The combinatorics of minimal unsatisfiability: connecting to graph theory

Minimally Unsatisfiable CNFs (MUs) are unsatisfiable CNFs where removing any clause destroys unsatisfiability. MUs are the building blocks of unsatisfia-bility, and our understanding of them can be very helpful in answering various algorithmic and structural questions relating to unsatisfiability. In this thesis we study MUs from a combinatorial point of view, with the aim of extending the understanding of the structure of MUs. We show that some important classes of MUs are very closely related to known classes of digraphs, and using arguments from logic and graph theory we characterise these MUs.Two main concepts in this thesis are isomorphism of CNFs and the implica-tion digraph of 2-CNFs (at most two literals per disjunction). Isomorphism of CNFs involves renaming the variables, and flipping the literals. The implication digraph of a 2-CNF F has both arcs (¬a → b) and (¬b → a) for every binary clause (a ∨ b) in F .In the first part we introduce a novel connection between MUs and Minimal Strong Digraphs (MSDs), strongly connected digraphs, where removing any arc destroys the strong connectedness. We introduce the new class DFM of special MUs, which are in close correspondence to MSDs. The known relation between 2-CNFs and implication digraphs is used, but in a simpler and more direct way, namely that we have a canonical choice of one of the two arcs. As an application of this new framework we provide short and intuitive new proofs for two im-portant but isolated characterisations for nonsingular MUs (every literal occurs at least twice), both with ingenious but complicated proofs: Characterising 2-MUs (minimally unsatisfiable 2-CNFs), and characterising MUs with deficiency 2 (two more clauses than variables).In the second part, we provide a fundamental addition to the study of 2-CNFs which have efficient algorithms for many interesting problems, namely that we provide a full classification of 2-MUs and a polytime isomorphism de-cision of this class. We show that implication digraphs of 2-MUs are “Weak Double Cycles” (WDCs), big cycles of small cycles (with possible overlaps). Combining logical and graph-theoretical methods, we prove that WDCs have at most one skew-symmetry (a self-inverse fixed-point free anti-symmetry, re-versing the direction of arcs). It follows that the isomorphisms between 2-MUs are exactly the isomorphisms between their implication digraphs (since digraphs with given skew-symmetry are the same as 2-CNFs). This reduces the classifi-cation of 2-MUs to the classification of a nice class of digraphs.Finally in the outlook we discuss further applications, including an alter-native framework for enumerating some special Minimally Unsatisfiable Sub-clause-sets (MUSs)

Clinical coding of long Covid in Wales: A cohort study of 3.5 million people using linked health and demographic data

Objectives ‘Long COVID’ (LC) is broadly defined as signs and symptoms that continue or develop after the acute phase of COVID-19, and can affect cardiovascular, respiratory and other organ systems. Using electronic health records, we investigated clinical coding of LC in primary and secondary care for the population of Wales. Methods We conducted a cohort study for the population of Wales, using anonymised individual-level linked data in the Secure Anonymised Information Linkage (SAIL) Databank. We used the Welsh COVID-19 e-cohort (doi:10.1136/bmjopen-2020-043010), which consists of all people (adults and children) alive and resident in Wales from 1st January 2020. To this e-cohort we linked primary and secondary care, COVID-19 testing, and ethnic group data. We then calculated the proportion of people with a LC diagnosis code (in primary and secondary care data) overall and stratified by demographic variables. Results Of 3.5m residents, 7,696 (0.2%) had a LC clinical diagnosis. Compared with the general population, a higher proportion of people with LC were female, middle age, white, and hospitalised within 28 days of a confirmed COVID-19 infection. LC affected all socioeconomic groups, as assessed using the Welsh Index of Multiple Deprivation. When looking at LC diagnosis codes in primary care, 30.9% of practices in SAIL have not used these codes at all. And the number of recorded events was low until the end of January 2021, after which there was an increase in coding. These findings are likely a substantial underestimate of LC prevalence in Wales. Earlier estimates from self-reported surveys, such as the Office for National Statistics, are much higher, ranging anywhere between 3-5%. Conclusion Low recording rates of LC and variation between practices could be due to a delay in introducing clinical coding and lack of presentation/recording. Understanding prevalence of LC is vital for addressing the scale of the problem. Therefore developing additional data-driven approaches is necessary to obtain an accurate prevalence estimate

Harmonising electronic health records for reproducible research: challenges, solutions and recommendations from a UK-wide COVID-19 research collaboration

