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

Tyler Cassidy; Morgan Craig; Antony R. Humphries

doi:10.3934/mbe.2019270

Mathematical Biosciences and Engineering, 2019, 16(5): 5419-5450. doi: 10.3934/mbe.2019270. Research article Special Issues

Export file:

Format

RIS(for EndNote,Reference Manager,ProCite)
BibTex
Text

Content

Citation Only
Citation and Abstract

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

Tyler Cassidy, Morgan Craig, Antony R. Humphries

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

Received: , Accepted: , Published:

Special Issues: Recent Advances in Mathematical Population Dynamics

Abstract Full Text(HTML) Figure/Table Related pages

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.

Figure/Table

Supplementary

Article Metrics

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.

show all references

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

Reader Comments

your name: * your email: *

Download full text in PDF

Associated material

Metrics