We consider the threshold-one contact process, the threshold-one voter model and the threshold-one voter model with positive spontaneous death on homogeneous trees $\mathbb{T}_d$, $d\ge 2$. Mainly inspired by the corresponding arguments for the contact process, we prove that the complete convergence theorem holds for these three systems under strong survival. When the system survives weakly, complete convergence may also hold under certain transition and/or initial conditions.

This paper studies the scaling of the expected total queue size in an $n\times n$ input-queued switch, as a function of both the load $\rho$ and the system scale n. We provide a new class of scheduling policies under which the expected total queue size scales as $O\big( n(1-\rho)^{-4/3} \log \big(\!\max\big\{\frac{1}{1-\rho}, n\big\}\big)\big)$, over all n and $\rho<1$, when the arrival rates are uniform. This improves on the best previously known scalings in two regimes: $O\big(n^{1.5}(1-\rho)^{-1} \log \frac{1}{1-\rho}\big)$ when $\Omega\big(n^{-1.5}\big) \le 1-\rho \le O\big(n^{-1}\big)$ and $O\big(\frac{n\log n}{(1-\rho)^2}\big)$ when $1-\rho \geq \Omega(n^{-1})$. A key ingredient in our method is a tight characterization of the largest k-factor of a random bipartite multigraph, which may be of independent interest.

Perron–Frobenius theory developed for irreducible non-negative kernels deals with so-called R-positive recurrent kernels. If the kernel M is R-positive recurrent, then the main result determines the limit of the scaled kernel iterations $R^nM^n$ as $n\to\infty$. In Nummelin (1984) this important result is proven using a regeneration method whose major focus is on M having an atom. In the special case when $M=P$ is a stochastic kernel with an atom, the regeneration method has an elegant explanation in terms of an associated split chain. In this paper we give a new probabilistic interpretation of the general regeneration method in terms of multi-type Galton–Watson processes producing clusters of particles. Treating clusters as macro-individuals, we arrive at a single-type Crump–Mode–Jagers process with a naturally embedded renewal structure.

Enterotoxigenic Escherichia coli (ETEC) is a well-established cause of traveller's diarrhoea and occasional domestic foodborne illness outbreaks in the USA. Although ETEC are not detected by conventional stool culture methods used in clinical laboratories, syndromic culture-independent diagnostic tests (CIDTs) capable of detecting ETEC have become increasingly prevalent in the last decade. This study describes the epidemiology of ETEC infections reported to the Minnesota Department of Health (MDH) during 2016–2017. ETEC-positive stool specimens were submitted to MDH to confirm the presence of ETEC DNA by polymerase chain reaction (PCR). Cases were interviewed to ascertain illness and exposures. Contemporaneous Salmonella cases were used as a comparison group in a case-case comparison analysis of risk factors. Of 222 ETEC-positive specimens received by MDH, 108 (49%) were concordant by PCR. ETEC was the sixth most frequently reported bacterial enteric pathogen among a subset of CIDT-positive specimens. Sixty-nine (64%) laboratory-confirmed cases had an additional pathogen codetected with ETEC, including enteroaggregative E. coli (n = 40) and enteropathogenic E. coli (n = 39). Although travel is a risk factor for ETEC infection, only 43% of cases travelled internationally, providing evidence for ETEC as an underestimated source of domestically acquired enteric illness in the USA.

Bonus-malus systems (BMSs) are widely used in actuarial sciences. These systems are applied by insurance companies to distinguish the policyholders by their risks. The most known application of BMS is in automobile third-party liability insurance. In BMS, there are several classes, and the premium of a policyholder depends on the class he/she is assigned to. The classification of policyholders over the periods of the insurance depends on the transition rules. In general, optimization of these systems involves the calculation of an appropriate premium scale considering the number of classes and transition rules as external parameters. Usually, the stationary distribution is used in the optimization process. In this article, we present a mixed integer linear programming (MILP) formulation for determining the premium scale and the transition rules. We present two versions of the model, one with the calculation of stationary probabilities and another with the consideration of multiple periods of the insurance. Furthermore, numerical examples will also be given to demonstrate that the MILP technique is suitable for handling existing BMSs.

