Abstract. We study the susceptible-infective-recovered (SIR) epidemic on a random graph chosen uniformly subject to having given vertex degrees. In this model infective vertices infect each of their susceptible neighbours, and recover, at a constant rate.Suppose that initially there are only a few infective vertices. We prove there is a threshold for a parameter involving the rates and vertex degrees below which only a small number of infections occur. Above the threshold a large outbreak occurs with probability bounded away from zero. Our main result is that, conditional on a large outbreak, the evolutions of certain quantities of interest, such as the fraction of infective vertices, converge to deterministic functions of time.We also consider more general initial conditions for the epidemic, and derive criteria for a simple vaccination strategy to be successful.In contrast to earlier results for this model, our approach only requires basic regularity conditions and a uniformly bounded second moment of the degree of a random vertex.En route, we prove analogous results for the epidemic on the configuration model multigraph under much weaker conditions. Essentially, our main result requires only that the initial values for our processes converge, i.e. it is the best possible.

We give three examples of stochastic processes in the Gelfand-Tsetlin cone in which each component evolves independently apart from a blocking and pushing interaction. These processes give rise to couplings between certain conditioned Markov processes, last passage times and exclusion processes. In the first two examples, we deduce known identities in distribution between such processes whilst in the third example, the components of the process cannot escape past a wall at the origin and we obtain a new relation.

Abstract. These notes give a mathematical introduction to two seemingly unrelated topics: (i) quantum spin systems and their cycle and loop representations, due to Tóth and Aizenman-Nachtergaele; (ii) coagulation-fragmentation stochastic processes. These topics are nonetheless related, as we argue that the lengths of cycles and loops effectively perform a coagulation-fragmentation process. This suggests that their joint distribution is Poisson-Dirichlet. These ideas are far from being proved, but they are backed by several rigorous results, notably of Dyson-Lieb-Simon and Schramm.

Women vaccinated against HPV16/18 are approaching the age for cervical screening; however, an updated screening algorithm has not been agreed. We use a microsimulation model calibrated to real published data to determine the appropriate screening intensity for vaccinated women. Natural histories in the absence of vaccination were simulated for 300,000 women using 10,000 sets of transition probabilities. Vaccination with (i) 100% efficacy against HPV16/18, (ii) 15% cross‐protection, (iii) 22% cross‐protection, (iv) waning vaccine efficacy and (v) 100% efficacy against HPV16/18/31/33/45/52/58 was added, as were a range of screening scenarios appropriate to the UK. To benchmark cost‐benefits of screening for vaccinated women, we evaluated the proportion of cancers prevented per additional screen (incremental benefit) of current cytology and likely HPV screening scenarios in unvaccinated women. Slightly more cancers are prevented through vaccination with no screening (70.3%, 95% CR: 65.1–75.5) than realistic compliance to the current UK screening programme in the absence of vaccination (64.3%, 95% CR: 61.3–66.8). In unvaccinated women, when switching to HPV primary testing, there is no loss in effectiveness when doubling the screening interval. Benchmarking supports screening scenarios with incremental benefits of ≥2.0%, and rejects scenarios with incremental benefits ≤0.9%. In HPV16/18‐vaccinated women, the incremental benefit of offering a third lifetime screen was at most 3.3% (95% CR: 2.2–4.5), with an incremental benefit of 1.3% (−0.3–2.8) for a fourth screen. For HPV16/18/31/33/45/52/58‐vaccinated women, two lifetime screens are supported. It is important to know women's vaccination status; in these simulations, HPV16/18‐vaccinated women require three lifetime screens, HPV16/18/31/33/45/52/58‐vaccinated women require two lifetime screens, yet unvaccinated women require seven lifetime screens.

SummaryBackgroundIn the next 25 years, the epidemiology of cervical cancer in England, UK, will change: human papillomavirus (HPV) screening will be the primary test for cervical cancer. Additionally, the proportion of women screened regularly is decreasing and women who received the HPV vaccine are due to attend screening for the first time. Therefore, we aimed to estimate how vaccination against HPV, changes to the screening test, and falling screening coverage will affect cervical cancer incidence in England up to 2040.MethodsWe did a data modelling study that combined results from population modelling of incidence trends, observable data from the individual level with use of a generalised linear model, and microsimulation of unobservable disease states. We estimated age-specific absolute risks of cervical cancer in the absence of screening (derived from individual level data). We used an age period cohort model to estimate birth cohort effects. We multiplied the absolute risks by the age cohort effects to provide absolute risks of cervical cancer for unscreened women in different birth cohorts. We obtained relative risks (RRs) of cervical cancer by screening history (never screened, regularly screened, or lapsed attender) using data from a population-based case-control study for unvaccinated women, and using a microsimulation model for vaccinated women. RRs of primary HPV screening were relative to cytology. We used the proportion of women in each 5-year age group (25–29 years to 75–79 years) and 5-year period (2016–20 to 2036–40) who have a combination of screening and vaccination history, and weighted to estimate the population incidence. The primary outcome was the number of cases and rates per 100 000 women under four scenarios: no changes to current screening coverage or vaccine uptake and HPV primary testing from 2019 (status quo), changing the year in which HPV primary testing is introduced, introduction of the nine-valent vaccine, and changes to cervical screening coverage.FindingsThe status quo scenario estimated that the peak age of cancer diagnosis will shift from the ages of 25–29 years in 2011–15 to 55–59 years in 2036–40. Unvaccinated women born between 1975 and 1990 were predicted to have a relatively high risk of cervical cancer throughout their lives. Introduction of primary HPV screening from 2019 could reduce age-standardised rates of cervical cancer at ages 25–64 years by 19%, from 15·1 in 2016 to 12·2 per 100 000 women as soon as 2028. Vaccination against HPV types 16 and 18 (HPV 16/18) could see cervical cancer rates in women aged 25–29 years decrease by 55% (from 20·9 in 2011–15 to 9·5 per 100 000 women by 2036–40), and introduction of nine-valent vaccination from 2019 compared with continuing vaccination against HPV 16/18 will reduce rates by a further 36% (from 9·5 to 6·1 per 100 000 women) by 2036–40. Women born before 1991 will not benefit directly from vaccination; therefore, despite vaccination and primary HPV screening with current screening coverage, European age-standardised ra...