Conditional mean risk sharing appears to be effective to distribute total losses amongst participants within an insurance pool. This paper develops analytical results for this allocation rule in the individual risk model with dependence induced by the respective position within a graph. Precisely, losses are modelled by zero-augmented random variables whose joint occurrence distribution and individual claim amount distributions are based on network structures and can be characterised by graphical models. The Ising model is adopted for occurrences and loss amounts obey decomposable graphical models that are specific to each participant. Two graphical structures are thus used: the first one to describe the contagion amongst member units within the insurance pool and the second one to model the spread of losses inside each participating unit. The proposed individual risk model is typically useful for modelling operational risks, catastrophic risks or cybersecurity risks.

The appearance on the skin of herpes virus lesions, concomitantly with the coronavirus disease 2019 (COVID-19) pandemic, leads us to suspect an underlying infection with severe acute respiratory syndrome coronavirus-2 (SARS-CoV-2). Diagnostic reverse transcriptase polymerase chain reaction tests and immunoglobulin M (IgM) and IgG seroconversion studies have therefore been carried out. We present three cases of herpes virus infections in immunocompetent patients: one of the infections was herpes simplex 1 in a 40-year-old woman, and the other two were herpes varicella-zoster infections in a 62-year-old man and a 25-year-old woman. The patients were in the care of the southern health district of Seville of the SAS (Andalusian Health Service) during the Spanish state of alarm over the COVID-19 pandemic. The SARS-CoV-2 infection was confirmed in only one of the three cases. In this study, we briefly review the etiopathogenic role of the COVID-19 pandemic situation, whereby immunodeficiencies are generated that favour the appearance of other viral infections, such as herpes virus infections.

Characteristics and research collaboration of registered systematic reviews (SRs) on treatment modalities for coronavirus disease-2019 (COVID-19) remain unclear. This study analysed research collaboration, interventions and outcome measures in registered SRs on COVID-19 treatments and pointed out the relevant problems. PROSPERO (international prospective register of systematic reviews) was searched for SRs on COVID-19 treatments as of 2 June 2020. Excel 2016 was used for descriptive analyses of the extracted data. VOSviewer 1.6.14 software was used to generate network maps for collaborations between countries and institutions. A total of 189 SRs were included, which were registered by 301 institutions from 39 countries. China (69, 36.50%) exhibited the highest output. Cooperation between countries was not close enough. As an institution, the Chengdu University of Traditional Chinese Medicine (7, 3.70%) had the highest output. There was close cooperation between institutions. Interventions included antiviral therapy (81, 42.86%), respiratory support (16, 8.47%), circulatory support (11, 5.82%), plasma therapy for convalescent patients (11, 5.82%), immunotherapy (9, 4.76%), TCM (traditional Chinese medicine) treatment (9, 4.76%), rehabilitation treatment (5, 2.65%), anti-inflammatory treatment (16, 8.47%) and other treatments (31, 16.40%). Concerning antiviral therapy (81, 42.86%), the most commonly used antiviral agents were chloroquine/hydroxychloroquine (26, 13.76%), followed by remdesivir (12, 6.35%), lobinavir/ritonavir (11, 5.82%), favipiravir (5, 2.65%), ribavirin (5, 2.65%), interferon (5, 2.65%), abiron (4, 2.12%) and abidor (4, 2.12%). The most frequently used primary and secondary outcomes were the mortality rate (92, 48.68%) and hospital stay length (48, 25.40%), respectively. The expression of the outcomes was not standardised. Many COVID-19 SRs on treatment modalities have been registered, with a low completion rate. Although there was some collaboration between countries and institutions in the currently registered SRs on treatment modalities for COVID-19 on PROSPERO, cooperation between countries should be further enhanced. More attention should be directed towards identifying deficiencies of outcome measures, and the standardisation of results should be maximised.

Consider the following experiment: a deck with m copies of n different card types is randomly shuffled, and a guesser attempts to guess the cards sequentially as they are drawn. Each time a guess is made, some amount of ‘feedback’ is given. For example, one could tell the guesser the true identity of the card they just guessed (the complete feedback model) or they could be told nothing at all (the no feedback model). In this paper we explore a partial feedback model, where upon guessing a card, the guesser is only told whether or not their guess was correct. We show in this setting that, uniformly in n, at most $m+O(m^{3/4}\log m)$ cards can be guessed correctly in expectation. This resolves a question of Diaconis and Graham from 1981, where even the $m=2$ case was open.

Consider a random $n\times n$ zero-one matrix with ‘sparsity’ p, sampled according to one of the following two models: either every entry is independently taken to be one with probability p (the ‘Bernoulli’ model) or each row is independently uniformly sampled from the set of all length-n zero-one vectors with exactly pn ones (the ‘combinatorial’ model). We give simple proofs of the (essentially best-possible) fact that in both models, if $\min(p,1-p)\geq (1+\varepsilon)\log n/n$ for any constant $\varepsilon>0$, then our random matrix is nonsingular with probability $1-o(1)$. In the Bernoulli model, this fact was already well known, but in the combinatorial model this resolves a conjecture of Aigner-Horev and Person.

A graph G arrows a graph H if in every 2-edge-colouring of G there exists a monochromatic copy of H. Schelp had the idea that if the complete graph $K_n$ arrows a small graph H, then every ‘dense’ subgraph of $K_n$ also arrows H, and he outlined some problems in this direction. Our main result is in this spirit. We prove that for every sufficiently large n, if $n = 3t+r$ where $r \in \{0,1,2\}$ and G is an n-vertex graph with $\delta(G) \ge (3n-1)/4$, then for every 2-edge-colouring of G, either there are cycles of every length $\{3, 4, 5, \dots, 2t+r\}$ of the same colour, or there are cycles of every even length $\{4, 6, 8, \dots, 2t+2\}$ of the samecolour.

Our result is tight in the sense that no longer cycles (of length $>2t+r$) can be guaranteed and the minimum degree condition cannot be reduced. It also implies the conjecture of Schelp that for every sufficiently large n, every $(3t-1)$-vertex graph G with minimum degree larger than $3|V(G)|/4$ arrows the path $P_{2n}$ with 2n vertices. Moreover, it implies for sufficiently large n the conjecture by Benevides, Łuczak, Scott, Skokan and White that for $n=3t+r$ where $r \in \{0,1,2\}$ and every n-vertex graph G with $\delta(G) \ge 3n/4$, in each 2-edge-colouring of G there exists a monochromatic cycle of length at least $2t+r$.

The Hodrick–Prescott (HP) filter has been a popular method of trend extraction from economic time series. However, it is impractical without modification if some observations are not available. This paper improves the HP filter so that it can be applied in such situations. More precisely, this paper introduces two alternative generalized HP filters that are applicable for this purpose. We provide their properties and a way of specifying those smoothing parameters that are required for their application. In addition, we numerically examine their performance. Finally, based on our analysis, we recommend one of them for applied studies.