The contact process seen from a typical infected site

边界修正接触过程的极限速度与波动

Limiting Speed and Fluctuations for the Boundary Modified Contact Process

研究边界修正接触过程在特定参数下的极限速度与波动特性，揭示了强数定律、中心极限定理及指数尾界，解决了开放性问题。

The effect of avoiding known infected neighbors on the persistence of a recurring infection process

We study a generalization of the classical contact process (SIS epidemic model) in a directed graph $G$. Our model is a continuous-time interacting particle system in which at every time, each vertex is either healthy or infected, and each oriented edge is either active or inactive. Infected vertices become healthy at rate $1$, and pass the infection along each active outgoing edge at rate $λ$. At rate $α$, healthy individuals deactivate each incoming edge from their infected neighbors. We study the persistence time of this epidemic model on the lattice $\mathbb{Z}$, the $n$-cycle $\mathbb{Z}_…

On the growth of one-dimensional reverse immunization contact processes

We are concerned with the supercritical contact process modified so that first infection occurs at a lower rate, it is known that the process survives with positive probability. Regarding the rightmost infected of the process started from one site infected and conditioned to survive, we specify a sequence of space-time points at which its behaviour regenerates and thus obtain the corresponding strong law and central limit theorem. We also extend complete convergence to this modified case.

Mean field limit for survival probability of the high-dimensional contact process

In this paper we are concerned with the contact process on the squared lattice. The contact process intuitively describes the spread of the infectious disease on a graph, where an infectious vertex becomes healthy at a constant rate while a healthy vertex is infected at rate proportional to the number of infectious neighbors. As the dimension of the lattice grows to infinity, we give a mean field limit for the survival probability of the process conditioned on the event that only the origin of the lattice is infected at t=0. The binary contact path process is a main auxiliary tool for our proo…

交换过程上的接触过程

Contact process on interchange process

该研究提出局部有限图上运动粒子间的流行病模型，结合接触传播与交换过程的粒子运动机制，定义感染持续的阈值并分析其渐近行为。

On the threshold of spread-out contact process percolation

We study the stationary distribution of the (spread-out) $d$-dimensional contact process from the point of view of site percolation. In this process, vertices of $\mathbb{Z}^d$ can be healthy (state 0) or infected (state 1). With rate one infected sites recover, and with rate $λ$ they transmit the infection to some other vertex chosen uniformly within a ball of radius $R$. The classical phase transition result for this process states that there is a critical value $λ_c(R)$ such that the process has a non-trivial stationary distribution if and only if $λ> λ_c(R)$. In configurations sampled from…

修正的边界接触过程：不变测度与临界灭绝

The modified boundary contact process: invariant measures and critical extinction

该研究针对一维修正边界接触过程，在非吸引区证明其从半无限初态收敛到不变测度，在吸引区临界曲线上证明感染几乎必然灭绝，完善了接触过程的相图分析。

带病毒载量的接触过程

Contact process with viral load

本文提出了两种新的接触过程变体，分别涉及病毒载量和潜伏状态，研究了其相变和不变分布特性。

A simple proof of exponential decay of subcritical contact processes

This paper gives a new, simple proof of the known fact that for contact processes on general lattices, in the subcritical regime the expected number of infected sites decays exponentially fast as time tends to infinity. The proof also yields an explicit bound on the survival probability below the critical recovery rate, which shows that the critical exponent associated with this function is bounded from below by its mean-field value. The main idea of the proof is that if the expected number of infected sites decays slower than exponentially, then this implies the existence of a harmonic functi…

Subcritical Contact Process Seen from the Edge: Convergence to Quasi-Equilibrium

The subcritical contact process seen from the rightmost infected site has no invariant measures. We prove that nevertheless it converges in distribution to a quasi-stationary measure supported on finite configurations.

Convergence rates for subcritical threshold-one contact processes on lattices

In this paper we are concerned with threshold-one contact processes on lattices. We show that the probability that the origin is infected converges to 0 at an exponential rate I in the subcritical case. Furthermore, we give a limit theorem for I as the degree of the lattice grows to infinity. Our results also hold for classic contact processes on lattices.

