Search

We consider Galton–Watson branching processes with countable typeset X. We study the vectors (Formula Presented) recording the conditional probabilities of extinction in subsets of types (Formula Presented), given that the type of the initial individual is x. We first investigate the location of the vectors q(A) in the set of fixed points of the progeny generating vector and prove that (Formula Presented) is larger than or equal to the xth entry of any fixed point, whenever it is different from 1. Next, we present equivalent conditions for (Formula Presented) for any initial type x and (Formula Presented). Finally, we develop a general framework to characterise all distinct extinction probability vectors, and thereby to determine whether there are finitely many, countably many, or uncountably many distinct vectors. We illustrate our results with examples, and conclude with open questions.

The Caregiver Skills Training programme is an evidence-informed group intervention developed by the World Health Organization to counter the treatment gap for neurodevelopmental disorders in children. We examined feasibility and acceptability of Caregiver Skills Training in public child neuropsychiatry services in Italy. Following a formative adaptation process and a pre-pilot implementation, the programme was piloted in six centres. Caregivers of children (2–5 years) with autism spectrum disorder were randomised to either Caregiver Skills Training (n = 43) or treatment as usual (n = 43). Professionals’ and caregivers’ experiences in the treatment arm were examined with focus groups and questionnaires. The programme was largely considered acceptable, relevant and feasible. Specific implementation challenges and suggestions for optimising the package are discussed. Lay abstract: Across the globe, children with neurodevelopmental disorders, including autism, have limited access to care through public services. To improve access to care the World Health Organization developed a novel, open-access programme: the Caregiver Skills Training programme. The Caregiver Skills Training consists of nine group sessions and three individual home visits, focused on training the caregiver on how to use everyday play and home routines as opportunities for learning and development. We implemented Caregiver Skills Training in public child neuropsychiatry services in Italy and examined with questionnaires and focus groups how feasible it was to deliver the programme in public health settings and how acceptable and relevant it was for caregiver users. We found that the Caregiver Skills Training was largely considered acceptable and relevant for families and feasible to deliver. We discuss potential solutions to address the specific implementation challenges that were identified, such as strategies to improve training of interventionists and overcome barriers to caregiver participation.

This paper deals with branching processes in varying environment with selection, where the offspring distribution depends on the generation and every particle has a random fitness which can only increase along genealog-ical lineages (descendants with small fitness do not survive). We view the branching process in varying environment (BPVE) as a particular example of branching random walk. We obtain conditions for the survival or extinction of a BPVE (with or without selection), using fixed point techniques for branching random walks. These conditions rely only on the first and second moments of the offspring distributions. Our results can be interpreted in terms of accessibility percolation on Galton-Watson trees. In particular, we obtain that there is no accessibility percolation on almost every Galton-Watson tree where the expected number of offspring grows sublinearly in time, while su-perlinear growths allows percolation. This result is in agreement with what was found for deterministic trees in Nowak and Krug (Europhysics Letters 101 (2013) 66004).

The current work deals with an epidemic model on the complete graph Kn on n vertices in a non-homogeneous setting, where the vertices may have distinct types. Different types differ in the probability of getting infected, and/or in the capacity of infecting other vertices. This generalizes the model in [4]. We prove in Theorem 5.1 and Theorem 6.3 laws of large numbers and central limit theorems for the the total duration of the process and for the number of infected vertices, respectively, when n tends to infinity. By coupling the epidemic model with a Poisson process, we also obtain continuous-time counterparts of the above-mentioned limit results. Moreover, we also prove that when all individuals have the same spread capacity, then a population with inhomogeneous susceptibility is less affected by the epidemics than a homogeneous population.

We investigated the mechanism of impact of poor communication skills and emotional and behavioural problems in children with ASD (22–61 months) on parental psychological distress. Participants were dyads enrolled in two pilot intervention studies; the dataset includes cross-sectional data at baseline (N = 82). We postulated an indirect effect of child expressive and receptive communication on parent psychological distress, through child emotional and behavioural problems. The effect of receptive skills on parent psychological distress was fully mediated by child emotional problems: lower receptive skills were associated with higher levels of emotional symptoms, which in turn predicted higher parent psychological distress. Expressive skills did not show either direct or indirect effects. Findings are discussed in light of children’s marked ‘receptive disadvantage’ communication profile.

Given a branching random walk on a set X, we study its extinction probability vectors q(., A). Their components are the probability that the process goes extinct in a fixed A subset of X, when starting from a vertex x is an element of X. The set of extinction probability vectors (obtained letting A vary among all subsets of X) is a subset of the set of the fixed points of the generating function of the branching random walk. In particular here we are interested in the cardinality of the set of extinction probability vectors. We prove results which allow to understand whether the probability of extinction in a set A is different from the one of extinction in another set B. In many cases there are only two possible extinction probability vectors and so far, in more complicated examples, only a finite number of distinct extinction probability vectors had been explicitly found. Whether a branching random walk could have an infinite number of distinct extinction probability vectors was not known. We apply our results to construct examples of branching random walks with uncountably many distinct extinction probability vectors.

It is well known that the behaviour of a branching process is completely described by the generating function of the offspring law and its fixed points. Branching random walks are a natural generalization of branching processes: a branching process can be seen as a one-dimensional branching random walk. We define a multidimensional generating function associated to a given branching random walk. The present paper investigates the similarities and the differences of the generating functions, their fixed points and the implications on the underlying stochastic process, between the one-dimensional (branching process) and the multidimensional case (branching random walk). In particular, we show that the generating function of a branching random walk can have uncountably many fixed points and a fixed point may not be an extinction probability, even in the irreducible case (extinction probabilities are always fixed points). Moreover, the generating function might not be a convex function. We also study how the behaviour of a branching random walk is affected by local modifications of the process. As a corollary, we describe a general procedure with which we can modify a continuous-time branching random walk which has a weak phase and turn it into a continuous-time branching random walk which has strong local survival for large or small values of the parameter and non-strong local survival for intermediate values of the parameter.

We generalize the evolution model introduced by Guiol, Machado and Schinazi (2010). In our model at odd times a random number X of species is created. Each species is endowed with a random fitness with arbitrary distribution on [0, 1]. At even times a random number Y of species is removed, killing the species with lower fitness. We show that there is a critical fitness f c below which the number of species hits zero i.o. and above of which this number goes to infinity. We prove uniform convergence for the fitness distribution of surviving species and describe the phenomena which could not be observed in previous works with uniformly distributed fitness.

In this paper we prove that, under the assumption of quasi-transitivity, if a branching random walk on ℤd 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 C∞ 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 ℤd also implies that for k sufficiently large the associated k-type contact process survives on C∞. This implies that the strong critical parameters of the branching random walk on ℤd and on C∞ 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.

In this paper we prove that, under the assumption of quasi-transitivity, if a branching random walk on Zd 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 C∞ 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 Zd also implies that for k sufficiently large the associated k-type contact process survives on C∞. This implies that the strong critical parameters of the branching random walk on Zd and on C∞ 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.

In this paper we study 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 for 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 quasitransitive). 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 nonstrong local survival. Finally, we show that the generating function of an irreducible branching random walk can have more than two fixed points; this disproves a previously known result