We consider the extinction time of the contact process on increasing sequences of finite graphs obtained from a variety of random graph models. Under the assumption that the infection rate is above the critical value for the process on the integer line, in each case we prove that the logarithm of the extinction time divided by the size of the graph converges in probability to a (model-dependent) positive constant. The graphs we treat include various percolation models on increasing boxes of Z d or R d in their supercritical or percolative regimes (Bernoulli bond and site percolation, the occupied and vacant sets of random interlacements, excursion sets of the Gaussian free field, random geometric graphs) as well as supercritical Galton-Watson trees grown up to finite generations.
In this paper, we prove lower and upper bounds for the extinction time of the contact process on random geometric graphs with connecting radius tending to infinity. We obtain that for any infection rate $λ>0$, the contact process on these graphs survives a time super-exponential in the number of vertices.
We give a construction of a tree in which the contact process with any positive infection rate survives but, if a certain privileged edge $e^*$ is removed, one obtains two subtrees in which the contact process with infection rate smaller than $1/4$ dies out.
We study the extinction time $\uptau$ of the contact process on finite trees of bounded degree. We show that, if the infection rate is larger than the critical rate for the contact process on $\Z$, then, uniformly over all trees of degree bounded by a given number, the expectation of $\uptau$ grows exponentially with the number of vertices. Additionally, for any sequence of growing trees of bounded degree, $\uptau$ divided by its expectation converges in distribution to the unitary exponential distribution. These also hold if one considers a sequence of graphs having spanning trees with unifor…
We introduce a method to prove metastability of the contact process on Erdős-Rényi graphs and on configuration model graphs. The method relies on uniformly bounding the total infection rate from below, over all sets with a fixed number of nodes. Once this bound is established, a simple comparison with a well chosen birth-and-death process will show the exponential growth of the extinction time. Our paper complements recent results on the metastability of the contact process: under a certain minimal edge density condition, we give explicit lower bounds on the infection rate needed to get metast…
We consider the contact process on finite and connected graphs and study the behavior of the extinction time, that is, the amount of time that it takes for the infection to disappear in the process started from full occupancy. We prove, without any restriction on the graph $G$, that if the infection rate $λ$ is larger than the critical rate of the one-dimensional process, then the extinction time grows faster than $\exp\{|G|/(\log|G|)^κ\}$ for any constant $κ> 1$, where $|G|$ denotes the number of vertices of $G$. Also for general graphs, we show that the extinction time divided by its expecta…
In this note we give a new method for getting a series of approximations for the extinction probability of the one-dimensional contact process by using the Gröbner basis.
We consider a two-type contact process on $\Z$ in which both types have equal finite range and supercritical infection rate. We show that a given type becomes extinct with probability 1 if and only if, in the initial configuration, it is confined to a finite interval $[-L,L]$ and the other type occupies infinitely many sites both in $(-\infty, L)$ and $(L, \infty)$. We also show that, starting from the configuration in which all sites in $(-\infty, 0]$ are occupied by type 1 particles and all sites in $(0, \infty)$ are occupied by type 2 particles, the process $ρ_t$ defined by the size of the …
We show that the contact process on the rank-one inhomogeneous random graphs and Erdos-R{é}nyi graphs with mean degree large enough survives a time exponential in the size of these graphs for any positive infection rate. In addition, a metastable result for the extinction time is also proved.
We prove the exponential estimate \begin{equation*} P \{ s < τ< \infty \} \leq C e^{-q s}, \quad s \geq 0, \end{equation*} where $C, q >0$ are constants and $ τ$ is the extinction time of the supercritical branching random walk (BRW) on a cube. We cover both the discrete-space and continuous-space BRWs.
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.
In this paper, we establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $λ_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution $ξ$ has an exponential tail, i.e., $\mathbb{E} e^{cξ}<\infty$ for some $c>0$, settling a conjecture by Huang and Durrett [12]. On the random graph with degree distribution $μ$, we show that if $μ$ has an exponential tail, then for small enough $λ$ the contact process with t…
We consider a supercritical branching process and define a contact tracing mechanism on its genealogical tree. We calculate the growth rate of the post tracing process, and give conditions under which the tracing is strong enough to drive the process to extinction.
