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Motivated by a practical problem, this paper investigates the integrated planning of maintenance operations and workload allocation on a set of machines in a workshop. Given quantities of products to be produced per period on a planning horizon must be processed on unrelated flexible machines. Moreover, each machine has to undergo one or more maintenance operations that must be planned within a given time window and impact products differently. The main goal is to find a feasible plan that satisfies the machine capacity by allocating the production quantities to machines and assigning maintenance operations as late as possible in their time windows. Various original mathematical models are presented. In particular, we propose models that allow maintenance operations and some production quantities to overlap two consecutive periods. Computational experiments based on industrial data show that allowing this overlapping helps the earliness of maintenance operations to be significantly reduced in the most difficult instances, going for example from a total of 14 periods to only 1 period, and by more than 35% on average. [ABSTRACT FROM AUTHOR]

Background: After abandoning the zero-case strategy for COVID-19 in December 2022 in mainland China, the Hong Kong Special Administrative Region of China (HKSAR) has undergone an anticipated surge of the prevalence of COVID-19, as well as other influenzas, such as influenza A/H1N1, influenza A/H3N2, and influenza B as well. Noteworthy, people are usually concerned about the mutual influences between two families of respiratory viruses, like co-circulation or co-infection. Methods: We fitted a unified model to weekly reported severe COVID-19 cases and the confirmed influenza A laboratory cases in HKSAR, respectively, using the R package POMP to obtain the best fitting and parameter estimates. The reconstructed transmission rates of the COVID-19 (/influenza A) versus the weekly reported influenza A (/COVID-19) confirmations between April 2022 and April 2024 were also compared. Results: Our numerical results suggest that influenza confirmations remained either at a very low level or were absent before 2023, while starting from 2023, the influenza epidemic re-emerged as expected because of the resumption of international travels and other social communications. Besides, the peak of influenza cases in April 2023 favored the form of the peak of COVID-19 between May–June, 2023. Additionally, during the sudden abolishment of the zero-case policy in mainland China (December 2022 to January 2023), we estimated that there were approximately 381 cases imported from mainland China into HKSAR. Conclusions: We estimated the potential number of imported COVID-19 severe cases from mainland China to Hong Kong and revealed some potential population-level interference between the two families of respiratory viruses. [ABSTRACT FROM AUTHOR]

Based on expectancy-value theory, this study adopted a person-centred approach to explore the heterogeneous profiles of secondary German and Taiwanese students' mathematical modelling task values, and examined the differences in their mathematical modelling performance, controlling for the variable of intra-mathematical knowledge among the heterogeneous profiles. Authors conducted a survey study of 452 ninth graders (201 German students and 251 Taiwanese students). The results showed that German and Taiwanese students respectively displayed three profiles of mathematical modelling task values: a) moderate utility and moderate interest/attainment, b) high utility but low interest/attainment, and c) low utility but high interest/attainment. Furthermore, different profiles of mathematical modelling task values showed significant differences in mathematical modelling performance for Taiwanese students but not for German students, even after removing the variable of intra-mathematical knowledge. This study advances the understanding of students' mathematical modelling task values and its relation with their mathematical modelling performance by the expectancy-value model of achievement motivation and person-centred analyses, and sheds light on the learning and teaching of mathematical modelling. [ABSTRACT FROM AUTHOR]

Working with approximations is a common practice in physics. This paper presents an exploratory study of student understanding of some of the elementary aspects of approximations encountered in college physics. For this purpose, a questionnaire (a set of 14 multiple-choice questions) was developed to probe how students dealt with these aspects in a variety of contexts. The questionnaire was finalized after a trial of its pilot version on 7 students on the basis of their written scripts and interviews. After content validation by three external experts, the test was administered to a sample of 141 senior undergraduate physics students from a diverse set of colleges and national institutes. Data analysis reveals several significant student pitfalls in the topic related principally to (i) the identification of a dimensionless ratio to characterize an approximation, (ii) the notion of the order of approximation and consistent handling of a problem up to the required order, and (iii) the notion of the order of infinitesimal in calculus-based derivations. The findings are of direct pedagogic relevance. [ABSTRACT FROM AUTHOR]

Fibrous dysplasia (FD) is a mosaic non-inheritable genetic disorder of the skeleton in which normal bone is replaced by structurally unsound fibro-osseous tissue. There is no curative treatment for FD, partly because its pathophysiology is not yet fully known. We present a simple mathematical model of the disease incorporating its basic known biology, to gain insight on the dynamics of the involved bone-cell populations, and shed light on its pathophysiology. We develop an analytical study of the model and study its basic properties. The existence and stability of steady states are studied, an analysis of sensitivity on the model parameters is done, and different numerical simulations provide findings in agreement with the analytical results. We discuss the model dynamics match with known facts on the disease, and how some open questions could be addressed using the model. [ABSTRACT FROM AUTHOR]

