A linear equation $E$ is said to be sparse if there is $c\gt 0$ so that every subset of $[n]$ of size $n^{1-c}$ contains a solution of $E$ in distinct integers. The problem of characterising the sparse equations, first raised by Ruzsa in the 90s, is one of the most important open problems in additive combinatorics. We say that $E$ in $k$ variables is abundant if every subset of $[n]$ of size $\varepsilon n$ contains at least $\text{poly}(\varepsilon )\cdot n^{k-1}$ solutions of $E$. It is clear that every abundant $E$ is sparse, and Girão, Hurley, Illingworth, and Michel asked if the converse implication also holds. In this note, we show that this is the case for every $E$ in four variables. We further discuss a generalisation of this problem which applies to all linear equations.

Syphilis remains a serious public health problem in mainland China that requires attention, modelling to describe and predict its prevalence patterns can help the government to develop more scientific interventions. The seasonal autoregressive integrated moving average (SARIMA) model, long short-term memory network (LSTM) model, hybrid SARIMA-LSTM model, and hybrid SARIMA-nonlinear auto-regressive models with exogenous inputs (SARIMA-NARX) model were used to simulate the time series data of the syphilis incidence from January 2004 to November 2023 respectively. Compared to the SARIMA, LSTM, and SARIMA-LSTM models, the median absolute deviation (MAD) value of the SARIMA-NARX model decreases by 352.69%, 4.98%, and 3.73%, respectively. The mean absolute percentage error (MAPE) value decreases by 73.7%, 23.46%, and 13.06%, respectively. The root mean square error (RMSE) value decreases by 68.02%, 26.68%, and 23.78%, respectively. The mean absolute error (MAE) value decreases by 70.90%, 23.00%, and 21.80%, respectively. The hybrid SARIMA-NARX and SARIMA-LSTM methods predict syphilis cases more accurately than the basic SARIMA and LSTM methods, so that can be used for governments to develop long-term syphilis prevention and control programs. In addition, the predicted cases still maintain a fairly high level of incidence, so there is an urgent need to develop more comprehensive prevention strategies.

Pharmaceutical distribution routing problem is a key problem for pharmaceutical enterprises, since efficient schedules can enhance resource utilization and reduce operating costs. Meanwhile, it is a complicated combinatorial optimization problem. Existing research mainly focused on delivery route lengths or distribution costs minimization, while seldom considered customer priority and carbon emissions simultaneously. However, considering the customer priority and carbon emissions simultaneously will not only help to enhance customer satisfaction, but also help to reduce the carbon emissions. In this article, we consider the customer priority and carbon emission minimization simultaneously in the pharmaceutical distribution routing problem, the corresponding problem is named pharmaceutical distribution routing problem considering customer priority and carbon emissions. A corresponding mathematical model is formulated, the objectives of which are minimizing fixed cost, refrigeration cost, fuel consumption cost, carbon emission cost, and penalty cost for violating time windows. Moreover, a hybrid genetic algorithm (HGA) is proposed to solve the problem. The framework of the proposed HGA is genetic algorithm (GA), where an effective local search based on variable neighborhood search (VNS) is specially designed and incorporated to improve the intensification abilities. In the proposed HGA, crossover with adaptive probability and mutation with adaptive probability are utilized to enhance the algorithm performance. Finally, the proposed HGA is compared with four optimization algorithms, and experimental results have demonstrated the effectiveness of the HGA.

International travel is thought to be a major risk factor for developing gastrointestinal (GI) illness for UK residents. Here, we present an analysis of routine laboratory and exposure surveillance data from North East (NE) England, describing the destination-specific contribution that international travel makes to the regional burden of GI infection.

Laboratory reports of common notifiable enteric infections were linked to exposure data for cases reported between 1 January 2013 and 31 December 2022. Demographic characteristics of cases were described, and rates per 100,000 visits were determined using published estimates of overseas visits from the Office for National Statistics (ONS) International Passenger Survey (IPS).

About 34.9% of cases reported international travel during their incubation period between 2013 and 2022, although travel-associated cases were significantly reduced (>80%) during the COVID-19 pandemic. Between 2013 and 2019, half of Shigella spp. and non-typhoidal Salmonella infections and a third of Giardia sp., Cryptosporidium spp., and Shiga toxin-producing Escherichia coli (STEC) infections were reported following travel. Rates of illness were highest in travellers returning from Africa and Asia (107.8 and 61.1 per 100,000 visits), with high rates also associated with tourist resorts like Turkey, Egypt, and the Dominican Republic (386.4–147.9 per 100,000 visits).

