Stochastic Mathematical Model Research Articles

Stochastic Model of Emotional Stress Development in the Educational Process

The aim of the study is the development of mathematical models that describe the reaction of the individual to the occurrence of stresses of various nature, including those that appear during the implementation of the educational process. The complexity of developing such a model is confirmed by the lack of theoretical results to substantiate the classic experiments of Holmes and Rae on the effect of stress on personality. In this regard, the task is to develop such a mathematical model that would allow not only to give a theoretical explanation of the experimental results used in the Holmes and Rae stress calculator, but also to become a tool for studying the effect of stress on a person in other conditions, including in the process of educational activities of both the teacher and the learner.The research method consists in a mathematical description of the process of occurrence of stresses that develop over time, and it is believed that stresses occur at random times and are characterized by relative stress values indicated in the classical table of Holmes and Rae. The need to involve these results is that they allow us to confirm the correspondence of the developed theoretical mathematical models to already known practical results. The following main assumptions are accepted in the paper. It is believed that a person is exposed to stresses that can occur at random localized points in time and are interpreted as a sequence of points on the time axis, the number and location of which is random. The response of a person to a particular stress is described by a decreasing exponential function of three arguments – current time, random time of stress occurrence, and stress magnitude. The reaction of the individual to a sequence of stresses is the sum of the responses of the person to individual stresses, i.e. it is assumed that the personality exhibits the properties of linearity. In the process of developing a mathematical model, the distribution of the number of random stresses is substantiated according to the Poisson law, which is used to describe the occurrence of random events with a distinct discreteness. The paper introduces one of the key indexes - the coefficient of emotional load, equal to the ratio of the mathematical expectation of everyday stressful background and the mathematical expectation of the sum of this background and additional stress. The response of a person to a particular stress is described by an exponential response function widely used in natural science applications. The total relative value of processed, experienced, random relative values of stress, as well as their non-random mathematical expectations, is introduced into consideration.The new results of the study are: – development of a stochastic mathematical model for the development of stresses over time, depending on the parameters included in the model. It is shown that the behavior of a person’s reaction to stress, predicted by a mathematical model, corresponds to the previously mentioned experimental results. – a study of the behavior in time of a person’s reaction to stress for a situation that was not previously considered and in which the person is affected by a single stress of great intensity, as well as regular emotional imposition.In conclusion, it is noted that the developed model not only makes it possible to theoretically explain the experimental data, but also significantly expand the scope of the studied effect of stress on personality. Thus, it turned out to be possible to predict the impact of a single impact, as well as to indicate a way to account for periodic intentional exposure (emotional suppression). In addition, the results can be used in the study of emotional stresses in the educational process in order to predict and consider them in practical activities.

DYNAMICS OF DELAYED NICHOLSON’S BLOWFLIES MODELS

The extinction and the persistence of the population of the harmful sheep blowfly (Lucilia cuprina) are discussed in this paper through a stochastic mathematical model. Using appropriate Lyapunov functionals, the extinction of these flies depends on the time to oviposition and the time between generations. The oviposition period must be sufficiently short. Another species of blowfly, called Lucilia sericata, also attacks sheep in many areas. Both blowflies are ectoparasites of warm-blooded vertebrates, particularly domestic sheep. These two blowflies are related to sharing the same mitochondrial DNA sequences, although the two species are distinct. A sufficiently small egg-laying delay then leads to the rapid extinction of both blowfly populations.

