Kostochka and Thomason independently showed that any graph with average degree $\Omega(r\sqrt{\log r})$ contains a $K_r$ minor. In particular, any graph with chromatic number $\Omega(r\sqrt{\log r})$ contains a $K_r$ minor, a partial result towards Hadwiger’s famous conjecture. In this paper, we investigate analogues of these results in the directed setting. There are several ways to define a minor in a digraph. One natural way is as follows. A strong $\overrightarrow{K}_{\!\!r}$ minor is a digraph whose vertex set is partitioned into r parts such that each part induces a strongly connected subdigraph, and there is at least one edge in each direction between any two distinct parts. We investigate bounds on the dichromatic number and minimum out-degree of a digraph that force the existence of strong $\overrightarrow{K}_{\!\!r}$ minors as subdigraphs. In particular, we show that any tournament with dichromatic number at least 2r contains a strong $\overrightarrow{K}_{\!\!r}$ minor, and any tournament with minimum out-degree $\Omega(r\sqrt{\log r})$ also contains a strong $\overrightarrow{K}_{\!\!r}$ minor. The latter result is tight up to the implied constant and may be viewed as a strong-minor analogue to the classical result of Kostochka and Thomason. Lastly, we show that there is no function $f\;:\;\mathbb{N} \rightarrow \mathbb{N}$ such that any digraph with minimum out-degree at least f(r) contains a strong $\overrightarrow{K}_{\!\!r}$ minor, but such a function exists when considering dichromatic number.

This paper studies the optimal allocation policy of a coherent system with independent heterogeneous components and dependent subsystems, the systems are assumed to consist of two groups of components whose lifetimes follow proportional hazard (PH) or proportional reversed hazard (PRH) models. We investigate the optimal allocation strategy by finding out the number $k$ of components coming from Group A in the up-series system. First, some sufficient conditions are provided in the sense of the usual stochastic order to compare the lifetimes of two-parallel–series systems with dependent subsystems, and we obtain the hazard rate and reversed hazard rate orders when two subsystems have independent lifetimes. Second, similar results are also obtained for two-series–parallel systems under certain conditions. Finally, we generalize the corresponding results to parallel–series and series–parallel systems with multiple subsystems in the viewpoint of the minimal path and the minimal cut sets, respectively. Some numerical examples are presented to illustrate the theoretical findings.

Escherichia coli concentration levels in recreational water are used by beach managers to evaluate the risk of gastrointestinal illness among beachgoers. We examined the relationship between specific environmental factors and E. coli concentration in recreational beaches in the Niagara Region. We analysed E. coli geometric means collected from eight beaches from two of the Great Lakes in the Niagara Region in Ontario, between 2011 and 2019. We applied path analysis to evaluate the relationship between the environmental factors and E. coli concentrations, including whether effects were direct or indirect via a mediator. Turbidity was found to be an important mediator for the indirect effect of environmental variables overall and in beach-specific models. Rainfall and streamflow had a positive indirect effect on E. coli via turbidity and a direct effect in five out of seven beach models. Streamflow was also a mediator for the indirect effect of previous day air temperature in five out of seven models. In three subset models, outfall E. coli concentration was a mediator for the effect of the environmental factors. Using a novel methodological approach, this study identifies important relationships and pathways that predict beach E. coli concentration in freshwater beaches located on two of the Great Lakes.

This study aims at providing estimates on the transmission risk of SARS-CoV-2 in schools and day-care centres. We calculated secondary attack rates (SARs) using individual-level data from state-wide mandatory notification of index cases in educational institutions, followed by contact tracing and PCR-testing of high-risk contacts. From August to December 2020, every sixth of overall 784 independent index cases was associated with secondary cases in educational institutions. Monitoring of 14 594 institutional high-risk contacts (89% PCR-tested) of 441 index cases during quarantine revealed 196 secondary cases (SAR 1.34%, 0.99–1.78). SARS-CoV-2 infection among high-risk contacts was more likely around teacher-indexes compared to student-/child-indexes (incidence rate ratio (IRR) 3.17, 1.79–5.59), and in day-care centres compared to secondary schools (IRR 3.23, 1.76–5.91), mainly due to clusters around teacher-indexes in day-care containing a higher mean number of secondary cases per index case (142/113 = 1.26) than clusters around student-indexes in schools (82/474 = 0.17). In 2020, SARS-CoV-2 transmission risk in educational settings was low overall, but varied strongly between setting and role of the index case, indicating the chance for targeted intervention. Surveillance of SARS-CoV-2 transmission in educational institutions can powerfully inform public health policy and improve educational justice during the pandemic.

