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  184      Downloads  56
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.
Keywords
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
Copyright
Copyright © 2019 Authors retain the copyright of this article.
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.
Reference
[1]
Arruda, E. D. (2015). An Optimal Control Approach to HIV Immunology. Applied Mathematics, 1115-1130.
[2]
Bonhoeffer, R. M. (2013). Production of resistant HIV mutants during antiretroviral therapy. Journal of Apllied Mathematics.
[3]
Fisher, K. A. (2013). Stability conditions of HIV co-infection model using Next generation method. Journal of Applied Mathematics Science, 2815-2832.
[4]
Hatta, K. A. (2012). Optimal control of a Delayed HIV infection model with immune Response using an efficient Numerical method. Journal of ISRN Biomathematics, 1-7.
[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.
[11]
Ngina Purity M., R. W., L. S. L. (2018). Modelling Optimal Control of In-Host Dynamics Using Different Control Strategies. Journal of Applied Mathematics. https://doi.org/10.1155/2018/9385080.
[12]
Perelson, S. (1989). modelling the interaction of immune system with HIV in mathematical and stastical approaches to AIS epidemiology. Journal of Mathematical and Statistics Appraoches on AIDs Epidemiology, 350-370.
[13]
Picker, A. (2013). CD4+ T-cells depletion in HIV infection. Journal of Mathematical Biology, 54-64.
[14]
Prashant K. Srivastava, M. B. (2012). Dynamic model of in-host HIV infection with drug therapy and multiviral strains. Journal of Biology Systems, 303-325.
[15]
Salantes, D. B. (2018). Rebound Relationships. An Investigation of HIV-1 Rebound Dynamics and Host Immune Response During Analytical Treatment Interruption. Publicly Accessible Penn Dissertations. 3179.
[16]
Assone T., A. (2016). Genetic makers of the host in persons living with HTLV-1, HIV and HCV infections. Journal of Mathematical Biology.
[17]
Tianlei Wang, Z. H. (2015). Global stability analysis for delayed virus infection model with general incidence rate and humoral immunity. Journal of Mathematics and computers in simulation 89, 13-22.
[18]
Waema. R. M., L. L. (2013). Stochastic model for in-host HIV dynamics with therapeutic intervention. Journal of ISRN Biomathematics, 11 pages.
Browse journals by subject