On predicting heterogeneity in nanoparticle dosage

Nanoparticles are increasingly employed as a vehicle for the targeted delivery of therapeutics to specific cell types. However, much remains to be discovered about the fundamental biology that dictates the interactions between nanoparticles and cells. Accordingly, few nanoparticle-based targeted therapeutics have succeeded in clinical trials. One element that hinders our understanding of nanoparticle–cell interactions is the presence of heterogeneity in nanoparticle dosage data obtained from standard experiments. It is difficult to distinguish between heterogeneity that arises from stochasticity in nanoparticle-cell interactions, and that which arises from heterogeneity in the cell population. Mathematical investigations have revealed that both sources of heterogeneity contribute meaningfully to the heterogeneity in nanoparticle dosage. However, these investigations have relied on simplified models of nanoparticle internalisation. Here we present a stochastic mathematical model of nanoparticle internalisation that incorporates a suite of relevant biological phenomena such as multistage internalisation, cell division, asymmetric nanoparticle inheritance and nanoparticle saturation. Critically, our model provides information about nanoparticle dosage at an individual cell level. We perform model simulations to examine the influence of specific biological phenomena on the heterogeneity in nanoparticle dosage in the absence of heterogeneity in the cell population. Under certain modelling assumptions, we derive analytic approximations of the nanoparticle dosage distribution. We demonstrate that the analytic approximations are accurate, and show that nanoparticle dosage can be described by a Poisson mixture distribution with rate parameters that are a function of Beta-distributed random variables. We discuss the implications of the analytic results with respect to parameter estimation and model identifiability from standard experimental data. Finally, we highlight extensions and directions for future research.

Study on the compaction space of coarse aggregates of asphalt mixtures

The initial accumulation state of coarse aggregates in asphalt mixtures has a compactable space. In this paper, the theoretical compaction space is studied for three groups of grades in the design process of skeletal compactness type Marshall grading and the range of grades in Marshall experimental specifications, the full-scale theoretical upper limit of crushing for each group of grades is projected using stochastic mathematical models, and the experimental images are combined to make an analysis and comparison. The study has shown that: the design grade of asphalt mixture and the specification grade in theoretical compaction differences, the average value of the preferred grade compaction space significantly deviated from the range of the guide grade, and the preferred grade compaction space stability is also less than the guide grade. The compaction space has been determined at the beginning of the grade design, and different grades should correspond to different compaction times.

Stochastic programming-based mathematical model and solution strategy for chemical production scheduling with processing time uncertainty

Scheduling of multipurpose batch chemical plant is always affected by uncertain factors, including processing time of tasks. When the processing time deviates from its nominal value, the task sequence and executing time based upon the original schedule may become suboptimal or even infeasible. To address this issue, an optimization model based on stochastic programming is proposed for the short-term scheduling of multipurpose batch chemical plant, by introducing task sequence variables and new logical constraints relating multiple binary variables. Additionally, a network-based decomposition solution strategy, accounting for different situations of profits and shared units, is proposed to solve the large-scale problems, which has been shown to provide high quality solutions while consuming substantially less solution time than solving the entire process directly.

Effects of noise on bursting phenomenon and bifurcation in glucose-insulin regulation system

In this paper, the stochastic time-delay mathematical model of the feedback system of glucose-insulin endocrine regulation with noise effects is proposed. The hopf bifurcation of the system is obtained with the delay as the parameter. Considering the influence of time delay and noise, the random bifurcation of the system is obtained, and through numerical analysis, the changement figure of the numerical solution of the model before and after the occurrence of random bifurcation is obtained. The results show that a high-noise and high-sugar diet can lead to the occurrence of random bifurcations, thereby destabilizing the system. In addition, we found the possible bursting phenomenon in the model, and through the analysis, it is concluded that bursting phenomenon is more likely to occur in the noise environment than in the no-noise environment.

Stationary distribution and persistence of a stochastic mathematical model for prostate cancer with pulsed therapy

Intermittent androgen deprivation therapy is one of the most commonly used therapeutic regimens for treating prostate cancer. Immunotherapy with dendritic cells, which act as the most robust antigen-presenting cells, are regarded as an effective method for the treatment of advanced prostate cancer. This paper utilizes impulsive differential equations to describe the combinations of a dendritic cell vaccine and intermittent androgen therapy with white noise. The tumour-free solution is obtained and the unique global positive solution of the system is explored. Then it is proved that the solutions of the system are stochastic ultimately bounded and stochastically permanent. In addition, threshold conditions for the extinction and persistence of prostate cancer cells are derived, and the stationary distribution and ergodicity of the system are investigated. Finally, numerical studies and the biological significance of the results are discussed.

