Mathematical Modelling and Applications

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Modelling the Effects of Immune Response and Time Delay on HIV-1 in Vivo Dynamics in the Presence of Chemotherapy

Received: Jun. 10, 2019    Accepted: Jul. 11, 2019    Published: Jul. 22, 2019
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Abstract

Numerous models of mathematics have existed to pronounce the immunological response to contagion by human immunodeficiency virus (HIV-1). The models have been used to envisage the regression of HIV-1 in vitro and in vivo dynamics. Ordinarily the studies have been on the interface of HIV virions, CD4+T-cells and Antiretroviral (ARV). In this study, time delay, chemotherapy and role of CD8+T-cells is considered in the HIV-1 in-vivo dynamics. The delay is used to account for the latent time that elapses between exposure of a host cell to HIV-1 and the production of contagious virus from the host cell. This is the period needed to cause HIV-1 to replicate within the host cell in adequate number to become transmittable. Chemotherapy is by use of combination of Reverse transcriptase inhibitor and Protease inhibitor. CD8+T-cells is innate immune response. The model has six variables: Healthy CD4+T-cells, Sick CD4+T-cells, Infectious virus, Non-infectious virus, used CD8+T-cells and unused CD8+T-cells. Positivity and boundedness of the solutions to the model equations is proved. In addition, Reproduction number (R0) is derived from Next Generation Matrix approach. The stability of disease free equilibrium is checked by use of linearization of the model equation. We show that the Disease Free Equilibrium is locally stable if and only if R0<1 and unstable otherwise. Of significance is the effect of CD8+ T- cells, time delay and drug efficacy on stability of Disease Free Equilibrium (DFE). From analytical results it is evident that for all τ > 0, Disease Free Equilibrium is stable when τ =0.67. This stability is only achieved if drug efficacy is administered. The results show that when drug efficacy of α1=0.723 and α2=0.723 the DFE is achieved.

DOI 10.11648/j.mma.20190402.11
Published in Mathematical Modelling and Applications ( Volume 4, Issue 2, June 2019 )
Page(s) 15-21
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

HIV, Reproduction Number, Delay, Stability, Disease Free Equilibrium, Chemotherapy

References
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[5] Jinliang Wang, J. (2015). Analysis of an age structured HIV infection model with virus-to-cell infection and cell-to-cell transmission. Journal of Apllied Mathematics.
[6] Kasia A Pawelek, S. L. (2012). A model of HIV-1 infection with two time delays: Mathematics analysis and comparison with patient data. Journal of Mathematics Biology, 98-109.
[7] Kirui Wesley, R. K. (2015). Modelling the effects of time delay on HIV-1 in vivo dynamics in the presence of ARVs. Science Journal of Applied Mathematics and Statistics, 204-213.
[8] Michael Y Li, H. (2012). Global dynamics of a mathematical model for HTLV-1 infection of CD4+ T cells with delayed with CTL response. Journal of Mathematics Biology, 1080-1092.
[9] Nakul, C. (2011). The basic reproduction number. Swiss: Swiss Tropical and Public Health Insitute.
[10] Ngina Purity M., R. W. (2017). Mathematical modelling of in vivo dynamics of HIV subject to the influence of the CD8+ T-cells. Journal of Applied Mathematics, 1153-1179.
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  • APA Style

    Cherono Pela, Kirui Wesley, Adicka Daniel. (2019). Modelling the Effects of Immune Response and Time Delay on HIV-1 in Vivo Dynamics in the Presence of Chemotherapy. Mathematical Modelling and Applications, 4(2), 15-21. https://doi.org/10.11648/j.mma.20190402.11

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    ACS Style

    Cherono Pela; Kirui Wesley; Adicka Daniel. Modelling the Effects of Immune Response and Time Delay on HIV-1 in Vivo Dynamics in the Presence of Chemotherapy. Math. Model. Appl. 2019, 4(2), 15-21. doi: 10.11648/j.mma.20190402.11

