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Optimal Perturbations of Systems with Delayed Independent Variables for Control of Dynamics of Infectious Diseases Based on Multicomponent Actions

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Abstract

In this paper, we apply optimal perturbations to control mathematical models of infectious diseases expressed as systems of nonlinear differential equations with delayed independent variables. We develop the method for calculation of perturbations of the initial state of a dynamical system with delayed independent variable 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 viral load, corresponding to different variants of chronic virus infection flow.

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References

  1. E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, Dongarra J., J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users Guide, SIAM, Philadelphia (1999).

  2. R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York–London (1963).

    MATH  Google Scholar 

  3. G. A. Bocharov, “Modelling the dynamics of LCMV infection in mice: conventional and exhaustive CTL responses,” J. Theor. Biol., 192, No. 3, 283–308 (1998).

    Article  MathSciNet  Google Scholar 

  4. G. A. Bocharov and G. I. Marchuk, “Applied problems of mathematical modelling in immunology,” Zh. Vychisl. Mat. Mat. Fiz., 40, 1905–1920 (2000).

    MATH  Google Scholar 

  5. G. A. Bocharov, G. I. Marchuk, and A. A. Romanyukha, “Numerical solution by LMMs of stiff delay differential systems modelling an immune response,” Numer. Math., 73, No. 2, 131–148 (1996).

    Article  MathSciNet  Google Scholar 

  6. G. A. Bocharov, Yu. M. Nechepurenko, M. Yu. Khristichenko, and D. S. Grebennikov, “Maximum response perturbation-based control of virus infection model with time-delays,” Russ. J. Numer. Anal. Math. Model., 32, 275–291 (2017).

    Article  MathSciNet  Google Scholar 

  7. A. V. Boiko, A. V. Dovgal, G. R. Grek, and V. V. Kozlov, Physics of Transitional Shear Flows: Instability and Laminar–Turbulent Transition in Incompressible Near-Wall Shear Layers, Springer, Berlin (2011).

    Google Scholar 

  8. A. V. Boiko, Y. M. Nechepurenko, and M. Sadkane, “Fast computation of optimal disturbances for duct flows with a given accuracy,” Comput. Math. Math. Phys., 50, No. 11, 1914–1924 (2010).

    Article  MathSciNet  Google Scholar 

  9. A. V. Boiko, Yu. M. Nechepurenko, and M. Sadkane, “Computing the maximum amplification of the solution norm of differential-algebraic systems,” Comput. Math. Model., 23, No. 2, 216–227 (2012).

    Article  MathSciNet  Google Scholar 

  10. V. A. Chereshnev, G. Bocharov, S. Bazhan, B. Bachmetyev, I. Gainova, V. Likhoshvai, J. M. Argilaguet, J. P. Martinez, J. A. Rump, B. Mothe et al., “Pathogenesis and treatment of HIV infection: the cellular, the immune system and the neuroendocrine systems perspective,” Int. Rev. Immunol., 32, No. 3, 282–306 (2013).

    Article  Google Scholar 

  11. 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, and R. M. Tret’yakova, Introduction to Problems of Modelling and Control of Dynamics of HIV Infection [in Russian], ANO Izhevskiy Inst. Komp. Issl., Moscow–Izhevsk (2016).

  12. A. Ciurea, P. Klenerman, L. Hunziker, E. Horvath, B. Odermatt, A. F. Ochsenbein, H. Hengartner, and R. M. Zinkernagel, “Persistence of lymphocytic choriomeningitis virus at very low levels in immune mice,” Proc. Natl. Acad. Sci. USA, 96, No. 21, 11964–11969 (1999).

    Article  Google Scholar 

  13. A. Crawford, J. M. Angelosanto, C. Kao, T. A. Doering, P. M. Odorizzi, B. E. Barnett, and E. J. Wherry, “Molecular and transcriptional basis of CD4+ T cell dysfunction during chronic infection,” Immunity, 40, No. 2, 289–302 (2014).

    Article  Google Scholar 

  14. M. E. Csete and J. C. Doyle, “Reverse engineering of biological complexity,” Science, 295, No. 5560, 1664–1669 (2002).

    Article  Google Scholar 

  15. V. G. Dement’ev, V. I. Lebedev, and Yu. M. Nechepurenko, “Spectral analysis of a model of nuclear reactor with delayed neutrons,” In: Algorithms and Programs for Neutron-Physical Calculations of Nuclear Reactors, FEI, Obninsk (1999), pp. 143–150.