Spatial units typically vary over many of their characteristics, introducing potential unobserved heterogeneity which invalidates commonly used homoskedasticity conditions. In the presence of unobserved heteroskedasticity, methods based on the quasi-likelihood function generally produce inconsistent estimates of both the spatial parameter and the coefficients of the exogenous regressors. A robust generalized method of moments estimator as well as a modified likelihood method have been proposed in the literature to address this issue. The present paper constructs an alternative indirect inference (II) approach which relies on a simple ordinary least squares procedure as its starting point. Heteroskedasticity is accommodated by utilizing a new version of continuous updating that is applied within the II procedure to take account of the parameterization of the variance–covariance matrix of the disturbances. Finite-sample performance of the new estimator is assessed in a Monte Carlo study. The approach is implemented in an empirical application to house price data in the Boston area, where it is found that spatial effects in house price determination are much more significant under robustification to heterogeneity in the equation errors.

We propose a novel tie-oriented model for longitudinal event network data. The generating mechanism is assumed to be a multivariate Poisson process that governs the onset and repetition of yearly observed events with two separate intensity functions. We apply the model to a network obtained from the yearly dyadic number of international deliveries of combat aircraft trades between 1950 and 2017. Based on the trade gravity approach, we identify economic and political factors impeding or promoting the number of transfers. Extensive dynamics as well as country heterogeneities require the specification of semiparametric time-varying effects as well as random effects. Our findings reveal strong heterogeneous as well as time-varying effects of endogenous and exogenous covariates on the onset and repetition of aircraft trade events.

This paper investigates the benefits of incorporating diversification effects into the pricing process of insurance policies from two different business lines. The paper shows that, for the same risk reduction, insurers pricing policies jointly can have a competitive advantage over those pricing them separately. However, the choice of competitiveness constrains the underwriting flexibility of joint pricers. The paper goes a step further by modelling explicitly the relationship between premiums and the number of customers in each line. Using the total collected premiums as a criterion to compare the competing strategies, the paper provides conditions for the optimal pricing decision based on policyholders’ sensitivity to price discounts. The results are illustrated for a portfolio of annuities and assurances. Further, using non-life data from the Brazilian insurance market, an empirical exploration shows that most pairs satisfy the condition for being priced jointly, even when pairwise correlations are high.

Copula is one of the widely used techniques to describe the dependency structure between components of a system. Among all existing copulas, the family of Archimedean copulas is the popular one due to its wide range of capturing the dependency structures. In this paper, we consider the systems that are formed by dependent and identically distributed components, where the dependency structures are described by Archimedean copulas. We study some stochastic comparisons results for series, parallel, and general $r$-out-of-$n$ systems. Furthermore, we investigate whether a system of used components performs better than a used system with respect to different stochastic orders. Furthermore, some aging properties of these systems have been studied. Finally, some numerical examples are given to illustrate the proposed results.

Hebei Province was affected by two coronavirus disease 2019 (COVID-19) outbreak waves during the period 22 January 2020 through 27 February 2020 (wave 1) and 2 January 2021 through 14 February 2021 (wave 2). To evaluate and compare the epidemiological characteristics, containment delay, cluster events and social activity, as well as non-pharmaceutical interventions of the two COVID-19 outbreak waves, we examined real-time update information on all COVID-19-confirmed cases from a publicly available database. Wave 1 was closely linked with the COVID-19 pandemic in Wuhan, whereas wave 2 was triggered, to a certain extent, by the increasing social activities such as weddings, multi-household gatherings and church events during the slack agricultural period. In wave 2, the epidemic spread undetected in the rural areas, and people living in the rural areas had a higher incidence rate than those living in the urban areas (5.3 vs. 22.0 per 1 000 000). Furthermore, Rt was greater than 1 in the early stage of the two outbreak waves, and decreased substantially after massive non-pharmaceutical interventions were implemented. In China's ‘new-normal’ situation, development of targeted and effective intervention remains key for COVID-19 control in consideration of the potential threat of new coronavirus strains.

