The well-known Erdős-Hajnal conjecture states that for any graph $F$, there exists $\epsilon \gt 0$ such that every $n$-vertex graph $G$ that contains no induced copy of $F$ has a homogeneous set of size at least $n^{\epsilon }$. We consider a variant of the Erdős-Hajnal problem for hypergraphs where we forbid a family of hypergraphs described by their orders and sizes. For graphs, we observe that if we forbid induced subgraphs on $m$ vertices and $f$ edges for any positive $m$ and $0\leq f \leq \binom{m}{2}$, then we obtain large homogeneous sets. For triple systems, in the first nontrivial case $m=4$, for every $S \subseteq \{0,1,2,3,4\}$, we give bounds on the minimum size of a homogeneous set in a triple system where the number of edges spanned by every four vertices is not in $S$. In most cases the bounds are essentially tight. We also determine, for all $S$, whether the growth rate is polynomial or polylogarithmic. Some open problems remain.

Men who have sex with men (MSM) who use injection drugs (MSM-IDU) are at high risk of sexually transmitted infections (STIs), but the long-term incidence is unclear. We conducted a single-centre retrospective cohort study using the clinical records of non-haemophilia men with human immunodeficiency virus (HIV) who visited the Institute of Medical Science, the University of Tokyo (IMSUT) Hospital, located in Tokyo, Japan, from 2013 to 2022. We analysed 575 patients including 62 heterosexual males and 513 MSM patients, of whom 6.8% (35/513) were injection drug use (IDU). Compared to non-IDU MSM, MSM-IDU had a higher incidence of hepatitis C virus (HCV) (44.8 vs 3.5 /1,000 person-years (PY); incidence rate ratio (IRR) [95% confidence interval (95% CI)], 12.8 [5.5–29.3], p < 0.001) and syphilis (113.8 vs 53.3 /1,000 PY; IRR, 2.1 [1.4–3.1], p < 0.001). The incidence of other symptomatic STIs (amoebiasis, chlamydia, and gonorrhoea infections) was <4/1,000 PY. In multivariable Poisson regression analysis, HCV incidence was associated with MSM (IRR, 1.8 × 106 [9.9 × 105–3.4 × 106], p < 0.001), IDU (IRR, 10.1 [4.0–25.6], p < 0.001), and syphilis infection during the study period (IRR, 25.0 [1.2–518.3]/time/year, p < 0.001). Among men with HIV, the prevalence of IDU in MSM and the long-term incidence of STIs in MSM-IDU were high. IDU and sexual contact are important modes of transmission of HCV among HIV-infected MSM in Tokyo.

Congenital Zika is a devastating consequence of maternal Zika virus infections. Estimates of age-dependent seroprevalence profiles are central to our understanding of the force of Zika virus infections. We set out to calculate the age-dependent seroprevalence of Zika virus infections in Brazil. We analyzed serum samples stratified by age and geographic location, collected from 2016 to 2019, from about 16,000 volunteers enrolled in a Phase 3 dengue vaccine trial led by the Institute Butantan in Brazil. Our results show that Zika seroprevalence has a remarkable age-dependent and geographical distribution, with an average age of the first infection varying from region to region, ranging from 4.97 (3.03–5.41) to 7.24 (6.98–7.90) years. The calculated basic reproduction number, $ {R}_0 $, varied from region to region, ranging from 1.18 (1.04–1.41) to 2.33 (1.54–3.85). Such data are paramount to determine the optimal age to vaccinate against Zika, if and when such a vaccine becomes available.

We show that for any $\varepsilon \gt 0$ and $\Delta \in \mathbb{N}$, there exists $\alpha \gt 0$ such that for sufficiently large $n$, every $n$-vertex graph $G$ satisfying that $\delta (G)\geq \varepsilon n$ and $e(X, Y)\gt 0$ for every pair of disjoint vertex sets $X, Y\subseteq V(G)$ of size $\alpha n$ contains all spanning trees with maximum degree at most $\Delta$. This strengthens a result of Böttcher, Han, Kohayakawa, Montgomery, Parczyk, and Person.

It is proven that a conjecture of Tao (2010) holds true for log-concave random variables on the integers: For every $n \geq 1$, if $X_1,\ldots,X_n$ are i.i.d. integer-valued, log-concave random variables, then

\begin{equation*} H(X_1+\cdots +X_{n+1}) \geq H(X_1+\cdots +X_{n}) + \frac {1}{2}\log {\Bigl (\frac {n+1}{n}\Bigr )} - o(1) \end{equation*}

$H(X_1) \to \infty$

$H(X_1)$

$U_1,\ldots,U_n$

$(0,1)$

\begin{equation*} h(X_1+\cdots +X_n + U_1+\cdots +U_n) = H(X_1+\cdots +X_n) + o(1), \end{equation*}

$H(X_1) \to \infty$

$h$

$o(1)$

A local COVID-19 outbreak with two community clusters occurred in a large industrial city, Shaoxing, China, in December 2021 after serial interventions were imposed. We aimed to understand the reason by analysing the characteristics of the outbreak and evaluating the effects of phase-adjusted interventions. Publicly available data from 7 December 2021 to 25 January 2022 were collected to analyse the epidemiological characteristics of this outbreak. The incubation period was estimated using Hamiltonian Monte Carlo method. A well-fitted extended susceptible-exposed-infectious-recovered model was used to simulate the impact of different interventions under various combination of scenarios. There were 387 SARS-CoV-2-infected cases identified, and 8.3% of them were initially diagnosed as asymptomatic cases. The estimated incubation period was 5.4 (95% CI 5.2–5.7) days for all patients. Strengthened measures of comprehensive quarantine based on tracing led to less infections and a shorter duration of epidemic. With a same period of incubation, comprehensive quarantine was more effective in containing the transmission than other interventions. Our findings reveal an important role of tracing and comprehensive quarantine in blocking community spread when a cluster occurred. Regions with tense resources can adopt home quarantine as a relatively affordable and low-impact intervention measure compared with centralized quarantine.

