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STABILITY ANALYSIS OF A NOVEL ODE MODEL FOR HIV INFECTION

Year 2021, Volume: 3 Issue: 1, 30 - 51, 29.04.2021

Hoang Ngo Hung Dang Nguyen Mehmet Dik

https://doi.org/10.47087/mjm.911431

Abstract

In this paper, we propose and investigate the stability of a novel
3-compartment ordinary differential equation (ODE) model of HIV infection
of CD4+ T-cells with a mass action term. Similar to various endemic models,
the dynamics within the model is fully determined by the basic reproduction
term R0. If R0 < 1, the disease-free (zero) equilibrium will be asymptotically
stable. On the other hand, if R0 > 1, there exists a positive equilibrium that
is globally/orbitally asymptotically stable under certain conditions within the
interior of a predefined region. Finally, numerical simulations are conducted to
illustrate and verify the results.

Keywords

HIV globally asymptotical stability periodic solution

References

  • [1] X. Song, and A. U. Neumann, Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl. 329 (2007) 281-297.
  • [2] R. V. Culshaw, and S. Ruan, A delay - differential equation model of HIV infection of CD4+ T-cells, Math. Biosci. 165 (2000) 27-39.
  • [3] G. Teschl, Ordinary Differential Equations and Dynamical Systems (2012), American Mathematical Society, Providence, ISBN 978-0-8218-8328-0.
  • [4] R. J. De Boer, and A. S. Perelson, Target Cell Limited and Immune Control Models of HIV Infection: A Comparison 190 (1998) 201-214.
  • [5] A. S. Perelson, Modelling the interaction of the immune system with HIV, In: C. Castillo-Chavez (Eds), Mathematical and Statistical Approaches to AIDS Epidemiology (1989) 250, Springer, Berlin.
  • [6] A. S. Perelson, Dynamics of HIV Infection of CD4+ T-cells, Math. Biosci. 114 (1993) 81.
  • [7] A. S. Perelson, A. U. Neumann, M. Markowitz, J. M. Leonard, and D.D. Ho, HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time, Science 271 (1996) 1582.
  • [8] A. S. Perelson, and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev. 41 (1999) 3.
  • [9] J. E. Mittler, B. Sulzer, A. U. Neumann, and A.S. Perelson, Influence of delayed viral production on viral dynamics in HIV-1 infected patients, Math. Biosci. 251 (1998) 143.
  • [10] J. E. Mittler, M. Markowitz, D. D. Ho, and A. S. Perelson, Improved estimates for HIV-1 clearance rate and intracellular delay, AIDS 13 (1999) 1415.
  • [11] R. M. Anderson, and R. M. May, Complex dynamical behavior in the interaction between HIV and the immune system, In: A. Goldbeter (Eds), Cell to Cell Signalling: From Experiments to Theoretical Models (1989) 335, Academic Press, New York.
  • [12] A. R. McLean, and T. B. L. Kirkwood, A model of human immunodeficiency virus infection in T-helper cell clones, J. Theoret. Biol. 147 (1990) 177.
  • [13] M. A. Nowak, and R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science 272 (1996) 74.
  • [14] D. E. Kirschner, Using mathematics to understand HIV immune dynamics, Notices Amer. Math. Soc. 43 (1996) 191.
  • [15] Y. Kuang, Delay-Differential Equations with Applications in Population Dynamics (1993), Academic Press, New York.
  • [16] A. V. M. Herz, S. Bonhoeffer, R. M. Anderson, R. M. May, and M. A. Nowak, Viral dynamics in vivo: limitations on estimates of intracellular delay and virus decay, Proc. Natl. Acad. Sci. USA 93 (1996) 7247.
  • [17] J. Tam, Delay effect in a model for virus replication, Math. Med. Biol. 16 (1999) 29.
  • [18] T. B. Kepler, and A.S. Perelson, Cyclic re-entry of germinal center B cells and the efficiency of affinity maturation, Immunology Today, 14 (1993) 412-415.
  • [19] J. K. Percus, O. E. Percus, and A. S. Perelson, Predicting the size of the T -cell receptor and antibody combining region from consideration of efficient self–nonself discrimination, Proc. Natl. Acad. Sci. USA 90 (1993) 2691-2695.
  • [20] F. R. Gantmacher, The Theory of Matrices (1959), Chelsea Publishing Company, New York.
  • [21] H. R. Zhu, H. L. Smith, and M. W. Hirsch, Stable periodic orbits for a class of three dimensional competitive systems, J. Differential Equations 110 (1994) 143-156.
  • [22] M. W. Hirsch, System of differential equations which are competitive or cooperative, IV, SIAM J. Math. Anal. 21 (1990) 1225-1234.
  • [23] G. Butler, H. I. Freedman, and P. Waltma, Uniform persistence system, Proc. Amer. Math. Soc. 96 (1986) 425-430.
  • [24] H. L. Smith, and H. Thieme, Convergence for strongly ordered preserving semiflows, SIAM J. Math. Anal. 22 (1991) 1081-1101.
  • [25] J. S. Muldowney, Compound matrices and ordinary differential equations, Rocky Mountain J. Math. 20 (1990) 857-872.
  • [26] Y. Li, and J. S. Muldowney, Global stability for the SEIR model in epidemiology, Math. Biosci. 125 (1995) 115-164.
  • [27] International Committee on Taxonomy of Viruses, Taxonomy (2019), National Institutes of Health, URL: https://talk.ictvonline.org/taxonomy/
  • [28] R. Harvey, Microbiology (3rd ed.) (2012) 295-306, Lippincott Williams and Wilkins, Philadelphia, PA.
  • [29] C. Rackauckas, and Q. Nie, DifferentialEquations.jl - A Performant and Feature-Rich Ecosystem for Solving Differential Equations in Julia, J. Open Res. Softw. 5(1) 2017 15.