    Google Scholar 

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

    Article  Google Scholar 

  17. G. H. Golub and C. F. Van Loan, Matrix Computations, The John Hopkins University Press, Baltimore (1996).

    MATH  Google Scholar 

  18. D. Gurdasani, L. Iles, D. G. Dillon, E. H. Young, A. D. Olson, V. Naranbhai, S. Fidler, E. Gkrania-Klotsas, F. A. Post, P. Kellam et al., “A systematic review of definitions of extreme phenotypes of HIV control and progression,” AIDS, 28, No. 2, 149–162 (2014).

    Article  Google Scholar 

  19. E. Hairer and G. Wanner, Solving Ordinary Differential Equations, Berlin (1996).

  20. D. S. Henningson and S. C. Reddy, “On the role of linear mechanisms in transition to turbulence,” Phys. Fluids A, 6, No. 3, 1396–1398 (1994).

    Article  MathSciNet  Google Scholar 

  21. A. V. Higham, “The scaling and squaring method for the matrix exponential revisited,” SIAM J. Matrix Anal. Appl., 26, No. 4, 1179–1193 (2005).

    Article  MathSciNet  Google Scholar 

  22. S. J. Kent, J. C. Reece, J. Petravic, A. Martyushev, M. Kramski, R. De Rose, D. A. Cooper, A. D. Kelleher, S. Emery, P. U. Cameron et al., “The search for an HIV cure: tackling latent infection,” Lancet Infect. Dis., 13, No. 7, 614–621 (2013).

    Google Scholar 

  23. H. Kitano, “Systems biology: a brief overview,” Science, 295, No. 5560, 1662–1664 (2002).

    Article  Google Scholar 

  24. H. Kitano, “Biological robustness,” Nat. Rev. Genet., 5, No. 11, 826–837 (2004).

    Article  Google Scholar 

  25. H. Kitano, “Biological robustness in complex host-pathogen systems,” In: Systems Biological Approaches in Infectious Diseases, Birkh¨auser, Basel (2007), pp. 239–263.

    Book  Google Scholar 

  26. A. Kotsarev, V. Lebedev, L. Shishkov, Yu. Nechepurenko, and V. Dementiev, “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, 453–468 (1999).

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

    Article  Google Scholar 

  28. T. Luzyanina, K. Engelborghs, S. Ehl, P. Klenerman, and G. Bocharov, “Low level viral persistence after infection with LCMV: a quantitative insight through numerical bifurcation analysis,” Math. Biosci., 173, No. 1, 1–23 (2001).

    Article  MathSciNet  Google Scholar 

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

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

  31. G. I. Marchuk, Mathematical Modelling of Immune Response in Infectious Diseases, Kluwer, Dordrecht (1997).

    Book  Google Scholar 

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

  33. G. I. Marchuk and N. I. Nisevich (eds.), Mathematical Methods in Clinical Practice, Pergamon Press Ltd., Oxford (1980).

    Google Scholar 

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

    Article  Google Scholar 

  35. Yu. M. Nechepurenko and M. Sadkane, “A low-rank approximation for computing the matrix exponential norm,” SIAM J. Matrix. Anal. Appl., 32, No. 2, 349–363 (2011).

    Article  MathSciNet  Google Scholar 

  36. Yu. M. Nechepurenko and M. Sadkane, “Computing humps of the matrix exponential,” J. Comput. Appl. Math., 319, 87–96 (2017).

    Article  MathSciNet  Google Scholar 

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

  38. E. Reshotko, “Transient growth: A factor in bypass transition,” Phys. Fluids, 13, No. 5, 1067–1075 (2001).

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  40. H.Weyl, The Classical Groups: Their Invariants and Representations, Princeton University Press, Princeton (2016).

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Authors and Affiliations

  1. Institute of Numerical Mathematics of the Russian Academy of Sciences, Moscow, Russia

    G. A. Bocharov, Yu. M. Nechepurenko, M. Yu. Khristichenko & D. S. Grebennikov

  2. Peoples’ Friendship University of Russia (RUDN University), Moscow, Russia

    G. A. Bocharov

  3. Keldysh Institute of Applied Mathematics of the Russian Academy of Sciences, Moscow, Russia

    Yu. M. Nechepurenko, M. Yu. Khristichenko & D. S. Grebennikov

Authors
  1. G. A. Bocharov
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  2. Yu. M. Nechepurenko
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  3. M. Yu. Khristichenko
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  4. D. S. Grebennikov
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Corresponding author

Correspondence to G. A. Bocharov.

Additional information

Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 63, No. 3, Differential and Functional Differential Equations, 2017.

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Bocharov, G.A., Nechepurenko, Y.M., Khristichenko, M.Y. et al. Optimal Perturbations of Systems with Delayed Independent Variables for Control of Dynamics of Infectious Diseases Based on Multicomponent Actions. J Math Sci 253, 618–641 (2021). https://doi.org/10.1007/s10958-021-05258-w

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  • DOI: https://doi.org/10.1007/s10958-021-05258-w

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