International travel is a major risk factor for the development of GI infections. High rates of illness were reported following travel to both destinations, which are typically regarded as high-risk and common tourist resorts. This work highlights the need to better understand risks while travelling to support the implementation of guidance and control measures to reduce the burden of illness in returning travellers.

Development of gastrointestinal illness after animal contact at petting farms is well described, as are factors such as handwashing and facility design that may modify transmission risk. However, further field evidence on other behaviours and interventions in the context of Cryptosporidium outbreaks linked to animal contact events is needed. Here, we describe a large outbreak of Cryptosporidium parvum (C. parvum) associated with a multi-day lamb petting event in the south-west of England in 2023 and present findings from a cohort study undertaken to investigate factors associated with illness. Detailed exposure questionnaires were distributed to email addresses of 647 single or multiple ticket bookings, and 157 complete responses were received. The outbreak investigation identified 23 laboratory-confirmed primary C. parvum cases. Separately, the cohort study identified 83 cases of cryptosporidiosis-like illness. Associations between illness and entering a lamb petting pen (compared to observing from outside the pen; odds ratio (OR) = 2.28, 95 per cent confidence interval (95% CI) 1.17 to 4.53) and self-reported awareness of diarrhoeal and vomiting disease transmission risk on farm sites at the time of visit (OR = 0.40, 95% CI 0.19 to 0.84) were observed. In a multivariable model adjusted for household clustering, awareness of disease transmission risk remained a significant protective factor (adjusted OR (aOR) = 0.07, 95% CI 0.01 to 0.78). The study demonstrates the likely under-ascertainment of cryptosporidiosis through laboratory surveillance and provides evidence of the impact that public health messaging could have.

We consider a Poisson autoregressive process whose parameters depend on the past of the trajectory. We allow these parameters to take negative values, modelling inhibition. More precisely, the model is the stochastic process $(X_n)_{n\ge0}$ with parameters $a_1,\ldots,a_p \in \mathbb{R}$, $p\in\mathbb{N}$, and $\lambda \ge 0$, such that, for all $n\ge p$, conditioned on $X_0,\ldots,X_{n-1}$, $X_n$ is Poisson distributed with parameter $(a_1 X_{n-1} + \cdots + a_p X_{n-p} + \lambda)_+$. This process can be regarded as a discrete-time Hawkes process with inhibition and a memory of length p. In this paper we initiate the study of necessary and sufficient conditions of stability for these processes, which seems to be a hard problem in general. We consider specifically the case $p = 2$, for which we are able to classify the asymptotic behavior of the process for the whole range of parameters, except for boundary cases. In particular, we show that the process remains stochastically bounded whenever the solution to the linear recurrence equation $x_n = a_1x_{n-1} + a_2x_{n-2} + \lambda$ remains bounded, but the converse is not true. Furthermore, the criterion for stochastic boundedness is not symmetric in $a_1$ and $a_2$, in contrast to the case of non-negative parameters, illustrating the complex effects of inhibition.

Suicide is a leading cause of death in the United States, particularly among adolescents. In recent years, suicidal ideation, attempts, and fatalities have increased. Systems maps can effectively represent complex issues such as suicide, thus providing decision-support tools for policymakers to identify and evaluate interventions. While network science has served to examine systems maps in fields such as obesity, there is limited research at the intersection of suicidology and network science. In this paper, we apply network science to a large causal map of adverse childhood experiences (ACEs) and suicide to address this gap. The National Center for Injury Prevention and Control (NCIPC) within the Centers for Disease Control and Prevention recently created a causal map that encapsulates ACEs and adolescent suicide in 361 concept nodes and 946 directed relationships. In this study, we examine this map and three similar models through three related questions: (Q1) how do existing network-based models of suicide differ in terms of node- and network-level characteristics? (Q2) Using the NCIPC model as a unifying framework, how do current suicide intervention strategies align with prevailing theories of suicide? (Q3) How can the use of network science on the NCIPC model guide suicide interventions?

We introduce a bivariate tempered space-fractional Poisson process (BTSFPP) by time-changing the bivariate Poisson process with an independent tempered $\alpha$-stable subordinator. We study its distributional properties and its connection to differential equations. The Lévy measure for the BTSFPP is also derived. A bivariate competing risks and shock model based on the BTSFPP for predicting the failure times of items that undergo two random shocks is also explored. The system is supposed to break when the sum of two types of shock reaches a certain random threshold. Various results related to reliability, such as reliability function, hazard rates, failure density, and the probability that failure occurs due to a certain type of shock, are studied. We show that for a general Lévy subordinator, the failure time of the system is exponentially distributed with mean depending on the Laplace exponent of the Lévy subordinator when the threshold has a geometric distribution. Some special cases and several typical examples are also demonstrated.