In this paper we consider a dynamic Erdős–Rényi graph in which edges, according to an alternating renewal process, change from present to absent and vice versa. The objective is to estimate the on- and off-time distributions while only observing the aggregate number of edges. This inverse problem is dealt with, in a parametric context, by setting up an estimator based on the method of moments. We provide conditions under which the estimator is asymptotically normal, and we point out how the corresponding covariance matrix can be identified. We also demonstrate how to adapt the estimation procedure if alternative subgraph counts are observed, such as the number of wedges or triangles.

The percentage of the global population aged over 65 is expected to rise to 16% by 2050(1). Community-dwelling older adults experience a high prevalence of malnutrition(2), which negatively impacts quality of life and increases mortality risk(3,4). Home-delivered meal programmes may improve the physical and mental well-being of older adults living independently. However, the effects of such programmes have not been comprehensively quantified. This systematic review and meta-analysis aimed to evaluate the impact of home-delivered meals on health-related outcomes among adults aged 60 and older residing in the community.

A systematic search of 6 databases was conducted to identify randomised controlled trials (RCTs) and non-RCTs reporting pre- and post-intervention changes in health-related outcomes. Only studies delivering meals were included; interventions involving snacks, oral supplements, or diet-specific meals were excluded. Studies were limited to participants living independently in the community; those in hospitals, care homes, or residential care units were excluded. Primary outcomes included body weight, energy intake, protein intake, nutrient status assessed by Mini Nutritional Assessment Scale, handgrip strength, and social loneliness. Standardised mean difference (Cohen’s d) estimated effect size.

Heterogeneity between trials was assessed using the I2 statistic. n denotes number of studies included.

A total of 15 studies out of 4414 were included (n = 15), including 5 RCTs and 10 non-RCTs and comprising 1206 and 1606 participants, respectively (mean pooled age: 77.3 ± 8.7 yrs). Meal delivery interventions varied in frequency (from one hot meal per day to five frozen meals delivered weekly) and duration (3 weeks to 24 months). No significant changes were identified between RCTs and non-RCTs for each outcome (P ≥ 0.42); pooled analyses are therefore presented. The meta-analysis showed no changes in body weight (d = -0.07; 95% CI: -0.20 to 0.05; P = 0.27; I2 = 0%; n = 5), energy intake (d = 0.16; 95% CI: -0.04 to 0.37, P = 0.12; I2 = 63%; n = 8), or protein intake (d = 0.11; 95% CI: -0.17 to 0.39, P = 0.44; I2 = 87%; n = 8). However, an improvement was observed in nutrient status (d = 0.57; 95% CI: 0.41 to 0.73; P < 0.01; I2 = 30%; n =6). No effects were detected for handgrip strength (d = 0.11; 95% CI: -0.12 to 0.35; P = 0.35; I2 = 0%; n =2). Social loneliness showed a small decrease (d = -0.22; 95% CI: -0.34 to -0.10; P < 0.01; I2 = 0%; n = 4).

Home-delivered meal interventions improve nutrient status and reduce social loneliness in community-dwelling older adults, though they show no significant effect on body weight or macronutrient intake. Further high-quality research is needed to explore broader health- related impacts and strengthen the evidence base.

This paper offers a formal analysis of continuity, welfarism, value satiability, lifeboat cases, along with their interconnectedness with sufficientarianism, with particular attention to the recent defences of sufficientarianism by Ben Davies and Lasse Nielsen in response to Hun Chung’s Prospect Utilitarianism (PU). It demonstrates how precise formal definitions help resolve conceptual ambiguities and sharpen philosophical argumentation in distributive ethics. Without such precision, one risks misidentifying or mischaracterizing important normative concepts and theories, leading to confusion or strawman critiques. By highlighting these risks, the paper underscores the methodological importance of precise definitions and formal analysis in ensuring clarity, consistency, and rigor in ethical theorizing.