Following the emergence of SARS-CoV-2, early outbreak response relied on behavioural interventions. In the USA, local governments implemented restrictions aimed at reducing movements and contacts to limit viral transmission. In Pennsylvania, restrictions closed schools and businesses in the spring of 2020 and interventions eased later through the summer. Here we use passive monitoring of vehicular traffic volume and mobile device-derived visits to points of interest as proxies for movements and contacts in a rural Pennsylvania county. Rural areas have limited health care resources, which magnifies the importance of disease prevention. These data show the lowest levels of movement occurred during the strictest phase of restrictions, indicating high levels of compliance with behavioural intervention. We find that increases in movement correlated with increases in reported SARS-CoV-2 cases 9–18 days later. The methodology used in this study can be adapted to inform outbreak management strategies for other locations and future outbreaks that use behavioural interventions to reduce pathogen transmission.

Little is known about the impact of COVID-19 on the outcomes of patients undergoing surgery and intervention. This study was conducted between 20 March and 20 May 2020 in six hospitals in Istanbul, and aimed to investigate the effects of surgery and intervention on COVID-19 disease progression, intensive care (ICU) need, mortality and virus transmission to patients and healthcare workers. Patients were examined in three groups: group I underwent emergency surgery, group II had an emergency non-operating room intervention, and group III received inpatient COVID-19 treatment but did not have surgery or undergo intervention. Mortality rates, mechanical ventilation needs and rates of admission to the ICU were compared between the three groups. During this period, patient and healthcare worker transmissions were recorded. In total, 1273 surgical, 476 non-operating room intervention patients and 1884 COVID-19 inpatients were examined. The rate of ICU requirement among patients who had surgery was nearly twice that for inpatients and intervention patients, but there was no difference in mortality between the groups. The overall mortality rates were 2.3% in surgical patients, 3.3% in intervention patients and 3% in inpatients. COVID-19 polymerase chain reaction positivity among hospital workers was 2.4%. Only 3.3% of infected frontline healthcare workers were anaesthesiologists. No deaths occurred among infected healthcare workers. We conclude that emergency surgery and non-operating room interventions during the pandemic period do not increase postoperative mortality and can be performed with low transmission rates.

Anonymous and aggregated statistics derived from mobile phone data have proven efficacy as a proxy for human mobility in international development work and as inputs to epidemiological modeling of the spread of infectious diseases such as COVID-19. Despite the widely accepted promise of such data for better development outcomes, challenges persist in their systematic use across countries. This is not only the case for steady-state development use cases such as in the transport or urban development sectors, but also for sudden-onset emergencies such as epidemics in the health sector or natural disasters in the environment sector. This article documents an effort to gain systematized access to and use of anonymized, aggregated mobile phone data across 41 countries, leading to fruitful collaborations in nine developing countries over the course of one year. The research identifies recurring roadblocks and replicable successes, offers lessons learned, and calls for a bold vision for future successes. An emerging model for a future that enables steady-state access to insights derived from mobile big data - such that they are available over time for development use cases - will require investments in coalition building across multiple stakeholders, including local researchers and organizations, awareness raising of various key players, demand generation and capacity building, creation and adoption of standards to facilitate access to data and their ethical use, an enabling regulatory environment and long-term financing schemes to fund these activities.

Nosocomial severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) outbreaks among health care workers have been scarcely reported so far. This report presents the results of an epidemiologic and molecular investigation of a SARS-CoV-2 outbreak among laundromat facility workers in a large tertiary centre in Israel. Following the first three reported cases of SARS-CoV-2 among laundromat workers, all 49 laundromat personnel were screened by qRT-PCR tests using naso- and oropharingeal swabs. Epidemiologic investigations included questionnaires, interviews and observations of the laundromat facility. Eleven viral RNA samples were then sequenced, and a phylogenetic analysis was performed using MEGAX.

The integrated investigation defined three genetic clusters and helped identify the index cases and the assumed routes of transmission. It was then deduced that shared commute and public showers played a role in SARS-CoV-2 transmission in this outbreak, in addition to improper PPE use and social gatherings (such as social eating and drinking). In this study, we present an integrated epidemiologic and molecular investigation may help detect the routes of SARS-CoV-2 transmission, emphasising such routes that are less frequently discussed. Our work reinforces the notion that person-to-person transmission is more likely to cause infections than environmental contamination (e.g. from handling dirty laundry).

