    September  2020, 13(9): 2365-2384. doi: 10.3934/dcdss.2020141

## Nonlinear effects in the dynamics of HIV-1 infection predicted by mathematical model with multiple delays

 1 Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Omsk Branch, Omsk, 644043, Russian Federation 2 Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, 119333, Russian Federation

* Corresponding author: Nikolay Pertsev

Received  January 2019 Revised  April 2019 Published  November 2019

Fund Project: The first and the last authors are supported by Russian Science Foundation grant 18-11-00171

We formulate a novel mathematical model to describe the development of acute HIV-1 infection and the antiviral immune response. The model is formulated using a system of delay differential- and integral equations. The model is applied to study the possibility of eradication of HIV infection during the primary acute phase of the disease. To this end a combination of analytical examination and computational treatment is used. The model belongs to the Wazewski differential equation systems with delay. The conditions of asymptotic stability of a trivial steady state solution are expressed in terms of the algebraic Sevastyanov–Kotelyanskii criterion. The results of the computational experiments with the model calibrated according to the available estimates of parameters suggest that a complete elimination of HIV-1 infection after acute phase of infection is feasible.

Citation: Nikolay Pertsev, Konstantin Loginov, Gennady Bocharov. Nonlinear effects in the dynamics of HIV-1 infection predicted by mathematical model with multiple delays. Discrete & Continuous Dynamical Systems - S, 2020, 13 (9) : 2365-2384. doi: 10.3934/dcdss.2020141
##### References:
  H. T. Banks and D. M. Bortz, A parameter sensitivity methodology in the context of HIV delay equation models, J. Math. Biol., 50 (2005), 607-625.  doi: 10.1007/s00285-004-0299-x.  Google Scholar  G. Bocharov, V. Chereshnev, I. Gainova, S. Bazhan, B. Bachmetyev, J. Argilaguet, J. Martinez and A. Meyerhans, Human Immunodeficiency Virus Infection: From Biological Observations to Mechanistic Mathematical Modelling, Math. Model. Nat. Phenom., 7 (2012), 78-104.  doi: 10.1051/mmnp/20127507.  Google Scholar  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. Modelling, 32 (2017), 275-291.  doi: 10.1515/rnam-2017-0027.  Google Scholar  V. A. Chereshnev, G. A. Bocharov, A. V. Kim, S. I. Bazhan, I. A. Gainova, A. N. Krasovskii, N. G. Shmagel', A. V. Ivanov, M. A. Safronov and R. M. Tret'yakova, Vvedenie V Zadachi Modelirovaniya I Upravleniya Dinamikoi VICH Infektsii, (Russian) [Introduction to the tasks of modeling and managing the dynamics of HIV infection], Ijevsk: Institut komp'yuternykh issledovanii, Moscow, 2016. Google Scholar  R. D. Driver, Ordinary and Delay Differential Equations, Applied Mathematical Sciences, Vol. 20, Springer-Verlag, New York-Heidelberg, 1977. doi: 10.1007/978-1-4684-9467-9.  Google Scholar  R. Eftimie, J. J. Gillard and D. A. Cantrell, Mathematical Models for Immunology: Current State of the Art and Future Research Directions, Bull. Math. Biol., 78 (2016), 2091-2134.  doi: 10.1007/s11538-016-0214-9.  