sciences. New York, NY: Springer New York, 1993.
  • 14. Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, vol. 1473 of Lecture Notes in Mathematics. Berlin, Heidelberg: Springer Berlin Heidelberg, 1991.
  • 15. W. Liu, T. Hillen and H. I. Freedman, A mathematical model for M-phase specific chemotherapy including the G0-phase and immunoresponse., Math. Biosci. Eng., 4 (2007), 239–259.
  • 16. F. Hartung, T. Krisztin, H. O. Walther, et al., Chapter 5 Functional Differential Equations with State-Dependent Delays: Theory and Applications, in Handb. Differ. Equations (A. Canada, P. Drabek, and A. Fonda, eds.), ch. 5, pp. 435–545, North Holland 2004: Elsevier, 3rd ed., 2006.
  • 17. Y. Yuan and J. Bélair, Stability and Hopf Bifurcation Analysis for Functional Differential Equa-tion with Distributed Delay, SIAM J. Appl. Dyn. Syst., 10 (2011), 551–581.
  • 18. H. L. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sci-ences, vol. 57 of Texts in Applied Mathematics. New York, NY: Springer New York, 2011.
  • 19. A. Teslya, Predator-Prey Models With Distributed Time Delay. Doctoral dissertation, McMaster University, 2015.
  • 20. T. Vogel, Systèmes Déferlants, Systèmes Héréditaires, Systèmes Dynamiques, in Proc. Int. Symp. Nonlinear Vib., (Kiev), pp. 123–130, Academy of Sciences USSR, 1961.
  • 21. N. MacDonald, Time Lags in Biological Models. Berlin: Springer, 1978.
  • 22. W. Krzyzanski, Interpretation of transit compartments pharmacodynamic models as lifespan based indirect response models., J. Pharmacokinet. Pharmacodyn., 38 (2011),179–204.
  • 23. W. Gurney, R. Nisbet and S. Blythe, The systematic formulation of models of stage-structured populations, in Dyn. Physiol. Struct. Popul. (J. A. J. Metz and O. Diekmann, eds.), ch. 11, 474–493, Berlin, Heidelberg: Springer Berlin Heidelberg, 3 ed., 1986.
  • 24. A. L. Quartino, M. O. Karlsson, H. Lindman, et al., Characterization of endogenous G-CSF and the inverse correlation to chemotherapy-induced neutropenia in patients with breast cancer using population modeling, Pharm. Res., 31 (2014), 3390–3403.
  • 25. L. Glass, Dynamical disease: Challenges for nonlinear dynamics and medicine, Chaos An Inter-discip. J. Nonlinear Sci., 25 (2015).
  • 26. S. Rubinow and J. Lebowitz, A mathematical model of neutrophil production and control in normal man, J. Math. Biol., 225 (1975), 187–225.
  • 27. M. C. Mackey, Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis, Blood, 51 (1978), 941–956.
  • 28. C. Colijn and M. C. Mackey, A mathematical model of hematopoiesis - I. Periodic chronic myelogenous leukemia, J. Theor. Biol., 237 (2005), 117–132.
  • 29. F. Crauste and M. Adimy, Modeling and asymptotic stability of a growth factor-dependent stem cell dynamics model with distributed delay, Discret. Contin. Dyn. Syst. - Ser. B, 8 (2007), 19–38.
  • 30. T. Hearn, C. Haurie and M. C. Mackey, Cyclical neutropenia and the peripherial control of white blood cell production, J. Theor. Biol., 192 (1998), 167–181.
  • 31. L. E. Friberg, A. Henningsson, H. Maas, et al., Model of chemotherapy-induced myelosuppres-sion with parameter consistency across drugs, J. Clin. Oncol., 20 (2002), 4713–4721.
  • 32. G. von Schulthess and N. Mazer, Cyclic neutropenia (CN): A clue to the control of granu-lopoiesis, Blood, 59 (1982), 27–37.
  • 33. W. Krzyzanski, P. Wiczling, P. Lowe, et al., Population modeling of filgrastim PK-PD in healthy adults following intravenous and subcutaneous administrations, J. Clin. Pharmacol., 50 (2010), 101S–112S.
  • 34. J. J. Pérez-Ruixo, W. Krzyzanski and J. Hing, Pharmacodynamic analysis of recombinant human erythropoietin effect on reticulocyte production rate and age distribution in healthy subjects., Clin. Pharmacokinet., 47 (2008), 399–415.
  • 35. O. Diekmann, M. Gyllenberg and J. A. J. Metz, Finite dimensional state representation of linear and nonlinear delay systems, J. Dyn. Differ. Equations, 30 (2018), 1439–1467.
  • 36. L. K. Roskos, P. Lum, P. Lockbaum, et al., Pharmacokinetic/pharmacodynamic modeling of pegfilgrastim in healthy subjects, J. Clin. Pharmacol., 46 (2006), 747–757.
  • 37. A. Roberts, G-CSF: A key regulator of neutrophil production, but that's not all!, Growth Factors, 23 (2005), 33–41.
  • 38. E. Shochat, V. Rom-Kedar and L. Segel, G-CSF control of neutrophils dynamics in the blood., Bull. Math. Biol., 69 (2007), 299–338.

 

This article has been cited by

  • 1. Tyler Cassidy, Morgan Craig, Aaron Goldman, Determinants of combination GM-CSF immunotherapy and oncolytic virotherapy success identified through in silico treatment personalization, PLOS Computational Biology, 2019, 15, 11, e1007495, 10.1371/journal.pcbi.1007495

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