A continuous-state branching process with immigration having branching mechanism $\Psi$ and immigration mechanism $\Phi$, a CBI$(\Psi,\Phi)$ process for short, may have either of two different asymptotic regimes, depending on whether $\int_{0}\frac{\Phi(u)}{|\Psi(u)|}\textrm{d} u<\infty$ or $\int_{0}\frac{\Phi(u)}{|\Psi(u)|}\textrm{d} u=\infty$. When $\int_{0}\frac{\Phi(u)}{|\Psi(u)|}\textrm{d} u<\infty$, the CBI process has either a limit distribution or a growth rate dictated by the branching dynamics. When $\scriptstyle\int_{0}\tfrac{\Phi(u)}{|\Psi(u)|}\textrm{d} u=\infty$, immigration overwhelms branching dynamics. Asymptotics in the latter case are studied via a nonlinear time-dependent renormalization in law. Three regimes of weak convergence are exhibited. Processes with critical branching mechanisms subject to a regular variation assumption are studied. This article proves and extends results stated by M. Pinsky in ‘Limit theorems for continuous state branching processes with immigration’ (Bull. Amer. Math. Soc. 78, 1972).

We correct typographical errors in our original paper.

The generalized perturbative approach is an all-purpose variant of Stein’s method used to obtain rates of normal approximation. Originally developed for functions of independent random variables, this method is here extended to functions of the realization of a hidden Markov model. In this dependent setting, rates of convergence are provided in some applications, leading, in each instance, to an extra log-factor vis-à-vis the rate in the independent case.

We solve non-Markovian optimal switching problems in discrete time on an infinite horizon, when the decision-maker is risk-aware and the filtration is general, and establish existence and uniqueness of solutions for the associated reflected backward stochastic difference equations. An example application to hydropower planning is provided.

This study deals with the ruin problem when an insurance company having two business branches, life insurance and non-life insurance, invests its reserves in a risky asset with the price dynamics given by a geometric Brownian motion. We prove a result on the smoothness of the ruin probability as a function of the initial capital, and obtain for it an integro-differential equation understood in the classical sense. For the case of exponentially distributed jumps we show that the survival (as well as the ruin) probability is a solution of an ordinary differential equation of the fourth order. Asymptotic analysis of the latter leads to the conclusion that the ruin probability decays to zero in the same way as in the already studied cases of models with one-sided jumps.

The increasing number of diversified small-scale farms (DSSF) that raise outdoor-based livestock in the USA reflects growing consumer demand for sustainably produced food. Diversified farms are small scale and raise a combination of multiple livestock species and numerous produce varieties. This 2015–2016 cross-sectional study aimed to describe the unique characteristics of DSSF in California, estimate the prevalence of Shiga toxin-producing Escherichia coli (STEC) in livestock and evaluate the association between risk factors and the presence of STEC in livestock, using generalised linear mixed models. STEC prevalence was 13.62% (76/558). Significant variables in the mixed-effect logistic regression model included daily maximum temperature (OR 0.95; CI95% 0.91–0.98), livestock sample source (cattle (OR 4.61; CI95% 1.64–12.96) and sheep (OR 5.29; CI95% 1.80–15.51)), multiple species sharing the same barn (OR 6.23; CI95% 1.84–21.15) and livestock having contact with wild areas (OR 3.63; CI95% 1.37–9.62). Identification of STEC serogroups of public health concern (e.g. O157:H7, O26, O103) in this study indicated the need for mitigation strategies to ensure food safety by evaluating risk factors and management practices that contribute to the spread and prevalence of foodborne pathogens in a pre-harvest environment on DSSF.

In this paper, we construct operator fractional Lévy motion (ofLm), a broad class of infinitely divisible stochastic processes that are covariance operator self-similar and have wide-sense stationary increments. The ofLm class generalizes the univariate fractional Lévy motion as well as the multivariate operator fractional Brownian motion (ofBm). OfLm can be divided into two types, namely, moving average (maofLm) and real harmonizable (rhofLm), both of which share the covariance structure of ofBm under assumptions. We show that maofLm and rhofLm admit stochastic integral representations in the time and Fourier domains, and establish their distinct small- and large-scale limiting behavior. We also characterize time-reversibility for ofLm through parametric conditions related to its Lévy measure. In particular, we show that, under non-Gaussianity, the parametric conditions for time-reversibility are generally more restrictive than those for the Gaussian case (ofBm).

