Optimal perturbations of systems with delayed argument for control of dynamics of infectious diseases based on multicomponent actions

G. A. Bocharov ; Yu. M. Nechepurenko ; M. Yu. Khristichenko ; D. S. Grebennikov

Contemporary Mathematics. Fundamental Directions, Differential and functional differential equations, Tome 63 (2017) no. 3, pp. 392-417.

Résumé

In this paper, we apply optimal perturbations to control mathematical models of infectious diseases expressed as systems of nonlinear differential equations with delayed argument. We develop the method for calculation of perturbations of the initial state of a dynamical system with delayed argument producing maximal amplification in the given local norm taking into account weights of perturbation components. For the model of experimental virus infection, we construct optimal perturbation for two types of stationary states, with low or high virus load, corresponding to different variants of chronic virus infection flow.

Bibliographie

[1] R. Bellman, K. L. Cooke, Differential-Difference Equations, Mir, Moscow, 1967, (Russian translation) | MR

[2] G. A. Bocharov, G. I. Marchuk, “Applied problems of mathematical modelling in immunology”, J. Comput. Math. Math. Phys., 40 (2000), 1905–1920 (in Russian) | MR | Zbl

[3] V. G. Dement'ev, V. I. Lebedev, Yu. M. Nechepurenko, “Spectral analysis of a model of nuclear reactor with delayed neutrons”, Algorithms and Programs for Neutron-Physical Calculations of Nuclear Reactors, FEI, Obninsk, 1999, 143–150 (in Russian)

[4] G. I. Marchuk, Mathematical Models in Immunology, Nauka, Moscow, 1980 (in Russian) | MR

[5] G. I. Marchuk, Mathematical Models in Immunology. Numerical Methods and Experiments, Nauka, Moscow, 1991 (in Russian) | MR

[6] G. I. Marchuk, E. P. Berbentsova, Acute Pneumonias. Immunology, Severity Estimate, Clinics, Treatment, Nauka, Moscow, 1989 (in Russian)

[7] B. T. Polyak, M. V. Khlebnikov, P. S. Shcherbakov, Control of Linear Systems under Outer Perturbations, LENAND, Moscow, 2014 (in Russian)

[8] V. A. Chereshnev, G. A. Bocharov, A. V. Kim, S. I. Bazhan, I. A. Gaynova, A. N. Krasovskiy, N. G. Shmagel', A. V. Ivanov, M. A. Safronov, R. M. Tret'yakova, Introduction to Problems of Modelling and Control of Dynamics of HIV Infection, ANO Izhevskiy in-t komp. issl., M.–Izhevsk, 2016 (in Russian)

[9] Anderson E., Bai Z., Bischof C., Blackford S., Demmel J., Dongarra J., Du Croz J., Greenbaum A., Hammarling S., McKenney A., Sorensen D., LAPACK users guide, SIAM, Philadelphia, 1999 | MR

[10] Bocharov G. A., “Modelling the dynamics of LCMV infection in mice: conventional and exhaustive CTL responses”, J. Theor. Biol., 192:3 (1998), 283–308 | DOI

[11] Bocharov G. A., Marchuk G. I., Romanyukha A. A., “Numerical solution by LMMs of stiff delay differential systems modelling an immune response”, Numer. Math., 73:2 (1996), 131–148 | DOI | MR

[12] Bocharov G. A., Nechepurenko Yu. M., Khristichenko M. Yu., Grebennikov D. S., “Maximum response perturbation-based control of virus infection model with time-delays”, Russ. J. Numer. Anal. Math. Modelling, 32 (2017), 275–291 | DOI | MR

[13] Boiko A. V., Dovgal A. V., Grek G. R., Kozlov V. V., Physics of Transitional Shear Flows: Instability and Laminar-Turbulent Transition in Incompressible Near-Wall Shear Layers, Springer, Berlin, 2011 | MR

[14] Boiko A. V., Nechepurenko Y. M., Sadkane M., “Fast computation of optimal disturbances for duct flows with a given accuracy”, Comput. Math. Math. Phys., 50:11 (2010), 1914–1924 | DOI | MR

[15] Boiko A. V., Nechepurenko Yu. M., Sadkane M., “Computing the maximum amplification of the solution norm of differential-algebraic systems”, Comput. Math. Model., 23:2 (2012), 216–227 | DOI | MR

[16] Chereshnev V. A., Bocharov G., Bazhan S., Bachmetyev B., Gainova I., Likhoshvai V., Argilaguet J. M., Martinez J. P., Rump J. A., Mothe B. et al., “Pathogenesis and treatment of {HIV} infection: the cellular, the immune system and the neuroendocrine systems perspective”, Int. Rev. Immunol., 32:3 (2013), 282–306 | DOI

