Given a finite strongly connected directed graph $G=(V, E)$, we study a Markov chain taking values on the space of probability measures on V. The chain, motivated by biological applications in the context of stochastic population dynamics, is characterized by transitions between states that respect the structure superimposed by E: mass (probability) can only be moved between neighbors in G. We provide conditions for the ergodicity of the chain. In a simple, symmetric case, we fully characterize the invariant probability.

Determining accurate capital requirements is a central activity across the life insurance industry. This is computationally challenging and often involves the acceptance of proxy errors that directly impact capital requirements. Within simulation-based capital models, where proxies are being used, capital estimates are approximations that contain both statistical and proxy errors. Here, we show how basic error analysis combined with targeted exact computation can entirely eliminate proxy errors from the capital estimate. Consideration of the possible ordering of losses, combined with knowledge of their error bounds, identifies an important subset of scenarios. When these scenarios are calculated exactly, the resulting capital estimate can be made devoid of proxy errors. Advances in the handling of proxy errors improve the accuracy of capital requirements.

This is the first report on a population-based prospective study of invasive group B streptococcus (GBS) disease among children aged <15 years conducted over a period of 11 years in Japan. This study investigated the incidence and clinical manifestations of invasive GBS disease in children in Chiba Prefecture, Japan, and analysed the serotypes and drug susceptibility of GBS strains isolated during the study period. Overall, 127 episodes of invasive GBS disease were reported in 123 patients. Of these, 124 were observed in 120 patients aged <1 year, and the remaining three episodes were reported in a 9-year-old child and two 14-year-old children with underlying disease. For patients aged <1 year, the incidence rate per 1000 live births was 0.24 (0.15–0.36). The incidences of early-onset disease and late-onset disease were 0.04 (0.0–0.09) and 0.17 (0.08–0.25), respectively. The rate of meningitis was 45.2%, and the incidence of GBS meningitis was higher than that of other invasive diseases among children in Japan. Of the 109 patients for whom prognosis was available, 7 (6.4%) died and 21 (19.3%) had sequelae. In total, 68 strains were analysed. The most common were serotype III strains (n = 42, 61.8%), especially serotype III/ST17 strains (n = 22, 32.4%). This study showed that the incidence of invasive GBS disease among Japanese children was constant during the study period. Because of the high incidence of meningitis and disease burden, new preventive strategies, such as GBS vaccine, are essential.

We present a recurrence–transience classification for discrete-time Markov chains on manifolds with negative curvature. Our classification depends only on geometric quantities associated to the increments of the chain, defined via the Riemannian exponential map. We deduce that a recurrent chain that has zero average drift at every point cannot be uniformly elliptic, unlike in the Euclidean case. We also give natural examples of zero-drift recurrent chains on negatively curved manifolds, including on a stochastically incomplete manifold.

Bovine tuberculosis (bTB) is a chronic, infectious and zoonotic disease of domestic and wild animals caused mainly by Mycobacterium bovis. This study investigated farm management factors associated with recurrent bTB herd breakdowns (n = 2935) disclosed in the period 23 May 2016 to 21 May 2018 and is a follow-up to our 2020 paper which looked at long duration bTB herd breakdowns. A case control study design was used to construct an explanatory set of farm-level management factors associated with recurrent bTB herd breakdowns. In Northern Ireland, a Department of Agriculture Environment and Rural Affairs (DAERA) Veterinarian investigates bTB herd breakdowns using standardised guidelines to allocate a disease source. In this study, source was strongly linked to carryover of infection, suggesting that the diagnostic tests had failed to clear herd infection during the breakdown period. Other results from this study associated with recurrent bTB herd breakdowns were herd size and type (dairy herds 43% of cases), with both these variables intrinsically linked. Other associated risk factors were time of application of slurry, badger access to silage clamps, badger setts in the locality, cattle grazing silage fields immediately post-harvest, number of parcels of land the farmer associated with bTB, number of land parcels used for grazing and region of the country.

We study multivariate polynomials over ‘structured’ grids. Firstly, we propose an interpretation as to what it means for a finite subset of a field to be structured; we do so by means of a numerical parameter, the nullity. We then extend several results – notably, the Combinatorial Nullstellensatz and the Coefficient Theorem – to polynomials over structured grids. The main point is that the structure of a grid allows the degree constraints on polynomials to be relaxed.

Household transmission plays a key role in the spread of COVID-19 through populations. In this paper, we report on the transmission of COVID-19 within households in a metropolitan area in Australia, examine the impact of various factors and highlight priority areas for future public health responses. We collected and reviewed retrospective case report data and follow-up interview responses from households with a positive case of the Delta COVID-19 variant in Queensland in 2021. The overall secondary attack rate (SAR) among household contacts was 29.6% and the mean incubation period for secondary cases was 4.3 days. SAR was higher where the index case was male (57.9% vs. 14.3%) or aged ≤12 years (38.7% vs. 17.4%) but similar for adult contacts that were double vaccinated (35.7%) and unvaccinated (33.3%). Most interview participants emphasised the importance of clear, consistent and compassionate health advice as a key priority for managing outbreaks in the home. The overall rate of household transmission was slightly higher than that reported in previous studies on the wild COVID-19 variant and secondary infections developed more rapidly. While vaccination did not appear to affect the risk of transmission to adult subjects, uptake in the sample was ultimately high.

