Global stability of HIV/HTLV co-infection model with CTL-mediated immunity

A. M. Elaiw; N. H. AlShamrani

doi:10.3934/dcdsb.2021108

Article Contents

2022, Volume 27, Issue 3: 1725-1764. Doi: 10.3934/dcdsb.2021108

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Global stability of HIV/HTLV co-infection model with CTL-mediated immunity

A. M. Elaiw^1,2, , and
N. H. AlShamrani^1,3,

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 Branch, Assiut, Egypt
3.
Department of Mathematics, Faculty of Science, University of Jeddah, P. O. Box 80327, Jeddah 21589, Saudi Arabia

^* Corresponding author: A. M. Elaiw
^* Corresponding author: A. M. Elaiw

Received: June 30, 2020

Revised: October 31, 2020

Early access: April 2021

Published: March 2022

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Abstract

Mathematical modeling of human immunodeficiency virus (HIV) and human T-lymphotropic virus type Ⅰ (HTLV-I) mono-infections has received considerable attention during the last decades. These two viruses share the same way of transmission between individuals; through direct contact with certain contaminated body fluids. Therefore, a person can be co-infected with both viruses. In the present paper, we construct and analyze a new HIV/HTLV-I co-infection model under the effect of Cytotoxic T lymphocytes (CTLs) immune response. The model describes the interaction between susceptible CD$ 4^{+} $T cells, silent HIV-infected cells, active HIV-infected cells, silent HTLV-infected cells, Tax-expressing (active) HTLV-infected cells, free HIV particles, HIV-specific CTLs and HTLV-specific CTLs. The HIV can spread by two routes of transmission, virus-to-cell (VTC) and cell-to-cell (CTC). Both active and silent HIV-infected cells can infect the susceptible CD$ 4^{+} $T cells by CTC mechanism. On the other side, HTLV-I has only one mode of transmission via direct cell-to-cell contact. The well-posedness of the model is established by showing that the solutions of the model are nonnegative and bounded. We calculate all possible equilibria and define the key threshold parameters which govern the existence and stability of all equilibria of the model. We explore the global asymptotic stability of all equilibria by utilizing Lyapunov function and LaSalle's invariance principle. We have discussed the influence of CTL immune response on the co-infection dynamics. We have presented numerical simulations to justify the applicability and effectiveness of the theoretical results. In addition, we evaluate the effect of HTLV-I infection on the HIV dynamics and vice versa.

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Mathematics Subject Classification: Primary: 34D20, 34D23, 37N25, 92B05.

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Figure 1. The schematic diagram of the HIV/HTLV-I co-infection dynamics in vivo

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Figure 2. The behavior of solution trajectories of system (4) when $\Re_{1}\leq1$ and $\Re_{2}\leq1$

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Figure 3. The behavior of solution trajectories of system (4) when $\Re_{1} > 1$, $\Re_{2}/\Re_{1}\leq1$ and $\Re_{3}\leq1$

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Figure 4. The behavior of solution trajectories of system (4) when $\Re_{2} > 1$, $\Re_{1}/\Re_{2}\leq1$ and $\Re_{4}\leq1$

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Figure 5. The behavior of solution trajectories of system (4) when $\Re_{3} > 1$ and $\Re_{5}\leq1$

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Figure 6. The behavior of solution trajectories of system (4) when $\Re_{4} > 1$ and $\Re_{6}\leq1$

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Figure 7. The behavior of solution trajectories of system (4) when $\Re_{5}^{\ast} > 1$, $\Re_{5} > 1$, $\Re_{8}\leq1$ and $\Re_{1}/\Re_{2} > 1$

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Figure 8. The behavior of solution trajectories of system (4) when $\Re_{6} > 1$, $\Re_{7}\leq1$ and $\Re_{2}/\Re_{1} > 1$

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Figure 9. The behavior of solution trajectories of system (4) when $\Re_{7} > 1$ and $\Re_{8} > 1$

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Figure 10. The influence of HTLV-I infection rate ($\eta_{4}\neq0$) on HIV mono-infection dynamics (31) will cause a chronic HIV/HTLV-I co-infection

