Russian version English version
Volume 18   Issue 1   Year 2023
Olga Krivorotko1,2, Sergey Kabanikhin1, Victoriya Petrakova3

The Identifiability of Mathematical Models in Epidemiology: Tuberculosis, HIV, COVID-19

Mathematical Biology & Bioinformatics. 2023;18(1):177-214.

doi: 10.17537/2023.18.177.

References

  1. Kabanikhin S.I. Definitions and examples of inverse and ill-posed problems. Journal of Inverse and Ill-posed Problems. 2008;16(4):317-357. doi: 10.1515/JIIP.2008.019
  2. Avdeenko T.V., Gorskii V.G. Postroenie dinamicheskikh modelei v prostranstve sostoianii: Analiz strukturnoi identifitsiruemosti (Construction of dynamic models in state space: Analysis of structural identifiability): monograph. Novosibirsk; 2007. 292 p. (in Russ.).
  3. Miao H., Xia X., Perelson A.S., Wu H. On identifiability of nonlinear ODE models and applications in viral dynamics. SIAM Review. 2011;53(1):3-39. doi: 10.1137/090757009
  4. Hamelin F., Iggidr A., Rapaport A., Sallet G. Observability, Identifiability and Epidemiology. A survey. Arxiv. https://arxiv.org/abs/2011.12202 (accessed 07.10.2023).
  5. Glover K., Willems J. Parametrization of linear dynamical systems: canonical forms and identifiability. IEEE Trans on Automatic Control. 1974;19:640-646. doi: 10.1109/TAC.1974.1100711
  6. Bellman R., Astrom K. On structural identifiability. Math. Biosci. 1970;30(4):65-74.
  7. Grewal M., Glover K. Identifiability of linear and nonlinear dynamical systems. IEEE Trans on Automatic Control. 1976;21(6):833-837. doi: 10.1109/TAC.1976.1101375
  8. Cobelli C., DiStefano J. Parameter and structural identifiability concepts and ambiguities: a Critical review and analysis. Amer. J. Physiology-Regulatory, Integrative and Comparative Physiology. 1980;3:369-380.
  9. Tunali T., Tarn T. New Results for Identifiability of Nonlinear Systems. IEEE Transactions on Automatic Control. 1987;15:45-51.
  10. Vajda S. Identifiability of first order reaction systems. Reaction Kinetics and Catalysis Letters. 1979;11(1):39-43. doi: 10.1007/BF02098331
  11. Audoly S., D'angio L. On the identifiability of linear compartmental system: a revisited transfer function approach based on topological properties. Mathematical Biosciences. 1983;66(2):201-208. doi: 10.1016/0025-5564(83)90089-5
  12. Shcherbak V.F. IEEE Transactions on Automatic Control. 1983;34:105-108 (in Russ.).
  13. Levakov A.A. Identifiability of nonlinear systems. Differ. Uravn. 1983;19(6):1074-1078.
  14. Karelin V.V. Voprosy mekhaniki i protsessov upravleniia (Questions of mechanics and control processes). 1982;359(5) (in Russ.).
  15. Avdeenko T.V., Kargin S.A. Analysis of the identifiability of linear dynamic models using separators of a parametric space. J. Appl. Industr. Math. 2008;2(4):464-476. doi: 10.1134/S1990478908040030
  16. Lomov A.A. Joint identifiability of parameters of linear dynamic equations of a plant and disturbances. J. Math. Sci. 2017;221(6):857-871. doi: 10.1007/s10958-017-3274-y
  17. Yakimenko A.A. On the question of identification of simultaneous equations models. Proceedings of BSTU. 2022;3(2):10-13.(in Russ.).
  18. Saccomani M., Cobeli C. Qualitative Experiment Design in Physiological System Identification. IEEE Control System. 1992;3(12):18-23. doi: 10.1109/37.168813
  19. Brown R. Identifiability: role in design of pharmacokinetic experiments. IEEE Transactions on Biomedical Engineering. 1982;14(3):31-41. doi: 10.1109/TBME.1982.324963
  20. Saccomani M., Audoly S., Angio L. An Effective Automatic Procedure for Testing Parameter Identifiability of HIV/AIDS Models. Bull Math Biol. 1978;73:1734-1753. doi: 10.1007/s11538-010-9588-2
