Elaiw A M, AlShamrani N H
Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589 Saudi Arabia.
Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut Branch, Assiut, Egypt.
Bol Soc Mat Mex. 2021;27(2):38. doi: 10.1007/s40590-021-00330-6. Epub 2021 Mar 29.
Human immunodeficiency virus (HIV) and human T-lymphotropic virus type I (HTLV-I) are two retroviruses that attack the T cells and impair their functions. Both HIV and HTLV-I can be transmitted between individuals through direct contact with certain body fluids from infected individuals. Therefore, a person can be co-infected with both viruses. HIV causes acquired immunodeficiency syndrome (AIDS), while HTLV-I is the causative agent for adult T-cell leukemia (ATL) and HTLV-I-associated myelopathy/tropical spastic paraparesis (HAM/TSP). Several mathematical models have been developed in the literature to describe the within-host dynamics of HIV and HTLV-I mono-infections. However, modeling a within-host dynamics of HIV/HTLV-I co-infection has not been involved. The present paper is concerned with the formulation and investigation of 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 T cells, silent HIV-infected cells, active HIV-infected cells, silent HTLV-infected cells, Tax-expressing HTLV-infected cells, free HIV particles, HIV-specific CTLs and HTLV-specific CTLs. The HIV can spread by virus-to-cell transmission. On the other side, HTLV-I has two modes of transmission, (i) horizontal transmission via direct cell-to-cell contact through the virological synapse, and (ii) vertical transmission through the mitotic division of Tax-expressing HTLV-infected cells. The well-posedness of the model is established by showing that the solutions of the model are nonnegative and bounded. We define a set of 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 Lyapunov-LaSalle asymptotic stability theorem. 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.
人类免疫缺陷病毒（HIV）和I型人类嗜T淋巴细胞病毒（HTLV-I）是两种攻击T细胞并损害其功能的逆转录病毒。HIV和HTLV-I均可通过与受感染个体的某些体液直接接触在个体之间传播。因此，一个人可能同时感染这两种病毒。HIV会导致获得性免疫缺陷综合征（AIDS），而HTLV-I是成人T细胞白血病（ATL）和HTLV-I相关脊髓病/热带痉挛性截瘫（HAM/TSP）的病原体。文献中已经开发了几种数学模型来描述HIV和HTLV-I单一感染的宿主体内动态。然而，尚未涉及对HIV/HTLV-I合并感染的宿主体内动态进行建模。本文关注在细胞毒性T淋巴细胞（CTL）免疫反应作用下的一种新的HIV/HTLV-I合并感染模型的建立和研究。该模型描述了易感T细胞、潜伏感染HIV的细胞、活跃感染HIV的细胞、潜伏感染HTLV的细胞、表达Tax的HTLV感染细胞、游离HIV颗粒、HIV特异性CTL和HTLV特异性CTL之间的相互作用。HIV可通过病毒到细胞的传播进行扩散。另一方面，HTLV-I有两种传播方式，（i）通过病毒突触进行直接细胞间接触的水平传播，以及（ii）通过表达Tax的HTLV感染细胞的有丝分裂进行垂直传播。通过证明模型的解是非负且有界的，建立了模型的适定性。我们定义了一组阈值参数，这些参数决定了模型所有平衡点的存在性和稳定性。我们利用李雅普诺夫函数和李雅普诺夫 - 拉萨尔渐近稳定性定理探索所有平衡点的全局渐近稳定性。我们进行了数值模拟以证明理论结果的适用性和有效性。此外，我们评估了HTLV-I感染对HIV动态的影响，反之亦然。