The aim of the present article is to evaluate the use of the Autoregressive Fractionally Integrated Moving Average (ARFIMA) model in predicting spatially and temporally localized political violent events using the Integrated Crisis Early Warning System (ICEWS). The performance of the ARFIMA model is compared to that of a naïve model in reference to two common relevant hypotheses: the ARFIMA model would outperform a naïve model and the rate of outperformance would deteriorate the higher the level of spatial aggregation. This analytical strategy is used to predict political violent events in Afghanistan. The analysis consists of three parts. The first is a replication of Yonamine’s study for the period beginning in April 2010 and ending in March 2012. The second part compares the results to those of Yonamine. The comparison was used to assess the validity of the conclusions drawn in the original study, which was based on the Global Database of Events, Language, and Tone, for the implementation of this approach to ICEWS data. Building on the conclusions of this comparison, the third part uses Yonamine’s approach to predict violent events in Afghanistan over a significantly longer period of time (January 1995–August 2021). The conclusions provide an assessment of the utility of short-term localized forecasting.

Motivated by problems from compressed sensing, we determine the threshold behaviour of a random $n\times d \pm 1$ matrix $M_{n,d}$ with respect to the property ‘every $s$ columns are linearly independent’. In particular, we show that for every $0\lt \delta \lt 1$ and $s=(1-\delta )n$, if $d\leq n^{1+1/2(1-\delta )-o(1)}$ then with high probability every $s$ columns of $M_{n,d}$ are linearly independent, and if $d\geq n^{1+1/2(1-\delta )+o(1)}$ then with high probability there are some $s$ linearly dependent columns.

We consider an (R, Q) inventory model with two types of orders, normal orders and emergency orders, which are issued at different inventory levels. These orders are delivered after exponentially distributed lead times. In between deliveries, the inventory level decreases in a state-dependent way, according to a release rate function $\alpha({\cdot})$. This function represents the fluid demand rate; it could be controlled by a system manager via price adaptations. We determine the mean number of downcrossings $\theta(x)$ of any level x in one regenerative cycle, and use it to obtain the steady-state density f (x) of the inventory level. We also derive the rates of occurrence of normal deliveries and of emergency deliveries, and the steady-state probability of having zero inventory.

Strengthening Hadwiger’s conjecture, Gerards and Seymour conjectured in 1995 that every graph with no odd $K_t$-minor is properly $(t-1)$-colourable. This is known as the Odd Hadwiger’s conjecture. We prove a relaxation of the above conjecture, namely we show that every graph with no odd $K_t$-minor admits a vertex $(2t-2)$-colouring such that all monochromatic components have size at most $\lceil \frac{1}{2}(t-2) \rceil$. The bound on the number of colours is optimal up to a factor of $2$, improves previous bounds for the same problem by Kawarabayashi (2008, Combin. Probab. Comput. 17 815–821), Kang and Oum (2019, Combin. Probab. Comput. 28 740–754), Liu and Wood (2021, arXiv preprint, arXiv:1905.09495), and strengthens a result by van den Heuvel and Wood (2018, J. Lond. Math. Soc. 98 129–148), who showed that the above conclusion holds under the more restrictive assumption that the graph is $K_t$-minor-free. In addition, the bound on the component-size in our result is much smaller than those of previous results, in which the dependency on $t$ was given by a function arising from the graph minor structure theorem of Robertson and Seymour. Our short proof combines the method by van den Heuvel and Wood for $K_t$-minor-free graphs with some additional ideas, which make the extension to odd $K_t$-minor-free graphs possible.

This article derives quantitative limit theorems for multivariate Poisson and Poisson process approximations. Employing the solution of the Stein equation for Poisson random variables, we obtain an explicit bound for the multivariate Poisson approximation of random vectors in the Wasserstein distance. The bound is then utilized in the context of point processes to provide a Poisson process approximation result in terms of a new metric called $d_\pi$, stronger than the total variation distance, defined as the supremum over all Wasserstein distances between random vectors obtained by evaluating the point processes on arbitrary collections of disjoint sets. As applications, the multivariate Poisson approximation of the sum of m-dependent Bernoulli random vectors, the Poisson process approximation of point processes of U-statistic structure, and the Poisson process approximation of point processes with Papangelou intensity are considered. Our bounds in $d_\pi$ are as good as those already available in the literature.

This paper provides a stochastic model, consistent with Solvency II and the Delegated Regulation, to quantify the capital requirement for demographic risk. In particular, we present a framework that models idiosyncratic and trend risks exploiting a risk theory approach in which results are obtained analytically. We apply the model to non-participating policies and quantify the Solvency Capital Requirement for the aforementioned risks in different time horizons.