Infected Vertices Research Articles

The large graph limit of a stochastic epidemic model on a dynamic multilayer network

ABSTRACTWe consider a Markovian SIR-type (Susceptible → Infected → 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 of infectious nodes. We derive a large graph limit theorem that gives a system of ordinary differential equations (ODEs) describing the evolution of quantities of interest, such as the proportions of infected and susceptible vertices, as the number of nodes tends to infinity. Analysis of the limiting system elucidates how the coupling of edge activation and deactivation to infection status affects disease dynamics, as illustrated by a two-layer network example with edge types corresponding to community and healthcare contacts. Our theorem extends some earlier results describing the deterministic limit of stochastic SIR processes on static, single-layer configuration model graphs. We also describe precisely the conditions for equivalence between our limiting ODEs and the systems obtained via pair approximation, which are widely used in the epidemiological and ecological literature to approximate disease dynamics on networks. The flexible modeling framework and asymptotic results have potential application to many disease settings including Ebola dynamics in West Africa, which was the original motivation for this study.

A phase transition in the evolution of bootstrap percolation processes on preferential attachment graphs

AbstractThe theme of this paper is the analysis of bootstrap percolation processes on random graphs generated by preferential attachment. This is a class of infection processes where vertices have two states: they are either infected or susceptible. At each round every susceptible vertex which has at least infected neighbours becomes infected and remains so forever. Assume that initially vertices are randomly infected, where is the total number of vertices of the graph. Suppose also that , where is the average degree. We determine a critical function such that when , complete infection occurs with high probability as , but when , then with high probability the process evolves only for a bounded number of rounds and the final set of infected vertices is asymptotically equal to .

A contact process with a semi-infected state on the complete graph

ABSTRACTIn this paper, we are concerned with a contact process with a semi-infected state on the complete graph Cn with n vertices. Our model is a special case of a general model introduced by Schinazi in 2003. In our model, each vertex is in one of three states, namely, “healthy,” “semi-infected,” or “fully-infected.” Only fully-infected vertices can infect others. A healthy vertex becomes semi-infected when being infected while a semi-infected vertex becomes fully-infected when being further infected. Each (semi- and fully-) infected vertex becomes healthy at constant rate. Our main result shows a phase transition for the waiting time until extinction of the fully-infected vertices. Conditioned on all the vertices are fully-infected when t = 0, we show that fully-infected vertices survive for exp {O(n)} units of time when the infection rate λ > 4 while they die out in O(log n) units of time when λ < 4.

How Complex Contagions Spread Quickly in Preferential Attachment Models and Other Time-Evolving Networks

The $k$ -complex contagion model is a social contagion model which describes the diffusion of behaviors in networks where the successful adoption of a behavior requires influence from multiple contacts. It has been argued that complex contagions better model behavioral changes such as adoption of new beliefs, fashion trends or expensive technology innovations. A contagion in this model starts from a set of initially infected seeds and progresses in rounds. In any round any node with at least $k>1$ infected neighbors becomes infected. Previous work on $k$ -complex contagions was focused on networks with uniform degree distributions. However, many real-world network topologies have non-uniform degree distribution and evolve over time. We analyze the spreading rate of a $k$ -complex contagion in a general family of time-evolving networks which includes the preferential attachment (PA) model. We prove that if the initial seeds are chosen as the $k$ earliest nodes in a network of this family, a $k$ -complex contagion covers the entire network of $n$ nodes in $O(\log n)$ rounds with high probability (w.h.p). We prove that the choice of the seeds is crucial: in the PA model, even if a much larger number of seeds are chosen uniformly randomly , the contagion stops prematurely w.h.p. Although the earliest nodes in a PA model are likely to have high degrees, it is actually the evolutionary graph structure of such models that facilitates fast spreading of complex contagions. The general family of time-evolving graphs with this property even contains networks without a power law degree distribution. Finally, we prove that when a $k$ -complex contagion starts from an arbitrary set of initial seeds on a general graph, determining if the number of infected vertices is above a given threshold is ${\mathbf {P}}$ -complete. Thus, one cannot hope to categorize all the settings in which $k$ -complex contagions percolate in a graph.

Contact processes on random regular graphs

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 lower 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 any $\varepsilon>0$, when the number $n$ of vertices is large then (a) at times $t<(C-\varepsilon)\log n$ the fraction of infected vertices is vanishingly small, but (b) at time $(C+\varepsilon)\log n$ the fraction of infected vertices is within $\varepsilon$ of $p$, with probability $p$.