Year 2021, Volume: 3 Issue: 1, 30 - 51, 29.04.2021

Hoang Ngo Hung Dang Nguyen Mehmet Dik

https://doi.org/10.47087/mjm.911431

Abstract

References

  • [1] X. Song, and A. U. Neumann, Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl. 329 (2007) 281-297.
  • [2] R. V. Culshaw, and S. Ruan, A delay - differential equation model of HIV infection of CD4+ T-cells, Math. Biosci. 165 (2000) 27-39.
  • [3] G. Teschl, Ordinary Differential Equations and Dynamical Systems (2012), American Mathematical Society, Providence, ISBN 978-0-8218-8328-0.
  • [4] R. J. De Boer, and A. S. Perelson, Target Cell Limited and Immune Control Models of HIV Infection: A Comparison 190 (1998) 201-214.
  • [5] A. S. Perelson, Modelling the interaction of the immune system with HIV, In: C. Castillo-Chavez (Eds), Mathematical and Statistical Approaches to AIDS Epidemiology (1989) 250, Springer, Berlin.
  • [6] A. S. Perelson, Dynamics of HIV Infection of CD4+ T-cells, Math. Biosci. 114 (1993) 81.
  • [7] A. S. Perelson, A. U. Neumann, M. Markowitz, J. M. Leonard, and D.D. Ho, HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time, Science 271 (1996) 1582.
  • [8] A. S. Perelson, and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev. 41 (1999) 3.
  • [9] J. E. Mittler, B. Sulzer, A. U. Neumann, and A.S. Perelson, Influence of delayed viral production on viral dynamics in HIV-1 infected patients, Math. Biosci. 251 (1998) 143.
  • [10] J. E. Mittler, M. Markowitz, D. D. Ho, and A. S. Perelson, Improved estimates for HIV-1 clearance rate and intracellular delay, AIDS 13 (1999) 1415.
  • [11] R. M. Anderson, and R. M. May, Complex dynamical behavior in the interaction between HIV and the immune system, In: A. Goldbeter (Eds), Cell to Cell Signalling: From Experiments to Theoretical Models (1989) 335, Academic Press, New York.
  • [12] A. R. McLean, and T. B. L. Kirkwood, A model of human immunodeficiency virus infection in T-helper cell clones, J. Theoret. Biol. 147 (1990) 177.
  • [13] M. A. Nowak, and R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science 272 (1996) 74.
  • [14] D. E. Kirschner, Using mathematics to understand HIV immune dynamics, Notices Amer. Math. Soc. 43 (1996) 191.
  • [15] Y. Kuang, Delay-Differential Equations with Applications in Population Dynamics (1993), Academic Press, New York.
  • [16] A. V. M. Herz, S. Bonhoeffer, R. M. Anderson, R. M. May, and M. A. Nowak, Viral dynamics in vivo: limitations on estimates of intracellular delay and virus decay, Proc. Natl. Acad. Sci. USA 93 (1996) 7247.
  • [17] J. Tam, Delay effect in a model for virus replication, Math. Med. Biol. 16 (1999) 29.
  • [18] T. B. Kepler, and A.S. Perelson, Cyclic re-entry of germinal center B cells and the efficiency of affinity maturation, Immunology Today, 14 (1993) 412-415.
  • [19] J. K. Percus, O. E. Percus, and A. S. Perelson, Predicting the size of the T -cell receptor and antibody combining region from consideration of efficient self–nonself discrimination, Proc. Natl. Acad. Sci. USA 90 (1993) 2691-2695.
  • [20] F. R. Gantmacher, The Theory of Matrices (1959), Chelsea Publishing Company, New York.
  • [21] H. R. Zhu, H. L. Smith, and M. W. Hirsch, Stable periodic orbits for a class of three dimensional competitive systems, J. Differential Equations 110 (1994) 143-156.
  • [22] M. W. Hirsch, System of differential equations which are competitive or cooperative, IV, SIAM J. Math. Anal. 21 (1990) 1225-1234.
  • [23] G. Butler, H. I. Freedman, and P. Waltma, Uniform persistence system, Proc. Amer. Math. Soc. 96 (1986) 425-430.
  • [24] H. L. Smith, and H. Thieme, Convergence for strongly ordered preserving semiflows, SIAM J. Math. Anal. 22 (1991) 1081-1101.
  • [25] J. S. Muldowney, Compound matrices and ordinary differential equations, Rocky Mountain J. Math. 20 (1990) 857-872.
  • [26] Y. Li, and J. S. Muldowney, Global stability for the SEIR model in epidemiology, Math. Biosci. 125 (1995) 115-164.
  • [27] International Committee on Taxonomy of Viruses, Taxonomy (2019), National Institutes of Health, URL: https://talk.ictvonline.org/taxonomy/
  • [28] R. Harvey, Microbiology (3rd ed.) (2012) 295-306, Lippincott Williams and Wilkins, Philadelphia, PA.
  • [29] C. Rackauckas, and Q. Nie, DifferentialEquations.jl - A Performant and Feature-Rich Ecosystem for Solving Differential Equations in Julia, J. Open Res. Softw. 5(1) 2017 15.
There are 29 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Hoang Ngo