Joint Optimization of Ticket Pricing and Allocation on High-Speed Railway Based on Dynamic Passenger Demand during Pre-Sale Period: A Case Study of Beijing–Shanghai HSR

Against the background of the gradual deepening of China Railway’s market-oriented reform, and in order to improve the revenue and competitiveness for high-speed railway (HSR) passenger transport, this paper studies the joint optimization problem of the high-speed railway ticket pricing and allocation considering the dynamic demand characteristics of passengers during the pre-sale period. Firstly, we use the compound non-homogeneous Poisson process to describe the passengers’ ticket-purchasing process and use the sparse method to simulate the passengers’ ticket demand during the pre-sale period. Secondly, taking the ticket pricing and allocation as the decision variables and considering the full utilization of the train seat capacity, a stochastic nonlinear-programming mathematical model is established with the goal of maximizing the train revenue. A particle swarm algorithm is designed to solve the model. Finally, this study takes the G19 train running on the Beijing–Shanghai HSR in China as a case study to verify the effectiveness of the model and algorithm. The results show that the joint optimization scheme of ticket pricing and allocation considering dynamic demand yields a revenue of CNY 601,881, which increases the revenue by 1.01% with a small adjustment of the price compared with the fixed ticket price and pre-allocation scheme. This study provides scientific support for the decisions made by railway transportation enterprises, which is conducive to further increasing the potential ticket revenue and promoting sustainable development.

Ellipsoidal Design of Robust Stabilization for Markov Jump Power Systems under Normal and Contingency Conditions

The essential prerequisites for secure customer service are power system stability and reliability. This work shows how to construct a robust switching control for studying power system load changes using an invariant ellipsoid method. Furthermore, the suggested control ensures stability when the system is subjected to random stochastic external disturbances, and functions randomly in two conditions: normal and contingency. The extreme (least) reliability state is chosen as the most severe scenario (corresponding to a transmission line outage). As a two-state Markov random chain, the transition probabilities are utilized to simulate the switching between normal and contingency modes (or processes). To characterize the dynamics of the studied system, a stochastic mathematical model is developed. The effect of stochastic disturbances and random normal/contingency operations is taken into account during the design stage. For a stochastic power system, a novel excitation control is designed. The attractive ellipsoid approach and linear matrix inequalities (LMIs) optimization are used to build the best two-controller gains. Therefore, the proposed modeling/design technique can be employed for the power system under load changes, stochastic topological changes, and random disturbances. Finally, the system’s random dynamics simulation indicates the effectiveness of the designed control law.

Data-Driven Discovery of Stochastic Differential Equations

Stochastic differential equations (SDEs) are mathematical models that are widely used to describe complex processes or phenomena perturbed by random noise from different sources. The identification of SDEs governing a system is often a challenge because of the inherent strong stochasticity of data and the complexity of the system’s dynamics. The practical utility of existing parametric approaches for identifying SDEs is usually limited by insufficient data resources. This study presents a novel framework for identifying SDEs by leveraging the sparse Bayesian learning (SBL) technique to search for a parsimonious, yet physically necessary representation from the space of candidate basis functions. More importantly, we use the analytical tractability of SBL to develop an efficient way to formulate the linear regression problem for the discovery of SDEs that requires considerably less time-series data. The effectiveness of the proposed framework is demonstrated using real data on stock and oil prices, bearing variation, and wind speed, as well as simulated data on well-known stochastic dynamical systems, including the generalized Wiener process and Langevin equation. This framework aims to assist specialists in extracting stochastic mathematical models from random phenomena in the natural sciences, economics, and engineering fields for analysis, prediction, and decision making.

STOCHASTIC OPTIMAL CONTROL ANALYSIS FOR THE COVID-19 EPIDEMIC MODEL UNDER REAL STATISTICS

The COVID-19 pandemic started, a global effort to develop vaccines and make them available to the public, has prompted a turning point in the history of vaccine development. In this study, we formulate a stochastic COVID-19 epidemic mathematical model with a vaccination effect. First, we present the model equilibria and basic reproduction number. To indicate that our stochastic model is well-posed, we prove the existence and uniqueness of a positive solution at the beginning. The sufficient conditions of the extinction and the existence of a stationary probability measure for the disease are established. For controlling the transmission of the disease by the application of external sources, the theory of stochastic optimality is established. The nonlinear least-squares procedure is utilized to parametrize the model from actual cases reported in Pakistan. The numerical simulations are carried out to demonstrate the analytical results.