  21. Bellu G., Saccomani M., Audoly S. Comput Methods Programs. Biomed. Mathematical Biosciences. 2007;5:67-75.
  22. Meshkat N., Anderson C., III J.D. Alternative to Ritt's Pseudodivision for finding the input-output equations of multi-output models. Mathematical Biosciences. 2012;239(1):117-123. doi: 10.1016/j.mbs.2012.04.008
  23. Meshkat N., Er-zhen Kuo C., DiStefano J. Finding and Using Identifiable Parameter Combinations in Nonlinear Dynamic Systems Biology Models and COMBOS: A NovelWeb Implementation. Plos One. 2014;9(10). Article No. e110261. doi: 10.1371/journal.pone.0110261
  24. Carson E., Cobelli C. Modelling Methodology for Physiology and Medicine. New-York: Academic Press, 2001.
  25. Carson E., Cobelli C. Introduction to Modelling in Physiology and Medicine. New-York: Academic Press, 2008. doi: 10.1016/B978-012160240-6.50002-8
  26. Raue A., Karlsson J., Saccomani M.P., Jirstrand M., Timmer J. Comparison of approaches for parameter identifiability analysis of biological systems. Bioinformatics. 2014;30(10):1440-1448. doi: 10.1093/bioinformatics/btu006
  27. Kabanikhin S.I., Voronov D.A., Grodz A.A., Krivorotko O.I. Identifiability of mathematical models in medical biology. Vavilov Journal of Genetics and Breeding. 2015;19(6):738-744.(in Russ.). doi: 10.18699/VJ15.097
  28. Villaverde A.F., Barreiroc A. Identifiability of large nonlinear biochemical networks. MATCH Commun. Math. Comput. Chem. 2016;76:259-296.
  29. Saltelli A., Chan K., Scott M. Sensitivity Analysis. Wiley, Hoboken, 2000.
  30. Raue A., Kreutz C., Maiwald T. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics. 2009;25(15):1923-1929. doi: 10.1093/bioinformatics/btp358
  31. Adams B.M., Banks H.T., Davidian M., Hee-Dae Kwon, Tran H.T., Wynne S.N., Rosenberg E.S. On Hiv dynamics: modeling, data analysis, and optimal treatment protocols. Journal of Computational and Applied Mathematics. 2005;184(1):10-49. doi: 10.1016/j.cam.2005.02.004
  32. Bellu G., Saccomani M.P., Audoly S., D’Angiò L. DAISY: A new software tool to test global identifiability of biological and physiological systems. Computer Methods and Programs in Biomedicine. 2007;88(1):52-61. doi: 10.1016/j.cmpb.2007.07.002
  33. Krivorotko O.I., Andornaya D.V., Kabanikhin S.I. Sensitivity analysis and practical identifiability of some mathematical models in biology. J. Appl. Industr. Math. 2020;14(1):115-130. doi: 10.1134/S1990478920010123
  34. Raue A., Becker V., Klingmüller U., Timmer J. Identifiability and observability analysis for experimental design in nonlinear dynamical models. Chaos. 2010;20(4):045105. doi: 10.1063/1.3528102
  35. Kermack W.O., McKendrick A.G. A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character. 1927;115(772):700-721. doi: 10.1098/rspa.1927.0118
  36. Magal P., Webb G. The parameter identification problem for SIR epidemic models: identifying unreported cases. Journal of Mathematical Biology. 2018;77(6-7):1629-1648. doi: 10.1007/s00285-017-1203-9
  37. Evans N.D., White L.J., Chapman M.J., Godfrey K.R., Chappell M.J. The structural identifiability of the susceptible infected recovered model with seasonal forcing. Mathematical Biosciences. 2005;194(2):175-197. doi: 10.1016/j.mbs.2004.10.011
  38. Diop S., Fliess M. Nonlinear observability, identifiability, and persistent trajectories. In: [1991] Proceedings of the 30th IEEE Conference on Decision and Control. 1991;1:714-719.
  39. Kolchin E.R. Differential Algebra, Algebraic Groups. Cambridge: Academic Press, 1973.
  40. Ljung L., Glad T. On global identifiability for arbitrary model parametrizations. Automatica. 1994;30(2):265-276. doi: 10.1016/0005-1098(94)90029-9
