Stability of an adaptive immunity delayed HIV infection model with active and silent cell-to-cell spread

A. M. Elaiw; N. H. AlShamrani; A. D. Hobiny; A. M. Elaiw; N. H. AlShamrani; A. D. Hobiny

doi:10.3934/mbe.2020337

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 6: 6401-6458. doi: 10.3934/mbe.2020337

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Stability of an adaptive immunity delayed HIV infection model with active and silent cell-to-cell spread

1.
Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut, Egypt
3.
Department of Mathematics, Faculty of Science, University of Jeddah, P. O. Box 80327, Jeddah 21589, Saudi Arabia

Received: 01 June 2020 Accepted: 07 September 2020 Published: 24 September 2020
MSC : 34D20, 34D23, 37N25, 92B05

This paper investigates an adaptive immunity HIV infection model with three types of distributed time delays. The model describes the interaction between healthy CD4⁺T cells, silent infected cells, active infected cells, free HIV particles, Cytotoxic T lymphocytes (CTLs) and antibodies. The healthy CD4⁺T cells can be infected when they contacted by free HIV particles or silent infected cells or active infected cells. The incidence rates of the healthy CD4⁺T cells with free HIV particles, silent infected cells, and active infected cells are given by general functions. Moreover, the production/proliferation and removal/death rates of the virus and cells are represented by general functions. The model is an improvement of the existing HIV infection models which have neglected the infection due to the incidence between the silent infected cells and healthy CD4⁺T cells. We show that the model is well posed and it has five equilibria and their existence are governed by five threshold parameters. Under a set of conditions on the general functions and the threshold parameters, we have proven the global asymptotic stability of all equilibria by using Lyapunov method. We have illustrated the theoretical results via numerical simulations. We have studied the effect of cell-to-cell (CTC) transmission and time delays on the dynamical behavior of the system. We have shown that the inclusion of time delay can significantly increase the concentration of the healthy CD4⁺ T cells and reduce the concentrations of the infected cells and free HIV particles. While the inclusion of CTC transmission decreases the concentration of the healthy CD4⁺ T cells and increases the concentrations of the infected cells and free HIV particles.
- HIV infection,
- cell-to-cell spread,
- intracellular delay,
- global stability,
- silent infected cells,
- adaptive immune response,
- Lyapunov function
Citation: A. M. Elaiw, N. H. AlShamrani, A. D. Hobiny. Stability of an adaptive immunity delayed HIV infection model with active and silent cell-to-cell spread[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 6401-6458. doi: 10.3934/mbe.2020337

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Abstract

This paper investigates an adaptive immunity HIV infection model with three types of distributed time delays. The model describes the interaction between healthy CD4⁺T cells, silent infected cells, active infected cells, free HIV particles, Cytotoxic T lymphocytes (CTLs) and antibodies. The healthy CD4⁺T cells can be infected when they contacted by free HIV particles or silent infected cells or active infected cells. The incidence rates of the healthy CD4⁺T cells with free HIV particles, silent infected cells, and active infected cells are given by general functions. Moreover, the production/proliferation and removal/death rates of the virus and cells are represented by general functions. The model is an improvement of the existing HIV infection models which have neglected the infection due to the incidence between the silent infected cells and healthy CD4⁺T cells. We show that the model is well posed and it has five equilibria and their existence are governed by five threshold parameters. Under a set of conditions on the general functions and the threshold parameters, we have proven the global asymptotic stability of all equilibria by using Lyapunov method. We have illustrated the theoretical results via numerical simulations. We have studied the effect of cell-to-cell (CTC) transmission and time delays on the dynamical behavior of the system. We have shown that the inclusion of time delay can significantly increase the concentration of the healthy CD4⁺ T cells and reduce the concentrations of the infected cells and free HIV particles. While the inclusion of CTC transmission decreases the concentration of the healthy CD4⁺ T cells and increases the concentrations of the infected cells and free HIV particles.

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