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Figure 11. The influence of HIV infection rates ($\eta_{1}, \eta_{2}, \eta_{3} \neq 0$) on HTLV mono-infection dynamics (32) will cause a chronic HIV/HTLV-I co-infection

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Table 1. Parameter description

Parameter	Description
$ \rho $	Recruitment rate for the susceptible CD4⁺T cells
$ \alpha $	Natural mortality rate constant for the susceptible CD4⁺T cells
$ \eta_{1} $	Virus-cell incidence rate constant between free HIV particles and susceptible CD4⁺T cells
$ \eta_{2} $	Cell-cell incidence rate constant between silent HIV-infected cells and susceptible CD4⁺T cells
$ \eta_{3} $	Cell-cell incidence rate constant between active HIV-infected cells and susceptible CD4⁺T cells
$ \eta_{4} $	Cell-cell incidence rate constant between Tax-expressing HTLV-infected cells and susceptible CD4⁺T cells
$ \beta \in \left( 0,1\right) $	Fraction coefficient accounts for the probability of new HIV-infected cells could be active, and the remaining part 1-β will be latent
$ \gamma $	Death rate constant of silent HIV-infected cells
$ a $	Death rate constant of active HIV-infected cells
$ \mu_{1} $	Killing rate constant of active HIV-infected cells due to HIV-specific CTLs
$ \mu_{2} $	Killing rate constant of Tax-expressing HTLV-infected cells due to HTLV-specific CTLs
$ \varphi \in \left( 0,1\right) $	Probability of new HTLV infections could be enter a silent period
$ \lambda $	Transmission rate constant of silent HIV-infected cells that become active HIV-infected cells
$ \psi $	Transmission rate constant of silent HTLV-infected cells that become Tax-expressing HTLV-infected cells
$ \omega $	Death rate constant of silent HTLV-infected cells
$ \delta $	Death rate constant of Tax-expressing HTLV-infected cells
$ b $	Generation rate constant of new HIV particles
$ \varepsilon $	Death rate constant of free HIV particles
$ \sigma_{1} $	Proliferation rate constant of HIV-specific CTLs
$ \sigma_{2} $	Proliferation rate constant of HTLV-specific CTLs
$ \pi_{1} $	Decay rate constant of HIV-specific CTLs
$ \pi_{2} $	Decay rate constant of HTLV-specific CTLs

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Table 2. Model (4) equilibria and their existence conditions

Equilibrium point	Definition	Existence conditions
Ð$ _{0}=(S_{0},0,0,0,0,0,0,0) $	Infection-free equilibrium	None
Ð$ _{1}=(S_{1},L_{1},I_{1},0,0,V_{1},0,0) $	Chronic HIV mono-infection equilibrium with inactive immune response	$ \Re_{1}>1 $
Ð$ _{2}=(S_{2},0,0,E_{2},Y_{2},0,0,0) $	Chronic HTLV mono-infection equilibrium with inactive immune response	$ \Re_{2}>1 $
Ð$ _{3}=(S_{3},L_{3},I_{3},0,0,V_{3},C_{3}^{I},0) $	Chronic HIV mono-infection equilibrium with only active HIV-specific CTL	$ \Re_{3}>1 $
Ð$ _{4}=(S_{4},0,0,E_{4},Y_{4},0,0,C_{4}^{Y}) $	Chronic HTLV mono-infection equilibrium with only active HTLV-specific CTL	$ \Re_{4}>1 $
Ð$ _{5}=(S_{5},L_{5},I_{5},E_{5},Y_{5},V_{5},C_{5}^{I},0) $	Chronic HIV/HTLV co-infection equilibrium with only active HIV-specific CTL	$ \Re_{5}^{\ast}, $ $ \Re_{5}>1 $ and $ \Re_{1}/\Re_{2}>1 $
Ð$ _{6}=(S_{6},L_{6},I_{6},E_{6},Y_{6},V_{6},0,C_{6}^{Y}) $	Chronic HIV/HTLV co-infection equilibrium with only active HTLV-specific CTL	$ \Re_{6}>1 $ and $ \Re_{2}/\Re_{1}>1 $
Ð$ _{7}=(S_{7},L_{7},I_{7},E_{7},Y_{7},V_{7},C_{7}^{I},C_{7}^{Y}) $	Chronic HIV/HTLV co-infection equilibrium with active HIV-specific CTL and HTLV-specific CTL	$ \Re_{7}>1 $ and $ \Re_{8}>1 $