  41. Godunov S.K., Antonov A.G., Kiriliuk O.P., Kostin V.I. Garantirovannaia tochnost' resheniia sistem lineinykh uravnenii v evklidovykh prostranstvakh (Guaranteed accuracy of solving systems of linear equations in Euclidean spaces). Novosibirsk; 1992 (in Russ.). doi: 10.1007/978-94-011-1952-8_2
  42. Tcheverda V.A., Kostin V.I. r-pseudoinverse for a compact operator. Sib. Èlektron. Mat. Izv. 2010;7:258-282.(in Russ.).
  43. Banks H.T., Kabanikhin S.I., Krivorotko O.I., Yermolenko D.V. A numerical algorithm for constructing an individual mathematical model of HIV dynamics at cellular level. Journal of Inverse and Ill-posed Problems. 2018;26(6):859-873. doi: 10.1515/jiip-2018-0019
  44. Kabanihin S.I., Krivorotko O.I., Ermolenko D.V., Kashtanova V.N., Latyshenko V.A. Inverse problems of immunology and epidemiology. Eurasian Journal of Mathematical and Computer Applications. 2017;5(2):14-35. doi: 10.32523/2306-6172-2017-5-2-14-35
  45. Wieland F.-G., Hauber A.L., Rosenblatt M., Tönsing C., Timmer J. On structural and practical identifiability. Current Opinion in Systems Biology. 2021;25:60-69. doi: 10.1016/j.coisb.2021.03.005
  46. Rodriguez-Fernandez M., Egea J.A., Banga J. R. Novel metaheuristic for parameter estimation in nonlinear dynamic biological systems. BMC Bioinformatics. 2006;7(483). doi: 10.1186/1471-2105-7-483
  47. Metropolis N., Ulam S. The Monte Carlo method. Journal of the American Statistical Association. 1949;44(247):335-341. doi: 10.1080/01621459.1949.10483310
  48. Cacuci D.G. Sensitivity and uncertainty analysis: theory. New York: Chapman & Hall/CRC, 2003. 304 p. doi: 10.1201/9780203498798
  49. Latyshenko V., Krivorotko O.I., Kabanikhin S.I, Zhang S., Kashtanova V.N., Yermolenko D.V. Identifiability analysis of inverse problems in biology. In: Proceedings of the 2nd International Conference on Computational Modeling, Simulation and Applied Mathematics (CMSAM2017). 2017:567-571. doi: 10.12783/dtcse/cmsam2017/16435
  50. Quaiser T., Monnigmann M. Systematic identifiability testing for unambiguous mechanistic modeling {textendash application to JAK-STAT, MAP kinase, and NF-κ B signaling pathway models. BMC Systems Biology. 2009;3(1). doi: 10.1186/1752-0509-3-50
  51. Cukier R.I., Fortuin C.M., Schuler K.E., Petschek A.G., Schaibly J.H. Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. The Journal of Chemical Physics. 1973;59:3873-3878. doi: 10.1063/1.1680571
  52. Sobol' I.M. On sensitivity estimation for nonlinear mathematical models. Matem. Mod. 1990;2(1):112-118 (in Russ.).
  53. Saltelli A., Annoni P., Azzini I., Campolongo F., Ratto M., Tarantola S. Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index. Computer Physics Communications. 2010;180:259-270. doi: 10.1016/j.cpc.2009.09.018
  54. Unlu E., Leger H., Motornyi O., Rukubayihunga A., Ishacian T., Chouiten M. Epidemic analysis of COVID-19 Outbreak and Counter-Measures in France. Medrxiv.  doi: 10.1101/2020.04.27.20079962
  55. Krivorotko O.I., Kabanikhin S.I., Zyatkov N.Yu., Prikhodko A.Yu., Prokhoshin N.M., Shishlenin M.A. Mathematical modeling and forecasting of COVID-19 in Moscow and Novosibirsk region. Num. Anal. Appl . 2020;13(4):332-348. doi: 10.1134/S1995423920040047
  56. Krivorotko O.I., Zyatkov N.Y. Data-driven regularization of inverse problem for SEIR-HCD model of COVID-19 propagation in Novosibirsk region. Eurasian Journal of Mathematical and Computer Applications. 2022;10(1):51-68. doi: 10.32523/2306-6172-2022-10-1-51-68
  57. Roeger L.I., Feng Z., Castillo-Chavez C. Modeling TB and HIV co-infections. Mathematical Biosciences and Engineering. 2009;6(4):815-837. doi: 10.3934/mbe.2009.6.815