We study an N-player game where a pure action of each player is to select a nonnegative function on a Polish space supporting a finite diffuse measure, subject to a finite constraint on the integral of the function. This function is used to define the intensity of a Poisson point process on the Polish space. The processes are independent over the players, and the value to a player is the measure of the union of her open Voronoi cells in the superposition point process. Under randomized strategies, the process of points of a player is thus a Cox process, and the nature of competition between the players is akin to that in Hotelling competition games. We characterize when such a game admits Nash equilibria and prove that when a Nash equilibrium exists, it is unique and consists of pure strategies that are proportional in the same proportions as the total intensities. We give examples of such games where Nash equilibria do not exist. A better understanding of the criterion for the existence of Nash equilibria remains an intriguing open problem.

A moment constraint that limits the number of dividends in an optimal dividend problem is suggested. This leads to a new type of time-inconsistent stochastic impulse control problem. First, the optimal solution in the precommitment sense is derived. Second, the problem is formulated as an intrapersonal sequential dynamic game in line with Strotz’s consistent planning. In particular, the notions of pure dividend strategies and a (strong) subgame-perfect Nash equilibrium are adapted. An equilibrium is derived using a smooth fit condition. The equilibrium is shown to be strong. The uncontrolled state process is a fairly general diffusion.

We provide constructions of age-structured branching processes without or with immigration as pathwise-unique solutions to stochastic integral equations. A necessary and sufficient condition for the ergodicity of the model with immigration is also given.

We make the first steps towards generalising the theory of stochastic block models, in the sparse regime, towards a model where the discrete community structure is replaced by an underlying geometry. We consider a geometric random graph over a homogeneous metric space where the probability of two vertices to be connected is an arbitrary function of the distance. We give sufficient conditions under which the locations can be recovered (up to an isomorphism of the space) in the sparse regime. Moreover, we define a geometric counterpart of the model of flow of information on trees, due to Mossel and Peres, in which one considers a branching random walk on a sphere and the goal is to recover the location of the root based on the locations of leaves. We give some sufficient conditions for percolation and for non-percolation of information in this model.

Prisons are susceptible to outbreaks. Control measures focusing on isolation and cohorting negatively affect wellbeing. We present an outbreak of coronavirus disease 2019 (COVID-19) in a large male prison in Wales, UK, October 2020 to April 2021, and discuss control measures.

We gathered case-information, including demographics, staff-residence postcode, resident cell number, work areas/dates, test results, staff interview dates/notes and resident prison-transfer dates. Epidemiological curves were mapped by prison location. Control measures included isolation (exclusion from work or cell-isolation), cohorting (new admissions and work-area groups), asymptomatic testing (case-finding), removal of communal dining and movement restrictions. Facemask use and enhanced hygiene were already in place. Whole-genome sequencing (WGS) and interviews determined the genetic relationship between cases plausibility of transmission.

Of 453 cases, 53% (n = 242) were staff, most aged 25–34 years (11.5% females, 27.15% males) and symptomatic (64%). Crude attack-rate was higher in staff (29%, 95% CI 26–64%) than in residents (12%, 95% CI 9–15%).

Whole-genome sequencing can help differentiate multiple introductions from person-to-person transmission in prisons. It should be introduced alongside asymptomatic testing as soon as possible to control prison outbreaks. Timely epidemiological investigation, including data visualisation, allowed dynamic risk assessment and proportionate control measures, minimising the reduction in resident welfare.

Several papers have highlighted the potential of network science to appeal to a younger audience of high school children and provided lesson material on network science for high school children. However, network science also provides a great topic for outreach activities for primary school children. Therefore, this article gives a short summary of an outreach activity on network science for primary school children aged 8–12 years. The material provided in this article contains presentation material for a lesson of approximately 1 hour, including experiments, exercises, and quizzes, which can be used by other scientists interested in popularizing network science. We then discuss the lessons learned from this material.

Machine learning applications on large-scale network-structured data commonly encode network information in the form of node embeddings. Network embedding algorithms map the nodes into a low-dimensional space such that the nodes that are “similar” with respect to network topology are also close to each other in the embedding space. Real-world networks often have multiple versions or can be “multiplex” with multiple types of edges with different semantics. For such networks, computation of Consensus Embeddings based on the node embeddings of individual versions can be useful for various reasons, including privacy, efficiency, and effectiveness of analyses. Here, we systematically investigate the performance of three dimensionality reduction methods in computing consensus embeddings on networks with multiple versions: singular value decomposition, variational auto-encoders, and canonical correlation analysis (CCA). Our results show that (i) CCA outperforms other dimensionality reduction methods in computing concensus embeddings, (ii) in the context of link prediction, consensus embeddings can be used to make predictions with accuracy close to that provided by embeddings of integrated networks, and (iii) consensus embeddings can be used to improve the efficiency of combinatorial link prediction queries on multiple networks by multiple orders of magnitude.

Hadwiger’s conjecture asserts that every graph without a $K_t$-minor is $(t-1)$-colourable. It is known that the exact version of Hadwiger’s conjecture does not extend to list colouring, but it has been conjectured by Kawarabayashi and Mohar (2007) that there exists a constant $c$ such that every graph with no $K_t$-minor has list chromatic number at most $ct$. More specifically, they also conjectured that this holds for $c=\frac{3}{2}$.

