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Nonlinear effects in the dynamics of HIV-1 infection predicted by mathematical model with multiple delays
September  2020, 13(9): 2385-2401. doi: 10.3934/dcdss.2020166

## Bistability analysis of virus infection models with time delays

 1 Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, 119333, Russian Federation 2 Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, 125047, Russia

* Corresponding author: Yuri Nechepurenko

Received  January 2019 Revised  April 2019 Published  September 2020 Early access  December 2019

Mathematical models with time delays are widely used to analyze the mechanisms of the immune response to virus infections and predict various therapeutic effects. Using the lymphocytic choriomeningitis virus infection model as an example, this work describes an original computational technology for searching the bistable regimes of such models. This technology includes numerical methods for finding all possible steady states at fixed values of parameters, for tracing these states along the parameters and for analyzing their stability.

Citation: Yuri Nechepurenko, Michael Khristichenko, Dmitry Grebennikov, Gennady Bocharov. Bistability analysis of virus infection models with time delays. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2385-2401. doi: 10.3934/dcdss.2020166
##### References:
 [1] D. Angeli, J. E. Ferrell Jr. and E. D. Sontag, Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems, Proc. Natl. Acad. Sci. U.S.A., 101 (2004), 1822-1827. [2] G. Bocharov, J. Argilaguet and A. Meyerhans, Understanding experimental LCMV infection of mice: The role of mathematical models, J. Immunol. Res., 2015 (2015), 1-10. [3] G. Bocharov, V. Volpert, B. Ludewig and A. Meyerhans, Mathematical Immunology of Virus Infections, Springer, Cham, 2018. doi: 10.1007/978-3-319-72317-4. [4] G. A. Bocharov, Modelling the dynamics of LCMV infection in mice: Conventional and exhaustive CTL responses, J. Theor. Biol., 192 (1998), 283-308. [5] G. A. Bocharov, Yu. M. Nechepurenko, M. Yu. Khristichenko and D. S. Grebennikov, Optimal perturbations of systems with delayed argument for control of dynamics of infectious diseases based on multicomponent actions, Sovrem. Mat. Fundam. Napravl., 63 (2017), 392-417. [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, Russian J. Num. Anal. Math. Model., 32 (2017), 275-291.  doi: 10.1515/rnam-2017-0027. [7] G. A. Bocharov, Yu. M. Nechepurenko, M. Yu. Khristichenko and D. S. Grebennikov, Optimal disturbances of bistable time-delay systems modeling virus infections, Doklady Mathrmatics, 98 (2018), 313-316. [8] G. Bocharov, A. Kim, A. Krasovskii, V. Chereshnev, V. Glushenkova and A. Ivanov, An extremal shift method for control of HIV infection dynamics, Russian J. Numer. Anal. Math. Modeling., 30 (2015), 11-25.  doi: 10.1515/rnam-2015-0002. [9] G. A. Bocharov and G. I. Marchuk, Applied problems of mathematical modeling in immunology, Comput. Math. Math. Phys., 40 (2000), 1830-1844. [10] A. V. Boǐko and Yu. M. Nechepurenko, A technique for the numerical analysis of the riblet effect on temporal stability of plane flows, Computational Mathematics and Mathematical Physics, 50 (2010), 1055-1070.  doi: 10.1134/S0965542510060114. [11] D. Breda, S. Maset and R. Vermiglio, TRACE-DDE: A tool for robust analysis and characteristic equations for delay differential equations, Topics in Time Delay Systems: Analysis, Algorithms and Control. Lecture Notes in Control and Information Sciences, Springer Berlin Heidelberg, Berlin, Heidelberg, (2009), 145–155. [12] S. M. Ciupe, C. J. Miller and J. E. Forde, A bistable switch in virus dynamics can explain the differences in disease outcome following SIV infections in rhesus macaques, Front. Microbiol., 9 (2018), 1216. [13] 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. U.S.A, 96 (1999), 11964-11969. [14] C. Effenberger, Robust successive computation of eigenpairs for nonlinear eigenvalue problems, SIAM J. Matrix Anal., 34 (2013), 1231-1256.  doi: 10.1137/120885644. [15] K. Engelborghs, T. Luzyanina and D. Roose, Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM Trans. Math. Softw., 28 (2002), 1-21.  doi: 10.1145/513001.513002. [16] J. E. Ferrell, Bistability, bifurcations, and Waddington's epigenetic landscape, Curr. Biol., 22 (2012), 458-466. [17] G. E. Forsythe, M. A. Malcolm and C. B. Moler, Computer Methods for Mathematical Computations, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Englewood Cliffs, 1977. [18] G. H. Golub and C. F. Van Loan, Matrix Computations, Second edition, Johns Hopkins Series in the Mathematical Sciences, 3. Johns Hopkins University Press, Baltimore, MD, 1989. [19] E. Hairer and G. Wanner, Solving Ordinary Differential Equations. II. Stiff and Differential-Algebraic Problems, Second edition. Springer Series in Computational Mathematics, 14. Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-642-05221-7. [20] H. Hilton, Plane Algebraic Curves, Oxford University Press, London, 1920. [21] D. Kahaner, C. Moler and S. Nash, Numerical Methods and Software, Prentice-Hall, Englewood Cliffs, 1977. [22] G. I. Marchuk, Mathematical Models in Immunology, Optimization Software Inc. Publications Division, New York, 1983. [23] G. I. Marchuk, Mathematical Modelling of Immune Response in Infectious Diseases, Mathematics and its Applications, 395. Kluwer Academic Publishers Group, Dordrecht, 1997. doi: 10.1007/978-94-015-8798-3. [24] 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 (1993), 758-761.  doi: 10.1038/362758a0. [25] Yu. M. Nechepurenko and M. Yu. Khristichenko, Development and analysis of algorithms for computing optimal disturbances for delay systems, Keldysh Institute Preprints, 120 (2018), 1-26. [26] D. Roose and R. Szalai, Continuation and bifurcation analysis of delay differential equations, Numerical Continuation Methods for Dynamical Systems: Path Following and Boundary Value Problems, Underst. Complex Syst., Springer, Dordrecht, (2007), 359–399. doi: 10.1007/978-1-4020-6356-5_12. [27] J. Sieber, K. Engelborghs, T. Luzyanina, G. Samaey and D. Roose, DDE-BIFTOOL Manual: Bifurcation analysis of delay differential equations, arXiv: 1406.7144.