Stochastic domination for a hidden Markov chain with applications to the contact process in a randomly evolving environment

The ordinary contact process is used to model the spread of a disease in a population. In this model, each infected individual waits an exponentially distributed time with parameter 1 before becoming healthy. In this paper, we introduce and study the contact process in a randomly evolving environment. Here we associate to every individual an independent two-state, $\{0,1\},$ background process. Given $δ_0<δ_1,$ if the background process is in state $0,$ the individual (if infected) becomes healthy at rate $δ_0,$ while if the background process is in state $1,$ it becomes healthy at rate $δ_1.$…

Phase transition for a non-attractive infection process in heterogeneous environment

We consider a non-attractive three state contact process on $\mathbb Z$ and prove that there exists a regime of survival as well as a regime of extinction. In more detail, the process can be regarded as an infection process in a dynamic environment, where non-infected sites are either healthy or passive. Infected sites can recover only if they have a healthy site nearby, whereas non-infected sites may become infected only if there is no healthy and at least one infected site nearby. The transition probabilities are governed by a global parameter $q$: for large $q$, the infection dies out, and …

具有社会簇和无症状状态的接触过程

The contact process with social clusters and asymptomatic states

本文研究接触过程的变体，在每个格点放置社会簇并区分有症状与无症状个体，证明确定性平均场与随机空间模型在四个感染率下具有相同相变，并发现簇大小可独立驱动相变。

运动助力接触过程：泊松布朗运动上逾渗阈值以下的存活

Motion helps the contact process: survival below the percolation threshold on Poisson Brownian motions

该研究针对泊松布朗运动上的SIS接触过程，证明当恢复速率足够小时，在低于静态布尔模型临界强度的条件下，感染可存活并以正速度传播。

The survival probability of the high-dimensional contact process with random vertex weights on the oriented lattice

This paper is a further study of Reference \cite{Xue2015}. We are concerned with the contact process with random vertex weights on the oriented lattice. Our main result gives the asymptotic behavior of the survival probability of the process conditioned on only one vertex is infected at $t=0$ as the dimension grows to infinity. A SIR model and a branching process with random vertex weights are the main auxiliary tools for the proof of the main result.

具有无症状状态的接触过程的灭绝相

On the extinction phase of the contact process with an asymptomatic state

本文研究具有无症状状态的接触过程，通过证明 Galton-Watson 分支过程后代的指数衰减，并利用块构造和扰动论证，将灭绝相从 $λ_1=0$ 扩展到小的 $λ_1>0$。

Metastability for the contact process on the preferential attachment graph

We consider the contact process on the preferential attachment graph. The work of Berger, Borgs, Chayes and Saberi [BBCS1] confirmed physicists predictions that the contact process starting from a typical vertex becomes endemic for an arbitrarily small infection rate $λ$ with positive probability. More precisely, they showed that with probability $λ^{Θ(1)}$, it survives for a time exponential in the largest degree. Here we obtain sharp bounds for the density of infected sites at a time close to exponential in the number of vertices (up to some logarithmic factor).

有限图上重尾更新下的接触过程

Contact Process under heavy-tailed renewals on finite graphs

本文研究有限图上具有重尾更新康复时间的非马尔可夫接触过程，通过图规模与重尾指数α的阈值条件，刻画了感染存活与否的相变行为。

A Metastability Result for the Contact Process on a Random Regular Graph

In this paper we study the metastability of the contact process on a random regular graph. We show that the extinction time of the contact process, when initialized so that all vertices are infected at time 0, grows exponentially with the vertex number. Moreover, we show that the extinction time divided by its mean converges to a unit exponential distribution in law.

Metastable Densities for Contact Processes on Power Law Random Graphs

We consider the contact process on a random graph with fixed degree distribution given by a power law. We follow the work of Chatterjee and Durrett, who showed that for arbitrarily small infection parameter $λ$, the survival time of the process is larger than a stretched exponential function of the number of vertices, $n$. We obtain sharp bounds for the typical density of infected sites in the graph, as $λ$ is kept fixed and $n$ tends to infinity. We exhibit three different regimes for this density, depending on the tail of the degree law.

