arXiv subjects

Fabio Zucca

Publications and source records attributed to Fabio Zucca.

At least 37 records · Page 2Linked to original sources

arXiv2013

Rumor processes in random environment on N and on Galton-Watson trees

The aim of this paper is to study rumor processes in random environment. In a rumor process a signal starts from the stations of a fixed vertex (the root) and travels on a graph from vertex to vertex. We consider two rumor processes. In the firework process each station, when reached by the signal, transmits it up to a random distance. In the reverse firework process, on the other hand, stations do not send any signal but they "listen" for it up to a random distance. The first random environment that we consider is the deterministic 1-dimensional tree N with a random number of stations on each vertex; in this case the root is the origin of N. We give conditions for the survival/extinction on almost every realization of the sequence of stations. Later on, we study the processes on Galton-Watson trees with random number of stations on each vertex. We show that if the probability of survival is positive, then there is survival on almost every realization of the infinite tree such that there is at least one station at the root. We characterize the survival of the process in some cases and we give sufficient conditions for survival/extinction.

math.PR↗

arXiv2013

Strong local survival of branching random walks is not monotone

The aim of this paper is the study of the strong local survival property for discrete-time and continuous-time branching random walks. We study this property by means of an infinite dimensional generating function G and a maximum principle which, we prove, is satisfied by every fixed point of G. We give results about the existence of a strong local survival regime and we prove that, unlike local and global survival, in continuous time, strong local survival is not a monotone property in the general case (though it is monotone if the branching random walk is quasi transitive). We provide an example of an irreducible branching random walk where the strong local property depends on the starting site of the process. By means of other counterexamples we show that the existence of a pure global phase is not equivalent to nonamenability of the process, and that even an irreducible branching random walk with the same branching law at each site may exhibit non-strong local survival. Finally we show that the generating function of a irreducible BRW can have more than two fixed points; this disproves a previously known result.

math.PR↗

arXiv2013

Combination versus sequential monotherapy in chronic HBV infection: a mathematical approach

Sequential monotherapy is the most widely used therapeutic approach in the treatment of HBV chronic infection. Unfortunately, under therapy, in some patients the hepatitis virus mutates and gives rise to variants which are drug resistant. We conjecture that combination therapy is able to delay drug resistance for a longer time than sequential monotherapy. To study the action of these two therapeutic approaches in the event of unknown mutations and to explain the emergence of drug resistance, we propose a stochastic model for the infection within a patient which is treated with two drugs, either sequentially or contemporaneously, and develops a two-step mutation which is resistant to both drugs. We study the deterministic approximation of our stochastic model and give a biological interpretation of its asymptotic behaviour. We compare the time when this new strain first reaches detectability in the serum viral load. Our results show that the best choice is to start an early combination therapy, which allows to stay drug-resistance free for a longer time.

math.PR↗

arXiv2013

Persistent and susceptible bacteria with individual deaths

The aim of this paper is to study two models for a bacterial population subject to antibiotic treatments. It is known that some bacteria are sensitive to antibiotics. These bacteria are in a state called persistence and each bacterium can switch from this state to a non-persistent (or susceptible) state and back. Our models extend those introduced in [6] by adding a (random) natural life cycle for each bacterium and by allowing bacteria in the susceptible state to escape the action of the antibiotics with a fixed probability 1-p (while every bacterium in a persistent state survives with probability 1). In the first model we "inject" the antibiotics in the system at fixed, deterministic times while in the second one the time intervals are random. We show that, in order to kill eventually the whole bacterial population, these time intervals cannot be "too large". The maximum admissible length is increasing with respect to p and it decreases rapidly when p<1.

math.PR↗

arXiv2012

Local and global survival for nonhomogeneous random walk systems on Z

We study an interacting random walk system on Z where at time 0 there is an active particle at 0 and one inactive particle on each site $n \ge 1$. Particles become active when hit by another active particle. Once activated, the particle starting at n performs an asymmetric, translation invariant, nearest neighbor random walk with left jump probability $l_n$. We give conditions for global survival, local survival and infinite activation both in the case where all particles are immortal and in the case where particles have geometrically distributed lifespan (with parameter depending on the starting location of the particle). More precisely, once activated, the particle at n survives at each step with probability $p_n \in [0, 1]$. In particular, in the immortal case, we prove a 0-1 law for the probability of local survival when all particles drift to the right. Besides that, we give sufficient conditions for local survival or local extinction when all particles drift to the left. In the mortal case, we provide sufficient conditions for global survival, local survival and local extinction (which apply to the immortal case with mixed drifts as well). Analysis of explicit examples is provided: we describe completely the phase diagram in the cases $1/2-l_n \sim \pm 1/n^α$, $p_n = 1$ and $1_n-1/2 \sim \pm 1/n^α$, $1-p_n \sim 1/n^β$ (where $α,β> 0$).

