Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.04323

The contact process with social clusters and asymptomatic states

Karima Belatreche·Lamia Belhadji·Nicolas Lanchier
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper is concerned with a natural variant of the contact process that places a social cluster of individuals, rather than a single individual, at each lattice point, and distinguishes between symptomatic and asymptomatic individuals. In particular, the epidemic model depends on four infection parameters: the external (inter-cluster) and internal (intra-cluster) infection rates for both symptomatic and asymptomatic individuals. Newly infected individuals begin as asymptomatic before transitioning to the symptomatic state at a spontaneous mutation rate, and infected individuals recover at a baseline rate of one. We study the phase structure of the deterministic nonspatial mean-field model when each cluster reduces to a pair of individuals, and prove that the stochastic spatial model exhibits the same phase transitions with respect to the four infection rates for all cluster sizes. We also prove the existence of a phase transition driven by the cluster size itself for the spatial model.

Explore related subjects

math.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Karima Belatreche, Lamia Belhadji, Nicolas Lanchier. 2026-10-03. The contact process with social clusters and asymptomatic states. https://arxiv.org/abs/2610.04323

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Representation-free Densities of Ratios and Powers of Stable Variables

We derive representation-free densities of ratios of the positive stable random variable and obtain their Laplace/Stieltjes transforms. We state representation-free densities of occupation times (Lamperti, 1958) and Bessel process excursions (Bertoin et al., 2006) and their transforms. We also obtain simple representations of these and other densities using a known infinite series representation of the stable density.

math.PR↗
arXiv2026

The Right Wing of the SABR Smile with an Absorbing Boundary: Resolution of Conjectures of Henry-Labordère

We determine the leading logarithmic asymptotics of the fixed-maturity right tail of the SABR model with $β\in(0,1)$ and absorption at zero, for every correlation $ρ\in(-1,1)$. If $P(k)$ is the probability that the terminal forward is at least $f_0e^k$, then $-k^{-2}\ln P(k)\to(1-β)^2/(2ν^2T(1-(ρ\wedge0)^2))$ as $k\to\infty$. For $ρ\ge0$ this is the rate predicted by the unrestricted hyperbolic geometry. For $ρ<0$ it is strictly larger: along the volatility excursion that produces the tail, survival forces the noise orthogonal to volatility to stay above a moving square-root barrier, at an additional cost of the same order $k^2$. Consequently the Black-Scholes implied volatility tends to $ν\sqrt{1-(ρ\wedge0)^2}/(1-β)$: Henry-Labordère's conjectured wing limit $ν/(1-β)$ holds for $ρ\ge0$ and fails for every $ρ<0$. For his second conjecture, whose unrestricted-distance Gaussian estimate does not hold for $ρ<0$, we also identify the correct replacement for this model: at the level of tail probabilities, the rate is given by the squared intrinsic Riemannian distance, within the survival domain, from the initial state to the terminal target set. The proofs are direct, combining a stopped Lamperti transform, a conditional time change, Gaussian moving-barrier bounds and change-of-measure constructions, and use no heat-kernel estimates.

math.PR↗
arXiv2026

Classification of commutation relation for multi-chordal SLE

We classify the linear solution spaces of the chordal Belavin--Polyakov--Zamolodchikov (BPZ) equations, which arise in Dubédat's commutation relations for multiple Schramm--Loewner evolutions (SLE). Without imposing growth conditions, we determine the subspaces selected by successive conformal Ward identities and their exact dimensions. For every $κ>0$ and BPZ spectral parameter $λ\in\mathbb{R}$, we determine the spectrum and Jordan structure of the translation operator. When $λ=0$ and $κ\in(0,8)$, we further determine all admissible scaling exponents and the dimensions of the corresponding translation-invariant solution spaces. A triangular change of derivative coordinates yields a first-order system of rational Knizhnik--Zamolodchikov type, allowing us to represent the Ward operators explicitly and reduce the classification to finite-dimensional linear algebra. For $2N$ boundary points and $κ\in(0,8)$, translation-invariant BPZ solutions with $λ=0$ and the homogeneity required by Möbius covariance automatically satisfy the third Ward identity and the usual power-law bound. This recovers, under weaker assumptions, the Catalan dimension formula previously established by Flores and Kleban. At $κ=8$, the third Ward identity imposes an additional constraint when $N\ge2$, but the full Möbius-covariant solution space still has Catalan dimension, establishing completeness of the partition functions constructed from uniform spanning trees. Finally, at $κ=4$ and $λ>0$, we construct an explicit basis of positive BPZ solutions and identify its elements as partition functions for level lines of a Gaussian free field with suitable harmonic means.

math.PR↗