The contact process as seen from a random walk

We consider a random walk on top of the contact process on $\mathbb{Z}^d$ with $d\geq 1$. In particular, we focus on the "contact process as seen from the random walk". Under the assumption that the infection rate of the contact process is large or the jump rate of the random walk is small, we show that this process has at most two extremal measures. Moreover, the convergence to these extremal measures is characterised by whether the contact process survives or dies out, similar to the complete convergence theorem known for the ordinary contact process. Using this, we furthermore provide a law…

Contact processes on random graphs with power law degree distributions have critical value 0

If we consider the contact process with infection rate $λ$ on a random graph on $n$ vertices with power law degree distributions, mean field calculations suggest that the critical value $λ_c$ of the infection rate is positive if the power $α>3$. Physicists seem to regard this as an established fact, since the result has recently been generalized to bipartite graphs by Gómez-Gardeñes et al. [Proc. Natl. Acad. Sci. USA 105 (2008) 1399--1404]. Here, we show that the critical value $λ_c$ is zero for any value of $α>3$, and the contact process starting from all vertices infected, with a probability…

Contact process under renewals I

Motivated by questions regarding long range percolation, we investigate a non-Markovian analogue of the Harris contact process in $\mathbb{Z}^d$: an individual is attached to each site $x \in \mathbb{Z}^d$, and it can be infected or healthy; the infection propagates to healthy neighbors just as in the usual contact process, according to independent exponential times with a fixed rate $λ$; nevertheless, the possible recovery times for an individual are given by the points of a renewal process with heavy tail; the renewal processes are assumed to be independent for different sites. We show that …

Exponential growth and continuous phase transitions for the contact process on trees

We study the supercritical contact process on Galton-Watson trees and periodic trees. We prove that if the contact process survives weakly then it dominates a supercritical Crump-Mode-Jagers branching process. Hence the number of infected sites grows exponentially fast. As a consequence we conclude that the contact process dies out at the critical value $λ_1$ for weak survival, and the survival probability $p(λ)$ is continuous with respect to the infection rate $λ$. Applying this fact, we show the contact process on a general periodic tree experiences two phase transitions in the sense that $λ…

Uniformity of hitting times of the contact process

For the supercritical contact process on the hyper-cubic lattice started from a single infection at the origin and conditioned on survival, we establish two uniformity results for the hitting times $t(x)$, defined for each site $x$ as the first time at which it becomes infected. First, the family of random variables $(t(x)-t(y))/|x-y|$, indexed by $x \neq y$ in $\mathbb{Z}^d$, is stochastically tight. Second, for each $\varepsilon >0$ there exists $x$ such that, for infinitely many integers $n$, $t(nx) < t((n+1)x)$ with probability larger than $1-\varepsilon$. A key ingredient in our proofs is…

度惩罚接触过程

Degree-penalized contact processes

本文研究了度惩罚接触过程的相变，分析了不同参数μ和度分布D对过程存活的影响，揭示了在不同条件下过程的灭绝或存活特性。

接触过程可以在略微亚临界动态渗透簇上生存

The Contact Process Can Survive on a Slightly Subcritical Dynamical Percolation Cluster

该研究证明在略微亚临界动态渗透簇上，接触过程可以生存，扩展了一维情况下的结果。

关于具有增强作用的一维接触过程

On the One-Dimensional Contact Process with Enhancements

研究具有两个感染参数的一维接触过程，证明参数临界值是相互的严格单调连续函数，指出一个参数为标准临界值且另一个更大时单点感染有存活正概率，还对比了半直线上增强感染率临界过程的灭绝情况。

个体隔离与社区影响隔离对有限随机图上SIS流行病持续性的影响

The effects of individual versus community-influenced isolation on SIS epidemic persistence on finite random graphs

本文通过引入个体隔离和社区影响隔离两种模型，研究接触过程（SIS流行病模型）在有限随机图上的持久时间，发现隔离模型中感染持久时间至少为拉伸指数时间，而警戒模型中存在相变。

具有混合群体的流行病模型中的侧向接触追踪

Sideward contact tracing in an epidemic model with mixing groups

本研究提出一种具有侧向接触追踪的随机流行病模型，通过宏观分支过程近似早期传播，推导有效繁殖数阈值，并分析其受混合事件大小、诊断率和追踪概率的影响。

Metastability for the contact process on the configuration model with infinite mean degree