Google Scholar  L. E. El'sgol'ts and S. B. Norkin, Introduction to the Theory and Application of Differential Equations with Deviating Arguments, Translated from the Russian by John L. Casti. Mathematics in Science and Engineering, Vol. 105, Academic Press [A Subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1973. Google Scholar  F. R. Gantmacher, The Theory of Matrices, AMS Chelsea Publishing, Providence, RI, 1998. Google Scholar  J. K. Hale, Theory of Functional Differential Equations, Applied Mathematical Sciences, Vol. 3, Springer-Verlag, New York-Heidelberg, 1977. doi: 10.1007/978-1-4612-9892-2.  Google Scholar  S. I. Kabanikhin, O. I. Krivorotko, D. V. Ermolenko, V. N. Kashtanova and V. A. Latyshenko, Inverse problems of immunology and epidemiology, Eurasian Journal of Mathematical and Computer Applications, 5 (2017), 14-35.   Google Scholar  S. I. Kabanikhin, O. I. Krivorotko, D. A. Voronov and D. V. Ermolenko, Optimization approach for solving the inverse problem of the simplest model of an infectious disease, Siberian Electronic Mathematical Reports, 12 (2015), 154-162.   Google Scholar  V. B. Kolmanovskii and V. R. Nosov, Ustoichivost' i Periodicheskie Rejimy Sistem Upravleniya s Posledeistviem, (Russian) [Stability and periodic regimes of control systems with aftereffect], [Theoretical Foundations of Engineering Cybernetics] Nauka, Moscow, 1981. Google Scholar  A. A. Lackner, M. M. Lederman and B. Rodriguez, HIV Pathogenesis: The Host, Cold Spring Harb Perspect Med, 2 (2012), a007005. doi: 10.1101/cshperspect.a007005. Google Scholar  G. I. Marchuk, Matematicheskie Modeli v Immunologii, (Russian) [Mathematical models in immunology], 2nd edition, With a preface by R. V. Petrov., Nauka, Moscow, 1985. Google Scholar  G. I. Marchuk, Matematicheskie Modeli V Immunologii. Vychislitel'nye Metody I Eksperimenty, (Russian) [Mathematical models in immunology. Computational methods and experiments], 3rd edition, Nauka, Moscow, 1991. Google Scholar  S. Nakaoka, I. Shingo and K. Sato, Dynamics of HIV infection in lymphoid tissue network, J. Math. Biol., 72 (2016), 909-938.  doi: 10.1007/s00285-015-0940-x.  Google Scholar  P. W. Nelson and A. S. Perelson, Mathematical Analysis of delay differential equation models of HIV-1 infection, Math. Biosci., 179 (2002), 73-94.  doi: 10.1016/S0025-5564(02)00099-8.  Google Scholar  A. Yu. Obolenskii, Stability of solutions of autonomous Wazewski systems with delayed action, Ukr. Math. J., 35 (1983), 486-492.  doi: 10.1007/BF01061640. Google Scholar  S. Pankavich, The effects of latent infection on the dynamics of HIV, Differ. Equ. Dyn. Syst., 24 (2016), 281-303.  doi: 10.1007/s12591-014-0234-6.  Google Scholar  K. A. Pawelek, S. Liu, F. Pahlevani and L. Rong, A model of HIV-1 infection with two time delays: Mathematical analysis and comparison with patient data, Math. Biosci., 235 (2012), 98-109.  doi: 10.1016/j.mbs.2011.11.002.  Google Scholar  A. S. Perelson and R. M. Ribeiro, Modeling The Within-Host Dynamics of HIV Infection, BMC Biology, 2013. doi: 10.1186/1741-7007-11-96. Google Scholar  N. V. Pertsev, Global Solvability and Estimates of Solutions to the Cauchy Problem for the Retarded Functional Differential Equations That Are Used to Model Living Systems, Sib. Math. J., 59 (2018), 113-125.  doi: 10.1134/S0037446618010135. Google Scholar  N. V. Pertsev, B. Yu. Pichugin and K. K. Loginov, Stochastic Analog of the Dynamic Model of HIV-1 Infection Described by Delay Differential Equations, J. Appl. Ind. Math., 13 (2019), 103-117.  