Consider a superdiffusion $X$ on $\mathbb R^d$ corresponding to the semilinear operator $\mathcal{A}(u)=Lu+βu-ku^2,$ where $L$ is a second order elliptic operator, $β(\cdot)$ is in the Kato class and bounded from above, and $k(\cdot)\ge 0$ is bounded on compact subsets of $\R^d$ and is positive on a set of positive Lebesgue measure. The main purpose of this paper is to complement the results obtained in \cite{Englander:2004}, in the following sense. Let $λ_\infty $ be the $L^\infty$-growth bound of the semigroup corresponding to the Schrödinger operator $L+β$. If $λ_\infty \neq0$, then we prov…
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…
This article is concerned with a version of the contact process with sexual reproduction on a graph with two levels of interactions modeling metapopulations. The population is spatially distributed into patches and offspring are produced in each patch at a rate proportional to the number of pairs of individuals in the patch (sexual reproduction) rather than simply the number of individuals as in the basic contact process. Offspring produced at a given patch either stay in their parents' patch or are sent to a nearby patch with some fixed probabilities. Specifically, we prove lower and upper bo…
We introduce and study the mutating contact process, a variant of the multitype contact process, where one type mutates at a constant rate to the other type. We prove that on $\mathbb{Z}$ a single mutant cannot survive while on $\mathbb{T}_{d}$ there are distinct weak survival and extinction of a single mutant phases, yet the limiting distribution concentrates on configurations with no mutants of the first type for any values of the parameters.
Motivated by a model of an area-wide integrated pest management, we develop an interacting particle system evolving in a random environment. It is a generalised contact process in which the birth rate takes two possible values, determined either by a dynamic or a static random environment. Our goal is to understand the phase diagram of both models by identifying the mechanisms that permit coexistence or extinction of the process.
Results on the contact process with dynamic edges or under renewals
Universidade Federal de Minas Gerais（米纳斯吉拉斯联邦大学）；Universidade de São Paulo（圣保罗大学）；University of Groningen（格罗宁根大学）；Universidade Federal do Rio de Janeiro（里约热内卢联邦大学）
We investigate the contact process on four different types of scale-free inhomogeneous random graphs evolving according to a stationary dynamics, where each potential edge is updated with a rate depending on the strength of the adjacent vertices. Depending on the type of graph, the tail exponent of the degree distribution and the updating rate, we find parameter regimes of fast and slow extinction and in the latter case identify metastable exponents that undergo first order phase transitions.
This article studies the quasi-stationary behaviour of population processes with unbounded absorption rate, including one-dimensional birth and death processes with catastrophes and multi-dimensional birth and death processes, modeling biological populations in interaction. To handle this situation, we develop original non-linear Lyapunov criteria. We obtain the exponential convergence in total variation of the conditional distributions to a unique quasi-stationary distribution, uniformly with respect to the initial distribution. Our results cover all one-dimensional birth and death processes …
In this paper, we study the speed of extinction of continuous state branching processes in subcritical Lévy environments. More precisely, when the associated Lévy process to the environment drifts to $-\infty$ and, under a suitable exponential martingale change of measure (Esscher transform), the environment either drifts to $-\infty$ or oscillates. We extend recent results of Palau et al. (2016) and Li and Xu (2018), where the branching term is associated to a spectrally positive stable Lévy process and complement the recent article of Bansaye et al. (2021) where the critical case was studied…
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…
This article is concerned with a version of the contact process with sexual reproduction on a graph with two levels of interactions modeling metapopulations. The population is spatially distributed into patches and offspring are produced in each patch at a rate proportional to the number of pairs of individuals in the patch (sexual reproduction) rather than simply the number of individuals as in the basic contact process. Offspring produced at a given patch either stay in their parents' patch or are sent to a nearby patch with some fixed probabilities. As the patch size tends to infinity, we i…
Let $T$ be the extinction moment of a critical branching process $Z=(Z_{n},n\geq 0) $ in a random environment specified by iid probability generating functions. We study the asymptotic behavior of the probability of extinction of the process $Z$ at moment $n\to \infty$, and show that if the logarithm of the (random) expectation of the offspring number belongs to the domain of attraction of a non-gaussian stable law then the extinction occurs owing to very unfavorable environment forcing the process, having at moment $T-1$ exponentially large population, to die out. We also give an interpretati…
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…
We study the speed of extinction of continuous state branching processes in a Lévy environment, where the associated Lévy process oscillates. Assuming that the Lévy process satisfies the Spitzer's condition and the existence of some exponential moments, we extend recent results where the associated branching mechanism was stable. Our study relies on the path analysis of the process together with its environment, when this latter is conditioned to have a non negative running infimum. This approach is inspired from the discrete setting with i.i.d. environment studied in (Afanasyev et al. 2005).