Let (Y, Z) denote the solution to a forward-backward stochastic differential equation (FBSDE). If one constructs a random walk $B^n$ from the underlying Brownian motion B by Skorokhod embedding, one can show $L_2$-convergence of the corresponding solutions $(Y^n,Z^n)$ to $(Y, Z).$ We estimate the rate of convergence based on smoothness properties, especially for a terminal condition function in $C^{2,\alpha}$. The proof relies on an approximative representation of $Z^n$ and uses the concept of discretized Malliavin calculus. Moreover, we use growth and smoothness properties of the partial differential equation associated to the FBSDE, as well as of the finite difference equations associated to the approximating stochastic equations. We derive these properties by probabilistic methods.

In this paper we study first passage percolation on a random graph model, the configuration model. We first introduce the notions of weighted diameter, which is the maximum of the weighted lengths of all optimal paths between any two vertices in the graph, and the flooding time, which represents the time (weighted length) needed to reach all the vertices in the graph starting from a uniformly chosen vertex. Our result consists in describing the asymptotic behavior of the diameter and the flooding time, as the number of vertices n tends to infinity, in the case where the weight distribution G has an exponential tail behavior, and proving that this category of distributions is the largest possible for which the asymptotic behavior holds.

In a classical, continuous-time, optimal stopping problem, the agent chooses the best time to stop a stochastic process in order to maximise the expected discounted return. The agent can choose when to stop, and if at any moment they decide to stop, stopping occurs immediately with probability one. However, in many settings this is an idealistic oversimplification. Following Strack and Viefers we consider a modification of the problem in which stopping occurs at a rate which depends on the relative values of stopping and continuing: there are several different solutions depending on how the value of continuing is calculated. Initially we consider the case where stopping opportunities are constrained to be event times of an independent Poisson process. Motivated by the limiting case as the rate of the Poisson process increases to infinity, we also propose a continuous-time formulation of the problem where stopping can occur at any instant.

We prove a ‘resilience’ version of Dirac’s theorem in the setting of random regular graphs. More precisely, we show that whenever d is sufficiently large compared to $\epsilon > 0$, a.a.s. the following holds. Let $G'$ be any subgraph of the random n-vertex d-regular graph $G_{n,d}$ with minimum degree at least $$(1/2 + \epsilon )d$$. Then $G'$ is Hamiltonian.

This proves a conjecture of Ben-Shimon, Krivelevich and Sudakov. Our result is best possible: firstly the condition that d is large cannot be omitted, and secondly the minimum degree bound cannot be improved.

We report key learning from the public health management of the first two confirmed cases of COVID-19 identified in the UK. The first case imported, and the second associated with probable person-to-person transmission within the UK. Contact tracing was complex and fast-moving. Potential exposures for both cases were reviewed, and 52 contacts were identified. No further confirmed COVID-19 cases have been linked epidemiologically to these two cases. As steps are made to enhance contact tracing across the UK, the lessons learned from earlier contact tracing during the country's containment phase are particularly important and timely.

Let M be an n × m matrix of independent Rademacher (±1) random variables. It is well known that if $n \leq m$, then M is of full rank with high probability. We show that this property is resilient to adversarial changes to M. More precisely, if $m \ge n + {n^{1 - \varepsilon /6}}$, then even after changing the sign of (1 – ε)m/2 entries, M is still of full rank with high probability. Note that this is asymptotically best possible as one can easily make any two rows proportional with at most m/2 changes. Moreover, this theorem gives an asymptotic solution to a slightly weakened version of a conjecture made by Van Vu in [17].

India is one of the severely affected countries by the Covid-19 pandemic at present. Within the stochastic framework of the SEQIR model, we studied publicly available data of the Covid-19 patients in India and analysed possible impacts of quarantine and social distancing as controlling strategies for the pandemic. Our stochastic simulation results clearly show that proper quarantine and social distancing should be maintained right from the start of the pandemic and continued until its end for effective control. This calls for a more disciplined social lifestyle in the future. However, only social distancing and quarantine of the exposed population are found not sufficient enough to end the pandemic in India. Therefore, implementation of other stringent policies like complete lockdown as well as increased testing of susceptible populations is necessary. The demographic stochasticity, which is quite visible in the system dynamics, has a critical role in regulating and controlling the pandemic.