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Table 3. Global stability conditions of the equilibria of model (4)

Equilibrium point	Global stability conditions
Ð$ _{0}=(S_{0},0,0,0,0,0,0,0) $	$ \Re_{1}\leq1 $ and $ \Re_{2}\leq1 $
Ð$ _{1}=(S_{1},L_{1},I_{1},0,0,V_{1},0,0) $	$ \Re_{1}>1 $, $ \Re_{2}/\Re _{1}\leq1 $ and $ \Re_{3}\leq1 $
Ð$ _{2}=(S_{2},0,0,E_{2},Y_{2},0,0,0) $	$ \Re_{2}>1 $, $ \Re_{1}/\Re_{2}\leq1 $ and $ \Re_{4}\leq1 $
Ð$ _{3}=(S_{3},L_{3},I_{3},0,0,V_{3},C_{3}^{I},0) $	$ \Re_{3}>1 $ and $ \Re_{5}\leq1 $
Ð$ _{4}=(S_{4},0,0,E_{4},Y_{4},0,0,C_{4}^{Y}) $	$ \Re_{4}>1 $ and $ \Re _{6}\leq1 $
Ð$ _{5}=(S_{5},L_{5},I_{5},E_{5},Y_{5},V_{5},C_{5}^{I},0) $	$ \Re_{5}^{\ast}, $ $ \Re_{5}>1 $, $ \Re_{8}\leq1 $ and $ \Re_{1}/\Re_{2}>1 $
Ð$ _{6}=(S_{6},L_{6},I_{6},E_{6},Y_{6},V_{6},0,C_{6}^{Y}) $	$ \Re_{6}>1 $, $ \Re_{7}\leq1 $ and $ \Re_{2}/\Re_{1}>1 $
Ð$ _{7}=(S_{7},L_{7},I_{7},E_{7},Y_{7},V_{7},C_{7}^{I},C_{7}^{Y}) $	$ \Re_{7}>1 $ and $ \Re_{8}>1 $

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Table 4. The data of model (4)

Parameter	Value	Parameter	Value	Parameter	Value
$ \rho $	$ 10 $	$ \varphi $	$ 0.2 $	$ \varepsilon $	$ 2 $
$ \alpha $	$ 0.01 $	$ \delta $	$ 0.2 $	$ \gamma $	$ 0.02 $
$ \eta_{1} $	Varied	$ b $	$ 5 $	$ \sigma_{1} $	Varied
$ \eta_{2} $	Varied	$ \pi_{1} $	$ 0.1 $	$ \sigma_{2} $	Varied
$ \eta_{3} $	Varied	$ \pi_{2} $	$ 0.1 $	$ \lambda $	$ 0.2 $
$ \eta_{4} $	Varied	$ \mu_{1} $	$ 0.2 $	$ \omega $	$ 0.01 $
$ a $	$ 0.5 $	$ \mu_{2} $	$ 0.2 $	$ \psi $	$ 0.003 $

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Table 5. Local stability of positive equilibria Ð$ _{i} $, $ i = 0,1,...,7 $

