Li Chin-Lung, Cheng Chang-Yuan, Li Chun-Hsien
Institute of Computational and Modeling Science, National Tsing Hua University, Hsinchu 30013, Taiwan.
Department of Applied Mathematics, National Pingtung University, Pingtung, 90003, Taiwan.
Nonlinear Anal Real World Appl. 2023 Feb;69:103738. doi: 10.1016/j.nonrwa.2022.103738. Epub 2022 Aug 26.
Contagious pathogens, such as influenza and COVID-19, are known to be represented by multiple genetic strains. Different genetic strains may have different characteristics, such as spreading more easily, causing more severe diseases, or even evading the immune response of the host. These facts complicate our ability to combat these diseases. There are many ways to prevent the spread of infectious diseases, and vaccination is the most effective. Thus, studying the impact of vaccines on the dynamics of a multi-strain model is crucial. Moreover, the notion of complex networks is commonly used to describe the social contacts that should be of particular concern in epidemic dynamics. In this paper, we investigate a two-strain epidemic model using a single-strain vaccine in complex networks. We first derive two threshold quantities, and , for each strain. Then, by using the basic tools for stability analysis in dynamical systems (i.e., Lyapunov function method and LaSalle's invariance principle), we prove that if and , then the disease-free equilibrium is globally asymptotically stable in the two-strain model. This means that the disease will die out. Furthermore, the global stability of each strain dominance equilibrium is established by introducing further critical values. Under these stability conditions, we can determine which strain will survive. Particularly, we find that the two strains can coexist under certain condition; thus, a coexistence equilibrium exists. Moreover, as long as the equilibrium exists, it is globally stable. Numerical simulations are conducted to validate the theoretical results.
已知诸如流感和新冠病毒等传染性病原体由多种基因菌株构成。不同的基因菌株可能具有不同的特性，比如更容易传播、引发更严重的疾病，甚至逃避宿主的免疫反应。这些事实使我们对抗这些疾病的能力变得复杂。预防传染病传播有多种方法，而接种疫苗是最有效的。因此，研究疫苗对多菌株模型动态的影响至关重要。此外，复杂网络的概念通常用于描述在疫情动态中应特别关注的社会接触。在本文中，我们在复杂网络中研究使用单菌株疫苗的双菌株流行病模型。我们首先为每个菌株推导出两个阈值量， 和 。然后，通过使用动力系统稳定性分析的基本工具（即李雅普诺夫函数法和拉萨尔不变性原理），我们证明如果 和 ，那么在双菌株模型中无病平衡点是全局渐近稳定的。这意味着疾病将会消亡。此外，通过引入进一步的临界值建立了每个菌株优势平衡点的全局稳定性。在这些稳定性条件下，我们可以确定哪种菌株会存活。特别地，我们发现两种菌株在一定条件下可以共存；因此，存在一个共存平衡点。而且，只要平衡点存在，它就是全局稳定的。进行了数值模拟以验证理论结果。
