Faculty of Engineering, Department of Engineering Mathematics and Physics, Alexandria University, Alexandria, Egypt.
High Institute of Public Health, Alexandria University, Alexandria, Egypt.
PLoS One. 2021 Oct 12;16(10):e0257975. doi: 10.1371/journal.pone.0257975. eCollection 2021.
In this paper, a new mathematical model is formulated that describes the interaction between uninfected cells, infected cells, viruses, intracellular viral RNA, Cytotoxic T-lymphocytes (CTLs), and antibodies. Hence, the model contains certain biological relations that are thought to be key factors driving this interaction which allow us to obtain precise logical conclusions. Therefore, it improves our perception, that would otherwise not be possible, to comprehend the pathogenesis, to interpret clinical data, to control treatment, and to suggest new relations. This model can be used to study viral dynamics in patients for a wide range of infectious diseases like HIV, HPV, HBV, HCV, and Covid-19. Though, analysis of a new multiscale HCV model incorporating the immune system response is considered in detail, the analysis and results can be applied for all other viruses. The model utilizes a transformed multiscale model in the form of ordinary differential equations (ODE) and incorporates into it the interaction of the immune system. The role of CTLs and the role of antibody responses are investigated. The positivity of the solutions is proven, the basic reproduction number is obtained, and the equilibrium points are specified. The stability at the equilibrium points is analyzed based on the Lyapunov invariance principle. By using appropriate Lyapunov functions, the uninfected equilibrium point is proven to be globally asymptotically stable when the reproduction number is less than one and unstable otherwise. Global stability of the infected equilibrium points is considered, and it has been found that each equilibrium point has a specific domain of stability. Stability regions could be overlapped and a bistable equilibria could be found, which means the coexistence of two stable equilibrium points. Hence, the solution converges to one of them depending on the initial conditions.
本文构建了一个新的数学模型，用于描述未感染细胞、感染细胞、病毒、细胞内病毒 RNA、细胞毒性 T 淋巴细胞（CTL）和抗体之间的相互作用。因此，该模型包含了一些被认为是驱动这种相互作用的关键因素的生物学关系，使我们能够获得精确的逻辑结论。因此，它提高了我们的认识，否则我们无法理解发病机制、解释临床数据、控制治疗和提出新的关系。该模型可用于研究广泛的传染病（如 HIV、HPV、HBV、HCV 和 COVID-19）患者中的病毒动力学。虽然详细考虑了包含免疫系统反应的新 HCV 多尺度模型的分析，但分析和结果可应用于所有其他病毒。该模型利用转化的多尺度模型，以常微分方程（ODE）的形式，并将免疫系统的相互作用纳入其中。研究了 CTL 的作用和抗体反应的作用。证明了解的正定性，获得了基本再生数，并指定了平衡点。基于李雅普诺夫不变性原理分析了平衡点的稳定性。通过使用适当的李雅普诺夫函数，当再生数小于 1 时，证明了未感染平衡点是全局渐近稳定的，否则是不稳定的。考虑了感染平衡点的全局稳定性，发现每个平衡点都有特定的稳定域。稳定性区域可能会重叠，并可能发现双稳态平衡点，这意味着两个稳定平衡点的共存。因此，根据初始条件，解将收敛到其中之一。