doi: 10.1134/S1990478919010125. Google Scholar  N. V. Pertsev, B. Yu. Pichugin and A. N. Pichugina, Application of M-Matrices in studies of mathematical models of living systems, Mathematical Biology and Bioinformatics, 13 (2018), t104–t131. doi: 10.17537/2018.13.t104. Google Scholar  M. Pitchaimani and C. Monica, Global stability analysis of HIV-1 infection model with three time delays, J. Appl. Math. Comput., 48 (2015), 293-319.  doi: 10.1007/s12190-014-0803-4.  Google Scholar  D. Sanchez-Taltavull, A. Vieiro and T. Alarcon, Stochastic modelling of the eradication of the HIV-1 infection by stimulation of latently infected cells in patients under highly active anti-retroviral therapy, J. Math. Biol., 73 (2016), 919-946.  doi: 10.1007/s00285-016-0977-5.  Google Scholar  A. Shet, P. Nagaraja and N. M. Dixit, Viral Decay Dynamics and Mathematical Modeling of Treatment Response: Evidence of Lower in vivo Fitness of HIV-1 Subtype C, J. Acquir. Immune Defic. Syndr., 73 (2016), 245-251.  doi: 10.1097/QAI.0000000000001101. Google Scholar  R. F. Siliciano and W. C. Greene, HIV Latency, Cold Spring Harb Perspect Med, 1 (2011), a007096. doi: 10.1101/cshperspect.a007096. Google Scholar  W. I. Sundquist and H.-G. Kraüsslich, HIV-1 Assembly, Budding, and Maturation, Cold Spring Harb Perspect Med, 2 (2012), a006924. doi: 10.1101/cshperspect.a006924. Google Scholar  J. Wang, J. Lang and X. Zou, Analysis of an age structured HIV infection model with virus-to-cell infection and cell-to-cell transmission, Nonlinear Anal. Real World Appl., 34 (2017), 75-96.  doi: 10.1016/j.nonrwa.2016.08.001.  Google Scholar

show all references

##### References:
  H. T. Banks and D. M. Bortz, A parameter sensitivity methodology in the context of HIV delay equation models, J. Math. Biol., 50 (2005), 607-625.  doi: 10.1007/s00285-004-0299-x.  Google Scholar  G. Bocharov, V. Chereshnev, I. Gainova, S. Bazhan, B. Bachmetyev, J. Argilaguet, J. Martinez and A. Meyerhans, Human Immunodeficiency Virus Infection: From Biological Observations to Mechanistic Mathematical Modelling, Math. Model. Nat. Phenom., 7 (2012), 78-104.  doi: 10.1051/mmnp/20127507.  Google Scholar  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. Modelling, 32 (2017), 275-291.  doi: 10.1515/rnam-2017-0027.  Google Scholar  V. A. Chereshnev, G. A. Bocharov, A. V. Kim, S. I. Bazhan, I. A. Gainova, A. N. Krasovskii, N. G. Shmagel', A. V. Ivanov, M. A. Safronov and R. M. Tret'yakova, Vvedenie V Zadachi Modelirovaniya I Upravleniya Dinamikoi VICH Infektsii, (Russian) [Introduction to the tasks of modeling and managing the dynamics of HIV infection], Ijevsk: Institut komp'yuternykh issledovanii, Moscow, 2016. Google Scholar  R. D. Driver, Ordinary and Delay Differential Equations, Applied Mathematical Sciences, Vol. 20, Springer-Verlag, New York-Heidelberg, 1977. doi: 10.1007/978-1-4684-9467-9.  Google Scholar  R. Eftimie, J. J. Gillard and D. A. Cantrell, Mathematical Models for Immunology: Current State of the Art and Future Research Directions, Bull. Math. Biol., 78 (2016), 2091-2134.  doi: 10.1007/s11538-016-0214-9.  Google Scholar  L. E. El'sgol'ts and S. B. Norkin, Introduction to the Theory and Application of Differential Equations with Deviating Arguments, Translated from the Russian by John L. Casti. Mathematics in Science and Engineering, Vol. 105, Academic Press [A Subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1973. Google Scholar  F. R. Gantmacher, The Theory of Matrices, AMS Chelsea Publishing, Providence, RI, 1998. Google Scholar  J. K. Hale, Theory of Functional Differential Equations, Applied Mathematical Sciences, Vol. 3, Springer-Verlag, New York-Heidelberg, 1977. doi: 10.1007/978-1-4612-9892-2.  