BackgroundThe CVD-COVID-UK consortium was formed to understand the relationship between COVID-19 and cardiovascular diseases through analyses of harmonised electronic health records (EHRs) across the four UK nations. Beyond COVID-19, data harmonisation and common approaches enable analysis within and across independent Trusted Research Environments. Here we describe the reproducible harmonisation method developed using large-scale EHRs in Wales to accommodate the fast and efficient implementation of cross-nation analysis in England and Wales as part of the CVD-COVID-UK programme. We characterise current challenges and share lessons learnt.MethodsServing the scope and scalability of multiple study protocols, we used linked, anonymised individual-level EHR, demographic and administrative data held within the SAIL Databank for the population of Wales. The harmonisation method was implemented as a four-layer reproducible process, starting from raw data in the first layer. Then each of the layers two to four is framed by, but not limited to, the characterised challenges and lessons learnt. We achieved curated data as part of our second layer, followed by extracting phenotyped data in the third layer. We captured any project-specific requirements in the fourth layer.ResultsUsing the implemented four-layer harmonisation method, we retrieved approximately 100 health-related variables for the 3.2 million individuals in Wales, which are harmonised with corresponding variables for > 56 million individuals in England. We processed 13 data sources into the first layer of our harmonisation method: five of these are updated daily or weekly, and the rest at various frequencies providing sufficient data flow updates for frequent capturing of up-to-date demographic, administrative and clinical information.ConclusionsWe implemented an efficient, transparent, scalable, and reproducible harmonisation method that enables multi-nation collaborative research. With a current focus on COVID-19 and its relationship with cardiovascular outcomes, the harmonised data has supported a wide range of research activities across the UK

Staff-Pupil SARS-CoV-2 Infection Pathways in Schools: A Population Level Linked Data Approach

Background Better understanding of the role that children and school staff play in the transmission of SARS-CoV-2 is essential to guide policy development on controlling infection while minimising disruption to children’s education and well-being.Methods Our national e-cohort (n=464531) study used anonymised linked data for pupils, staff and associated households linked via educational settings in Wales. We estimated the odds of testing positive for SARS-CoV-2 infection for staff and pupils over the period August– December 2020, dependent on measures of recent exposure to known cases linked to their educational settings.Results The total number of cases in a school was not associated with a subsequent increase in the odds of testing positive (staff OR per case: 0.92, 95% CI 0.85 to 1.00; pupil OR per case: 0.98, 95% CI 0.93 to 1.02). Among pupils, the number of recent cases within the same year group was significantly associated with subsequent increased odds of testing positive (OR per case: 1.12, 95% CI 1.08 to 1.15). These effects were adjusted for a range of demographic covariates, and in particular any known cases within the same household, which had the strongest association with testing positive (staff OR: 39.86, 95% CI 35.01 to 45.38; pupil OR: 9.39, 95% CI 8.94 to 9.88).Conclusions In a national school cohort, the odds of staff testing positive for SARS-CoV-2 infection were not significantly increased in the 14-day period after case detection in the school. However, pupils were found to be at increased odds, following cases appearing within their own year group, where most of their contacts occur. Strong mitigation measures over the whole of the study period may have reduced wider spread within the school environment

Risk of thrombocytopenic, haemorrhagic and thromboembolic disorders following COVID-19 vaccination and positive test: a self-controlled case series analysis in Wales

There is a need for better understanding of the risk of thrombocytopenic, haemorrhagic, thromboembolic disorders following first, second and booster vaccination doses and testing positive for SARS-CoV-2. Self-controlled cases series analysis of 2.1 million linked patient records in Wales between 7th December 2020 and 31st December 2021. Outcomes were the first diagnosis of thrombocytopenic, haemorrhagic and thromboembolic events in primary or secondary care datasets, exposure was defined as 0–28 days post-vaccination or a positive reverse transcription polymerase chain reaction test for SARS-CoV-2. 36,136 individuals experienced either a thrombocytopenic, haemorrhagic or thromboembolic event during the study period. Relative to baseline, our observations show greater risk of outcomes in the periods post-first dose of BNT162b2 for haemorrhagic (IRR 1.47, 95%CI: 1.04–2.08) and idiopathic thrombocytopenic purpura (IRR 2.80, 95%CI: 1.21–6.49) events; post-second dose of ChAdOx1 for arterial thrombosis (IRR 1.14, 95%CI: 1.01–1.29); post-booster greater risk of venous thromboembolic (VTE) (IRR-Moderna 3.62, 95%CI: 0.99–13.17) (IRR-BNT162b2 1.39, 95%CI: 1.04–1.87) and arterial thrombosis (IRR-Moderna 3.14, 95%CI: 1.14–8.64) (IRR-BNT162b2 1.34, 95%CI: 1.15–1.58). Similarly, post SARS-CoV-2 infection the risk was increased for haemorrhagic (IRR 1.49, 95%CI: 1.15–1.92), VTE (IRR 5.63, 95%CI: 4.91, 6.4), arterial thrombosis (IRR 2.46, 95%CI: 2.22–2.71). We found that there was a measurable risk of thrombocytopenic, haemorrhagic, thromboembolic events after COVID-19 vaccination and infection

Impact of COVID-19 pandemic on community medication dispensing: a national cohort analysis in Wales, UK