show all references

##### References:
 [1] D. Angeli, J. E. Ferrell Jr. and E. D. Sontag, Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems, Proc. Natl. Acad. Sci. U.S.A., 101 (2004), 1822-1827. [2] G. Bocharov, J. Argilaguet and A. Meyerhans, Understanding experimental LCMV infection of mice: The role of mathematical models, J. Immunol. Res., 2015 (2015), 1-10. [3] G. Bocharov, V. Volpert, B. Ludewig and A. Meyerhans, Mathematical Immunology of Virus Infections, Springer, Cham, 2018. doi: 10.1007/978-3-319-72317-4. [4] G. A. Bocharov, Modelling the dynamics of LCMV infection in mice: Conventional and exhaustive CTL responses, J. Theor. Biol., 192 (1998), 283-308. [5] G. A. Bocharov, Yu. M. Nechepurenko, M. Yu. Khristichenko and D. S. Grebennikov, Optimal perturbations of systems with delayed argument for control of dynamics of infectious diseases based on multicomponent actions, Sovrem. Mat. Fundam. Napravl., 63 (2017), 392-417. [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, Russian J. Num. Anal. Math. Model., 32 (2017), 275-291.  doi: 10.1515/rnam-2017-0027. [7] G. A. Bocharov, Yu. M. Nechepurenko, M. Yu. Khristichenko and D. S. Grebennikov, Optimal disturbances of bistable time-delay systems modeling virus infections, Doklady Mathrmatics, 98 (2018), 313-316. [8] G. Bocharov, A. Kim, A. Krasovskii, V. Chereshnev, V. Glushenkova and A. Ivanov, An extremal shift method for control of HIV infection dynamics, Russian J. Numer. Anal. Math. Modeling., 30 (2015), 11-25.  doi: 10.1515/rnam-2015-0002. [9] G. A. Bocharov and G. I. Marchuk, Applied problems of mathematical modeling in immunology, Comput. Math. Math. Phys., 40 (2000), 1830-1844. [10] A. V. Boǐko and Yu. M. Nechepurenko, A technique for the numerical analysis of the riblet effect on temporal stability of plane flows, Computational Mathematics and Mathematical Physics, 50 (2010), 1055-1070.  doi: 10.1134/S0965542510060114. [11] D. Breda, S. Maset and R. Vermiglio, TRACE-DDE: A tool for robust analysis and characteristic equations for delay differential equations, Topics in Time Delay Systems: Analysis, Algorithms and Control. Lecture Notes in Control and Information Sciences, Springer Berlin Heidelberg, Berlin, Heidelberg, (2009), 145–155. [12] S. M. Ciupe, C. J. Miller and J. E. Forde, A bistable switch in virus dynamics can explain the differences in disease outcome following SIV infections in rhesus macaques, Front. Microbiol., 9 (2018), 1216. [13] 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. U.S.A, 96 (1999), 11964-11969. [14] C. Effenberger, Robust successive computation of eigenpairs for nonlinear eigenvalue problems, SIAM J. Matrix Anal., 34 (2013), 1231-1256.  doi: 10.1137/120885644. [15] K. Engelborghs, T. Luzyanina and D. Roose, Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM Trans. Math. Softw., 28 (2002), 1-21.  doi: 10.1145/513001.513002. [16] J. E. Ferrell, Bistability, bifurcations, and Waddington's epigenetic landscape, Curr. Biol., 22 (2012), 458-466. [17] G. E. Forsythe, M. A. Malcolm and C. B. Moler, Computer Methods for Mathematical Computations, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Englewood Cliffs, 1977. [18] G. H. Golub and C. F. Van Loan, Matrix Computations, Second edition, Johns Hopkins Series in the Mathematical Sciences, 3. Johns Hopkins University Press, Baltimore, MD, 1989. [19] E. Hairer and G. Wanner, Solving Ordinary Differential Equations. II. Stiff and Differential-Algebraic Problems, Second edition. Springer Series in Computational Mathematics, 14. Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-642-05221-7. [20] H. Hilton, Plane Algebraic Curves, Oxford University Press, London, 1920. [21] D. Kahaner, C. Moler and S. Nash, Numerical Methods and Software, Prentice-Hall, Englewood Cliffs, 1977. [22] G. I. Marchuk, Mathematical Models in Immunology, Optimization Software Inc. Publications Division, New York, 1983. [23] G. I. Marchuk, Mathematical Modelling of Immune Response in Infectious Diseases, Mathematics and its Applications, 395. Kluwer Academic Publishers Group, Dordrecht, 1997. doi: 10.1007/978-94-015-8798-3. [24] 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 (1993), 758-761.  doi: 10.1038/362758a0. [25] Yu. M. Nechepurenko and M. Yu. Khristichenko, Development and analysis of algorithms for computing optimal disturbances for delay systems, Keldysh Institute Preprints, 120 (2018), 1-26. [26] D. Roose and R. Szalai, Continuation and bifurcation analysis of delay differential equations, Numerical Continuation Methods for Dynamical Systems: Path Following and Boundary Value Problems, Underst. Complex Syst., Springer, Dordrecht, (2007), 359–399. doi: 10.1007/978-1-4020-6356-5_12. [27] J. Sieber, K. Engelborghs, T. Luzyanina, G. Samaey and D. Roose, DDE-BIFTOOL Manual: Bifurcation analysis of delay differential equations, arXiv: 1406.7144.
The first step of constructing elementary components
Steady states (Ⅰ — black, Ⅱ — red, Ⅲ — green, Ⅳ — blue) and their stability (solid line) and instability (dashed line) as functions of $b_p$
Leading eigenvalues in steady states Ⅰ, Ⅱ, Ⅲ and Ⅳ at $b_p = 10^{-5}$ ("x"), $3.5\cdot10^{-4}$ ("+"), $6.7\cdot10^{-4}$ ("o")
Steady states (Ⅰ — black, Ⅱ — red, Ⅲ — green, Ⅳ — blue) and their stability (solid line) and instability (dashed line) as functions of $\beta$
Steady state values of $V$ as a function of parameter $b_p$ for the following values of $\beta$: $1.72$ (A), $1.69$ (B), $1.671$ (C), $1.67$ (D)
Biological meaning of the model (1) parameters
 Parameter Biological meaning $\beta$ Viruses replication rate constant $\gamma_{VE}$ Rate constant of virus clearance due to effector CTLs $V_{mvc}$ Maximum possible virus concentration in spleen $\tau$ Typical duration of CTL division cycle $b_p$ Rate constant of CTL stimulation $b_d$ Rate constant of CTL differentiation $\theta_p$ Cumulative viral load threshold for anergy induction in precursor CTLs $\theta_E$ Cumulative viral load threshold for anergy induction in effector CTLs $\alpha_{E_{p}}$ Precursor CTL natural death rate constant $\alpha_{E_{e}}$ Effector CTL natural death rate constant $E_p^0$ Concentration of precursor CTLs in spleen of unprimed mouse $\tau_A$ Typical duration of CTLs commitment for apoptosis $\alpha_{AP}$ Precursor CTL apoptosis rate constant $\alpha_{AE}$ Effector CTL apoptosis rate constant $b_{W}$ Rate constant of cumulative viral load increase $\alpha_{W}$ Rate constant of restoration from the inhibitory effect of cumulative viral load
 Parameter Biological meaning $\beta$ Viruses replication rate constant $\gamma_{VE}$ Rate constant of virus clearance due to effector CTLs $V_{mvc}$ Maximum possible virus concentration in spleen $\tau$ Typical duration of CTL division cycle $b_p$ Rate constant of CTL stimulation $b_d$ Rate constant of CTL differentiation $\theta_p$ Cumulative viral load threshold for anergy induction in precursor CTLs $\theta_E$ Cumulative viral load threshold for anergy induction in effector CTLs $\alpha_{E_{p}}$ Precursor CTL natural death rate constant $\alpha_{E_{e}}$ Effector CTL natural death rate constant $E_p^0$ Concentration of precursor CTLs in spleen of unprimed mouse $\tau_A$ Typical duration of CTLs commitment for apoptosis $\alpha_{AP}$ Precursor CTL apoptosis rate constant $\alpha_{AE}$ Effector CTL apoptosis rate constant $b_{W}$ Rate constant of cumulative viral load increase $\alpha_{W}$ Rate constant of restoration from the inhibitory effect of cumulative viral load
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