During the coronavirus disease 2019 (COVID-19) pandemic, a new phenomenon manifesting as a multisystem inflammatory syndrome in children (MIS-C) which has a similar clinical presentation to Kawasaki disease, toxic shock syndrome and severe sepsis has emerged. Although the number of MIS-C reports is increasing, rare reports in Asia is still available. To our knowledge, this study is the largest series of published MIS-C cases in Iran. We performed a retrospective study of all patients with case definition for MIS-C admitted to the three paediatric hospitals in Iran. All of these hospitals are located within the most active COVID-19 pandemic areas (Tehran, Qom and Mazandaran) in Iran. Demographic characteristics, clinical data, laboratory findings, imaging and echocardiographic findings, treatment and outcomes were collected. Between 7 March and 23 June 2020, 45 children were included in the study. The median age of children was 7 years (range between 10 months and 17 years). Common presenting symptoms include fever (91%), abdominal pain (58%), nausea/vomiting (51%), mucocutaneous rash (53%), conjunctivitis (51%) and hands and feet oedema (40%) with median duration of symptoms prior to presentation of 5 (interquartile range (IQR) 3, 7) days. Fifty-three percent of children showed lymphopaenia. Overall, the majority of cases at admission had markedly elevated inflammatory markers erythrocyte sedimentation rate (ESR) (95.5%) and C-reactive protein (CRP) (97%). Ferritin was abnormal in 11 out of 14 tested patients (73%), and it was highly elevated (>500 ng/ml) in 47% of cases. Median fibrinogen level was 210 (IQR 165, 291) mg/dl, D-dimer was 3909 (IQR 848, 4528) ng/ml and troponin was 0.6 (IQR 0.1, 26) ng/ml, respectively. Twenty out of 31 patients (64.5%) had hypoalbuminaemia. In addition, hyponatraemia was found in 64% of cases. Twenty-five patients (56%) presented with cardiac involvement and acute renal failure was observed in 13 cases (29%). Pleural, ascitic, ileitis and pericardial effusions were found in 18%, 11%, 4% and 2% of cases, respectively. In conclusion, this is a first large case series of hospitalised children who met criteria for MIS-C in Iran. There was a wide spectrum of presenting signs and symptoms; evidence of inflammation with abnormal values of CRP, ESR, D-dimer, ferritin and albumin; and multi-organ involvement.

As the Middle East respiratory syndrome coronavirus (MERS-CoV) continues to occur in small outbreaks in Saudi Arabia, we aimed to assess the knowledge, attitudes and intended practices of healthcare workers (HCWs) during the early stage of the COVID-19 pandemic and compare worry levels with previous findings during the MERS-CoV outbreak in 2015. We sent an adapted version of our previously published MERS-CoV questionnaire to the same cohort of HCWs at a tertiary hospital in Saudi Arabia. About 40% of our sample had previous experience with confirmed or suspected MERS-CoV patients, and those had a significantly higher knowledge score (13.16 ± 2.02 vs. 12.58 ± 2.27, P = 0.002) and higher adherence to protective hygienic practices (2.95 ± 0.80 vs. 2.74 ± 0.92, P = 0.003). The knowledge scores on COVID-19 were higher in the current cohort than the previous MERS-CoV outbreak cohort (68% vs. 79.7%, P < 0.001). HCWs from the current cohort who felt greater anxiety from COVID-19 compared to MERS-CoV were less likely to have been exposed to MERS-CoV infected/suspected cases (odds ratio (OR) = 0.646, P = 0.042) and were less likely to have attended the hospital awareness campaign on COVID-19 (OR = 0.654, P = 0.035). We concluded that previous experience with MERS-CoV was associated with increased knowledge and adherence to protective hygienic practices, and reduction of anxiety towards COVID-19.