BackgroundPopulation-level information on dispensed medication provides insight on the distribution of treated morbidities, particularly if linked to other population-scale data at an individual-level.ObjectiveTo evaluate the impact of COVID-19 on dispensing patterns of medications.MethodsRetrospective observational study using population-scale, individual-level dispensing records in Wales, UK. Total dispensed drug items for the population between 1st January 2016 and 31st December 2019 (3-years, pre-COVID-19) were compared to 2020 with follow up until 27th July 2021 (COVID-19 period). We compared trends across all years and British National Formulary (BNF) chapters and highlighted the trends in three major chapters for 2019-21: 1-Cardiovascular system (CVD); 2-Central Nervous System (CNS); 3-Immunological & Vaccine. We developed an interactive dashboard to enable monitoring of changes as the pandemic evolves.ResultAmongst all BNF chapters, 73,410,543 items were dispensed in 2020 compared to 74,121,180 items in 2019 demonstrating -0.96% relative decrease in 2020. Comparison of monthly patterns showed average difference (D) of -59,220 and average Relative Change (RC) of -0.74% between the number of dispensed items in 2020 and 2019. Maximum RC was observed in March 2020 (D= +1,224,909 and RC= +20.62%), followed by second peak in June 2020 (D= +257,920, RC= +4.50%). A third peak was observed in September 2020 (D= +264,138, RC= +4.35%). Large increases in March 2020 were observed for CVD and CNS medications across all age groups. The Immunological and Vaccine products dropped to very low levels across all age groups and all months (including the March dispensing peak).ConclusionsReconfiguration of routine clinical services during COVID-19 led to substantial changes in community pharmacy drug dispensing. This change may contribute to a long-term burden of COVID-19, raising the importance of a comprehensive and timely monitoring of changes for evaluation of the potential impact on clinical care and outcomes

COVID-19 mitigation measures in primary schools and association with infection and school staff wellbeing: An observational survey linked with routine data in Wales, UK

IntroductionSchool-based COVID-19 mitigation strategies have greatly impacted the primary school day (children aged 3–11) including: wearing face coverings, two metre distancing, no mixing of children, and no breakfast clubs or extra-curricular activities. This study examines these mitigation measures and association with COVID-19 infection, respiratory infection, and school staff wellbeing between October to December 2020 in Wales, UK.MethodsA school staff survey captured self-reported COVID-19 mitigation measures in the school, participant anxiety and depression, and open-text responses regarding experiences of teaching and implementing measures. These survey responses were linked to national-scale COVID-19 test results data to examine association of measures in the school and the likelihood of a positive (staff or pupil) COVID-19 case in the school (clustered by school, adjusted for school size and free school meals using logistic regression). Linkage was conducted through the SAIL (Secure Anonymised Information Linkage) Databank.ResultsResponses were obtained from 353 participants from 59 primary schools within 15 of 22 local authorities. Having more direct non-household contacts was associated with a higher likelihood of COVID-19 positive case in the school (1–5 contacts compared to none, OR 2.89 (1.01, 8.31)) and a trend to more self-reported cold symptoms. Staff face covering was not associated with a lower odds of school COVID-19 cases (mask vs. no covering OR 2.82 (1.11, 7.14)) and was associated with higher self-reported cold symptoms. School staff reported the impacts of wearing face coverings on teaching, including having to stand closer to pupils and raise their voices to be heard. 67.1% were not able to implement two metre social distancing from pupils. We did not find evidence that maintaining a two metre distance was associated with lower rates of COVID-19 in the school.ConclusionsImplementing, adhering to and evaluating COVID-19 mitigation guidelines is challenging in primary school settings. Our findings suggest that reducing non-household direct contacts lowers infection rates. There was no evidence that face coverings, two metre social distancing or stopping children mixing was associated with lower odds of COVID-19 or cold infection rates in the school. Primary school staff found teaching challenging during COVID-19 restrictions, especially for younger learners and those with additional learning needs

Trends in SARS-CoV-2 infection and vaccination in school staff, students and their household members from 2020 to 2022 in Wales, UK: an electronic cohort study

Objectives We investigated SARS-CoV-2 infection trends, risk of SARS-CoV-2 infection and COVID-19 vaccination uptake among school staff, students and their household members in Wales, UK. Design Seven-day average of SARS-CoV-2 infections and polymerase chain reaction tests per 1000 people daily, cumulative incidence of COVID-19 vaccination uptake and multi-level Poisson models with time-varying covariates. Setting National electronic cohort between September 2020 and May 2022 when several variants were predominant in the UK (Alpha, Delta and Omicron). Participants School students aged 4 to 10/11 years (primary school and younger middle school, n = 238,163), and 11 to 15/16 years (secondary school and older middle school, n = 182,775), school staff in Wales (n = 47,963) and the household members of students and staff (n = 697,659). Main outcome measures SARS-CoV-2 infection and COVID-19 vaccination uptake. Results School students had a sustained period of high infection rates compared with household members after August 2021. Primary schedule vaccination uptake was highest among staff (96.3%) but lower for household members (72.2%), secondary and older middle school students (59.8%), and primary and younger middle school students (3.3%). Multi-level Poisson models showed that vaccination was associated with a lower risk of SARS-CoV-2 infection. The Delta variant posed a greater infection risk for students than the Alpha variant. However, Omicron was a larger risk for staff and household members. Conclusions Public health bodies should be informed of the protection COVID-19 vaccines afford, with more research being required for younger populations. Furthermore, schools require additional support in managing new, highly transmissible variants. Further research should examine the mechanisms between child deprivation and SARS-CoV-2 infection