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In this work, we introduce a novel methodology for modeling discrete count variables within the framework of stochastic processes. Our approach integrates two statistical areas: Non-Homogeneous Poisson Processes for the estimation and prediction of intensity functions based on explanatory variables and functional data estimation techniques. Through a comprehensive case study focusing on an infectious disease with viral characteristics, we demonstrate the potential of our methodology. We provide empirical evidence that our methodology offers a robust alternative for modeling count variables. Our findings support the utility of our approach in capturing the complex dynamics inherent in count data in infectious disease epidemiological phenomena.

Infectious diseases still remain one of the major causes of death worldwide, despite the fact that various treatments, such as antibiotics, antiviral drugs, and vaccines for some diseases, are more available to people. Factors such as drug resistance, lack of access to health care, and environmental changes contribute to their persistence and spread. Motivated by this fact, in this study, the stochastic susceptible-infectious-recovered (SIR) epidemiological model with treatment and non-linear incidence rate is extended, by introducing coloured noise, to model which takes into account the seasonal nature of the disease, as well as the fact that the disease is constantly changing through mutations, which leads to the appearance of new disease strains. For the model formulated in this way, we first prove the existence and uniqueness of the global positive solution. Then, we provide conditions under which the disease persists in the population, as well as sufficient conditions for the disease to die out. The theoretical results of the current study are validated by numerical simulations. For that purpose, we use data on the spread of the Ebola epidemic in Sierra Leone and the coronavirus disease 2019 (COVID-19) pandemic in Pakistan. Both theoretical and numerical results can lead us to conclusion that our model represents the solid research base for further investigation in the field of epidemiological modelling.

Background The aim of this work was to create and evaluate a preoperative non-contrast-enhanced (CE) magnetic resonance imaging (MRI)/angiography (MRA) protocol to assess renal function and visualize renal arteries and any abnormalities in potential living kidney donors. Methods In total, 28 subjects were examined using scintigraphy to determine renal function. In addition, 3D-pseudocontinuous arterial spin labeling (pCASL), a 2D-non-CE electrocardiogram-triggered radial quiescent interval slice-selective (QISS-MRA), and 4D-CE time-resolved angiography with interleaved stochastic trajectories (CE-MRA) were performed to assess renal perfusion, visualize renal arteries and detect any abnormalities. Two glomerular filtration rates [described by Gates (GFRG) and according to the Chronic Kidney Disease Epidemiology Collaboration formula (GFRCKD-EPI)]. The renal volumes were determined using both MRA techniques. Results The mean value of regional renal blood flow (rRBF) on the right side was significantly higher than that on the left. The agreements between QISS-MRA and CE-MRA concerning the assessment of absence or presence of an aberrant artery and renal arterial stenosis were perfect. The mean renal volumes measured in the right kidney with QISS-MRA were lower than the corresponding values of CE-MRA. In contrast, the mean renal volumes measured in the left kidney with both MRA techniques were similar. The correlation between the GFRG and rRBF was compared in the same manner as that between GFRCKD-EPI and rRBF. Conclusion The combination of pCASL and QISS-MRA constitute a reliable preoperative protocol with a total measurement time of <10 min without the potential side effects of gadolinium-based contrast agents or radiation exposure. Graphical Abstract Graphical Abstract

This paper presents the concept of operations, architecture and trajectory optimisation algorithms of a Next Generation Flight Management System (NG-FMS). The NG-FMS is developed for Four Dimensional (4D) Intent Based Operations (IBO) in the next generation Communications, Navigation, Surveillance and Air Traffic Management system (CNS+A) context. The NG-FMS, primarily responsible for the aircraft navigation and guidance task, acts as a key enabler for achieving higher level of operational efficiency and mitigating environmental impacts both in manned and unmanned aircraft applications. The NG-FMS is interoperable with the future ground based 4DT Planning, Negotiation and Validation (4-PNV) systems, enabling automated Trajectory/Intent Based Operations (TBO/IBO). After the NG-FMS architecture is presented, the key mathematical models describing the trajectory generation and optimisation modes are introduced. A detailed error analysis is performed and the uncertainties affecting the nominal trajectories are studied to obtain the total NG-FMS error budgets. These are compared with the Required Navigation Performance (RNP) values for the various operational flight tasks considered.

Trajectory prediction constitutes a key technology for intelligent systems to forecast future movements of dynamic agents, yet it faces significant challenges due to the uncertainty of motion behavior. We propose iDMa, a stochastic trajectory prediction framework that pioneers the integration of diffusion model with Mamba architecture to achieve high-precision and high-efficiency trajectory generation. Our approach introduces two key innovations: (1) a dual-parameter learning mechanism that optimizes noise estimation of mean and variance space, unlike conventional diffusion methods that employ fixed variance during the denoising process, so as to constrain the feasible domain more accurately; (2) a hybrid denoising backbone network that incorporates Transformer encoders and Mamba blocks. Compared to existing state-of-the-art methods, iDMa reduces the average displacement error (ADE) by 4.76% (0.20 vs. 0.21) on the ETH-UCY dataset and 1.85% (7.95 vs. 8.10) on the SDD dataset.