Evolutionary optimisation of antibiotic dosing regimens for bacteria with different levels of resistance

Antimicrobial resistance is one of the biggest threats to global health, food security, and development. Antibiotic overuse and misuse are the main drivers for the emergence of resistance. It is crucial to optimise the use of existing antibiotics in order to improve medical outcomes, decrease toxicity and reduce the emergence of resistance. We formulate the design of antibiotic dosing regimens as an optimisation problem, and use an evolutionary algorithm suited to continuous optimisation (differential evolution) to solve it. Regimens are represented as vectors of real numbers encoding daily doses, which can vary across the treatment duration. A stochastic mathematical model of bacterial infections with tuneable resistance levels is used to evaluate the effectiveness of evolved regimens. The objective is to minimise the treatment failure rate, subject to a constraint on the maximum total antibiotic used. We consider simulations with different levels of bacterial resistance, two ways of administering the drug (orally and intravenously), as well as coinfections with two strains of bacteria. Our approach produced effective dosing regimens, with an average improvement in lowering the failure rate 30%, when compared with standard fixed-daily-dose regimens with the same total amount of antibiotic.

A stochastic mathematical model for coupling the cytosolic and sarcoplasmic calcium movements in diseased cardiac myocytes

Several computational studies have been undertaken to explore the Ca2+-induced Ca2+ release (CICR) events in cardiac myocytes and along with experimental studies it has given us invaluable insight into the mechanism of CICR from spark/blink initiation to termination and regulation, and their interplay under normal and pathological conditions. The computational modelling of this mechanism has mainly been investigated using coupled differential equations (DEs). However, there is a lack of computational investigation into (1) how the different formulation of coupled DEs capture the Ca2+ movement in the cytosol and sarcoplasmic reticulum (SR), (2) the buffer and dye inclusion in both compartments, and (3) the effect of buffer and dye properties on the calcium behaviour. This work is set out to explore (1) the effect of different coupled formulation of DEs on spark/blink occurrence, (2) the inclusion of improved sarcoplasmic buffering properties, and (3) the effects of cytosolic and sarcoplasmic dye and buffer properties on Ca2+ movement. The simulation results show large discrepancies between different formulations of the governing equations. Additionally, extension of the model to include sarcoplasmic buffering properties show normalised fluorescent dye profiles to be in good agreement with experimental and amongst its one- and two-dimensional representations.

A two-stage stochastic programming model for collaborative asset protection routing problem enhanced with machine learning: a learning-based matheuristic algorithm

ABSTRACT In this paper, a two-stage stochastic mathematical model is developed for an asset protection routing problem under a wildfire. The main aim of this study is to reduce the negative impact of a wildfire. Some parameters, such as travel and service times, obtaining profit by protecting an asset, and upper bounds of time windows, are considered as stochastic parameters. Generating proper scenarios for uncertain parameters has a large impact on the accuracy of the obtained solutions. Therefore, artificial neural networks are employed to extract possible scenarios according to previous actual wildfire events. The problem cannot be solved by exact solvers for large instances, so two matheuristic algorithms are proposed in this study to solve the problem in a reasonable time. In the first algorithm, a set of feasible routes is generated based on a heuristic approach, then a route-based mathematical model is used to obtain the final solution. Also, another matheuristic algorithm based on adaptive large neighbourhood search (ALNS) is proposed. In this algorithm, routing decisions are determined using the ALNS algorithm while other decisions are achieved by solving an intermediate mathematical model. The numerical analysis confirms the efficiency of both proposed algorithms; however, the first algorithm performs more efficiently.

Macrophage polarization as a potential therapeutic target for atherosclerosis: a dynamic stochastic modelling study.