[17] Ciurea A., Klenerman P., Hunziker L., Horvath E., Odermatt B., Ochsenbein A. F., Hengartner H., Zinkernagel R. M., “Persistence of lymphocytic choriomeningitis virus at very low levels in immune mice”, Proc. Natl. Acad. Sci. USA, 96:21 (1999), 11964–11969 | DOI

[18] Crawford A., Angelosanto J. M., Kao C., Doering T. A., Odorizzi P. M., Barnett B. E., Wherry E. J., “Molecular and transcriptional basis of CD4+T cell dysfunction during chronic infection”, Immunity, 40:2 (2014), 289–302 | DOI

[19] Csete M. E., Doyle J. C., “Reverse engineering of biological complexity”, Science, 295:5560 (2002), 1664–1669 | DOI

[20] Germain R. N., Meier-Schellersheim M., Nita-Lazar A., Fraser I. D., “Systems biology in immunology: a computational modeling perspective”, Annu. Rev. Immunol., 29 (2011), 527–585 | DOI

[21] Golub G. H., Van Loan C. F., Matrix Computations, The John Hopkins University Press, Baltimore, 1996 | MR

[22] Gurdasani D., Iles L., Dillon D. G., Young E. H., Olson A. D., Naranbhai V., Fidler S., Gkrania-Klotsas E., Post F. A., Kellam P. et al., “A systematic review of definitions of extreme phenotypes of HIV control and progression”, AIDS, 28:2 (2014), 149–162 | DOI

[23] Hairer E., Wanner G., Solving ordinary differential equations, Springer-Verlag, Berlin, 1996 | MR

[24] Henningson D. S., Reddy S. C., “On the role of linear mechanisms in transition to turbulence”, Phys. Fluids A, 6:3 (1994), 1396–1398 | DOI | MR

[25] Higham A. V., “The scaling and squaring method for the matrix exponential revisited”, SIAM J. Matrix Anal. Appl., 26:4 (2005), 1179–1193 | DOI | MR

[26] Kent S. J., Reece J. C., Petravic J., Martyushev A., Kramski M., De Rose R., Cooper D. A., Kelleher A. D., Emery S., Cameron P. U. et al., “The search for an HIV cure: tackling latent infection”, Lancet Infect. Dis., 13:7 (2013), 614–621 | DOI

[27] Kitano H., “Systems biology: a brief overview”, Science, 295:5560 (2002), 1662–1664 | DOI

[28] Kitano H., “Biological robustness”, Nat. Rev. Genet., 5:11 (2004), 826–837 | DOI

[29] Kitano H., Biological robustness in complex host-pathogen systems, Birkhäuser, Basel, 2007

[30] Kotsarev A., Lebedev V., Shishkov L., Nechepurenko Yu., Dementiev V., “Spectral analysis of VVER-1000 reactor model at high negative reactivities”, Proceedings of the Ninth Symposium of AER (4–8 October, Slovakia, 1999), Kiadja a KFKI Atomenergia Kutato Intezet, Budapest, 1999, 453–468

[31] Ludewig B., Stein J. V., Sharpe J., Cervantes-Barragan L., Thiel V., Bocharov G., “A global imaging view on systems approaches in immunology”, Eur. J. Immunol., 42 (2012), 3116–3125 | DOI

[32] Luzyanina T., Engelborghs K., Ehl S., Klenerman P., Bocharov G., “Low level viral persistence after infection with LCMV: a quantitative insight through numerical bifurcation analysis”, Math. Biosci., 173:1 (2001), 1–23 | DOI | MR

[33] Marchuk G. I., Mathematical Modelling of Immune Response in Infectious Diseases, Kluwer, Dordrecht, 1997 | MR

[34] Marchuk G. I., Nisevich N. I. (ed.), Mathematical Methods in Clinical Practice, Pergamon Press Ltd., Oxford, 1980

[35] Moskophidis D., Lechner F., Pircher H., Zinkernagel R. M., “Virus persistence in acutely infected immunocompetent mice by exhaustion of antiviral cytotoxic effector T cells”, Nature, 362 (1993), 758–758 | DOI

[36] Nechepurenko Yu. M., Sadkane M., “A low-rank approximation for computing the matrix exponential norm”, SIAM J. Matrix. Anal. Appl., 32:2 (2011), 349–363 | DOI | MR

[37] Nechepurenko Yu. M., Sadkane M., “Computing humps of the matrix exponential”, J. Comput. Appl. Math., 319 (2017), 87–96 | DOI | MR

[38] Reshotko E., “Transient growth: A factor in bypass transition”, Phys. Fluids, 13:5 (2001), 1067–1075 | DOI | MR

[39] Takeuchi Y., Adachi N., Tokumaru H., “The stability of generalized Volterra equations”, J. Math. Anal. Appl., 62 (1978), 453–473 | DOI | MR

[40] Weyl H., The Classical Groups: Their Invariants and Representations, Princeton University Press, Princeton, 2016 | MR