Surface diffusion of small adsorbates is analyzed in terms of the so-called intermediate scattering function and dynamic structure factor, observables in experiments using the well-known quasielastic Helium atom scattering and Helium spin echo techniques. The linear theory applied is an extension of the neutron scattering due to van Hove and considers the time evolution of the position of the adsorbates in the surface. This approach allows us to use a stochastic trajectory description following the classical, quantum and Bohmian frameworks. Three different regimes of motion are clearly identified in the diffusion process: ballistic, Brownian and intermediate which are well characterized, for the first two regimes, through the mean square displacements and Einstein relation for the diffusion constant. The Langevin formalism is used by considering Ohmic friction, moderate surface temperatures and small coverages. In the Bohmian framework, analyzed here, the starting point is the so-called Schrödinger-Langevin equation which is a nonlinear, logarithmic differential equation. By assuming a Gaussian function for the probability density, the corresponding quantum stochastic trajectories are given by a dressing scheme consisting of a classical stochastic trajectory followed by the center of the Gaussian wave packet, and issued from solving the Langevin equation (particle property), plus the time evolution of its width governed by the damped Pinney differential equation (wave property). The Bohmian velocity autocorrelation function is the same as the classical one when the initial spread rate is assumed to be zero. If not, in the diffusion regime, the Brownian-Bohmian motion shows a weak anomalous diffusion.

In this review we deal with open (dissipative and stochastic) quantum systems within the Bohmian mechanics framework which has the advantage to provide a clear picture of quantum phenomena in terms of trajectories, originally in configuration space. The gradual decoherence process is studied from linear and nonlinear Schrödinger equations through Bohmian trajectories as well as by using the so-called quantum-classical transition differential equation through scaled trajectories. This transition is governed by a continuous parameter, the transition parameter, covering these two extreme open dynamical regimes. Thus, two sources of decoherence of different nature are going to be considered. Several examples will be presented and discussed in order to illustrate the corresponding theory behind each case, namely: the so-called Brownian–Bohmian motion leading to quantum diffusion coefficients, dissipative diffraction in time, dissipative tunnelling for a parabolic barrier under the presence of an electric field and stochastic early arrivals for the same type of barrier. In order to simplify the notations and physical discussion, the theoretical developments will be carried out in one dimension throughout all this wok. One of the main goals is to analyze the gradual decoherence process existing in these open dynamical regimes in terms of trajectories, leading to a more intuitive way of understanding the underlying physics in order to gain new insights.

We propose and solve a stochastic mathematical model of general applicability to interbreeding populations which share the same habitat. Resources are limited so that the total population size is fixed by environmental factors. Interbreeding occurs during all the time of coexistence until one of the two population disappears by a random fluctuation. None of the two populations has a selective advantage. We answer the following questions: How long the two populations coexist and how genetically similar they become before the extinction of one of the two? how much the genetic makeup of the surviving population changes by the contribution of the disappearing one? what it is the number of interbreeding events given the observed introgression of genetic material? The model was originally motivated by a paleoanthropological problem concerning the interbreeding of Neanderthals and African modern humans in Middle East which is responsible for the fraction of Neanderthal genes (1–4 %) in present Eurasian population.

The aim of this paper is to provide a realistic stochastic trajectory generation method for unmanned aerial vehicles that offers a tool for the emulation of trajectories in typical flight scenarios. Three scenarios are defined in this paper. The trajectories for these scenarios are implemented with quintic B-splines that grant smoothness in the second-order derivatives of Euler angles and accelerations. In order to tune the parameters of the quintic B-spline in the search space, a multi-objective optimization method called particle swarm optimization (PSO) is used. The proposed technique satisfies the constraints imposed by the configuration of the unmanned aerial vehicle (UAV). Further particular constraints can be introduced such as: obstacle avoidance, speed limitation, and actuator torque limitations due to the practical feasibility of the trajectories. Finally, the standard rapidly-exploring random tree (RRT*) algorithm, the standard (A*) algorithm and the genetic algorithm (GA) are simulated to make a comparison with the proposed algorithm in terms of execution time and effectiveness in finding the minimum length trajectory.

The sign of six Interactions between microscopic particles are usually described as two-body interactions, although it has been shown that higher order multi-body interactions could give rise to novel quantum phases with intriguing properties. This paper demonstrates effective six-body interactions in a system of ultracold bosonic atoms in a three-dimensional optical lattice. The coherent multi-particle interactions observed here open a new window for simulations of effective field theories and may help to enable the realization of novel topologically ordered many-body quantum phases. Interactions between microscopic particles are usually described as two-body interactions, although it has been shown that higher-order multi-body interactions could give rise to new quantum phases with intriguing properties. Here, effective six-body interactions are demonstrated in a system of ultracold bosonic atoms in a three-dimensional optical lattice. Interactions lie at the heart of correlated many-body quantum phases 1 , 2 , 3 . Typically, the interactions between microscopic particles are described as two-body interactions. However, it has been shown that higher-order multi-body interactions could give rise to novel quantum phases with intriguing properties. So far, multi-body interactions have been observed as inelastic loss resonances in three- and four-body recombinations of atom–atom and atom–molecule collisions 4 , 5 , 6 . Here we demonstrate the presence of effective multi-body interactions 7 in a system of ultracold bosonic atoms in a three-dimensional optical lattice, emerging through virtual transitions of particles from the lowest energy band to higher energy bands. We observe such interactions up to the six-body case in time-resolved traces of quantum phase revivals 8 , 9 , 10 , 11 , using an atom interferometric technique that allows us to precisely measure the absolute energies of atom number states at a lattice site. In addition, we show that the spectral content of these time traces can reveal the atom number statistics at a lattice site, similar to foundational experiments in cavity quantum electrodynamics that yield the statistics of a cavity photon field 12 . Our precision measurement of multi-body interaction energies provides crucial input for the comparison of optical-lattice quantum simulators with many-body quantum theory.