We study a discrete Susceptible-Infected-Recovered (SIR) model for the spread of infectious disease on a homogeneous tree and the limit behavior of the model in the case when the tree vertex degree tends to infinity. We obtain the distribution of the time it takes for a susceptible vertex to get infected in terms of a solution of a non-linear integral equation under broad assumptions on the model parameters. Namely, infection rates are assumed to be time-dependent, and recovery times are given by random variables with a fairly arbitrary distribution. We then study the behavior of the model in the limit when the tree vertex degree tends to infinity, and infection rates are appropriately scaled. We show that in this limit the integral equation of the discrete model implies an equation for the susceptible population compartment. This is a master equation in the sense that both the infectious and the recovered compartments can be explicitly expressed in terms of its solution. In other words, the master equation implies a continuous SIR model for the joint time evolution of all three population compartments.
To better describe the spread of a disease, we extend a discrete time stochastic SIR-type epidemic model of Tuckwell and Williams. We assume the dependence on time of the number of daily encounters and include a parameter to represent a possible quarantine of the infectious individuals. We provide an analytic description of this Markovian model and investigate its dynamics. Both a diffusion approximation and the basic reproduction number are derived. Through several simulations, we show how the evolution of a disease is affected by the distribution of the number of daily encounters and its dep…
We consider Markovian susceptible-infectious-removed (SIR) dynamics on time-invariant weighted contact networks where the infection and removal processes are Poisson and where network links may be directed or undirected. We prove that a particular pair-based moment closure representation generates the expected infectious time series for networks with no cycles in the underlying graph. Moreover, this ``deterministic'' representation of the expected behaviour of a complex heterogeneous and finite Markovian system is straightforward to evaluate numerically.
The stochastic SIRS model is a continuous-time Markov chain modelling the spread of infectious diseases with temporary immunity, in a homogeneously-mixing population of fixed size $N$. We study the scaling behaviour of the extinction time of stochastic SIRS models as $N$ tends to infinity. When the initial size of infected population is small, we obtain the closed-form expression of the asymptotic distribution of this extinction time, and compare it with the data from Monte Carlo simulation.
We consider a very general stochastic model for an SIR epidemic on a network which allows an individual's infectious period, and the time it takes to contact each of its neighbours after becoming infected, to be correlated. We write down the message passing system of equations for this model and prove, for the first time, that it has a unique feasible solution. We also generalise an earlier result by proving that this solution provides a rigorous upper bound for the expected epidemic size (cumulative number of infection events) at any fixed time $t>0$. We specialise these results to a homogene…
In a previous paper Sharkey et al. [13] proved the exactness of closures at the level of triples for Markovian SIR (susceptible-infected-removed) dynamics on tree-like networks. This resulted in a deterministic representation of the epidemic dynamics on the network that can be numerically evaluated. In this paper, we extend this modelling framework to certain classes of networks exhibiting loops. We show that closures where the loops are kept intact are exact, and lead to a simplified and numerically solvable system of ODEs (ordinary-differential-equations). The findings of the paper lead us t…
We prove the kernel estimates related to subordinated semigroups on homogeneous trees. We study the long time propagation problem. We exploit this to show exit time estimates for (large) balls. We use an abstract setting of metric measure spaces. This enables us to give these results for trees end hyperbolic spaces as well. Finally, we show some estimates for the Poisson kernel of a ball.
In this paper, we consider the threshold-one contact process and the threshold-one voter model w/o spontaneous death on homogeneous trees $\mathbb{T}_d$, $d\ge 2$. Mainly inspired by the corresponding arguments for ordinary contact processes, we prove that the complete convergence theorem holds for these three systems under strong survival. When the systems survives weakly, complete convergence may also hold under certain transition and/or initial conditions.
We consider the interacting particle system on the homogeneous tree of degree $(d + 1)$, known as frog model. In this model, active particles perform independent random walks, awakening all sleeping particles they encounter, and dying after a random number of jumps, with geometric distribution. We prove an upper bound for the critical parameter of survival of the model, which improves the previously known results. This upper bound was conjectured in a paper by Lebensztayn et al. ($ J. Stat. Phys.$, 119(1-2), 331-345, 2005). We also give a closed formula for the upper bound.