We proposed a dynamic stochastic mathematical model to evaluate the role of macrophage polarization in plaque development. The dynamic process of macrophages from proliferation to death was simulated under different lipid microenvironments. The probability of macrophage phenotypic switching was described using a Bernoulli distribution where the stochastic variable was determined by the local lipid level. Moreover, the interactions between macrophages and microenvironmental factors vary with macrophage phenotype. We investigated the distribution of key microenvironmental factors, the dynamics of macrophage polarization and its influence on foam cell formation. M1 macrophages were found to predominate in advanced plaque corresponding to the exacerbated inflammation observed in mice experiments. The imbalance between the deposition of oxidized low-density lipoprotein and phagocytic effects of macrophages governed the formation of foam cells. Furthermore, we simulated targeted therapies by either directly inhibiting the polarization probability to M1 macrophages or indirectly regulating macrophage polarization due to high-density lipoprotein levels. Comparison of simulation results with experimental findings in both therapies indicated that the intervention and regulation of macrophage polarization could influence plaque microenvironment and subsequently induce plaque regression, especially in the early stage. The proposed modelling system can facilitate the evaluation of novel therapies targeting macrophage polarization.

Impact of decentralized microgrids optimal energy management on power system dynamics

This paper proposes a unified framework between the optimal energy management (OEM) problem of grid-connected microgrids and the frequency control of power systems through time-domain simulations. The proposed formulation is aimed at analyzing how different objectives for the microgrid OEM and control strategies affect the system frequency regulation. Each OEM model minimizes the microgrid local objective while ensuring energy reserves and operational limits by solving a two-stage stochastic optimization mathematical model. Two control approaches of microgrids are considered, namely, greedy and cooperative. The proposed framework is evaluated considering the IEEE 39-bus test system, including multiple microgrids with different energy storage capacities, and control strategies. Results show an improvement in system’s frequency deviation when MGs operate in the cooperative configuration while simultaneously reducing their operational costs.

A data-driven analytical roadmap to a sustainable 2030 in South Korea based on optimal renewable microgrids

Combined technological, economic, sociological, and environmental (TESE) models can play a unique role in leveraging renewable energies and supporting sustainable development. Yet, a multi-aspect TESE model has never been used in Korea to adapt to climate change through sustainable energy policies. This comprehensive TESE study addresses several sustainability challenges in the Korean energy sector. First, demand electricity is predicted using four deep and stacked neural networks to develop a smart demand-based model. Second, optimal hybrid renewable microgrids (HRMGs) are simulated at 17 sites to evaluate five renewable energy sources in three scenarios. Third, hybrid assessment results are clustered using a K-means algorithm to generate hybrid renewable energy maps for South Korea. Fourth, the TESE model is analyzed with more than 13 variables using a cascading multi-criteria decision-making approach to prorate a budget and develop a prioritized roadmap for a sustainable 2030 in Korea. Fifth, a stochastic linear mathematical model is developed to propose a sustainable energy policy that considers the water-energy-carbon nexus. The results show that a convolutional neural network can efficiently predict sequential demand electricity (R2 = 98.79%), with respective biogas, solar, hydrostatic, wind, and hydrokinetic energy fractions of 45.7%, 34.5%, 14%, 5.78%, and 0.01% under optimal conditions in Korea. The present free market-based policies are recommended to be revised in favor of domestic production of renewable energy facilities if new jobs can be created for more than $7500 each, and the carbon penalty cost can be kept below $83/ton CO2-eq in Korea.

Reliable and efficient parameter estimation using approximate continuum limit descriptions of stochastic models