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    AMA Style

    Cherono Pela, Kirui Wesley, Adicka Daniel. Modelling the Effects of Immune Response and Time Delay on HIV-1 in Vivo Dynamics in the Presence of Chemotherapy. Math Model Appl. 2019;4(2):15-21. doi: 10.11648/j.mma.20190402.11

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  • @article{10.11648/j.mma.20190402.11,
      author = {Cherono Pela and Kirui Wesley and Adicka Daniel},
      title = {Modelling the Effects of Immune Response and Time Delay on HIV-1 in Vivo Dynamics in the Presence of Chemotherapy},
      journal = {Mathematical Modelling and Applications},
      volume = {4},
      number = {2},
      pages = {15-21},
      doi = {10.11648/j.mma.20190402.11},
      url = {https://doi.org/10.11648/j.mma.20190402.11},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.mma.20190402.11},
      abstract = {Numerous models of mathematics have existed to pronounce the immunological response to contagion by human immunodeficiency virus (HIV-1). The models have been used to envisage the regression of HIV-1 in vitro and in vivo dynamics. Ordinarily the studies have been on the interface of HIV virions, CD4+T-cells and Antiretroviral (ARV). In this study, time delay, chemotherapy and role of CD8+T-cells is considered in the HIV-1 in-vivo dynamics. The delay is used to account for the latent time that elapses between exposure of a host cell to HIV-1 and the production of contagious virus from the host cell. This is the period needed to cause HIV-1 to replicate within the host cell in adequate number to become transmittable. Chemotherapy is by use of combination of Reverse transcriptase inhibitor and Protease inhibitor. CD8+T-cells is innate immune response. The model has six variables: Healthy CD4+T-cells, Sick CD4+T-cells, Infectious virus, Non-infectious virus, used CD8+T-cells and unused CD8+T-cells. Positivity and boundedness of the solutions to the model equations is proved. In addition, Reproduction number (R0) is derived from Next Generation Matrix approach. The stability of disease free equilibrium is checked by use of linearization of the model equation. We show that the Disease Free Equilibrium is locally stable if and only if R0 0, Disease Free Equilibrium is stable when τ =0.67. This stability is only achieved if drug efficacy is administered. The results show that when drug efficacy of α1=0.723 and α2=0.723 the DFE is achieved.},
     year = {2019}
    }
    

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    AU  - Cherono Pela
    AU  - Kirui Wesley
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    AB  - Numerous models of mathematics have existed to pronounce the immunological response to contagion by human immunodeficiency virus (HIV-1). The models have been used to envisage the regression of HIV-1 in vitro and in vivo dynamics. Ordinarily the studies have been on the interface of HIV virions, CD4+T-cells and Antiretroviral (ARV). In this study, time delay, chemotherapy and role of CD8+T-cells is considered in the HIV-1 in-vivo dynamics. The delay is used to account for the latent time that elapses between exposure of a host cell to HIV-1 and the production of contagious virus from the host cell. This is the period needed to cause HIV-1 to replicate within the host cell in adequate number to become transmittable. Chemotherapy is by use of combination of Reverse transcriptase inhibitor and Protease inhibitor. CD8+T-cells is innate immune response. The model has six variables: Healthy CD4+T-cells, Sick CD4+T-cells, Infectious virus, Non-infectious virus, used CD8+T-cells and unused CD8+T-cells. Positivity and boundedness of the solutions to the model equations is proved. In addition, Reproduction number (R0) is derived from Next Generation Matrix approach. The stability of disease free equilibrium is checked by use of linearization of the model equation. We show that the Disease Free Equilibrium is locally stable if and only if R0 0, Disease Free Equilibrium is stable when τ =0.67. This stability is only achieved if drug efficacy is administered. The results show that when drug efficacy of α1=0.723 and α2=0.723 the DFE is achieved.
    VL  - 4
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Author Information
  • Department of Mathematics and Computer Science, University of Kabianga, Kericho, Kenya

  • Department of Mathematics and Actuarial Science, South Eastern Kenya University, Kitui, Kenya

  • Department of Mathematics and Computer Science, University of Kabianga, Kericho, Kenya

  • Section