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

Cherono Pela; Kirui Wesley; Adicka Daniel

doi:doi:10.11648/j.mma.20190402.11

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

Cherono Pela, Kirui Wesley, Adicka Daniel

Published in Mathematical Modelling and Applications (Volume 4, Issue 2)

Received: 10 June 2019 Accepted: 11 July 2019 Published: 22 July 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.

Published in	Mathematical Modelling and Applications (Volume 4, Issue 2)
DOI	10.11648/j.mma.20190402.11
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), 2019. Published by Science Publishing Group

Keywords

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

References

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[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://article.sciencepublishinggroup.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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TY - JOUR
T1 - Modelling the Effects of Immune Response and Time Delay on HIV-1 in Vivo Dynamics in the Presence of Chemotherapy
AU - Cherono Pela
AU - Kirui Wesley
AU - Adicka Daniel
Y1 - 2019/07/22
PY - 2019
N1 - https://doi.org/10.11648/j.mma.20190402.11
DO - 10.11648/j.mma.20190402.11
T2 - Mathematical Modelling and Applications
JF - Mathematical Modelling and Applications
JO - Mathematical Modelling and Applications
SP - 15
EP - 21
PB - Science Publishing Group
SN - 2575-1794
UR - https://doi.org/10.11648/j.mma.20190402.11
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
IS - 2
ER -

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Author Information

Cherono Pela

Department of Mathematics and Computer Science, University of Kabianga, Kericho, Kenya
Kirui Wesley

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

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

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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://article.sciencepublishinggroup.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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