Stochastic individual-based mathematical models are attractive for modelling biological phenomena because they naturally capture the stochasticity and variability that is often evident in biological data. Such models also allow us to track the motion of individuals within the population of interest. Unfortunately, capturing this microscopic detail means that simulation and parameter inference can become computationally expensive. One approach for overcoming this computational limitation is to coarse-grain the stochastic model to provide an approximate continuum model that can be solved using far less computational effort. However, coarse-grained continuum models can be biased or inaccurate, particularly for certain parameter regimes. In this work, we combine stochastic and continuum mathematical models in the context of lattice-based models of two-dimensional cell biology experiments by demonstrating how to simulate two commonly used experiments: cell proliferation assays and barrier assays. Our approach involves building a simple statistical model of the discrepancy between the expensive stochastic model and the associated computationally inexpensive coarse-grained continuum model. We form this statistical model based on a limited number of expensive stochastic model evaluations at design points sampled from a user-chosen distribution, corresponding to a computer experiment design problem. With straightforward design point selection schemes, we show that using the statistical model of the discrepancy in tandem with the computationally inexpensive continuum model allows us to carry out prediction and inference while correcting for biases and inaccuracies due to the continuum approximation. We demonstrate this approach by simulating a proliferation assay, where the continuum limit model is the well-known logistic ordinary differential equation, as well as a barrier assay where the continuum limit model is closely related to the well-known Fisher-KPP partial differential equation. We construct an approximate likelihood function for parameter inference, both with and without discrepancy correction terms. Using maximum likelihood estimation, we provide point estimates of the unknown parameters, and use the profile likelihood to characterise the uncertainty in these estimates and form approximate confidence intervals. For the range of inference problems considered, working with the continuum limit model alone leads to biased parameter estimation and confidence intervals with poor coverage. In contrast, incorporating correction terms arising from the statistical model of the model discrepancy allows us to recover the parameters accurately with minimal computational overhead. The main tradeoff is that the associated confidence intervals are typically broader, reflecting the additional uncertainty introduced by the approximation process. All algorithms required to replicate the results in this work are written in the open source Julia language and are available at GitHub.

RSDM-AHSnet: Designing a robust stochastic dynamic model to allocating health service network under disturbance situations with limited capacity using algorithms NSGA-II and PSO

In the present study, health services networks were classified into low-level hospitals (provision of public health services) and high-level hospitals (providing specialized health services), which are at risk of being disrupted. They refer the patients to high-level hospitals for inpatient visits or emergencies by ambulance. In the present case, patients are divided into two categories: high priority (the category in which immediate service delivery is needed) and low priority. A stochastic robust dynamic mathematical model for location and allocation of health network regarding limited capacity and disturbance is developed to reduce the total costs and include the basic features of a real problem such as limited capacity. Regarding limited capacity for hospitals, the health network needs redefinition of different layers in the disturbance situation. In this study, we reduce the total costs by reducing hospital costs and costs such as transportation and service to patients. Two metaheuristic algorithms consisting of Non-dominated Sorting Genetic Algorithm II (NSGA-II) and Particle Swarm Optimization (PSO) have been applied to solve the model. Taguchi method design minimizes the cost of parameter tuning, including the level of factors related to the proposed. The results showed the method's applicability for large-scale problems that could evaluate different tools for decision-makers to select effective management strategies in constructing a dependable and robust healthcare network. For example, the total cost is minimized in conditions considered in the genetic algorithm, the population parameter at the highest level, 150, the intersection parameters, and the probability of mutation at the lowest level, 0.7 and 0.1.

Anomalous diffusion and asymmetric tempering memory in neutrophil chemotaxis.

The motility of neutrophils and their ability to sense and to react to chemoattractants in their environment are of central importance for the innate immunity. Neutrophils are guided towards sites of inflammation following the activation of G-protein coupled chemoattractant receptors such as CXCR2 whose signaling strongly depends on the activity of Ca2+ permeable TRPC6 channels. It is the aim of this study to analyze data sets obtained in vitro (murine neutrophils) and in vivo (zebrafish neutrophils) with a stochastic mathematical model to gain deeper insight into the underlying mechanisms. The model is based on the analysis of trajectories of individual neutrophils. Bayesian data analysis, including the covariances of positions for fractional Brownian motion as well as for exponentially and power-law tempered model variants, allows the estimation of parameters and model selection. Our model-based analysis reveals that wildtype neutrophils show pure superdiffusive fractional Brownian motion. This so-called anomalous dynamics is characterized by temporal long-range correlations for the movement into the direction of the chemotactic CXCL1 gradient. Pure superdiffusion is absent vertically to this gradient. This points to an asymmetric ‘memory’ of the migratory machinery, which is found both in vitro and in vivo. CXCR2 blockade and TRPC6-knockout cause tempering of temporal correlations in the chemotactic gradient. This can be interpreted as a progressive loss of memory, which leads to a marked reduction of chemotaxis and search efficiency of neutrophils. In summary, our findings indicate that spatially differential regulation of anomalous dynamics appears to play a central role in guiding efficient chemotactic behavior.