This note gives an exponential tail approximation for the extinction time of a subcritical multitype branching process arising from the SIR epidemic model on a random graph with given degrees, where the type corresponds to the vertex degree. As a corollary we obtain a Gumbel limit law for the extinction time, when beginning with a large population. Our contribution is to allow countably many types (this corresponds to unbounded degrees in the random graph epidemic model, as the number of vertices tends to infinity). We only require a second moment for the offspring-type distribution featuring …
We propose the following model for speciation and extinction. Birth and deaths occur according to spatially inhomogeneous contact rates. We assume that the ratio of the birth rate over the death rate at a site converges to some limit as the site goes to infinity. We show that the existence of a phase transition depends critically on the value of this limit.
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 study a model of growing population that competes for resources. At each time step, all existing particles reproduce and the offspring randomly move to neighboring sites. Then at any site with more than one offspring, the particles are annihilated. This is a nonmonotone model, which makes the analysis more difficult. We consider the extinction window of this model in the finite mean-field case, where there are $n$ sites but movement is allowed to any site (the complete graph). We show that although the system survives for exponential time, the extinction window is logarithmic.
In this paper, we consider time-inhomogeneous branching processes and time-inhomogeneous birth-and-death processes, in which the offspring distribution and birth and death rates (respectively) vary in time. A classical result of branching processes states that in the critical regime, a process conditioned on non-extinction and normalized will converge in distribution to a standard exponential. In a paper of Jagers, time-inhomogeneous branching processes are shown to exhibit this convergence as well. In this paper, the hypotheses of Jagers' result are relaxed, further hypotheses are presented f…
Recent progress in the study of the contact process [2] has verified that the extinction-survival threshold $λ_1$ on a Galton-Watson tree is strictly positive if and only if the offspring distribution $ξ$ has an exponential tail. In this paper, we derive the first-order asymptotics of $λ_1$ for the contact process on Galton-Watson trees and its corresponding analog for random graphs. In particular, if $ξ$ is appropriately concentrated around its mean, we demonstrate that $λ_1(ξ) \sim 1/\mathbb{E} ξ$ as $\mathbb{E}ξ\rightarrow \infty$, which matches with the known asymptotics on the $d$-regular…
We consider contact processes on the hierarchical group, where sites infect other sites at a rate depending on their hierarchical distance, and sites become healthy with a constant recovery rate. If the infection rates decay too fast as a function of the hierarchical distance, then we show that the critical recovery rate is zero. On the other hand, we derive sufficient conditions on the speed of decay of the infection rates for the process to exhibit a nontrivial phase transition between extinction and survival. For our sufficient conditions, we use a coupling argument that compares contact pr…
We consider a general discrete-time branching random walk on a countable set X. We relate local, strong local and global survival with suitable inequalities involving the first-moment matrix M of the process. In particular we prove that, while the local behavior is characterized by M, the global behavior cannot be completely described in terms of properties involving M alone. Moreover we show that locally surviving branching random walks can be approximated by sequences of spatially confined and stochastically dominated branching random walks which eventually survive locally if the (possibly f…
We consider the contact process with infection rate $λ$ on a random $(d+1)$-regular graph with $n$ vertices, $G_n$. We study the extinction time $τ_{G_n}$ (that is, the random amount of time until the infection disappears) as $n$ is taken to infinity. We establish a phase transition depending on whether $λ$ is smaller or larger than $λ_1(\mathbb{T}^d)$, the lower critical value for the contact process on the infinite, $(d+1)$-regular tree: if $λ< λ_1(\mathbb{T}^d)$, $τ_{G_n}$ grows logarithmically with $n$, while if $λ> λ_1(\mathbb{T}^d)$, it grows exponentially with $n$. This result differs f…
Let $L$ be a positive definite self-adjoint operator on the $L^2$-space associated to a $\si$-finite measure space. Let $H$ be the dual space of the domain of $L^{1/2}$ w.r.t. $L^2(μ)$. By using an Itô type inequality for the $H$-norm and an integrability condition for the hyperbound of the semigroup $P_t:=\e^{-Lt}$, general extinction results are derived for a class of continuous adapted processes on $H$. Main applications include stochastic and deterministic fast diffusion equations with fractional Laplacians. Furthermore, we prove exponential integrability of the extinction time for all spa…
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).