Google Scholar  S. I. Kabanikhin, O. I. Krivorotko, D. V. Ermolenko, V. N. Kashtanova and V. A. Latyshenko, Inverse problems of immunology and epidemiology, Eurasian Journal of Mathematical and Computer Applications, 5 (2017), 14-35.   Google Scholar  S. I. Kabanikhin, O. I. Krivorotko, D. A. Voronov and D. V. Ermolenko, Optimization approach for solving the inverse problem of the simplest model of an infectious disease, Siberian Electronic Mathematical Reports, 12 (2015), 154-162.   Google Scholar  V. B. Kolmanovskii and V. R. Nosov, Ustoichivost' i Periodicheskie Rejimy Sistem Upravleniya s Posledeistviem, (Russian) [Stability and periodic regimes of control systems with aftereffect], [Theoretical Foundations of Engineering Cybernetics] Nauka, Moscow, 1981. Google Scholar  A. A. Lackner, M. M. Lederman and B. Rodriguez, HIV Pathogenesis: The Host, Cold Spring Harb Perspect Med, 2 (2012), a007005. doi: 10.1101/cshperspect.a007005. Google Scholar  G. I. Marchuk, Matematicheskie Modeli v Immunologii, (Russian) [Mathematical models in immunology], 2nd edition, With a preface by R. V. Petrov., Nauka, Moscow, 1985. Google Scholar  G. I. Marchuk, Matematicheskie Modeli V Immunologii. Vychislitel'nye Metody I Eksperimenty, (Russian) [Mathematical models in immunology. Computational methods and experiments], 3rd edition, Nauka, Moscow, 1991. Google Scholar  S. Nakaoka, I. Shingo and K. Sato, Dynamics of HIV infection in lymphoid tissue network, J. Math. Biol., 72 (2016), 909-938.  doi: 10.1007/s00285-015-0940-x.  Google Scholar  P. W. Nelson and A. S. Perelson, Mathematical Analysis of delay differential equation models of HIV-1 infection, Math. Biosci., 179 (2002), 73-94.  doi: 10.1016/S0025-5564(02)00099-8.  Google Scholar  A. Yu. Obolenskii, Stability of solutions of autonomous Wazewski systems with delayed action, Ukr. Math. J., 35 (1983), 486-492.  doi: 10.1007/BF01061640. Google Scholar  S. Pankavich, The effects of latent infection on the dynamics of HIV, Differ. Equ. Dyn. Syst., 24 (2016), 281-303.  doi: 10.1007/s12591-014-0234-6.  Google Scholar  K. A. Pawelek, S. Liu, F. Pahlevani and L. Rong, A model of HIV-1 infection with two time delays: Mathematical analysis and comparison with patient data, Math. Biosci., 235 (2012), 98-109.  doi: 10.1016/j.mbs.2011.11.002.  Google Scholar  A. S. Perelson and R. M. Ribeiro, Modeling The Within-Host Dynamics of HIV Infection, BMC Biology, 2013. doi: 10.1186/1741-7007-11-96. Google Scholar  N. V. Pertsev, Global Solvability and Estimates of Solutions to the Cauchy Problem for the Retarded Functional Differential Equations That Are Used to Model Living Systems, Sib. Math. J., 59 (2018), 113-125.  doi: 10.1134/S0037446618010135. Google Scholar  N. V. Pertsev, B. Yu. Pichugin and K. K. Loginov, Stochastic Analog of the Dynamic Model of HIV-1 Infection Described by Delay Differential Equations, J. Appl. Ind. Math., 13 (2019), 103-117.  