Xinxuan Wang | Searcharxiv

arXiv2026

Synchronized disease and behavioural dynamics in weakly coupled populations

The spread of infectious disease is strongly influenced by social dynamics. In addition to infection risk, individuals vaccination decisions depend on prevailing social behavior: high infection levels and widespread vaccination can increase vaccine uptake, which in turn suppresses infection. This feedback can generate sustained oscillations in disease prevalence and vaccination behavior. Here, we study two such populations undergoing the same behavioral epidemiological limit cycle and introduce weak coupling between them through social influence. We show that coupling leads to synchronization of disease dynamics between the two groups. Moreover, we find that different payoff sensitivity may lead to synchronization or anti synchronization.

q-bio.PE↗

arXiv2025

Polynomiality of Subdimensions of Diagonal Harmonics and a Sharp Stability Bound

A sequence of $S_n$-representations $\{V_n\}_{n \ge 1}$ is representation stable if, writing $V(\lambda) = (n-|\lambda|, \lambda_1, \lambda_2, \dots)$ for each partition $\lambda$, the multiplicity of the irreducible indexed by $V(\lambda)$ in $V_n$ is eventually independent of $n$. In particular, Church, Ellenberg and Farb \cite{Church_2015} found that if we fix $a$ and $b$, then the space of diagonal harmonics $DH_n^{a,b}$ exhibits this behavior, and its dimension stabilizes to a polynomial in $n$ eventually. Building on this result, we use the Schedules Formula by Haglund and Loehr \cite{HAGLUND2005189} to get an explicit combinatorial polynomial for the dimension of the bigraded spaces $DH_n^{a,b}$. This derivation not only yields the dimension formula but also produces a new stability bound of $ a + b $ which is sharp, and determines the exact degree of the dimension polynomial, which is also $ a + b $.

math.CO↗