We introduce a new percolation model to describe and analyze the spread of an epidemic on a general directed and locally finite graph. We assign a two-dimensional random weight vector to each vertex of the graph in such a way that the weights of different vertices are i.i.d., but the two entries of the vector assigned to a vertex need not be independent. The probability for an edge to be open depends on the weights of its end vertices, but conditionally on the weights, the states of the edges are independent of each other. In an epidemiological setting, the vertices of a graph represent the individuals in a (social) network and the edges represent the connections in the network. The weights assigned to an individual denote its (random) infectivity and susceptibility, respectively. We show that one can bound the percolation probability and the expected size of the cluster of vertices that can be reached by an open path starting at a given vertex from above and below by the corresponding quantities for respectively independent bond and site percolation with certain densities; this generalizes a result of Kuulasmaa. Many models in the literature are special cases of our general model.
In this paper we establish a connection between epidemic models on random networks with general infection times considered in Barbour and Reinert 2013 and first passage percolation. Using techniques developed in Bhamidi, van der Hofstad, Hooghiemstra 2012, when each vertex has infinite contagious periods, we extend results on the epidemic curve in Barbour Reinert 2013 from bounded degree graphs to general sparse random graphs with degrees having finite third moments as the number of vertices tends to infinity. We also study the epidemic trail between the source and typical vertices in the grap…
We obtain tight thresholds for bond percolation on one-dimensional small-world graphs, and apply such results to obtain tight thresholds for the \emph{Independent Cascade} process and the \emph{Reed-Frost} process in such graphs. These are the first fully rigorous results establishing a phase transition for bond percolation and SIR epidemic processes in small-world graphs. Although one-dimensional small-world graphs are an idealized and unrealistic network model, a number of realistic qualitative epidemiological phenomena emerge from our analysis, including the epidemic spread through a sequen…
Human to human transmissible infectious diseases spread in a population using human interactions as its transmission vector. The early stages of such an outbreak can be modeled by a graph whose edges encode these interactions between individuals, the vertices. This article attempts to account for the case when each individual entails in different kinds of interactions which have therefore different probabilities of transmitting the disease. The majority of these results can be also stated in the language of percolation theory. The main contributions of the article are: (1) Extend to this setti…
We consider a stochastic SIR (susceptible $\to$ infective $\to$ recovered) epidemic defined on a configuration model random graph, in which infective individuals can infect only their neighbours in the graph during an infectious period which has an arbitrary but specified distribution. Central limit theorems for the final size (number of initial susceptibles that become infected) of such an epidemic as the population size $n$ tends to infinity, with explicit, easy to compute expressions for the asymptotic variance, are proved assuming that the degrees are bounded. The results are obtained for …
We prove a shape theorem for the set of infected individuals in a spatial epidemic model with 3 states (susceptible-infected-recovered) on ${\mathbb Z}^d,d\ge 3$, when there is no extinction of the infection. For this, we derive percolation estimates (using dynamic renormalization techniques) for a locally dependent random graph in correspondence with the epidemic model.
We use the connection between bond percolation and SIR epidemics to establish lower bounds for the critical percolation probability in $2$ and $3$ dimensions as the range becomes large. The bound agrees with the conjectured asymptotics for the long range critical probability, refines results of M. Penrose, and complements results of van der Hofstad and Sakai in dimensions greater than $6$.
An upper bound for the critical probability of long range bond percolation in $d=2$ and $d=3$ is obtained by connecting the bond percolation with the SIR epidemic model, thus complementing the lower bound result in Frei and Perkins arXiv:arch-ive/1603.04130. A key ingredient is that we establish a uniform bound for the local times of branching random walk by calculating their exponential moments and by using the discrete versions of Tanaka's formula and Garsia's Lemma.