doi: 10.1134/S1990478919010125. Google Scholar  N. V. Pertsev, B. Yu. Pichugin and A. N. Pichugina, Application of M-Matrices in studies of mathematical models of living systems, Mathematical Biology and Bioinformatics, 13 (2018), t104–t131. doi: 10.17537/2018.13.t104. Google Scholar  M. Pitchaimani and C. Monica, Global stability analysis of HIV-1 infection model with three time delays, J. Appl. Math. Comput., 48 (2015), 293-319.  doi: 10.1007/s12190-014-0803-4.  Google Scholar  D. Sanchez-Taltavull, A. Vieiro and T. Alarcon, Stochastic modelling of the eradication of the HIV-1 infection by stimulation of latently infected cells in patients under highly active anti-retroviral therapy, J. Math. Biol., 73 (2016), 919-946.  doi: 10.1007/s00285-016-0977-5.  Google Scholar  A. Shet, P. Nagaraja and N. M. Dixit, Viral Decay Dynamics and Mathematical Modeling of Treatment Response: Evidence of Lower in vivo Fitness of HIV-1 Subtype C, J. Acquir. Immune Defic. Syndr., 73 (2016), 245-251.  doi: 10.1097/QAI.0000000000001101. Google Scholar  R. F. Siliciano and W. C. Greene, HIV Latency, Cold Spring Harb Perspect Med, 1 (2011), a007096. doi: 10.1101/cshperspect.a007096. Google Scholar  W. I. Sundquist and H.-G. Kraüsslich, HIV-1 Assembly, Budding, and Maturation, Cold Spring Harb Perspect Med, 2 (2012), a006924. doi: 10.1101/cshperspect.a006924. Google Scholar  J. Wang, J. Lang and X. Zou, Analysis of an age structured HIV infection model with virus-to-cell infection and cell-to-cell transmission, Nonlinear Anal. Real World Appl., 34 (2017), 75-96.  doi: 10.1016/j.nonrwa.2016.08.001.  Google Scholar Computational Experiment 1. Dynamics of $y_1(t)$, $y_2(t)$, $y_3(t)$ for $R_0 = 0.9224$, $V^0 = 10^4$ Computational Experiment 2. Dynamics of $y_1(t)$, $y_2(t)$, $y_3(t)$ for $R_0 = 0.9073$, $V^0 = 10^4$ Computational Experiment 3. Dynamics of $y_1(t)$, $y_2(t)$, $y_3(t)$ for $R_0 = 2.2875$. (a) $V^0 = 10^4$, (b) $V^0 = 10^6$ Computational Experiment 4. Dynamics of $y_1(t)$, $y_2(t)$, $y_3(t)$ for $R_0 = 2.2875$. (a) $V^0 = 10^4$, (b) $V^0 = 10^6$ Computational Experiment 5. Dynamics of $y_1(t)$, $y_2(t)$, $y_3(t)$ for $R_0 = 13.2135$. (a) $V^0 = 10^2$, (b) $V^0 = 10^6$ Computational Experiment 6. Dynamics of $y_1(t)$, $y_2(t)$, $y_3(t)$ for $R_0 = 3.3789$. (a) $V^0 = 10^4$, (b) $V^0 = 10^6$
Time post infection needed for a complete elimination of HIV-1 infection
 $V_0$ Computational experiment number, the value $\tau_0$ (day) 1 2 3 4 5 6 $10$ 1 1 $-$ $-$ $-$ 46 $10^2$ 51 46 $-$ 51 $-$ 44 $10^4$ 60 115 $-$ 37 $-$ 42 $10^6$ 67 132 76 78 102 53 $10^7$ 72 140 96 94 $-$ 57
 $V_0$ Computational experiment number, the value $\tau_0$ (day) 1 2 3 4 5 6 $10$ 1 1 $-$ $-$ $-$ 46 $10^2$ 51 46 $-$ 51 $-$ 44 $10^4$ 60 115 $-$ 37 $-$ 42 $10^6$ 67 132 76 78 102 53 $10^7$ 72 140 96 94 $-$ 57
  Olivier Bonnefon, Jérôme Coville, Jimmy Garnier, Lionel Roques. Inside dynamics of solutions of integro-differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3057-3085. doi: 10.3934/dcdsb.2014.19.3057  Narcisa Apreutesei, Arnaud Ducrot, Vitaly Volpert. Travelling waves for integro-differential equations in population dynamics. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 541-561. doi: 10.3934/dcdsb.2009.11.541  Songbai Guo, Wanbiao Ma. Global behavior of delay differential equations model of HIV infection with apoptosis. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 103-119. doi: 10.3934/dcdsb.2016.21.103  Frederic Abergel, Remi Tachet. A nonlinear partial integro-differential equation from mathematical finance. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 907-917. doi: 10.3934/dcds.2010.27.907  Jean-Michel Roquejoffre, Juan-Luis Vázquez. Ignition and propagation in an integro-differential model for spherical flames. Discrete & Continuous Dynamical Systems - B, 2002, 2 (3) : 379-387. doi: 10.3934/dcdsb.2002.2.379  Liang Zhang, Bingtuan Li. Traveling wave solutions in an integro-differential competition model. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 417-428. doi: 10.3934/dcdsb.2012.17.417  Tomás Caraballo, P.E. Kloeden. Non-autonomous attractors for integro-differential evolution equations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 17-36. doi: 10.3934/dcdss.2009.2.17  Yi Cao, Jianhua Wu, Lihe Wang. Fundamental solutions of a class of homogeneous integro-differential elliptic equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1237-1256. doi: 10.3934/dcds.2019053  Yubo Chen, Wan Zhuang. The extreme solutions of PBVP for integro-differential equations with caratheodory functions. Conference Publications, 1998, 1998 (Special) : 160-166. doi: 10.3934/proc.1998.1998.160  Ramasamy Subashini, Chokkalingam Ravichandran, Kasthurisamy Jothimani, Haci Mehmet Baskonus. Existence results of Hilfer integro-differential equations with fractional order. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 911-923. doi: 10.3934/dcdss.2020053  Tonny Paul, A. Anguraj. Existence and uniqueness of nonlinear impulsive integro-differential equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1191-1198. doi: 10.3934/dcdsb.2006.6.1191  Sertan Alkan. A new solution method for nonlinear fractional integro-differential equations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1065-1077. doi: 10.3934/dcdss.2015.8.1065  Tianling Jin, Jingang Xiong. Schauder estimates for solutions of linear parabolic integro-differential equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5977-5998. doi: 10.3934/dcds.2015.35.5977  Patricio Felmer, Ying Wang. Qualitative properties of positive solutions for mixed integro-differential equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 369-393. doi: 10.3934/dcds.2019015  Sebti Kerbal, Yang Jiang. General integro-differential equations and optimal controls on Banach spaces. Journal of Industrial & Management Optimization, 2007, 3 (1) : 119-128. doi: 10.3934/jimo.2007.3.119  Eitan Tadmor, Prashant Athavale. Multiscale image representation using novel integro-differential equations. Inverse Problems & Imaging, 2009, 3 (4) : 693-710. doi: 10.3934/ipi.2009.3.693  Michel Chipot, Senoussi Guesmia. On a class of integro-differential problems. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1249-1262. doi: 10.3934/cpaa.2010.9.1249  Zindoga Mukandavire, Abba B. Gumel, Winston Garira, Jean Michel Tchuenche. Mathematical analysis of a model for HIV-malaria co-infection. Mathematical Biosciences & Engineering, 2009, 6 (2) : 333-362. doi: 10.3934/mbe.2009.6.333  Thanh-Anh Nguyen, Dinh-Ke Tran, Nhu-Quan Nguyen. Weak stability for integro-differential inclusions of diffusion-wave type involving infinite delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3637-3654. doi: 10.3934/dcdsb.2016114  Stephen Pankavich, Nathan Neri, Deborah Shutt. Bistable dynamics and Hopf bifurcation in a refined model of early stage HIV infection. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 2867-2893. doi: 10.3934/dcdsb.2020044

2019 Impact Factor: 1.233