We study the contact process on the configuration model with a power law degree distribution, when the exponent is smaller than or equal to two. We prove that the extinction time grows exponentially fast with the size of the graph and prove two metastability results. First the extinction time divided by its mean converges in distribution toward an exponential random variable with mean one, when the size of the graph tends to infinity. Moreover, the density of infected sites taken at exponential times converges in probability to a constant. This extends previous results in the case of an expone…

Contact process on a graph with communities

We are interested in the spread of an epidemic between two communities that have higher connectivity within than between them. We model the two communities as independent Erdos-Renyi random graphs, each with n vertices and edge probability p = n^{a-1} (0<a<1), then add a small set of bridge edges, B, between the communities. We model the epidemic on this network as a contact process (Susceptible-Infected-Susceptible infection) with infection rate λand recovery rate 1. If npλ= b > 1 then the contact process on the Erdos-Renyi random graph is supercritical, and we show that it survives for expon…

Spatial Epidemics and Local Times for Critical Branching Random Walks in Dimensions 2 and 3

The behavior at criticality of spatial SIR (susceptible/infected/recovered) epidemic models in dimensions two and three is investigated. In these models, finite populations of size N are situated at the vertices of the integer lattice, and infectious contacts are limited to individuals at the same or at neighboring sites. Susceptible individuals, once infected, remain contagious for one unit of time and then recover, after which they are immune to further infection. It is shown that the measure-valued processes associated with these epidemics, suitably scaled, converge, in the large-N limit, e…

正则树上的接触过程渗流

Percolation of the contact process on the regular tree

本文研究正则树上接触过程的渗流相变，证明在 $d\geq 3$ 时 $\lambda_1<\lambda_p$，并在 $d\geq 7$ 时给出完整的相变顺序，解决了长期未决问题。

The contact process on the complete graph with random vertex-dependent infection rates

We study the contact process on the complete graph on $n$ vertices where the rate at which the infection travels along the edge connecting vertices $i$ and $j$ is equal to $ λw_i w_j / n$ for some $λ>0$, where $w_i$ are i.i.d. vertex weights. We show that when $E[w_1^2] < \infty$ there is a phase transition at $λ_c > 0$ so that for $λ<λ_c$ the contact process dies out in logarithmic time, and for $λ>λ_c$ the contact process lives for an exponential amount of time. Moreover, we give a formula for $λ_c$ and when $λ>λ_c$ we are able to give precise approximations for the probability a given verte…

Richardson模型与带扰动的接触过程：长时间行为

Richardson's model and the contact process with stirring: long time behavior

本文研究了带扰动动力学的Richardson模型与接触过程的长时间行为。利用耦合与重启技术，证明了在感染率足够高时两者均满足渐近形状定理，且Richardson模型中每个位点最终几乎必然被永久感染；并在齐次无穷树上证明了两者存在两次相变。

Critical value for the contact process with random edge weights on regular tree

In this paper we are concerned with contact processes with random edge weights on rooted regular trees. We assign i.i.d weights on each edge on the tree and assume that an infected vertex infects its healthy neighbor at rate proportional to the weight on the edge connecting them. Under the annealed measure, we define the critical value λ_c as the maximum of the infection rate with which the process will die out and define λ_e as the maximum of the infection rate with which the process dies out at exponential rate. We show that these two critical values satisfy an identical limit theorem and gi…

具有确定性治愈的广义接触过程的灭绝与存活

Extinction and Survival for a Generalized Contact Process with Deterministic Cures

研究具有确定性治愈时间的广义接触过程，证明低感染率下灭绝、高感染率下存活，并给出单点密度性质。

具有无症状状态的接触过程

The contact process with an asymptomatic state

本文提出区分无症状与有症状个体的接触过程变体，通过均场模型与相互作用粒子系统分析发现，局部相互作用下无症状者低传染性可阻止高传染性有症状者引发流行病。

Contact and voter processes on the infinite percolation cluster as models of host-symbiont interactions

