Identifiability and numerical algebraic geometry

Autoři: Daniel J. Bates aff001;  Jonathan D. Hauenstein aff002;  Nicolette Meshkat aff003
Působiště autorů: Department of Mathematics, United States Naval Academy, Annapolis, MD, United States of America aff001;  Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN, United States of America aff002;  Department of Mathematics and Computer Science, Santa Clara University, Santa Clara, CA, United States of America aff003
Vyšlo v časopise: PLoS ONE 14(12)
Kategorie: Research Article
doi: 10.1371/journal.pone.0226299


A common problem when analyzing models, such as mathematical modeling of a biological process, is to determine if the unknown parameters of the model can be determined from given input-output data. Identifiable models are models such that the unknown parameters can be determined to have a finite number of values given input-output data. The total number of such values over the complex numbers is called the identifiability degree of the model. Unidentifiable models are models such that the unknown parameters can have an infinite number of values given input-output data. For unidentifiable models, a set of identifiable functions of the parameters are sought so that the model can be reparametrized in terms of these functions yielding an identifiable model. In this work, we use numerical algebraic geometry to determine if a model given by polynomial or rational ordinary differential equations is identifiable or unidentifiable. For identifiable models, we present a novel approach to compute the identifiability degree. For unidentifiable models, we present a novel numerical differential algebra technique aimed at computing a set of algebraically independent identifiable functions. Several examples are used to demonstrate the new techniques.

Klíčová slova:

Blood – Computing methods – Interpolation – Polynomials – Vector spaces – Algebraic geometry – Complex numbers – Compartment models


1. Bellu G, Saccomani MP, Audoly S, D’Angiò L. DAISY: A new software tool to test global identifiability of biological and physical systems. Computers in Biology and Medicine 2007;88:52–61.

2. Meshkat N, Eisenberg M, DiStefano JJ III. An algorithm for finding globally identifiable parameter combinations of nonlinear ODE models using Gröbner Bases. Math. Biosci. 2009;222:61–72. doi: 10.1016/j.mbs.2009.08.010 19735669

3. Bellman R, Astrom KJ. On structural identifiability. Math. Biosci. 1970;7:329–339. doi: 10.1016/0025-5564(70)90132-X

4. Chappell MJ, Gunn RN. A procedure for generating locally identifiable reparameterisations of unidentifiable non-linear systems by the similarity transformation approach. Math. Biosci. 1998;148:21–41. doi: 10.1016/s0025-5564(97)10004-9 9597823

5. Evans ND, Chappell MJ. Extensions to a procedure for generating locally identifiable reparameterisations of unidentifiable systems. Math. Biosci. 2000;168:137–159. doi: 10.1016/s0025-5564(00)00047-x 11121562

6. Bates DJ, Hauenstein JD, Sommese AJ, Wampler CW. Numerically solving polynomial systems with Bertini. Philadelphia: SIAM; 2013.

7. Sommese AJ, Wampler CW. The numerical solution of polynomial systems arising in engineering and science. Singapore: World Scientific, 2005.

8. Bearup DJ, Evans ND, Chappell MJ. The input-output relationship approach to structural identifiability analysis. Computer Methods and Programs in Biomedicine 2013:109:171–181. doi: 10.1016/j.cmpb.2012.10.012 23228562

9. Boulier F. Differential Elimination and Biological Modelling. Radon Series Comp. Appl. Math 2007;2:111–139.

10. Evans ND, Moyse H, Lowe D, Briggs D, Higgins R, Mitchell D, et al. Structural identifiability of surface binding reactions involving heterogenous analyte: application to surface plasmon resonance experiments. Automatica 2012;49:48–57. doi: 10.1016/j.automatica.2012.09.015

11. Ljung L, Glad T. On global identifiability for arbitrary model parameterization. Automatica 1994;30(2):265–276. doi: 10.1016/0005-1098(94)90029-9

12. Meshkat N, Anderson C, DiStefano JJ III. Alternative to Ritt’s Pseudodivision for finding the input-output equations of multi-output models. Math. Biosci. 2012;239:117–123. doi: 10.1016/j.mbs.2012.04.008 22626896

13. Ollivier F. Le probleme de l’identifiabilite structurelle globale: etude theoretique, methodes effectives and bornes de complexite. PhD thesis 1990; Ecole Polytechnique.

14. Pohjanpalo H. System identifiability based on the power series expansion of the solution. Math. Biosci. 1978;41:21–33. doi: 10.1016/0025-5564(78)90063-9

15. Saccomani MP, Audoly S, Bellu G, D’Angiò L. A new differential algebra algorithm to test identifiability of nonlinear systems with given initial conditions. Proceedings of the 40th IEEE Conference on Decision and Control 2001;3108-3113.

16. Audoly S, Bellu G, D’Angiò L, Saccomani MP, Cobelli C. Global identifiability of nonlinear models of biological systems. IEEE Trans. on Biomed. Eng. 2001;48:55–65.