Scenario	The equilibria	$ (\operatorname{Re}(\lambda_{i}), $ $ i=1,2,...,6) $	Stability
1	$ \begin{array} [c]{c} Ð_{0}=(1000,0,0,0,0,0,0,0) \end{array} $	$ \begin{array} [c]{c} \left( -1.98,-0.63,-0.2,-0.1,-0.1,-0.07,-0.01,-0.01\right) \end{array} $	$ \begin{array} [c]{l} \rm{stable} \end{array} $
2	$ \begin{array} [c]{l} Ð_{0}=(1000,0,0,0,0,0,0,0) \\ Ð_{1}=\left( 523.81,21.65,8.66,0,0,21.65,0,0\right) \end{array} $	$ \begin{array} [c]{l} \left( -1.96,-0.7,-0.2,0.15,-0.1,-0.1,-0.01,-0.01\right) \\ \left( -1.98,-0.64,-0.2,-0.1,-0.07,-0.01,-0.01,-0.01\right) \end{array} $	$ \begin{array} [c]{l} \rm{unstable} \\ \rm{stable} \end{array} $
3	$ \begin{array} [c]{l} Ð_{0}=(1000,0,0,0,0,0,0,0) \\ Ð_{2}=\left( 722.22,0,0,42.74,0.64,0,0,0\right) \end{array} $	$ \begin{array} [c]{l} \left( -1.96,-0.68,-0.22,-0.1,-0.1,-0.03,-0.01,0.004\right) \\ \left( -1.97,-0.64,-0.21,-0.1,-0.07,-0.07,-0.01,-0.01\right) \end{array} $	$ \begin{array} [c]{c} \rm{unstable} \\ {\rm{stable}} \end{array} $
4	$ \begin{array} [c]{l} Ð_{0}=(1000,0,0,0,0,0,0,0) \\ Ð_{1}=\left( 180.33,37.26,14.9,0,0,37.26,0,0\right) \\ Ð_{3}=\left( 569.59,19.56,2,0,0,5,7.28,0\right) \end{array} $	$ \begin{array} [c]{l} \left( -1.51,-1.51,0.41,-0.2,-0.1,-0.1,-0.01,-0.01\right) \\ \left( -1.92,-0.79,0.65,-0.2,-0.1,-0.02,-0.02,-0.01\right) \\ \left( -2.03,-2.03,-0.2,-0.03,-0.03,-0.1,-0.02,-0.01\right) \end{array} $	$ \begin{array} [c]{c} \rm{unstable} \\ \rm{unstable} \\ {\rm{stable}} \end{array} $
5	$ \begin{array} [c]{l} Ð_{0}=(1000,0,0,0,0,0,0,0) \\ Ð_{1}=\left( 785.71,9.74,3.9,0,0,9.74,0,0\right) \\ Ð_{2}=\left( 123.81,0,0,134.8,2.02,0,0,0\right) \\ Ð_{3}=\left( 870.39,5.89,2,0,0,5,0.45,0\right) \\ Ð_{4}=\left( 533.33,0,0,71.79,0.25,0,0,3.31\right) \end{array} $	$ \begin{array} [c]{l} \left( -1.96,-0.7,-0.28,-0.1,-0.1,0.07,0.04,-0.01\right) \\ \left( -1.97,-0.67,-0.27,-0.1,0.09,0.05,-0.01,-0.01\right) \\ \left( -2,0.71,-0.54,-0.22,-0.17,-0.1,-0.06,-0.01\right) \\ \left( -1.97,-0.75,-0.27,-0.1,0.06,-0.01,-0.01,-0.01\right) \\ \left( -1.98,-0.79,-0.63,-0.1,-0.06,-0.06,-0.03,-0.01\right) \end{array} $	$ \begin{array} [c]{c} \rm{unstable} \\ \rm{unstable} \\ \rm{unstable} \\ \rm{unstable} \\ {\rm{stable}} \end{array} $