In 2015, Public Health England challenged the food industry to reduce the free-sugars content of their sweetened products by 2020. However, whilst there has been some success in reformulating sugar-sweetened beverages, breakfast cereals and dairy products to reduce their sugars content, the contribution of sucrose to the taste and texture of sweetened baked goods and confectionery has been difficult to replace using readily available non-nutritive sweeteners(1). D-allulose, a C3-epimer of fructose with low available energy content(2) and similar physicochemical properties to sucrose, is a candidate for replacing sucrose and glucose in sweetened foods to reduce energy content and glycaemic load. However, the incorporation of D-allulose into sweetened high- carbohydrate food products could confer additional health benefits. In vitro, D-allulose appears to compete with glucose at GLUT2 on the enterocyte basolateral membrane(3) (slowing glucose permeation) and stimulate glucokinase translocation; increasing glycogen synthesis in hepatocytes(4) and insulin secretion in β- cells(5), with postprandial glycaemia being attenuated in humans after D-allulose consumption in a drink alongside foods and drinks containing glucose(6). However, there is limited information concerning glucose profile and insulin secretion when D-allulose is integrated into consumed foods. The study aimed to investigate the effects of D-allulose consumption (within a high-carbohydrate breakfast) on glycaemia, insulinaemia and C- peptide concentration in healthy adults.

This randomised, double-blinded, placebo-controlled, crossover study investigated the effects of D-allulose consumption (within a high-carbohydrate breakfast) on postprandial BG, serum insulin and C-peptide concentrations in twelve healthy young (19-26y), non- obese adults (6M:6F). A high glycaemic index porridge, sweetened with 10g glucose and either 15g of D-allulose or 0.06g sodium saccharin (matched-sweetness control), were consumed across two visits. BG, insulin and C-peptide were measured when fasted and at timepoints across 180 minutes following test meals. Single measurements were compared between visits using paired students t-test or Wilcoxon signed rank test, with comparisons in variables measured at more than one timepoint carried out using 2-way ANOVA with repeated measures.

Peak concentration, the postprandial incremental response and net incremental area- under-curves (iAUC180) across 180-minutes for BG and insulin did not differ between meals.However, incremental BG response differed in the second half of the measurement period (Time x Meal: P<0.001; ηp2= 0.386), with this variable being lower at 90min (Control: 057 ± 0.59 mmol.l-1, D-allulose: 0.12 ± 0.57 mmol.l-1) and 105min (Control: 0.54 ± 0.68 mmol.l-1, D- allulose: -0.02 ± 0.47 mmol.l-1) after D-allulose consumption (both P<0.05). Moreover, C-peptide iAUC180 was lower after D-allulose consumption (Control: 135 ± 62 nmol. l-1.180min, D-allulose: 83 ± 24 nmol.l-1.180min; P<0.05).

In young, healthy individuals, the addition of D-allulose to a standardised porridge breakfast reduced BG in the late postprandial period, but did not suppress the initial hyperglycaemic response to eating. Moreover, D-allulose consumption did not enhance insulin secretion, as described in vitro.

Parkinson’s disease, the second most prevalent neurological disorder, is a multisystem neurodegenerative disease characterized by both motor and non-motor symptoms. Transcranial direct current stimulation (tDCS) is a non-invasive brain neuromodulation technique that has been shown to be effective in some neurological conditions and for some clinical outcomes. To evaluate the efficacy of tDCS combined with gait training in Parkinson’s disease, compared to placebo, absence of treatment, conventional therapy, or other therapies. A systematic review and meta-analysis were performed in accordance with the PRISMA guidelines and registered in PROSPERO CRD42024542552. The literature search was conducted in PubMed, CINAHL, SPORT Discus, Web of Science, Scopus, MEDLINE, and Academic Search Ultimate (EBSCO) databases up to May 2024, limited to trials from the last 10 years. A total of 600 articles were identified; 9 were included in the systematic review and 8 in the meta-analysis. Significant intra-group changes were observed, but in the meta-analysis, no significant differences were seen between tDCS + gait training and tDCS placebo + gait training, although variables such as motor function slightly favored the combination (MD = −0.49; 95% CI [−1.55; 0.57], I2 = 0%). The combination of tDCS and gait training could provide significant motor benefits in terms of gait speed, functional mobility, cadence, motor function, quality of life, 6MWT, coordination and dynamic balance, flexibility, and stretch resistance in patients with Parkinson’s disease, but not in a more effective way than the same training without stimulation.