17. Meshkat N, Sullivant S, Eisenberg M. Identifiability results for several classes of linear compartment models. Bull. of Math. Bio. 2015;77(8):1620–1651. doi: 10.1007/s11538-015-0098-0

18. Berman M, Schoenfeld R. Invariants in experimental data on linear kinetics and the formulation of models, Journal of Applied Physics 1956;27:1361–1370. doi: 10.1063/1.1722264

19. Berman M, Shahn E, Weiss MF. Some formal approaches to the analysis of kinetic data in terms of linear compartmental systems. Biophysical Journal 1962;2:289–316. doi: 10.1016/s0006-3495(62)86856-8 13867976

20. DiStefano JJ III. Dynamic Systems Biology Modeling and Simulation. London: Elsevier; 2014.

21. Mulholland RJ, Keener MS. Analysis of linear compartment models for ecosystems. Journal of Theoretical Biology 1974;44:105–116. doi: 10.1016/s0022-5193(74)80031-7 4822909

22. Wagner JG. History of pharmacokinetics. Pharmacology & Therapeutics 1981;12:537–562. doi: 10.1016/0163-7258(81)90097-8

23. Meshkat N, Sullivant S. Identifiable reparameterizations of linear compartment models. J. Symb. Comp. 2014;63:46–67. doi: 10.1016/j.jsc.2013.11.002

24. Saccomani MP, Audoly S, Bellu G, D’Angiò L. Examples of testing global identifiability of biological and biomedical models with the DAISY software. Computers in Biology and Medicine 2010;40:402–407. doi: 10.1016/j.compbiomed.2010.02.004 20185123

25. Karlsson J, Anguelova M, Jirstrand M. An Efficient Method for Structural Identifiability Analysis of Large Dynamic Systems. IFAC Proceedings Volumes 2012;45(16):941–946. doi: 10.3182/20120711-3-BE-2027.00381

26. Reid GJ, Lin P, Wittkopf AD. Differential elimination-completion algorithms for DAE and PDAE. Stud. Appl. Math. 2001;106(1):1–45. doi: 10.1111/1467-9590.00159

27. Hauenstein JD, Sommese AJ. Witness sets of projections. Appl. Math. and Comput. 2010;217(7):3349–3354.

28. Saccomani MP, Audoly S, D’Angiò L. Parameter identifiability of nonlinear systems: the role of initial conditions. Automatica 2003;39:619–632. doi: 10.1016/S0005-1098(02)00302-3

29. Walter E. Identifiability of state space models. Lecture Notes in Biomathematics Volume 46. Berlin: Springer, 1982.

30. Meshkat N, Kuo CE, DiStefano JJ III. On finding and using identifiable parameter combinations in nonlinear dynamic systems biology models and COMBOS: A novel web implementation. PLoS ONE 2014;9(10):e110261. doi: 10.1371/journal.pone.0110261 25350289

31. Bates DJ, Hauenstein JD, Sommese AJ, Wampler CW. Bertini: Software for numerical algebraic geometry. Available for download at

32. Hauenstein JD, Oeding L, Ottaviani G, Sommese AJ. Homotopy techniques for tensor decomposition and perfect identifiability. J. Reine Angew. Math. 2019;2019(753), 1–22. doi: 10.1515/crelle-2016-0067

33. Sommese AJ, Verschelde J, Wampler CW. Using monodromy to decompose solution sets of polynomial systems into irreducible components. NATO Science Series 2001;36:297–315.

34. Hauenstein JD, Rodriguez JI. Multiprojective witness sets and a trace test. Adv. Geom., to appear.

35. Sommese AJ, Verschelde J, Wampler CW. Symmetric functions applied to decomposing solution sets of polynomial systems. SIAM J. Numer. Anal. 2002;40(6):2026–2046. doi: 10.1137/S0036142901397101

36. Leykin A, Rodriguez JI, Sottile F. Trace test. Arnold Math. J. 2018;4(1):113–125. doi: 10.1007/s40598-018-0084-3

37. Brake DA, Hauenstein JD, Liddell AC. Decomposing solution sets of polynomial systems using derivatives. Mathematical Software—ICMS 2016, LNCS 2016;9725:127–135.

38. Duff T, Hill C, Jensen A, Lee K, Leykin A, Sommars J. Solving polynomial systems via homotopy continuation and monodromy. IMA J. Num. An. 2019;39(3):1421–1446. doi: 10.1093/imanum/dry017

39. Miao H, Xia X, Perelson A, Wu H. On identifiability of nonlinear ODE models and applications in viral dynamics. SIAM Review 2011;53(1):3–39. doi: 10.1137/090757009 21785515

40. Bates DJ, Hauenstein JD, McCoy T, Peterson C, Sommese AJ. Recovering exact results from inexact numerical data in algebraic geometry. Experimental Math. 2013;22(1):38–50. doi: 10.1080/10586458.2013.737640

41. Manrai AK, Gunawardena J. The geometry of multisite phosphorylation. Biophys. J. 2008;95:5533–5543. doi: 10.1529/biophysj.108.140632 18849417

42. Maple documentation.

Článek vyšel v časopise


2019 Číslo 12