We study the contact process in the regime of small infection rates on finite scale-free networks with stationary dynamics based on simultaneous updating of all connections of a vertex. We allow the update rates of individual vertices to increase with the strength of a vertex, leading to a fast evolution of the network. We first develop an approach for inhomogeneous networks with general kernel and then focus on two canonical cases, the factor kernel and the preferential attachment kernel. For these specific networks we identify and analyse four possible strategies how the infection can surviv…
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…
We consider continuous state branching processes (CSBP) with additional multiplicative jumps modeling dramatic events in a random environment. These jumps are described by a Lévy process with bounded variation paths. We construct a process of this class as the unique solution of a stochastic differential equation. The quenched branching property of the process allows us to derive quenched and annealed results and to observe new asymptotic behaviors. We characterize the Laplace exponent of the process as the solution of a backward ordinary differential equation and establish the probability of …
In this article we provide new applications for exponential approximation using the framework of Peköz and Röllin (in press), which is based on Stein's method. We give error bounds for the nearly critical Galton-Watson process conditioned on non-extinction, and for the occupation times of Markov chains; for the latter, in particular, we give a new exponential approximation rate for the number of revisits to the origin for general two dimensional random walk, also known as the Erdős-Taylor theorem.
We consider the contact process with infection rate $λ$ on $\mathbb{T}_n^d$, the $d$-ary tree of height $n$. We study the extinction time $τ_{\mathbb{T}_n^d}$, that is, the random time it takes for the infection to disappear when the process is started from full occupancy. We prove two conjectures of Stacey regarding $τ_{\mathbb{T}_n^d}$. Let $λ_2$ denote the upper critical value for the contact process on the infinite $d$-ary tree. First, if $λ< λ_2$, then $τ_{\mathbb{T}_n^d}$ divided by the height of the tree converges in probability, as $n \to \infty$, to a positive constant. Second, if $λ>…
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 study threshold-one contact processes on lattices and regular trees. The asymptotic behavior of the critical infection rates as the degrees of the graphs growing to infinity are obtained. Defining λ_c as the supremum of infection rates which causes extinction of the process at equilibrium, we prove that nλ_c^{T^n}\rightarrow1 and 2dλ_c^{Z^d}\rightarrow1 as n,d\rightarrow+\infty. Our result is a development of the conclusion that λ_c^{Z^d}\leq\frac{2.18}{d} shown in \cite{Dur1991}. To prove our main result, a crucial lemma about the probability of a simple random walk on a latt…
Recent years have seen a large amount of interest in epidemics on networks as a way of representing the complex structure of contacts capable of spreading infections through the modern human population. The configuration model is a popular choice in theoretical studies since it combines the ability to specify the distribution of the number of contacts (degree) with analytical tractability. Here we consider the early real-time behaviour of the Markovian SIR epidemic model on a configuration model network using a multi-type branching process. We find closed-form analytic expressions for the mean…
We consider the super-critical contact process on $\mathbb{Z}^d$. It is known that measures which dominate the upper invariant measure $μ$ converge exponentially fast to $μ$. However, the same is not true for measures which are below $μ$, as the time to infect a large empty region is related to its diameter. The result of this paper is the existence of a spectral gap in $L^2(μ)$, that is, the spectrum of the generator is empty inside an open strip $\{z\in\mathbb{C}: -λ<\Im(z)<0\}$ of the complex plane. This is equivalent to the fact that the variance of the semi-group of the contact process de…
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.