Hung Dang Nguyen This is me 0000-0002-3169-6499

Mehmet Dik 0000-0003-0643-2771

Publication Date April 29, 2021
Acceptance Date April 15, 2021
Published in Issue Year 2021 Volume: 3 Issue: 1

Cite

APA Ngo, H., Dang Nguyen, H., & Dik, M. (2021). STABILITY ANALYSIS OF A NOVEL ODE MODEL FOR HIV INFECTION. Maltepe Journal of Mathematics, 3(1), 30-51. https://doi.org/10.47087/mjm.911431
AMA Ngo H, Dang Nguyen H, Dik M. STABILITY ANALYSIS OF A NOVEL ODE MODEL FOR HIV INFECTION. Maltepe Journal of Mathematics. April 2021;3(1):30-51. doi:10.47087/mjm.911431
Chicago Ngo, Hoang, Hung Dang Nguyen, and Mehmet Dik. “STABILITY ANALYSIS OF A NOVEL ODE MODEL FOR HIV INFECTION”. Maltepe Journal of Mathematics 3, no. 1 (April 2021): 30-51. https://doi.org/10.47087/mjm.911431.
EndNote Ngo H, Dang Nguyen H, Dik M (April 1, 2021) STABILITY ANALYSIS OF A NOVEL ODE MODEL FOR HIV INFECTION. Maltepe Journal of Mathematics 3 1 30–51.
IEEE H. Ngo, H. Dang Nguyen, and M. Dik, “STABILITY ANALYSIS OF A NOVEL ODE MODEL FOR HIV INFECTION”, Maltepe Journal of Mathematics, vol. 3, no. 1, pp. 30–51, 2021, doi: 10.47087/mjm.911431.
ISNAD Ngo, Hoang et al. “STABILITY ANALYSIS OF A NOVEL ODE MODEL FOR HIV INFECTION”. Maltepe Journal of Mathematics 3/1 (April 2021), 30-51. https://doi.org/10.47087/mjm.911431.
JAMA Ngo H, Dang Nguyen H, Dik M. STABILITY ANALYSIS OF A NOVEL ODE MODEL FOR HIV INFECTION. Maltepe Journal of Mathematics. 2021;3:30–51.
MLA Ngo, Hoang et al. “STABILITY ANALYSIS OF A NOVEL ODE MODEL FOR HIV INFECTION”. Maltepe Journal of Mathematics, vol. 3, no. 1, 2021, pp. 30-51, doi:10.47087/mjm.911431.
Vancouver Ngo H, Dang Nguyen H, Dik M. STABILITY ANALYSIS OF A NOVEL ODE MODEL FOR HIV INFECTION. Maltepe Journal of Mathematics. 2021;3(1):30-51.
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