  58. Kolmogorov A.N., Petrovskii I.G., Piskunov N.S. Biulleten' Moskovskogo gosudarstvennogo universiteta (Bulletin of Moscow State University). 1937;1(6):1-26.(in Russ.).
  59. Andrianakis I., Vernon I.R., McCreesh N., McKinley T.J., Oakley J.E., Nsubuga R.N., Goldstein M., White R.G. Bayesian history matching of complex infectiousdisease models using emulation: a tutorial and a case study on HIV in Uganda. PLOS Computational Biology. 2015;11(1). Article No. e1003968. doi: 10.1371/journal.pcbi.1003968
  60. Raue A., Karlsson J., Saccomani M.P., Jirstrand M., Timmer J. Comparison of approaches for parameter identifiability analysis of biological systems. Bioinformatics. 2014;30(10):1440-1448. doi: 10.1093/bioinformatics/btu006
  61. Krivorotko O.I., Kabanikhin S.I., Sosnovskaya M.I., Andornaya D.V. Sensitivity and identifiability analysis of COVID-19 pandemic models. Vavilov Journal of Genetics and Breeding. 2021;25(1):82-91. doi: 10.18699/VJ21.010
  62. Krivorotko O., Sosnovskaia M., Vashchenko I., Kerr C., Lesnic D. Agent-based modeling of COVID-19 outbreaks for New York state and UK: Parameter identification algorithm. Infect. Dis. Model. 2022;7(1):30-44. doi: 10.1016/j.idm.2021.11.004
  63. McKay M.D., Beckman R.J., Conover W.J. A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics. 1979;21(2):239-254. doi: 10.2307/1268522
  64. Malouf R. A comparison of algorithms for maximum entropy parameter estimation. In: Proceedings of the Sixth Conference on Natural Language Learning. 2002;20:49-55. doi: 10.3115/1118853.1118871
  65. Pukelsheim F. The three sigma rule. The American Statistician. 1994;48(2):88-91. doi: 10.1080/00031305.1994.10476030
  66. Lee W., Liu S., Tembine H., Li W., Osher S. Controlling Propagation of epidemics via mean-field control. SIAM J. Appl. Math. 2021;81(1):190–-207. doi: 10.1137/20M1342690
  67. Petrakova V., Krivorotko O. Mean field game for modeling of covid-19 spread. Journal of Mathematical Analysis and Applications. 2022;514(1). Article No. 126271. doi: 10.1016/j.jmaa.2022.126271
  68. Petrakova V., Shaydurov V. MFG SIRV-D Model with Managed Rates of Epidemic Spread (in print).
  69. Bensoussan A., Frehse J., Yam Ph. Mean Field Games and Mean Field Type Control Theory. New York: Springer, 2013. 128 p. doi: 10.1007/978-1-4614-8508-7
  70. Saltelli A., Tarantola S., Chan K.-S. A quantitative model-independent method for global sensitivity analysis of model output. Technometrics. 1999;41:39-56. doi: 10.1080/00401706.1999.10485594
  71. Cukier R.I., Fortuin C.M., Shuler K.E., Petschek A.G., Schaibly J.H. Study of the Sensitivity of Coupled Reaction Systems to Uncertainties in Rate Coefficients. I. Theory. The Journal of Chemical Physics. 1973;59:3873-3878. doi: 10.1063/1.1680571
  72. Saltelli A., Tarantola S., Campolongo F. Sensitivity Analysis as an Ingredient of Modeling. Statistical Science. 2000;15(4):377-395. doi: 10.1214/ss/1009213004
  73. Schaibly J.H., Shuler K.E. Study of the Sensitivity of Coupled Reaction Systems to Uncertainties in Rate Coefficients. Part II, Applications. Journal of Chemical Physics. 1973;59:3879-3888. doi: 10.1063/1.1680572
  74. Frey H., Patil S. Identification and Review of Sensitivity Analysis Methods. Risk Analysis. 2002;22(3):553-578. doi: 10.1111/0272-4332.00039
  75. Tembine H. COVID-19: Data-Driven Mean-Field-TypeGame Perspective. Games. 2020;11(151):107. doi: 10.3390/g11040051

 

Table of Contents Original Article
Math. Biol. Bioinf.
2023;18(1):177-214
doi: 10.17537/2023.18.177
published in Russian

Abstract (rus.)
Abstract (eng.)
Full text (rus., pdf)
References

 

  Copyright IMPB RAS © 2005-