We consider the frog model with geometric lifetime (parameter $1-p$) on homogeneous trees of dimension $d$. In 2002, \cite{alves2002-2} proved that there exists a critical lifetime parameter $p_c\in(0,1)$ above which infinitely many frogs are activated with positive probability, and they gave lower and upper bounds for $p_c$. Since then, the literature on this model focussed on refinements of the upper bound. In the present paper we improve the bounds for $p_c$ \emph{on both sides}. We also provide a discussion comparing the bounds of the literature and their proofs. Our proofs are based on co…
This paper belongs to a series of papers aiming to investigate scaling limits of Crump-Mode-Jagers (CMJ) trees. In the previous two papers we identified general conditions under which CMJ trees belong to the universality class of Galton-Watson and Bellman-Harris processes. In this paper we identify general conditions for CMJ trees to belong to the universality class of binary, homogeneous CMJ trees. These conditions state that the offspring process should 'look like' a renewal process, and also that it should not accumulate too many atoms near the origin. We show in particular that any renewal…
This paper is concerned with the continuous-time quantum walk on Z, Z^d, and infinite homogeneous trees. By using the generating function method, we compute the limit of the average probability distribution for the general isotropic walk on Z, and for nearest-neighbor walks on Z^d and infinite homogeneous trees. In addition, we compute the asymptotic approximation for the probability of the return to zero at time t in all these cases.
We study a rumour model from a percolation theory and branching process point of view. The existence of a giant component is related to the event where the rumour, which started from the root of a tree, spreads out through an infinite number of its vertices. We present lower and upper bounds for the probability of that event, according to the distribution of the random variables that defines the radius of influence of each individual. We work with Galton-Watson branching trees (homogeneous and non-homogeneous) and spherically symmetric trees which includes homogeneous and $k-$periodic trees.
There are two types of particles interacting on a homogeneous tree of degree d + 1. The particles of the first type colonize the empty space with exponential rate 1, but cannot take over the vertices that are occupied by the second type. The particles of the second type spread with exponential rate λ. They colonize the neighboring vertices that are either vacant or occupied by the representatives of the opposite type, and annihilate the particles of the type 1 as they reach them. There exists a critical value λ_c =(2d - 1) + \sqrt{(2d -1)^2 -1} such that the first type survives with positive p…
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>…
We investigate a basic immigration process where colonies grow, during a random time, according to a general counting process until collapse. Upon collapse a random amount of individuals survive. These survivors try independently establishing new colonies at neighbour sites. Here we consider this general process subject to two schemes, Poisson growth with geometric catastrophe and Yule growth with binomial catastrophe. Independent of everything else colonies growth, during an exponential time, as a Poisson (or Yule) process and right after that exponential time their size is reduced according …
The constant rate birth--death process is a popular null model for speciation and extinction. If one removes extinct and non-sampled lineages, this process induces `reconstructed trees' which describe the relationship between extant lineages. We derive the probability density of the length of a randomly chosen pendant edge in a reconstructed tree. For the special case of a pure-birth process with complete sampling, we also provide the probability density of the length of an interior edge, of the length of an edge descending from the root, and of the diversity (which is the sum of all edge leng…
We study the threshold $theta geq 2$ contact process on a homogeneous tree $T_b$ of degree $kappa = b + 1$, with infection parameter $lambda geq 0$ and started from a product measure with density $p$. The corresponding mean-field model displays a discontinuous transition at a critical point $lambda_c^{MF}(kappa,theta)$ and for $lambda geq lambda_c^{MF}(kappa,theta)$ it survives iff $p geq p_c^{MF}(kappa,theta,lambda)$, where this critical density satisfies $0 < p_c^{MF}(kappa,theta,lambda) < 1$, $lim_{lambda to infty} p_c^{MF}(kappa,theta,lambda) = 0$. For large $b$, we show that the process o…
The existence of a weak survival region is established for the anisotropic symmetric contact process on a homogeneous tree T_{2d} of degree 2d > 2: For parameter values in a certain connected region of positive Lebesgue measure, the population survives forever with positive probability but ultimately vacates every finite subset of the tree with probability one. In this phase, infection trails must converge to the geometric boundary Ωof the tree. The random subset Λof the boundary consisting of all ends of the tree in which the infection survives, called the limit set of the process, is shown t…
Asymptotics for Beta-Splitting Trees via Homogeneous Fragmentations and Meromorphic Potential Theory
Faculty of Mathematics and Informatics, Sofia University “St. Kliment Ohridski”, Bulgaria（索非亚大学“圣克利门特奥赫里德斯基”数学与信息学院）；Institute of Mathematics, University of Zurich, Switzerland（苏黎世大学数学研究所）；Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Bulgaria（保加利亚科学院数学与信息研究所）
We provide information about the asymptotic regimes for a homogeneous fragmentation of a finite set. We establish a phase transition for the asymptotic behaviours of the shattering times, defined as the first instants when all the blocks of the partition process have cardinality less than a fixed integer. Our results may be applied to the study of certain random split trees.
Let ${\cal G}$ be the incipient infinite cluster (IIC) for percolation on a homogeneous tree of degree $n_0+1$. We obtain estimates for the transition density of the continuous time simple random walk $Y$ on ${\cal G}$; the process satisfies anomalous diffusion and has spectral dimension 4/3.
