Global Properties of HIV-1 Dynamics Models with CTL Immune Impairment and Latent Cell-to-Cell Spread
This paper presents and analyzes two mathematical models for the human immunodeficiency virus type-1 (HIV-1) infection with Cytotoxic T Lymphocyte cell (CTL) immune impairment. These models describe the interactions between healthy CD<inline-formula><math xmlns="http://www.w3.org/1998/...
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MDPI AG
2023-08-01
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| author | Noura H. AlShamrani Reham H. Halawani Wafa Shammakh Ahmed M. Elaiw |
| author_facet | Noura H. AlShamrani Reham H. Halawani Wafa Shammakh Ahmed M. Elaiw |
| author_sort | Noura H. AlShamrani |
| collection | DOAJ |
| description | This paper presents and analyzes two mathematical models for the human immunodeficiency virus type-1 (HIV-1) infection with Cytotoxic T Lymphocyte cell (CTL) immune impairment. These models describe the interactions between healthy CD<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>4</mn><mo>+</mo></msup></semantics></math></inline-formula>T cells, latently and actively infected cells, HIV-1 particles, and CTLs. The healthy CD<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>4</mn><mo>+</mo></msup></semantics></math></inline-formula>T cells might be infected when they make contact with: (i) HIV-1 particles due to virus-to-cell (VTC) contact; (ii) latently infected cells due to latent cell-to-cell (CTC) contact; and (iii) actively infected cells due to active CTC contact. Distributed time delays are considered in the second model. We show the nonnegativity and boundedness of the solutions of the systems. Further, we derive basic reproduction numbers <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>ℜ</mo><mn>0</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mover accent="true"><mo>ℜ</mo><mo stretchy="false">˜</mo></mover><mn>0</mn></msub></semantics></math></inline-formula>, that determine the existence and stability of equilibria of our proposed systems. We establish the global asymptotic stability of all equilibria by using the Lyapunov method together with LaSalle’s invariance principle. We confirm the theoretical results by numerical simulations. The effect of immune impairment, time delay and CTC transmission on the HIV-1 dynamics are discussed. It is found that weak immunity contributes significantly to the development of the disease. Further, we have established that the presence of time delay can significantly decrease the basic reproduction number and then suppress the HIV-1 replication. On the other hand, the presence of latent CTC spread increases the basic reproduction number and then enhances the viral progression. Thus, neglecting the latent CTC spread in the HIV-1 infection model will lead to an underestimation of the basic reproduction number. Consequently, the designed drug therapies will not be accurate or sufficient to eradicate the viruses from the body. These findings may help to improve the understanding of the dynamics of HIV-1 within a host. |
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| spelling | doaj.art-fce716ffa09c43079a7264f9bbce8c3d2023-11-19T08:31:32ZengMDPI AGMathematics2227-73902023-08-011117374310.3390/math11173743Global Properties of HIV-1 Dynamics Models with CTL Immune Impairment and Latent Cell-to-Cell SpreadNoura H. AlShamrani0Reham H. Halawani1Wafa Shammakh2Ahmed M. Elaiw3Department of Mathematics, Faculty of Science, University of Jeddah, P.O. Box 80327, Jeddah 21589, Saudi ArabiaDepartment of Mathematics, Faculty of Science, University of Jeddah, P.O. Box 80327, Jeddah 21589, Saudi ArabiaDepartment of Mathematics, Faculty of Science, University of Jeddah, P.O. Box 80327, Jeddah 21589, Saudi ArabiaDepartment of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi ArabiaThis paper presents and analyzes two mathematical models for the human immunodeficiency virus type-1 (HIV-1) infection with Cytotoxic T Lymphocyte cell (CTL) immune impairment. These models describe the interactions between healthy CD<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>4</mn><mo>+</mo></msup></semantics></math></inline-formula>T cells, latently and actively infected cells, HIV-1 particles, and CTLs. The healthy CD<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>4</mn><mo>+</mo></msup></semantics></math></inline-formula>T cells might be infected when they make contact with: (i) HIV-1 particles due to virus-to-cell (VTC) contact; (ii) latently infected cells due to latent cell-to-cell (CTC) contact; and (iii) actively infected cells due to active CTC contact. Distributed time delays are considered in the second model. We show the nonnegativity and boundedness of the solutions of the systems. Further, we derive basic reproduction numbers <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>ℜ</mo><mn>0</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mover accent="true"><mo>ℜ</mo><mo stretchy="false">˜</mo></mover><mn>0</mn></msub></semantics></math></inline-formula>, that determine the existence and stability of equilibria of our proposed systems. We establish the global asymptotic stability of all equilibria by using the Lyapunov method together with LaSalle’s invariance principle. We confirm the theoretical results by numerical simulations. The effect of immune impairment, time delay and CTC transmission on the HIV-1 dynamics are discussed. It is found that weak immunity contributes significantly to the development of the disease. Further, we have established that the presence of time delay can significantly decrease the basic reproduction number and then suppress the HIV-1 replication. On the other hand, the presence of latent CTC spread increases the basic reproduction number and then enhances the viral progression. Thus, neglecting the latent CTC spread in the HIV-1 infection model will lead to an underestimation of the basic reproduction number. Consequently, the designed drug therapies will not be accurate or sufficient to eradicate the viruses from the body. These findings may help to improve the understanding of the dynamics of HIV-1 within a host.https://www.mdpi.com/2227-7390/11/17/3743HIV-1cell-to-cell infectionlatently infected cellsimmune impairmentglobal stabilitydistributed delays |
| spellingShingle | Noura H. AlShamrani Reham H. Halawani Wafa Shammakh Ahmed M. Elaiw Global Properties of HIV-1 Dynamics Models with CTL Immune Impairment and Latent Cell-to-Cell Spread Mathematics HIV-1 cell-to-cell infection latently infected cells immune impairment global stability distributed delays |
| title | Global Properties of HIV-1 Dynamics Models with CTL Immune Impairment and Latent Cell-to-Cell Spread |
| title_full | Global Properties of HIV-1 Dynamics Models with CTL Immune Impairment and Latent Cell-to-Cell Spread |
| title_fullStr | Global Properties of HIV-1 Dynamics Models with CTL Immune Impairment and Latent Cell-to-Cell Spread |
| title_full_unstemmed | Global Properties of HIV-1 Dynamics Models with CTL Immune Impairment and Latent Cell-to-Cell Spread |
| title_short | Global Properties of HIV-1 Dynamics Models with CTL Immune Impairment and Latent Cell-to-Cell Spread |
| title_sort | global properties of hiv 1 dynamics models with ctl immune impairment and latent cell to cell spread |
| topic | HIV-1 cell-to-cell infection latently infected cells immune impairment global stability distributed delays |
| url | https://www.mdpi.com/2227-7390/11/17/3743 |
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