The transmission process: A combinatorial stochastic process for the evolution of transmission trees over networks

We derive a combinatorial stochastic process for the evolution of the transmission tree over the infected vertices of a host contact network in a susceptible-infected (SI) model of an epidemic. Models of transmission trees are crucial to understanding the evolution of pathogen populations. We provide an explicit description of the transmission process on the product state space of (rooted planar ranked labelled) binary transmission trees and labelled host contact networks with SI-tags as a discrete-state continuous-time Markov chain. We give the exact probability of any transmission tree when the host contact network is a complete, star or path network – three illustrative examples. We then develop a biparametric Beta-splitting model that directly generates transmission trees with exact probabilities as a function of the model parameters, but without explicitly modelling the underlying contact network, and show that for specific values of the parameters we can recover the exact probabilities for our three example networks through the Markov chain construction that explicitly models the underlying contact network. We use the maximum likelihood estimator (MLE) to consistently infer the two parameters driving the transmission process based on observations of the transmission trees and use the exact MLE to characterize equivalence classes over the space of contact networks with a single initial infection. An exploratory simulation study of the MLEs from transmission trees sampled from three other deterministic and four random families of classical contact networks is conducted to shed light on the relation between the MLEs of these families with some implications for statistical inference along with pointers to further extensions of our models. The insights developed here are also applicable to the simplest models of “meme” evolution in online social media networks through transmission events that can be distilled from observable actions such as “likes”, “mentions”, “retweets” and “+1s” along with any concomitant comments.

Critical value for the contact process with random recovery rates and 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 \lambda_c as the maximum of the infection rate with which the process will die out and define \lambda_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 give an precise lower bound of \lambda_e. We also study the critical value under the quenched measure. We show that this critical value equals that under the annealed measure or infinity according to a dichotomy criterion. The contact process on a Galton-Watson tree with binomial offspring distribution is a special case of our model.

First Passage Percolation on the Newman–Watts Small World Model

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 $\rho/n$ for some $\rho>0$ 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 $\frac1\lambda \log n$ for a $\lambda>0$ 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 shortest weight path follows a Central Limit Theorem, and show that in a corresponding epidemic spread model the fraction of infected vertices follows a deterministic curve with a random shift.

Bootstrap percolation in random k-uniform hypergraphs

We investigate bootstrap percolation with infection threshold r>1 on the binomial k-uniform random hypergraph Hk(n,p) in the regime n−1≪nk−2p≪n−1/r, when the initial set of infected vertices is chosen uniformly at random from all sets of given size. We establish a threshold such that if there are less vertices in the initial set of infected vertices, then whp only a few additional vertices become infected, while if the initial set of infected vertices exceeds the threshold then whp almost every vertex becomes infected. In addition, for k=2, we show that the probability of failure decreases exponentially.

On the expected total number of infections for virus spread on a finite network

In this work we consider a simple SIR infection spread model on a finite population of $n$ agents represented by a finite graph $G$. Starting with a fixed set of initial infected vertices the infection spreads in discrete time steps, where each infected vertex tries to infect its neighbors with a fixed probability $\beta\in(0,1)$, independently of others. It is assumed that each infected vertex dies out after an unit time and the process continues till all infected vertices die out. This model was first studied by [Ann. Appl. Probab. 18 (2008) 359–378]. In this work we find a simple lower bound on the expected number of ever infected vertices using breath-first search algorithm and show that it asymptotically performs better for a fairly large class of graphs than the upper bounds obtained in [Ann. Appl. Probab. 18 (2008) 359–378]. As a by product we also derive the asymptotic value of the expected number of the ever infected vertices when the underlying graph is the random $r$-regular graph and $\beta<\frac{1}{r-1}$.

The maximum time of 2-neighbour bootstrap percolation: Algorithmic aspects

In 2-neighbourhood bootstrap percolation on a graph G, an infection spreads according to the following deterministic rule: infected vertices of G remain infected forever and in consecutive rounds healthy vertices with at least 2 already infected neighbours become infected. Percolation occurs if eventually every vertex is infected. In this paper, we are interested to calculate the maximal time t(G) the process can take, in terms of the number of times the interval function is applied, to eventually infect the entire vertex set. We prove that the problem of deciding if t(G)≥k is NP-complete for: (a) fixed k≥4; (b) bipartite graphs and fixed k≥7; and (c) planar graphs. Moreover, we obtain linear and polynomial time algorithms for trees and chordal graphs, respectively.

Multiple transitions of the susceptible-infected-susceptible epidemic model on complex networks.

The epidemic threshold of the susceptible-infected-susceptible (SIS) dynamics on random networks having a power law degree distribution with exponent γ>3 has been investigated using different mean-field approaches, which predict different outcomes. We performed extensive simulations in the quasistationary state for a comparison with these mean-field theories. We observed concomitant multiple transitions in individual networks presenting large gaps in the degree distribution and the obtained multiple epidemic thresholds are well described by different mean-field theories. We observed that the transitions involving thresholds which vanish at the thermodynamic limit involve localized states, in which a vanishing fraction of the network effectively contributes to epidemic activity, whereas an endemic state, with a finite density of infected vertices, occurs at a finite threshold. The multiple transitions are related to the activations of distinct subdomains of the network, which are not directly connected.