Under the European Union’s Solvency II regulations, insurance firms are required to use a one-year VaR (Value at Risk) approach. This involves a one-year projection of the balance sheet and requires sufficient capital to be solvent in 99.5% of outcomes. The Solvency II Internal Model risk calibrations require annual changes in market indices/term structure/transitions for the estimation of the risk distribution for each of the Internal Model risk drivers.

Transition and default risk are typically modelled using transition matrices. To model this risk requires a model of transition matrices and how these can change from year to year. In this paper, four such models have been investigated and compared to the raw data they are calibrated to. The models investigated are:

• A bootstrapping approach – sampling from an historical data set with replacement.

• The Vašíček model was calibrated using the Belkin approach.

• The K-means model – a new non-parametric model produced using the K-means clustering algorithm.

• A two-factor model – a new parametric model, using two factors (instead of a single factor with the Vašíček) to represent each matrix.

The models are compared in several ways:

1. 1. A principal components analysis (PCA) approach that compares how closely the models move compared to the raw data.

2. 2. A backtesting approach that compares how each model’s extreme percentile compares to regulatory backtesting requirements.

3. 3. A commentary on the amount of expert judgement in each model.

4. 4. Model simplicity and breadth of uses are also commented on.

People who inject drugs are at risk of acute bacterial and fungal injecting-related infections. There is evidence that incidence of hospitalizations for injecting-related infections are increasing in several countries, but little is known at an individual level. We aimed to examine injecting-related infections in a linked longitudinal cohort of people who inject drugs in Melbourne, Australia. A retrospective descriptive analysis was conducted to estimate the prevalence and incidence of injecting-related infections using administrative emergency department and hospital separation datasets linked to the SuperMIX cohort, from 2008 to 2018. Over the study period, 33% (95%CI: 31–36%) of participants presented to emergency department with any injecting-related infections and 27% (95%CI: 25–30%) were admitted to hospital. Of 1,044 emergency department presentations and 740 hospital separations, skin and soft tissue infections were most common, 88% and 76%, respectively. From 2008 to 2018, there was a substantial increase in emergency department presentations and hospital separations with any injecting-related infections, 48 to 135 per 1,000 person-years, and 18 to 102 per 1,000 person-years, respectively. The results emphasize that injecting-related infections are increasing, and that new models of care are needed to help prevent and facilitate early detection of superficial infection to avoid potentially life-threatening severe infections.

We examined the association between face masks and risk of infection with SARS-CoV-2 using cross-sectional data from 3,209 participants in a randomized trial exploring the effectiveness of glasses in reducing the risk of SARS-CoV-2 infection. Face mask use was based on participants’ response to the end-of-follow-up survey. We found that the incidence of self-reported COVID-19 was 33% (aRR 1.33; 95% CI 1.03–1.72) higher in those wearing face masks often or sometimes, and 40% (aRR 1.40; 95% CI 1.08–1.82) higher in those wearing face masks almost always or always, compared to participants who reported wearing face masks never or almost never. We believe the observed increase in the incidence of infection associated with wearing a face mask is likely due to unobservable and hence nonadjustable differences between those wearing and not wearing a mask. Observational studies reporting on the relationship between face mask use and risk of respiratory infections should be interpreted cautiously, and more randomized trials are needed.

This paper establishes bounds on the performance of empirical risk minimization for large-dimensional linear regression. We generalize existing results by allowing the data to be dependent and heavy-tailed. The analysis covers both the cases of identically and heterogeneously distributed observations. Our analysis is nonparametric in the sense that the relationship between the regressand and the regressors is not specified. The main results of this paper show that the empirical risk minimizer achieves the optimal performance (up to a logarithmic factor) in a dependent data setting.

A collection of graphs is nearly disjoint if every pair of them intersects in at most one vertex. We prove that if $G_1, \dots, G_m$ are nearly disjoint graphs of maximum degree at most $D$, then the following holds. For every fixed $C$, if each vertex $v \in \bigcup _{i=1}^m V(G_i)$ is contained in at most $C$ of the graphs $G_1, \dots, G_m$, then the (list) chromatic number of $\bigcup _{i=1}^m G_i$ is at most $D + o(D)$. This result confirms a special case of a conjecture of Vu and generalizes Kahn’s bound on the list chromatic index of linear uniform hypergraphs of bounded maximum degree. In fact, this result holds for the correspondence (or DP) chromatic number and thus implies a recent result of Molloy and Postle, and we derive this result from a more general list colouring result in the setting of ‘colour degrees’ that also implies a result of Reed and Sudakov.