In June 2019 the Health Protection Team in Yorkshire and Humber, England, was notified of cases of hepatitis A virus (HAV) infection in staff at a secondary school. Investigation revealed that an earlier case worked as a food handler in the school kitchen. Indirect transmission through food from the canteen was considered the most likely route of transmission. Cases were described according to setting of exposure. Oral fluid was obtained from students for serological testing. Environmental investigations were undertaken at settings where food handling was considered a potential transmission risk. Thirty-three confirmed cases were linked to the outbreak. All of those tested (n = 31) shared the same sequence with a HAV IB genotype. The first three cases were a household cluster and included the index case for the school. A further 19 cases (16 students, 3 staff) were associated with the school and consistent with indirect exposure to the food handler. One late onset case could not be ruled out as a secondary case within the school and resulted in vaccination of the school population. Five cases were linked to a bakery where a case from the initial household cluster worked as a food server. No concerns about hygiene standards were noted at either the school or the bakery. Oral fluid samples taken at the time of vaccination from asymptomatic students (n = 219, 11–16 years-old) showed no evidence of recent or current infection. This outbreak included household and foodborne transmission but limited (and possibly zero) person-to-person transmission among secondary school students. Where adequate hygiene exists, secondary transmission within older students may not occur.

This article describes the limiting distribution of the extremes of observations that arrive in clusters. We start by studying the tail behaviour of an individual cluster, and then we apply the developed theory to determine the limiting distribution of $\max\{X_j\,:\, j=0,\ldots, K(t)\}$, where K(t) is the number of independent and identically distributed observations $(X_j)$ arriving up to the time t according to a general marked renewal cluster process. The results are illustrated in the context of some commonly used Poisson cluster models such as the marked Hawkes process.

An old conjecture of Erdős and McKay states that if all homogeneous sets in an $n$-vertex graph are of order $O(\!\log n)$ then the graph contains induced subgraphs of each size from $\{0,1,\ldots, \Omega \big(n^2\big)\}$. We prove a bipartite analogue of the conjecture: if all balanced homogeneous sets in an $n \times n$ bipartite graph are of order $O(\!\log n)$, then the graph contains induced subgraphs of each size from $\{0,1,\ldots, \Omega \big(n^2\big)\}$.

Gambling marketing is frequently visible in the United Kingdom, especially around the national sport, soccer. Previous research has documented the frequency with which gambling marketing logos can be seen in domestic club soccer, and also the frequency of television advertising around international tournaments. The present research investigates the frequency and content of television advertising during the men’s 2020 Euro soccer tournament, a high-profile tournament shown since the industry’s voluntary “whistle-to-whistle ban” on gambling advertising came into effect. Overall, 113 gambling adverts were recorded (4.5 adverts per relevant match). Financial inducements were the most frequently shown category (56.6%), followed by adverts raising awareness of a given operator’s brand (19.5%), adverts featuring the odds on specific complex bets (18.6%), and adverts promoting safer gambling (5.3%). Adverts featured a range of safer gambling messages, with the “when the fun stops, stop” message featuring in 56.6% of adverts. This research indicates that gambling advertising remains a frequent part of the experience of watching live televised soccer in the UK, and shows how the content of this advertising was comparable to what has been seen in the previous literature.

Considering a representative agent in the market, we study the long-term optimal investment problem in a discrete-time financial market, introducing a set of restrictions in the admissible strategies. The drawdown constraints limit the size of possible losses of the portfolio and impose a floor-based performance measure. The optimal growth rate is characterized, and under suitable hypotheses it is proved that an optimal strategy exists. The approach to solving this problem is based on dynamic programming techniques and a fixed point argument adapted from the theory of Markov decision processes.

We consider the random splitting and aggregating of Hawkes processes. We present the random splitting schemes using the direct approach for counting processes, as well as the immigration–birth branching representations of Hawkes processes. From the second scheme, it is shown that random split Hawkes processes are again Hawkes. We discuss functional central limit theorems (FCLTs) for the scaled split processes from the different schemes. On the other hand, aggregating multivariate Hawkes processes may not necessarily be Hawkes. We identify a necessary and sufficient condition for the aggregated process to be Hawkes. We prove an FCLT for a multivariate Hawkes process under a random splitting and then aggregating scheme (under certain conditions, transforming into a Hawkes process of a different dimension).

Given a graphon $W$ and a finite simple graph $H$, with vertex set $V(H)$, denote by $X_n(H, W)$ the number of copies of $H$ in a $W$-random graph on $n$ vertices. The asymptotic distribution of $X_n(H, W)$ was recently obtained by Hladký, Pelekis, and Šileikis [17] in the case where $H$ is a clique. In this paper, we extend this result to any fixed graph $H$. Towards this we introduce a notion of $H$-regularity of graphons and show that if the graphon $W$ is not $H$-regular, then $X_n(H, W)$ has Gaussian fluctuations with scaling $n^{|V(H)|-\frac{1}{2}}$. On the other hand, if $W$ is $H$-regular, then the fluctuations are of order $n^{|V(H)|-1}$ and the limiting distribution of $X_n(H, W)$ can have both Gaussian and non-Gaussian components, where the non-Gaussian component is a (possibly) infinite weighted sum of centred chi-squared random variables with the weights determined by the spectral properties of a graphon derived from $W$. Our proofs use the asymptotic theory of generalised $U$-statistics developed by Janson and Nowicki [22]. We also investigate the structure of $H$-regular graphons for which either the Gaussian or the non-Gaussian component of the limiting distribution (but not both) is degenerate. Interestingly, there are also $H$-regular graphons $W$ for which both the Gaussian or the non-Gaussian components are degenerate, that is, $X_n(H, W)$ has a degenerate limit even under the scaling $n^{|V(H)|-1}$. We give an example of this degeneracy with $H=K_{1, 3}$ (the 3-star) and also establish non-degeneracy in a few examples. This naturally leads to interesting open questions on higher order degeneracies.