math.PR↗

arXiv2012

A self-regulating and patch subdivided population

We consider an interacting particle process on a graph which, from a macroscopic point of view, looks like $\Z^d$ and, at a microscopic level, is a complete graph of degree $N$ (called a patch). There are two birth rates: an inter-patch one $λ$ and an intra-patch one $ϕ$. Once a site is occupied, there is no breeding from outside the patch and the probability $c(i)$ of success of an intra-patch breeding decreases with the size $i$ of the population in the site. We prove the existence of a critical value $λ_{cr}(ϕ, c, N)$ and a critical value $ϕ_{cr}(λ, c, N)$. We consider a sequence of processes generated by the families of control functions $\{c_i\}_{i \in \N}$ and degrees $\{N_i\}_{i \in \N}$; we prove, under mild assumptions, the existence of a critical value $i_{cr}$. Roughly speaking we show that, in the limit, these processes behave as the branching random walk on $\Z^d$ with external birth rate $λ$ and internal birth rate $ϕ$. Some examples of models that can be seen as particular cases are given.

math.PR↗

arXiv2011

Survival, extinction and approximation of discrete-time branching random walks

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 finite) state space is large enough. An analogous result can be achieved by approximating a branching random walk by a sequence of multitype contact processes and allowing a sufficiently large number of particles per site. We compare these results with the ones obtained in the continuous-time case and we give some examples and counterexamples.

math.PR↗

arXiv2010

Moving closer: contractive maps on discrete metric spaces and graphs

We consider discrete metric spaces and we look for non-constant contractions. We introduce the notion of contractive map and we characterize the spaces with non-constant contractive maps. We provide some examples to discussion the possible relations between contractions, contractive maps and constant functions. Finally we apply the main result to the subgraphs of a non-oriented, connected graph.

math.CA↗

arXiv2008

Characterization of the critical values of branching random walks on weighted graphs through infinite-type branching processes

We study the branching random walk on weighted graphs; site-breeding and edge-breeding branching random walks on graphs are seen as particular cases. We describe the strong critical value in terms of a geometrical parameter of the graph. We characterize the weak critical value and relate it to another geometrical parameter. We prove that, at the strong critical value, the process dies out locally almost surely; while, at the weak critical value, global survival and global extinction are both possible.

math.PR↗

arXiv2008

Approximating critical parameters of branching random walks

Given a branching random walk on a graph, we consider two kinds of truncations: by inhibiting the reproduction outside a subset of vertices and by allowing at most $m$ particles per site. We investigate the convergence of weak and strong critical parameters of these truncated branching random walks to the analogous parameters of the original branching random walk. As a corollary, we apply our results to the study of the strong critical parameter of a branching random walk restricted to the cluster of a Bernoulli bond percolation.

math.PR↗

arXiv2007

An inequality for correlated measurable functions

A classical inequality, which is known for families of monotone functions, is generalized to a larger class of families of measurable functions. Moreover we characterize all the families of functions for which the equality holds. We apply this result to a problem arising from probability theory.

math.CA↗

arXiv2007

Ecological equilibrium for restrained branching random walks

We study a generalized branching random walk where particles breed at a rate which depends on the number of neighboring particles. Under general assumptions on the breeding rates we prove the existence of a phase where the population survives without exploding. We construct a nontrivial invariant measure for this case.

math.PR↗

arXiv2007

Weak survival for branching random walks on graphs

We study weak and strong survival for branching random walks on multigraphs. We prove that, for a large class of multigraphs, weak survival is related to a geometrical parameter of the multigraph and that the existence of a pure weak phase is equivalent to nonamenability. Finally we study weak and strong critical behaviors of the branching random walk.

math.PR↗

arXiv2004

A note on compact Markov operators

The analytic properties of the Markov operator associated to a random walk are common tools in the study of the behaviour and some probabilistic features related to the walk. In this paper we consider a class of Markov operators which generalizes the class of compact Markov operators and we study some probabilistic properties of the associated random walk.

math.PR↗

arXiv2002

Classification on the average of random walks

We introduce a new method for studying large scale properties of random walks. The new concepts of transience and recurrence on the average are compared with the ones introduced by R.Burioni, D.Cassi and A.Vezzani and with the usual ones; their relationships are analyzed and various examples are provided.

math.PR↗

arXiv2001

Strong and weak mean value properties on trees

We consider the mean value properties for finite variation measures with respect to a Markov operator in a discrete environnement. We prove equivalent conditions for the weak mean value property in the case of general Markov operators and for the strong mean value property in the case of transient Markov operators adapted to a tree structure. In this last case, conditions for the equivalence between weak and strong mean value properties are given.

math.PR↗

arXiv2001

On some properties of transitions operators

We study a general transition operator, generated by a random walk on a graph $X$; in particular we give necessary and sufficient condition on the matrix coefficient (1-step transition probablilities) to be a bounded operator from $l^\infty(X)$ into itself. Moreover we characterize compact operators and we relate this property to the behaviour of the associated random walk. We give a necessary and sufficient condition for the pre-adjoint of the discrete Laplace operator to be an injective map.

math.PR↗

arXiv2000

Uniform asymptotic estimates of transition probabilities on combs

We investigate the asymptotical behaviour of the transition probabilities of the simple random walk on the 2-comb. In particular we obtain space-time uniform asymptotical estimates which show the lack of symmetry of this walk better than local limit estimates. Our results also point out the impossibility of getting Jones-type non-Gaussian estimates.

math.PR↗