bibkey: espinosagarcia2026hivwheels authors: Manuel A. Espinosa-García, Ana Paulina Figueroa, Julián A. Fresán-Figueroa, Gerardo L. Maldonado, L. Ariadna Sánchez-Solís year: 2026 title: ‘Extinction thresholds in a graph-based model of HIV infection dynamics’ doi: 10.48550/arXiv.2608.00340 url: https://arxiv.org/abs/2608.00340v1 claim: “The paper studies Mukwembi’s cellular-automaton model of HIV infection on graphs, defines extinction sets and the thresholds hiv(G) and HIV(G), determines them for several graph families, and conjectures (Conjecture 1) from computations that the extinction set of the wheel W_n is {3} together with all R >= n - 1 for even n >= 12 and {4} together with all R >= n - 1 for odd n >= 17.” strata_touched:

• D5/S3/Combinatorics/WheelHivExtinctionRefutation license: citation-only triage: anchor

Extinction thresholds in a graph-based model of HIV infection dynamics

The paper works with Mukwembi’s graph model of HIV infection. A state of a graph G gives each vertex one of the values 0 (healthy), 1 (infected) and 2 (dead); an admissible initial state uses only 0 and 1. With d_{t,I}(v) the number of infected neighbours of v at time t and a positive replacement parameter R, a healthy vertex becomes infected when d_{t,I}(v) ≥ 1, an infected vertex dies, and a dead vertex is replaced by an infected one when d_{t,I}(v) ≥ R and by a healthy one otherwise. The extinction set 𝓔(G) collects the R for which every admissible initial state reaches the all-healthy state; hiv(G) = min 𝓔(G) and HIV(G) is one more than the largest R outside 𝓔(G).

For the wheels W_n = K₁ ∨ C_{n−1} the paper reports an exhaustive computation for 4 ≤ n ≤ 10 and, for 11 ≤ n ≤ 26, computations described as experimental evidence rather than an exhaustive determination. Its table and Conjecture 1 state:

The extinction sets of the wheel graphs satisfy $\mathcal{E}(W_n)=\{3\}\cup \{R\in {\mathbb{Z}}^{+}:R\ge n-1\}$ for every even $n\ge 12$, and $\mathcal{E}(W_n)=\{4\}\cup \{R\in {\mathbb{Z}}^{+}:R\ge n-1\}$ for every odd $n\ge 17$.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2608.00340
• URL: https://arxiv.org/abs/2608.00340v1
• Version and location: arXiv:2608.00340v1 (2026-07-31), the only version listed by the arXiv API on 2026-09-26; source file main.tex: Section 2 for the rules and the extinction sets, Section 4 (“Results for families of graphs”) for the wheel table and Conjecture 1, the only conjecture environment of the paper.