We introduce spatially explicit stochastic processes to model multispecies host-symbiont interactions. The host environment is static, modeled by the infinite percolation cluster of site percolation. Symbionts evolve on the infinite cluster through contact or voter type interactions, where each host may be infected by a colony of symbionts. In the presence of a single symbiont species, the condition for invasion as a function of the density of the habitat of hosts and the maximal size of the colonies is investigated in details. In the presence of multiple symbiont species, it is proved that th…

A sharper threshold for bootstrap percolation in two dimensions

Two-dimensional bootstrap percolation is a cellular automaton in which sites become 'infected' by contact with two or more already infected nearest neighbors. We consider these dynamics, which can be interpreted as a monotone version of the Ising model, on an n x n square, with sites initially infected independently with probability p. The critical probability p_c is the smallest p for which the probability that the entire square is eventually infected exceeds 1/2. Holroyd determined the sharp first-order approximation: p_c \sim π^2/(18 log n) as n \to \infty. Here we sharpen this result, prov…

动态随机环境中接触过程的形状定理

Shape Theorem for the Contact Process in a Dynamical Random Environment

本文研究动态随机环境中的接触过程，在满足增长条件时证明感染顶点集合H_t的渐近形状由凸集U张成的锥体刻画，且该形状不依赖环境初始构型，并将结果推广到动态图和切换顶点状态等非单调系统。

Gaussian scaling for the critical spread-out contact process above the upper critical dimension

We consider the critical spread-out contact process in $\Zd$ with $d\geq 1$, whose infection range is denoted by $L\geq1$. The two-point function $τ_t(x)$ is the probability that $x\in\Zd$ is infected at time $t$ by the infected individual located at the origin $o\in\Zd$ at time 0. We prove Gaussian behavior for the two-point function with $L\geq L_0$ for some finite $L_0=L_0(d)$ for $d>4$. When $d\leq 4$, we also perform a local mean-field limit to obtain Gaussian behaviour for $τ_{tT}$ with $t>0$ fixed and $T\to \infty$ when the infection range depends on $T$ such that $L_T=LT^b$ for any $b>…

Proof of a conjecture of N. Konno for the 1D contact process

Consider the one-dimensional contact process. About ten years ago, N. Konno stated the conjecture that, for all positive integers $n,m$, the upper invariant measure has the following property: Conditioned on the event that $O$ is infected, the events $\{$All sites $-n,...,-1$ are healthy$\}$ and $\{$All sites $1,...,m$ are healthy$\}$ are negatively correlated. We prove (a stronger version of) this conjecture, and explain that in some sense it is a dual version of the planar case of one of our results in \citeBHK.

Tightness for the interface of the one-dimensional contact process

We consider a symmetric, finite-range contact process with two types of infection; both have the same (supercritical) infection rate and heal at rate 1, but sites infected by Infection 1 are immune to Infection 2. We take the initial configuration where sites in $(-\infty,0]$ have Infection 1 and sites in $[1,\infty)$ have Infection 2, then consider the process $ρ_t$ defined as the size of the interface area between the two infections at time $t$. We show that the distribution of $ρ_t$ is tight, thus proving a conjecture posed by Cox and Durrett in [Bernoulli 1 (1995) 343--370].

A computational framework for evaluating the role of mobility on the propagation of epidemics on point processes

This paper is focused on SIS (Susceptible-Infected-Susceptible) epidemic dynamics (also known as the contact process) on populations modelled by homogeneous Poisson point processes of the Euclidean plane, where the infection rate of a susceptible individual is proportional to the number of infected individuals in a disc around it. The main focus of the paper is a model where points are also subject to some random motion. Conservation equations for moment measures are leveraged to analyze the stationary regime of the point processes of infected and susceptible individuals. A heuristic factoriza…

Elementary proof of convergence to the mean-field model for the SIR process

The susceptible-infected-recovered (SIR) model has been used extensively to model disease spread and other processes. Despite the widespread usage of this ordinary differential equation (ODE) based model which represents the mean-field approximation of the underlying stochastic SIR process on contact networks, only few rigorous approaches exist and these use complex semigroup and martingale techniques to prove that the expected fraction of the susceptible and infected nodes of the stochastic SIR process on a complete graph converges as the number of nodes increases to the solution of the mean-…