6	$ \begin{array} [c]{l} Ð_{0}=(1000,0,0,0,0,0,0,0) \\ Ð_{1}=\left( 186.44,36.98,14.79,0,0,36.98,0,0\right) \\ Ð_{2}=\left( 393.94,0,0,93.24,1.4,0,0,0\right) \\ Ð_{3}=\left( 714.37,12.98,1,0,0,2.5,10.48,0\right) \\ Ð_{5}=\left( 393.94,5.89,1,73.31,1.1,2.5,3.39,0\right) \end{array} $	$ \begin{array} [c]{l} \left( -1.51,-1.51,0.39,-0.23,-0.1,-0.1,0.02,-0.01\right) \\ \left( -1.92,1.38,-0.79,-0.21,-0.1,-0.02,-0.02,-0.01\right) \\ \left( -1.82,-1,-0.21,0.13,-0.1,-0.09,-0.01,-0.01\right) \\ \left( -2.34,-2.34,-0.22,-0.03,-0.03,-0.1,-0.01,0.01\right) \\ \left( -1.66,-1.66,-0.21,-0.02,-0.02,-0.09,-0.01,-0.01\right) \end{array} $	$ \begin{array} [c]{c} \rm{unstable} \\ \rm{unstable} \\ \rm{unstable} \\ \rm{unstable} \\ {\rm{stable}} \end{array} $
7	$ \begin{array} [c]{l} Ð_{0}=(1000,0,0,0,0,0,0,0) \\ Ð_{1}=\left( 282.05,32.63,13.05,0,0,32.63,0,0\right) \\ Ð_{2}=\left( 108.33,0,0,137.18,2.06,0,0,0\right) \\ Ð_{4}=\left( 636.36,0,0,55.94,0.14,0,0,4.87\right) \\ Ð_{6}=\left( 282.05,25.31,10.12,24.8,0.14,25.31,0,1.6\right) \end{array} $	$ \begin{array} [c]{l} \left( -1.65,-1.24,-0.29,0.27,-0.1,-0.1,0.07,-0.01\right) \\ \left( -1.93,-0.77,-0.23,-0.1,-0.09,-0.01,-0.01,0.02\right) \\ \left( -1.98,1.34,-0.62,-0.22,-0.11,-0.1,-0.07,-0.01\right) \\ \left( -1.83,-1.1,-0.98,0.15,-0.1,-0.07,-0.02,-0.01\right) \\ \left( -1.93,-0.77,-0.47,-0.09,-0.02,-0.02,-0.05,-0.02\right) \end{array} $	$ \begin{array} [c]{c} \rm{unstable} \\ \rm{unstable} \\ \rm{unstable} \\ \rm{unstable} \\ {\rm{stable}} \end{array} $
8	$ \begin{array} [c]{l} Ð_{0}=(1000,0,0,0,0,0,0,0) \\ Ð_{1}=\left( 282.05,32.63,13.05,0,0,32.63,0,0\right) \\ Ð_{2}=\left( 144.44,0,0,131.62,1.97,0,0,0\right) \\ Ð_{3}=\left( 627.17,16.95,2.5,0,0,6.25,4.28,0\right) \\ Ð_{4}=\left( 625,0,0,57.69,0.2,0,0,3.33\right) \\ Ð_{6}=\left( 282.05,24.94,9.98,26.04,0.2,24.94,0,0.95\right) \\ Ð_{7}=\left( 467.37,11.46,2.5,43.14,0.2,6.25,2.09,2.24\right) \end{array} $	$ \begin{array} [c]{l} \left( -1.65,-1.24,0.27,-0.27,-0.1,-0.1,0.06,-0.01\right) \\ \left( -1.93,-0.77,0.42,-0.22,-0.1,-0.01,-0.01,0.01\right) \\ \left( -1.97,0.89,-0.66,-0.22,-0.1,-0.08,-0.05,-0.01\right) \\ \left( -1.73,-1.73,-0.25,-0.1,-0.02,-0.02,0.04,-0.02\right) \\ \left( -1.83,-0.98,-0.8,0.15,-0.1,-0.06,-0.02,-0.01\right) \\ \left( -1.93,-0.77,-0.35,0.3,-0.02,-0.02,-0.03,-0.03\right) \\ \left( -1.81,-1.25,-0.59,-0.02,-0.02,-0.05,-0.03,-0.01\right) \end{array} $	$ \begin{array} [c]{c} \rm{unstable} \\ \rm{unstable} \\ \rm{unstable} \\ \rm{unstable} \\ \rm{unstable} \\ \rm{unstable} \\ {\rm{stable}} \end{array} $

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