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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  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 & Continuous Dynamical Systems - S, 2020, 13 (9) : 2385-2401. doi: 10.3934/dcdss.2020166
References:
[1]

D. AngeliJ. 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.   Google Scholar

[2]

G. BocharovJ. Argilaguet and A. Meyerhans, Understanding experimental LCMV infection of mice: The role of mathematical models, J. Immunol. Res., 2015 (2015), 1-10.   Google Scholar

[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.  Google Scholar

[4]

G. A. Bocharov, Modelling the dynamics of LCMV infection in mice: Conventional and exhaustive CTL responses, J. Theor. Biol., 192 (1998), 283-308.   Google Scholar

[5]

G. A. BocharovYu. M. NechepurenkoM. 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.   Google Scholar

[6]

G. A. BocharovYu. M. NechepurenkoM. 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.  Google Scholar

[7]

G. A. BocharovYu. M. NechepurenkoM. Yu. Khristichenko and D. S. Grebennikov, Optimal disturbances of bistable time-delay systems modeling virus infections, Doklady Mathrmatics, 98 (2018), 313-316.   Google Scholar

[8]

G. BocharovA. KimA. KrasovskiiV. ChereshnevV. 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.  Google Scholar

[9]

G. A. Bocharov and G. I. Marchuk, Applied problems of mathematical modeling in immunology, Comput. Math. Math. Phys., 40 (2000), 1830-1844.   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

A. CiureaP. KlenermanL. HunzikerE. HorvathB. OdermattA. F. OchsenbeinH. 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.   Google Scholar

[14]

C. Effenberger, Robust successive computation of eigenpairs for nonlinear eigenvalue problems, SIAM J. Matrix Anal., 34 (2013), 1231-1256.  doi: 10.1137/120885644.  Google Scholar

[15]

K. EngelborghsT. 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.  Google Scholar

[16]

J. E. Ferrell, Bistability, bifurcations, and Waddington's epigenetic landscape, Curr. Biol., 22 (2012), 458-466.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

H. Hilton, Plane Algebraic Curves, Oxford University Press, London, 1920. Google Scholar

[21]

D. Kahaner, C. Moler and S. Nash, Numerical Methods and Software, Prentice-Hall, Englewood Cliffs, 1977. Google Scholar

[22]

G. I. Marchuk, Mathematical Models in Immunology, Optimization Software Inc. Publications Division, New York, 1983.  Google Scholar

[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.  Google Scholar

[24]

D. MoskophidisF. LechnerH. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[27]

J. Sieber, K. Engelborghs, T. Luzyanina, G. Samaey and D. Roose, DDE-BIFTOOL Manual: Bifurcation analysis of delay differential equations, arXiv: 1406.7144. Google Scholar

show all references

References:
[1]

D. AngeliJ. 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.   Google Scholar

[2]

G. BocharovJ. Argilaguet and A. Meyerhans, Understanding experimental LCMV infection of mice: The role of mathematical models, J. Immunol. Res., 2015 (2015), 1-10.   Google Scholar

[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.  Google Scholar

[4]

G. A. Bocharov, Modelling the dynamics of LCMV infection in mice: Conventional and exhaustive CTL responses, J. Theor. Biol., 192 (1998), 283-308.   Google Scholar

[5]

G. A. BocharovYu. M. NechepurenkoM. 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.   Google Scholar

[6]

G. A. BocharovYu. M. NechepurenkoM. 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.  Google Scholar

[7]

G. A. BocharovYu. M. NechepurenkoM. Yu. Khristichenko and D. S. Grebennikov, Optimal disturbances of bistable time-delay systems modeling virus infections, Doklady Mathrmatics, 98 (2018), 313-316.   Google Scholar

[8]

G. BocharovA. KimA. KrasovskiiV. ChereshnevV. 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.  Google Scholar

[9]

G. A. Bocharov and G. I. Marchuk, Applied problems of mathematical modeling in immunology, Comput. Math. Math. Phys., 40 (2000), 1830-1844.   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

A. CiureaP. KlenermanL. HunzikerE. HorvathB. OdermattA. F. OchsenbeinH. 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.   Google Scholar

[14]

C. Effenberger, Robust successive computation of eigenpairs for nonlinear eigenvalue problems, SIAM J. Matrix Anal., 34 (2013), 1231-1256.  doi: 10.1137/120885644.  Google Scholar

[15]

K. EngelborghsT. 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.  Google Scholar

[16]

J. E. Ferrell, Bistability, bifurcations, and Waddington's epigenetic landscape, Curr. Biol., 22 (2012), 458-466.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

H. Hilton, Plane Algebraic Curves, Oxford University Press, London, 1920. Google Scholar

[21]

D. Kahaner, C. Moler and S. Nash, Numerical Methods and Software, Prentice-Hall, Englewood Cliffs, 1977. Google Scholar

[22]

G. I. Marchuk, Mathematical Models in Immunology, Optimization Software Inc. Publications Division, New York, 1983.  Google Scholar

[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.  Google Scholar

[24]

D. MoskophidisF. LechnerH. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[27]

J. Sieber, K. Engelborghs, T. Luzyanina, G. Samaey and D. Roose, DDE-BIFTOOL Manual: Bifurcation analysis of delay differential equations, arXiv: 1406.7144. Google Scholar

Figure 1.  The first step of constructing elementary components
Figure 2.  Steady states (Ⅰ — black, Ⅱ — red, Ⅲ — green, Ⅳ — blue) and their stability (solid line) and instability (dashed line) as functions of $ b_p $
Figure 3.  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")
Figure 4.  Steady states (Ⅰ — black, Ⅱ — red, Ⅲ — green, Ⅳ — blue) and their stability (solid line) and instability (dashed line) as functions of $ \beta $
Figure 5.  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)
Table 1.  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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