AI 中文总结
该研究针对基于图的HIV感染动力学模型，引入两个灭绝参数并证明相关界、刻画极端情况，还分析参数差值的无界性，结合状态转换图揭示灭绝与图结构及替换参数相关。
AI 中文摘要
我们研究了Mukwembi最初提出的、用于淋巴结网络中HIV感染动力学的基于图的元胞自动机。每个顶点代表一个细胞位点，可处于健康、感染或死亡状态，演化由替换参数$R$控制，$R$根据死亡细胞的感染邻居数量，决定该死亡细胞被感染细胞还是健康细胞替换。对于图$G$，我们引入两个灭绝参数：$\boldsymbol{\text{hiv}}(G)$是使所有可允许初始构型下都发生灭绝的最小$R$值，$\boldsymbol{\text{HIV}}(G)$是使所有大于等于它的替换参数下都发生灭绝的最小阈值。我们证明了一般界$2 \boldsymbol{\text{hiv}}(G) \boldsymbol{\text{HIV}}(G) \boldsymbol{\text{Δ}}(G)+1$，并刻画了$\boldsymbol{\text{HIV}}(G)=\boldsymbol{\text{Δ}}(G)+1$的极端情况。我们还证明$\boldsymbol{\text{HIV}}(G)-\boldsymbol{\text{hiv}}(G)$的差值无界，并确定了一些经典图族的这两个参数。最后，我们利用系统的状态转换有向图及轨迹收敛到非平凡周期轨道的构型，研究该模型的动力学，结果表明灭绝不仅取决于替换参数，还依赖于基础图的结构性质。
英文摘要
We study a graph-based cellular automaton for HIV infection dynamics in lymph-node networks, originally introduced by Mukwembi. Each vertex represents a cell site that may be healthy, infected, or dead, and the evolution is controlled by a replacement parameter $R$, which determines whether a dead cell is replaced by an infected or a healthy cell according to the number of its infected neighbors. For a graph $G$, we introduce two extinction parameters. The parameter $\mathbf{hiv}(G)$ is the smallest value of $R$ for which extinction occurs for every admissible initial configuration, whereas $\mathbf{HIV}(G)$ is the smallest threshold such that extinction occurs for every replacement parameter greater than or equal to it. We prove the general bounds $2\leq \mathbf{hiv}(G)\leq \mathbf{HIV}(G)\leq Δ(G)+1$ and characterize the extremal case $\mathbf{HIV}(G)=Δ(G)+1$. We also show that the gap $\mathbf{HIV}(G)-\mathbf{hiv}(G)$ is unbounded and determine both parameters for some classical families of graphs. Finally, we study the dynamics of the model using the state-transition digraph of the system and the configurations whose trajectories converge to nontrivial periodic orbits. The results show that extinction depends not only on the replacement parameter but also on the structural properties of the underlying graph.
Comments16 pages