The Williams–Bjerknes model on regular trees

We consider the Williams–Bjerknes model, also known as the biased voter model on the $d$-regular tree $\mathbb{T} ^{d}$, where $d\geq3$. Starting from an initial configuration of “healthy” and “infected” vertices, infected vertices infect their neighbors at Poisson rate $\lambda\geq1$, while healthy vertices heal their neighbors at Poisson rate $1$. All vertices act independently. It is well known that starting from a configuration with a positive but finite number of infected vertices, infected vertices will continue to exist at all time with positive probability if and only if $\lambda>1$. We show that there exists a threshold $\lambda_{c}\in(1,\infty)$ such that if $\lambda>\lambda_{c}$ then in the above setting with positive probability, all vertices will become eventually infected forever, while if $\lambda<\lambda_{c}$, all vertices will become eventually healthy with probability $1$. In particular, this yields a complete convergence theorem for the model and its dual, a certain branching coalescing random walk on $\mathbb{T} ^{d}$—above $\lambda_{c}$. We also treat the case of initial configurations chosen according to a distribution which is invariant or ergodic with respect to the group of automorphisms of $\mathbb{T} ^{d}$.

A Sharp Threshold for a Modified Bootstrap Percolation with Recovery

Bootstrap percolation is a type of cellular automaton on graphs, introduced as a simple model of the dynamics of ferromagnetism. Vertices in a graph can be in one of two states: `healthy' or `infected' and from an initial configuration of states, healthy vertices become infected by local rules. While the usual bootstrap processes are monotone in the sets of infected vertices, in this paper, a modification is examined in which infected vertices can return to a healthy state. Vertices are initially infected independently at random and the central question is whether all vertices eventually become infected. The model examined here is such a process on a square grid for which healthy vertices with at least two infected neighbours become infected and infected vertices with no infected neighbours become healthy. Sharp thresholds are given for the critical probability of initial infections for all vertices eventually to become infected.

Bootstrap Percolation in Power-Law Random Graphs

A bootstrap percolation process on a graph $$G$$ is an “infection” process which evolves in rounds. Initially, there is a subset of infected nodes and in each subsequent round each uninfected node which has at least $$r$$ infected neighbours becomes infected and remains so forever. The parameter $$r\ge 2$$ is fixed. Such processes have been used as models for the spread of ideas or trends within a network of individuals. We analyse this process in the case where the underlying graph is an inhomogeneous random graph, which exhibits a power-law degree distribution, and initially there are $$a(n)$$ randomly infected nodes. The main focus of this paper is the number of vertices that will have been infected by the end of the process. The main result of this work is that if the degree sequence of the random graph follows a power law with exponent $$\beta $$ , where $$2 < \beta < 3$$ , then a sublinear number of initially infected vertices is enough to spread the infection over a linear fraction of the nodes of the random graph, with high probability. More specifically, we determine explicitly a critical function $$a_c(n)$$ such that $$a_c(n) = o(n)$$ with the following property. Assuming that $$n$$ is the number of vertices of the underlying random graph, if $$a(n) \ll a_c(n)$$ , then the process does not evolve at all, with high probability as $$n$$ grows, whereas if $$a(n)\gg a_c(n)$$ , then there is a constant $$\varepsilon > 0$$ such that, with high probability, the final set of infected vertices has size at least $$\varepsilon n$$ . This behaviour is in sharp contrast with the case where the underlying graph is a $$G(n, p)$$ random graph with $$p=d/n$$ . It follows from an observation of Balogh and Bollobás that in this case if the number of initially infected vertices is sublinear, then there is lack of evolution of the process. It turns out that when the maximum degree is $$o(n^{1/(\beta - 1)})$$ , then $$a_c(n)$$ depends also on $$r$$ . But when the maximum degree is $$\Theta (n^{1/(\beta - 1)})$$ , then $$a_c (n) = n^{\beta - 2 \over \beta - 1}$$ .

