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.Comment: 14 pages, 5 figures, corrected some misprint

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.Comment: 9 page

A mathematical model for the atomic clock error in case of jumps

We extend the mathematical model based on stochastic differential equations describing the error gained by an atomic clock to the cases of anomalous behavior including jumps and an increase of instability. We prove an exact iterative solution that can be useful for clock simulation, prediction, and interpretation, as well as for the understanding of the impact of clock error in the overall system in which clocks may be inserted as, for example, the Global Satellite Navigation Systems

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.Comment: 19 pages. The paper has been deeply reorganized and two pictures have been added. arXiv admin note: substantial text overlap with arXiv:1104.508

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.Comment: 28 page

Branching random walks and multi-type contact-processes on the percolation cluster of Zd{\mathbb{Z}}^{d}

In this paper we prove that, under the assumption of quasi-transitivity, if a branching random walk on ${{\mathbb{Z}}^d}$ survives locally (at arbitrarily large times there are individuals alive at the origin), then so does the same process when restricted to the infinite percolation cluster ${{\mathcal{C}}_{\infty}}$ of a supercritical Bernoulli percolation. When no more than $k$ individuals per site are allowed, we obtain the $k$-type contact process, which can be derived from the branching random walk by killing all particles that are born at a site where already $k$ individuals are present. We prove that local survival of the branching random walk on ${{\mathbb{Z}}^d}$ also implies that for $k$ sufficiently large the associated $k$-type contact process survives on ${{\mathcal{C}}_{\infty}}$. This implies that the strong critical parameters of the branching random walk on ${{\mathbb{Z}}^d}$ and on ${{\mathcal{C}}_{\infty}}$ coincide and that their common value is the limit of the sequence of strong critical parameters of the associated $k$-type contact processes. These results are extended to a family of restrained branching random walks, that is, branching random walks where the success of the reproduction trials decreases with the size of the population in the target site.Comment: Published at http://dx.doi.org/10.1214/14-AAP1040 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org