We consider long-range percolation on $\mathbb{Z}^d$, where the probability that two vertices at distance $r$ are connected by an edge is given by $p(r)=1-\exp[-λ(r)]\in(0,1)$ and the presence or absence of different edges are independent. Here, $λ(r)$ is a strictly positive, nonincreasing, regularly varying function. We investigate the asymptotic growth of the size of the $k$-ball around the origin, $|\mathcal{B}_k|$, that is, the number of vertices that are within graph-distance $k$ of the origin, for $k\to\infty$, for different $λ(r)$. We show that conditioned on the origin being in the (un…
Majority bootstrap percolation on a graph $G$ is an epidemic process defined in the following manner. Firstly, an initially infected set of vertices is selected. Then step by step the vertices that have more infected than non-infected neighbours are infected. We say that percolation occurs if eventually all vertices in $G$ become infected. In this paper we study majority bootstrap percolation on the Erdős-Rényi random graph $G(n,p)$ above the connectivity threshold. Perhaps surprisingly, the results obtained for small $p$ are comparable to the results for the hypercube obtained by Balogh, Boll…
Random graphs have played an instrumental role in modelling real-world networks arising from the internet topology, social networks, or even protein-interaction networks within cells. Percolation, on the other hand, has been the fundamental model for understanding robustness and spread of epidemics on these networks. From a mathematical perspective, percolation is one of the simplest models that exhibits phase transition, and fascinating features are observed around the critical point. In this thesis, we prove limit theorems about structural properties of the connected components obtained from…
We consider the spread of a supercritical stochastic SIR (Susceptible, Infectious, Recovered) epidemic on a configuration model random graph. We mainly focus on the final stages of a large outbreak and provide limit results for the duration of the entire epidemic, while we allow for non-exponential distributions of the infectious period and for both finite and infinite variance of the asymptotic degree distribution in the graph. Our analysis relies on the analysis of some subcritical continuous time branching processes and on ideas from first-passage percolation. As an application we investiga…
We consider the Erdös--Rényi random graph $G_{n,p}$ and we analyze the simple irreversible epidemic process on the graph, known in the literature as bootstrap percolation. We give a quantitative version of some results by Janson et al. (2012), providing a fine asymptotic analysis of the final size $A_n^*$ of active nodes, under a suitable super-critical regime. More specifically, we establish large deviation principles for the sequence of random variables $\{\frac{n- A_n^*}{f(n)}\}_{n\geq 1}$ with explicit rate functions and allowing the scaling function $f$ to vary in the widest possible rang…
In this paper we are concerned with the SIR (Susceptible-Infective-Removed) epidemic on open clusters of bond percolation on the squared lattice. For the SIR model, a susceptible vertex is infected at rate proportional to the number of infective neighbors while an infective vertex becomes removed at a constant rate. A removed vertex will never be infected again. We assume that there is only one infective vertex at $t=0$ and define the critical value of the model as the maximum of the infection rates with which infective vertices die out with probability one, then we show that the critical valu…
\emph{Full-bond percolation} with parameter $p$ is the process in which, given a graph, for every edge independently, we delete the edge with probability $1-p$. Bond percolation is motivated by problems in mathematical physics and it is studied in parallel computing and network science to understand the resilience of distributed systems to random link failure and the spread of information in networks through unreliable links. Full-bond percolation is also equivalent to the \emph{Reed-Frost process}, a network version of \emph{SIR} epidemic spreading, in which the graph represents contacts amon…
In this paper, we derive nonasymptotic theoretical bounds for the influence in random graphs that depend on the spectral radius of a particular matrix, called the Hazard matrix. We also show that these results are generic and valid for a large class of random graphs displaying correlation at a local scale, called the LPC random graphs. In particular, they lead to tight and novel bounds in percolation, epidemiology and information cascades. The main result of the paper states that the influence in the sub-critical regime for LPC random graphs is at most of the order of $O(\sqrt{n})$ where $n$ i…
The Newman-Watts model is given by taking a cycle graph of n vertices and then adding each possible edge $(i,j), |i-j|\neq 1 \mod n$ with probability $ρ/n$ for some $ρ>0$ constant. In this paper we add i.i.d. exponential edge weights to this graph, and investigate typical distances in the corresponding random metric space given by the least weight paths between vertices. We show that typical distances grow as $\frac1λ\log n$ for a $λ>0$ and determine the distribution of smaller order terms in terms of limits of branching process random variables. We prove that the number of edges along the sho…
We study a simple model of epidemics where an infected node transmits the infection to its neighbors independently with probability $p$. This is also known as the independent cascade or Susceptible-Infected-Recovered (SIR) model with fixed recovery time. The size of an outbreak in this model is closely related to that of the giant connected component in ``edge percolation'', where each edge of the graph is kept independently with probability $p$, studied for a large class of networks including configuration model \cite{molloy2011critical} and preferential attachment \cite{bollobas2003,Riordan2…