Bootstrap percolation on Galton-Watson trees

Bootstrap percolation is a type of cellular automaton which has been used to model various physical phenomena, such as ferromagnetism. For each natural number $r$, the $r$-neighbour bootstrap process is an update rule for vertices of a graph in one of two states: `infected' or `healthy'. In consecutive rounds, each healthy vertex with at least $r$ infected neighbours becomes itself infected. Percolation is said to occur if every vertex is eventually infected. Usually, the starting set of infected vertices is chosen at random, with all vertices initially infected independently with probability $p$. In that case, given a graph $G$ and infection threshold $r$, a quantity of interest is the critical probability, $p_c(G,r)$, at which percolation becomes likely to occur. In this paper, we look at infinite trees and, answering a problem posed by Balogh, Peres and Pete, we show that for any $b \geq r$ and for any $\epsilon > 0$ there exists a tree $T$ with branching number $\operatorname{br}(T) = b$ and critical probability $p_c(T,r) 0$ such that if $T$ is a Galton- Watson tree with branching number $\operatorname{br}(T) = b \geq r$ then $$p_c(T,r) > \frac{c_r}{b} e^{-\frac{b}{r-1}}.$$ We also show that this bound is sharp up to a factor of $O(b)$ by giving an explicit family of Galton--Watson trees with critical probability bounded from above by $C_r e^{-\frac{b}{r-1}}$ for some constant $C_r>0$.

On Slowly Percolating Sets of Minimal Size in Bootstrap Percolation

Bootstrap percolation, one of the simplest cellular automata, can be seen as a model of the spread of infection. In $r$-neighbour bootstrap percolation on a graph $G$ we assign a state, infected or healthy, to every vertex of $G$ and then update these states in successive rounds, according to the following simple local update rule: infected vertices of $G$ remain infected forever and a healthy vertex becomes infected if it has at least $r$ already infected neighbours. We say that percolation occurs if eventually every vertex of $G$ becomes infected. A well known and celebrated fact about the classical model of $2$-neighbour bootstrap percolation on the $n \times n$ square grid is that the smallest size of an initially infected set which percolates in this process is $n$. In this paper we consider the problem of finding the maximum time a $2$-neighbour bootstrap process on $[n]^2$ with $n$ initially infected vertices can take to eventually infect the entire vertex set. Answering a question posed by Bollobás we compute the exact value for this maximum showing that, for $n \ge 4$, it is equal to the integer nearest to $(5n^2-2n)/8$.

Maximal Percolation Time in Hypercubes Under 2-Bootstrap Percolation

Bootstrap percolation is one of the simplest cellular automata. In $r$-bootstrap percolation on a graph $G$, an infection spreads according to the following deterministic rule: infected vertices of $G$ remain infected forever and in consecutive rounds healthy vertices with at least $r$ already infected neighbours become infected. Percolation occurs if eventually every vertex is infected. In this paper we prove that in the case of $2$-bootstrap percolation on the $n$-dimensional hypercube the maximal time the process can take to eventually infect the entire vertex set is $\lfloor \frac{n^2}{3} \rfloor$.

Epidemic dynamics behavior in some bus transport networks

We abstract bus transport networks (BTNs) to complex networks using the Space P approach. First, we select three actual BTNs in three major cities in China, namely, Beijing, Shanghai and Hangzhou. Using the SIS model, we simulate and study the epidemic spreading in the three BTNs. We obtain the density of infected vertices varying with time and the stationary density of infected vertices varying with infection rate. Second, we simulate and study the epidemic spreading in a recently introduced BTN evolution model, the network properties of which correspond well with those of actual BTNs. Third, we use mean-field theory to analyze the epidemic dynamics behavior of the BTN evolution model and obtain the theoretical epidemic threshold of this model. The theoretical value agrees well with the simulation results. Based on the work in this paper, we provide the following possible forecasts for epidemic dynamics in actual BTNs. An actual BTN should have a finite positive epidemic threshold. If the effective infection rate is above this threshold, the epidemic spread in the network and the density of infected vertices finally stabilizes in a balanced state. Below this threshold, the number of infected vertices decays exponentially fast and the epidemic cannot spread on a large scale.

Bootstrap Percolation in High Dimensions

In r-neighbour bootstrap percolation on a graph G, a set of initially infected vertices A ⊂ V(G) is chosen independently at random, with density p, and new vertices are subsequently infected if they have at least r infected neighbours. The set A is said to percolate if eventually all vertices are infected. Our aim is to understand this process on the grid, [n]d, for arbitrary functions n = n(t), d = d(t) and r = r(t), as t → ∞. The main question is to determine the critical probability pc([n]d, r) at which percolation becomes likely, and to give bounds on the size of the critical window. In this paper we study this problem when r = 2, for all functions n and d satisfying d ≫ log n.The bootstrap process has been extensively studied on [n]d when d is a fixed constant and 2 ⩽ r ⩽ d, and in these cases pc([n]d, r) has recently been determined up to a factor of 1 + o(1) as n → ∞. At the other end of the scale, Balogh and Bollobás determined pc([2]d, 2) up to a constant factor, and Balogh, Bollobás and Morris determined pc([n]d, d) asymptotically if d ≥ (log log n)2+ϵ, and gave much sharper bounds for the hypercube.Here we prove the following result. Let λ be the smallest positive root of the equation so λ ≈ 1.166. Then if d is sufficiently large, and moreover as d → ∞, for every function n = n(d) with d ≫ log n.