In this paper we are concerned with the contact process with semi-infected state on the complete graph $C_n$ with $n$ vertices. In our model, each vertex is in one of three states that `healthy', `semi-infected' or `wholly-infected'. Only wholly-infected vertices can infect others. A healthy vertex becomes semi-infected when being infected while a semi-infected vertex becomes wholly-infected when being further infected. Each (semi- and wholly-) infected vertex becomes healthy at constant rate. Our main result shows the phase transition for the time wholly-infected vertices wait for to die out. Conditioned on all the vertices are wholly-infected when $t=0$, we show that wholly-infected vertices survive for $\exp\{O(n)\}$ units of time when the infection rate $λ>4$ while die out in $O(\log n)$ units of time when $λ<4$.
This paper studies contact processes on general countable groups. It is shown that any such contact process has a well-defined exponential growth rate, and this quantity is used to study the process. In particular, it is proved that on any nonamenable group, the critical contact process dies out.
In this paper we describe the subcritical contact process on $\Z^d$ for large times, starting with all sites infected. The configuration is described in terms of the macroscopic locations of infected regions in space and the relative positions of infected sites in each such region.
What is the long-time behavior of the law of a contact process started with a single infected site, distributed according to counting measure on the lattice? This question is related to the configuration as seen from a typical infected site and gives rise to the definition of so-called eigenmeasures, which are possibly infinite measures on the set of nonempty configurations that are preserved under the dynamics up to a multiplicative constant. In this paper, we study eigenmeasures of contact processes on general countable groups in the subcritical regime. We prove that in this regime, the proc…
The basic contact process with parameter $μ$ altered so that infections of sites that have not been previously infected occur at rate proportional to $λ$ instead is considered. Emergence of an infinite epidemic starting out from a single infected site is not possible for $μ$ less than the contact process' critical value, whereas it is possible for $μ$ greater than that value. In the former case the space and time infected regions are shown to decay exponentially; in the latter case and for $λ$ greater than $μ$, the ratio of the endmost infected site's velocity to that of the contact process is…
The stacked contact process is a stochastic model for the spread of an infection within a population of hosts located on the $d$-dimensional integer lattice. Regardless of whether they are healthy or infected, hosts give birth and die at the same rate and in accordance to the evolution rules of the neutral multitype contact process. The infection is transmitted both vertically from infected parents to their offspring and horizontally from infected hosts to nearby healthy hosts. The population survives if and only if the common birth rate of healthy and infected hosts exceeds the critical value…
We show that the contact process on a random $d$-regular graph initiated by a single infected vertex obeys the "cutoff phenomenon" in its supercritical phase. In particular, we prove that when the infection rate is larger than the critical value of the contact process on the infinite $d$-regular tree there are positive constants $C, p$ depending on the infection rate such that for sufficiently small $ε> 0$, when the number $n$ of vertices is large then (a) at times $t< (C - ε) \log n$ the fraction of infected vertices is vanishingly small, but (b) at time $(C + ε) \log n$ the fraction of infec…
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}_…
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.
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…
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…
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…
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.
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 $δ_1.$…
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 …
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.
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).
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.
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.
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…
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 …
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…
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…
The Contact Process Can Survive on a Slightly Subcritical Dynamical Percolation Cluster
Univ Paris Est Creteil, Univ Gustave Eiffel, CNRS, LAMA UMR8050（巴黎东克雷泰伊大学，古斯塔夫·埃菲尔大学，法国国家科学研究中心，拉马实验室）；IRL CNRS IFUMI-2030（法国国家科学研究中心乌拉圭联合研究）；Université de Lorraine, Institut Élie Cartan de Lorraine, UMR 7502（洛林大学，洛林艾利嘉当研究所）；CNRS, Institut Élie Cartan de Lorraine, UMR 7502（法国国家科学研究中心，洛林艾利嘉当研究所）
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…
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…
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…
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…
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…
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>…
We consider an SIR-type (Susceptible $\to$ Infected $\to$ Recovered) stochastic epidemic process with multiple modes of transmission on a contact network. The network is given by a random graph following a multilayer configuration model where edges in different layers correspond to potentially infectious contacts of different types. We assume that the graph structure evolves in response to the epidemic via activation or deactivation of edges. We derive a large graph limit theorem that gives a system of ordinary differential equations (ODEs) describing the evolution of quantities of interest, s…
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.
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].
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-…
