Computational modelling with functional differential equations: Identification, selection, and sensitivity

Applied Numerical Mathematics

Volumn 53, Issue 2-4, 2005, Pages 107-129

  Baker, C T H ^a   Bocharov, G A ^a   Paul, C A H ^a   Rihan, F A ^a  

^a UNIVERSITY OF MANCHESTER   (United Kingdom)

Author keywords

Computation; Data; Differential equations; Identifiability; Information theoretic criteria; Modelling; Objective function; Parametric estimation; Sensitivity; Time lag; Well posedness

Indexed keywords

DATA REDUCTION; DIFFERENTIAL EQUATIONS; INFORMATION THEORY; INTEGRAL EQUATIONS; MATHEMATICAL MODELS; MAXIMUM LIKELIHOOD ESTIMATION; PARAMETER ESTIMATION; SENSITIVITY ANALYSIS;

IDENTIFIABILITY; INFORMATION-THEORETIC CRITERIA; OBJECTIVE FUNCTION; TIME-LAG; WELL-POSEDNESS;

COMPUTATIONAL METHODS;

EID: 14844354032     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apnum.2004.08.014     Document Type: Conference Paper

References (53)

1. H. Akaike A new look at the statistical model identification IEEE Trans. Automatic Control. 19 1974 716 723
2. D.H. Anderson Compartmental Modeling and Tracer Kinetics Lecture Notes in Biomathematics vol. 50 1983 Springer Berlin
3. R.M. Anderson, and R.M. May Infectious Diseases of Humans. Dynamics and Control 1991 Oxford University Press Oxford
4. D.R. Appleton An overview of models of cell proliferation J. Theor. Medicine 1 1997 53 62
5. P. Armitage, G. Berry, and J.N.S. Matthews Statistical Methods in Medical Research fourth ed. 2001 Blackwell Science Oxford
6. S. Audoly, G. Bellu, L. D'Angio, M. Saccomni, and C. Cobelli Global identifiability of nonlinear biological system IEEE Trans. Biomedical Engrg. 48 2001 55 65
7. S. Audoly, L. D'Angio, M. Saccomni, and C. Cobelli Global identifiability of linear compartmental models - A computer algebra algorithm IEEE Trans. Biomedical Engrg. 45 1998 33 47
8. N. Bailey The Mathematical Theory of Infectious Diseases second ed. 1975 Charles Griffin London
9. C.T.H. Baker, G.A. Bocharov, J.M. Ford, P.M. Lumb, S.J. Norton, C.A.H. Paul, T. Junt, P. Krebs, P. Ludewig, Computational approaches to parameter estimation and model selection in immunology, MCCM Rep. 451, University of Manchester, 2004, ISSN 1360-1725
10. C.T.H. Baker, G.A. Bocharov, C.A.H. Paul, and F.A. Rihan Modelling and analysis of time-lags in some basic patterns of cell proliferation J. Math. Biol. 37 1998 341 371
11. C.T.H. Baker, G.A. Bocharov, C.A.H. Paul, F.A. Rihan, Models with delays for cell population dynamics: Identification, selection and analysis - Part I, MCCM Rep. 425, University of Manchester, ISSN 1360-1725, Parts II & III, in preparation
12. C.T.H. Baker, E.I. Parmuzim, Identification of the initial function for delay differential equations: Parts I, II, III, MCCM Rep. 431, 443, & 444, University of Manchester, 2004, ISSN 1360-1275
13. C.T.H. Baker, and C.A.H. Paul Pitfalls in parameter estimation for delay differential equations SIAM J. Sci. Comput. 18 1997 305 314
14. C.T.H. Baker, C.A.H. Paul, Piecewise continuous solution of neutral delay differential equations, MCCM Rep. 417, University of Manchester, 2004, ISSN 1360-1725
15. H.T. Banks, J.A. Burns, and E.M. Cliff Parameter estimation and identification for systems with delays SIAM J. Control Optim. 19 1981 791 828
16. R.B. Banks Growth and Diffusion Phenomena. Mathematical Frameworks and Applications 1994 Springer Berlin
17. Y. Bard Nonlinear Parameter Estimation 1974 Academic Press New York
18. R. Bellman, and K.M. Åström On structural identifiability Math. Biosci. 7 1970 329 339
19. G.A. Bocharov, and F.A. Rihan Numerical modelling in biosciences using delay differential equations J. Comput. Appl. Math. 125 2000 183 199
20. J.A. Borghans, L.S. Taams, M.H.M. Wauben, and R.J. De Boer Competition for antigenic sites during T cell proliferation: A mathematical interpretation of in vitro data Proc. National Acad. Sci. USA 96 1999 10782 10787
21. H. Bozdogan Akaike's information criterion and recent developments in information complexity J. Math. Psych. 44 2000 62 91
22. K.P. Burnham, and D.R. Anderson Model Selection and Inference - A Practical Information-Theoretic Approach 1998 Springer New York