With the declaration of the coronavirus disease 2019 (COVID-19) pandemic in Nigeria in 2020, the Nigeria Governors’ Forum (NGF) instigated a collaboration with MTN Nigeria to develop data-driven insights, using mobile big data (MBD) and other data sources, to shape the planning and response to the pandemic. First, a model was developed to predict the worst-case scenario for infections in each state. This was used to support state-level health committees to make local resource planning decisions. Next, as containment interventions resulted in subsistence/daily paid workers losing their income and ability to buy essential food supplies, NGF and MTN agreed a second phase of activity, to develop insights to understand the population clusters at greatest socioeconomic risk from the impact of the pandemic. This insight was used to promote available financial relief to the economically vulnerable population clusters in Lagos state via the HelpNow crowdfunding initiative. This article discusses how anonymized and aggregated mobile network data (MBD), combined with other data sources, were used to create valuable insights and inform the government, and private business, response to the pandemic in Nigeria. Finally, we discuss lessons learnt. Firstly, how a collaboration with, and support from, the regulator enabled MTN to deliver critical insights at a national scale. Secondly, how the Nigeria Data Protection Regulation and the GSMA COVID-19 Privacy Guidelines provided an initial framework to open the discussion and define the approach. Thirdly, why stakeholder management is critical to the understanding, and application, of insights. Fourthly, how existing relationships ease new project collaborations. Finally, how MTN is developing future preparedness by creating a team that is focused on developing data-driven insights for social good.

In this paper, we present the work conducted by Vodafone to enrich the understanding of people movement in Italy during the outbreak of the Coronavirus in 2020, and the tool developed to support the decisions taken by the authorities during that period. We have developed a solution to anonymously monitor the daily movements of Vodafone SIMs in Italy, at aggregate level, at different spatial and temporal granularity, to provide insights into the movements of Italians.

Heterogeneity in the number of secondary tuberculosis (TB) cases per source case, the effective reproductive number, R, is important in modelling prevention strategies' impact on incidence.

We estimated mean R (Rm) and calculate the dispersion parameter of this distribution, k, using surveillance and genotyping data for U.S. cases during 2009–2018. We modelled transmission assuming cases in a cluster have matching genotypes and share characteristics related to geography, temporal proximity (i.e. serial interval) and time since U.S. arrival among non-U.S.-born persons.

Complete data were available for 55 330/85 958 cases. Varying the serial interval and geographic proximity used to derive clusters, we consistently estimated Rm<1.0 and k < 0.08; the low value of k indicates a small number of source cases produce a disproportionate number of secondary cases.

U.S. TB reproductive number has a highly skewed distribution, indicating a minority of source cases disproportionately contribute to transmission.

Cyclosporiasis is an illness characterised by watery diarrhoea caused by the food-borne parasite Cyclospora cayetanensis. The increase in annual US cyclosporiasis cases led public health agencies to develop genotyping tools that aid outbreak investigations. A team at the Centers for Disease Control and Prevention (CDC) developed a system based on deep amplicon sequencing and machine learning, for detecting genetically-related clusters of cyclosporiasis to aid epidemiologic investigations. An evaluation of this system during 2018 supported its robustness, indicating that it possessed sufficient utility to warrant further evaluation. However, the earliest version of CDC's system had some limitations from a bioinformatics standpoint. Namely, reliance on proprietary software, the inability to detect novel haplotypes and absence of a strategy to select an appropriate number of discrete genetic clusters would limit the system's future deployment potential. We recently introduced several improvements that address these limitations and the aim of this study was to reassess the system's performance to ensure that the changes introduced had no observable negative impacts. Comparison of epidemiologically-defined cyclosporiasis clusters from 2019 to analogous genetic clusters detected using CDC's improved system reaffirmed its excellent sensitivity (90%) and specificity (99%), and confirmed its high discriminatory power. This C. cayetanensis genotyping system is robust and with ongoing improvement will form the basis of a US-wide C. cayetanensis genotyping network for clinical specimens.