Volume 4, Issue 2, June 2019, Page: 15-21
Modelling the Effects of Immune Response and Time Delay on HIV-1 in Vivo Dynamics in the Presence of Chemotherapy
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
Received: Jun. 10, 2019;       Accepted: Jul. 11, 2019;       Published: Jul. 22, 2019
DOI: 10.11648/j.mma.20190402.11      View  46      Downloads  32
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.
HIV, Reproduction Number, Delay, Stability, Disease Free Equilibrium, Chemotherapy
To cite this article
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, Mathematical Modelling and Applications. Vol. 4, No. 2, 2019, pp. 15-21. doi: 10.11648/j.mma.20190402.11
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This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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