23. K.P. Burnham, and D.R. Anderson Model Selection and Multimodel Inference - A Practical Information-Theoretic Approach second ed. 2002 Springer New York
24. J. Chambers, and M. Hill Fitting nonlinear models: Numerical techniques Biometrika 60 1973 1 13
25. C. Cobelli, and J. DiStefano Parameter and structural identifiability concepts and ambiguities: A critical review and analysis Amer. J. Physiol. 239 1980 R7 R24
26. V.V. Fedorov Theory of Optimal Experiment Design 1972 Academic Press New York
27. K.W. Neves, and A. Feldstein Characterization of jump discontinuities for state dependent delay differential equations J. Math. Anal. 56 1976 689 707
28. P.D. Gingerich Arithmetic or geometric normality of biological variation: An empirical test of theory J. Theor. Biol. 204 2000 201 221
29. K. Gopalsamy Stability and Oscillations in Delay Differential Equations of Population Dynamics 1992 Kluwer Academic Dordrecht
30. F. Hartung, and J. Turi On differentiability of solutions with respect to parameters in state-dependent delay equations J. Differential Equations 135 1997 192 237
31. D.M. Himmelblau, C.R. Jones, and K.B. Bischoff Determination of rate constants for complex kinetic models I & EC Fundamentals 6 1967 539 543
32. R. Hilborn, and M. Mangel The Ecological Detective: Confronting Models with Data 1997 Princeton University Press Princeton, NJ
33. J.C. Hopkins, and R.J. Leipold On the dangers of adjusting the parameter values of mechanism based mathematical models J. Theor. Biol. 183 1996 417 427
34. S. Kullback, and R.A. Leibler On information and sufficiency Ann. Math. Statist. 22 1951 79 86
35. Y. Kuang Delay Differential Equations with Applications in Population Dynamics 1993 Academic Boston
36. E.M. Landaw, and J.J. DiStefano Multiexponential, multicompartmental, and noncompart-mental modeling. II Data analysis and statistical considerations Amer. J. Physiol. 5 1984 665 677
37. T.M. Ludden, S.L. Beal, and L.B. Sheiner Comparison of the Akaike information criterion, the Schwarz criterion and the F test as guides to model selection J. Pharmacokinetics & Biopharmaceutics 22 1994 431 445
38. G.I. Marchuk Adjoint Equations and Analysis of Complex Systems MIA vol. 295 1995 Kluwer Academic Dordrecht
39. G.I. Marchuk Mathematical Modelling of Immune Response in Infectious Diseases 1997 Kluwer Academic Dordrecht
40. I.J. Myung Tutorial on maximum likelihood estimation J. Math. Psych. 47 2003 90 100
41. I.J. Myung M.R. Forster M.W. Browne Special Issue on Model Selection J. Math. Psych. 44 2000 1 231
42. S. Omatu, and J.H. Seinfeld Distributed parameter systems. Theory and applications Oxford Mathematical Monographs 1989 Oxford University Press Oxford
43. C.A.H. Paul Designing efficient software for solving delay differential equations J. Comput. Appl. Math. 125 2000 287 295
44. C.A.H. Paul, A user-guide to Archi - An explicit Runge-Kutta code for solving delay and neutral differential equations and parameter estimation problems, MCCM Rep. 283, University of Manchester, 1997, ISSN 1360-1725
45. H. Rabitz Chemical sensitivity analysis theory with applications to molecular dynamics and kinetics Computers & Chemistry 5 1981 167 180
46. D.A. Ratkowsky Nonlinear Regression Modeling, A Unified Practical Approach 1983 Marcel Dekker New York
47. S.I. Rubinow Mathematical Problems in the Biological Sciences 1973 SIAM Philadelphia, PA
48. G. Schwarz Estimating the dimension of a model Ann. Statist. 6 1978 461 464
49. L.B. Sheiner, and S.L. Beal Pharmacokinetic parameter estimates from several least squares procedures: Superiority of extended least squares J. Pharmacokinetics & Biopharmaceutics 13 1985 185 201
50. A. Tikhonov, and V. Arsenin Solution of Ill-Posed Problems 1977 Wiley New York
51. D. Verotta, and F. Schaedeli Non-linear dynamics models characterizing long-term virological data from AIDS clinical trials Math. Biosci. 176 2002 163 183
52. E.O. Voit Computational Analysis of Biochemical Systems. A Practical Guide for Biochemists and Molecular Biologists 2000 Cambridge University Press Cambridge
53. L.M.M. Wolters, B.E. Hansen, H.G.M. Niesters, R.S.L. Levi-Drummer, A.U. Neumann, S.W. Schalm, and R.A. de Man The influence of baseline characteristics on viral dynamic parameters in chronic hepatitis B patients treated with lamivudine J. Hepatology 37 2002 253 258