Refuting the latter conjecture, we show that the maximum list chromatic number of graphs with no $K_t$-minor is at least $(2-o(1))t$, and hence $c \ge 2$ in the above conjecture is necessary. This improves the previous best lower bound by Barát, Joret and Wood (2011), who proved that $c \ge \frac{4}{3}$. Our lower-bound examples are obtained via the probabilistic method.

For a subgraph $G$ of the blow-up of a graph $F$, we let $\delta ^*(G)$ be the smallest minimum degree over all of the bipartite subgraphs of $G$ induced by pairs of parts that correspond to edges of $F$. Johansson proved that if $G$ is a spanning subgraph of the blow-up of $C_3$ with parts of size $n$ and $\delta ^*(G) \ge \frac{2}{3}n + \sqrt{n}$, then $G$ contains $n$ vertex disjoint triangles, and presented the following conjecture of Häggkvist. If $G$ is a spanning subgraph of the blow-up of $C_k$ with parts of size $n$ and $\delta ^*(G) \ge \left(1 + \frac 1k\right)\frac n2 + 1$, then $G$ contains $n$ vertex disjoint copies of $C_k$ such that each $C_k$ intersects each of the $k$ parts exactly once. A similar conjecture was also made by Fischer and the case $k=3$ was proved for large $n$ by Magyar and Martin.

In this paper, we prove the conjecture of Häggkvist asymptotically. We also pose a conjecture which generalises this result by allowing the minimum degree conditions in each bipartite subgraph induced by pairs of parts of $G$ to vary. We support this new conjecture by proving the triangle case. This result generalises Johannson’s result asymptotically.

For interventions that affect how individuals interact, social network data may aid in understanding the mechanisms through which an intervention is effective. Social networks may even be an intermediate outcome observed prior to end of the study. In fact, social networks may also mediate the effects of the intervention on the outcome of interest, and Sweet (2019) introduced a statistical model for social networks as mediators in network-level interventions. We build on their approach and introduce a new model in which the network is a mediator using a latent space approach. We investigate our model through a simulation study and a real-world analysis of teacher advice-seeking networks.

This study considers a network formation model in which each dyad of agents strategically determines the link status. Our model allows the agents to have unobserved group heterogeneity in the propensity of link formation. For the model estimation, we propose a three-step maximum likelihood method, in which the latent group structure is estimated using the binary segmentation algorithm in the second step. As an empirical illustration, we focus on the network data of international visa-free travels. The results indicate the presence of significant strategic complementarity and a certain level of degree heterogeneity in the network formation behavior.

In this article, we provide a comprehensive analyses of two continuous review lost sales inventory system based on different replenishment policies, namely $(s,S)$ and $(s,Q)$. We assume that the arrival times of demands form a Poisson process and that the demand sizes have i.i.d. exponential distribution. We assume that the items in stock may obsolete after an exponential time. The lead time for replenishment is exponential. We also assume that the excess demands and the demands that occurred during stock out periods are lost. Using the system point method of level crossing and integral equation method, we derive the steady-state probability distribution of inventory level explicitly. After deriving some system performance measures, we computed the total expected cost rate. We also provide numerical examples of sensitivity analyses involving different parameters and costs. As a result of our numerical analysis, we provide several insights on the optimal $(s,S)$ and $(s,Q)$ policies for inventory systems of obsolescence items with positive lead times. The better policy for maintaining inventory can be quantified numerically.

India has the third-largest burden of human immunodeficiency virus (HIV) infection in the world. The coronavirus disease 2019 (COVID-19) pandemic has only exposed the cracks in the Indian healthcare infrastructure concerning HIV. The prevalence of HIV in India is more among the destitute or sections of society shrouded by years of social stigma such as prostitutes, truck drivers, transsexuals and intravenous drug users. National AIDS Control Organisation and The Joint United Nations Programme on HIV/AIDS (UNAIDS) organisation have many several efforts over the years to set up counselling and testing centres all over the country and spread awareness about HIV among the masses. COVID-19 pandemic has reversed years of progress made by the same. HIV patients are biologically more susceptible to COVID-19, and the lockdown has led to the loss of jobs, economic crises, shortage of drugs and necessities such as food and housing among this vulnerable population, which can result in lowered CD4-T cell counts in the coming months and make way for dangerous opportunistic infection outbreaks in this population increasing the overall HIV burden of India. This article explores how COVID-19 has impacted India's already existing HIV epidemic and tries to put forth recommendations based on the evidence found to be better prepared in treating the HIV-positive population in India in the face of another catastrophe like the COVID-19.