Every year, the problem of environmental pollution from metallurgical waste is gaining momentum. In particular, these are metallurgical wastes containing residual metals. However, with the help of innovative methods of processing lead-containing slags, they can be turned into valuable resources with minimal impact on the environment. This article discusses the importance of recycling lead-containing slag and describes various methods of recycling. The scientific novelty of the article is to analyze the influence of the content of toxic metals, such as lead, chromium, zinc, and copper, in the products obtained after roasting around the waste storage. The results of the optimization of experimental data allowed to specify the composition of the mixture temperature of the process of ignition of residual and corrosion-protective metals chromium and nickel, with the indicators of dependencies in the three-dimensional model. The results of the mathematical planning of the processing of waste slag and galvanic sludge showed that the recovery of lead reached 93-95%, chromium-94-96% nickel-93%, and copper-90%. The obtained product is identified as a composite material by its elemental composition and microstructure. In the general structure of the surface, firing products are characterized by the presence of the aluminosilicate minerals melilite, merwinite, and ockermanite, as well as calcium monosilicates, in the form of irregular forms of light gray lamellar crystals. [ABSTRACT FROM AUTHOR]

Background: Throughout the SARS-CoV-2 pandemic, policymakers have had to navigate between recommending voluntary behaviour change and policy-driven behaviour change to mitigate the impact of the virus. While individuals will voluntarily engage in selfprotective behaviour when there is an increasing infectious disease risk, the extent to which this occurs and its impact on an epidemic is not known. Methods: This paper describes a deterministic disease transmission model exploring the impact of individual avoidance behaviour and policy-mediated avoidance behaviour on epidemic outcomes during the second wave of SARS-CoV-2 infections in Ontario, Canada (September 1, 2020 to February 28, 2021). The model incorporates an information feedback function based on empirically derived behaviour data describing the degree to which avoidance behaviour changed in response to the number of new daily cases COVID-19. Results: Voluntary avoidance behaviour alone was estimated to reduce the final attack rate by 23.1%, the total number of hospitalizations by 26.2%, and cumulative deaths by 27.5% over 6 months compared to a counterfactual scenario in which there were no interventions or avoidance behaviour. A provincial shutdown order issued on December 26, 2020 was estimated to reduce the final attack rate by 66.7%, the total number of hospitalizations by 66.8%, and the total number of deaths by 67.2% compared to the counterfactual scenario. Conclusion: Given the dynamics of SARS-CoV-2 in a pre-vaccine era, individual avoidance behaviour in the absence of government action would have resulted in a moderate reduction in disease however, it would not have been sufficient to entirely mitigate transmission and the associated risk to the population in Ontario. Government action during the second wave of the COVID-19 pandemic in Ontario reduced infections, protected hospital capacity, and saved lives. [ABSTRACT FROM AUTHOR]

This study summarizes the most recent findings on osmotic dehydration, a crucial step in food preservation. The many benefits of osmotic dehydration are listed, including longer shelf life and preserved nutritional value. Mass transfer dynamics, which are critical to understanding osmotic dehydration, are explored alongside mathematical models essential for comprehending this process. The effect of osmotic agents and process parameters on efficacy, such as temperature, agitation and osmotic agent concentration, is closely examined. Pre-treatment techniques are emphasized in order to improve process effectiveness and product quality. The increasing demand for sustainability is a critical factor driving research into eco-friendly osmotic agents, waste valorization, and energy-efficient methods. The review also provides practical insights into process optimization and discusses the energy consumption and viability of osmotic dehydration compared to other drying methods. Future applications and improvements are highlighted, making it an invaluable tool for the food industry. [ABSTRACT FROM AUTHOR]

The estimation of machine's centre of gravity (COG) dislocation is presented in this article. Various types of agricultural tractors' parameters were analysed. Their parameters such as track width in the range of 〈1222 mm, 1900 mm〉 and COG height in the range of 〈830 mm, 1340 mm〉 are known from the reports of OECD tests. The static stability of the machine is defined by the critical angle of labile equilibrium or critical overturning angle in static disposition, which is determined from measurement on the tilt table. The critical overturning angle can be calculated from the known parameters of COG dislocation. The critical overturning angle for each analysed machine can be evaluated from the obtained dataset. The mean and standard deviation of parameters were used as input parameters into Monte Carlo simulation (MCS). The MCS method was used for the estimation of the critical overturning angle and height of COG for 2500 data. Simulation was performed using a Mathcad® Prime 9.0 software. The basic statistical analysis was performed using a Spyder software in the Python language. The polynomial approximation equation was set up, and then COG heights were calculated. The critical overturning angle was calculated from the real and simulated parameters. The functions of the critical overturning angle were compared, and the fit of approximation was evaluated by the R2 parameter. The designed model was verified by the F-distribution test. [ABSTRACT FROM AUTHOR]

A fundamental analysis is presented to describe the dissolution behavior of lime (CaO) in FeO-containing slags. FeO present in steelmaking slag is seen to penetrate preferentially into the lime particle and form a (CaO–FeO) layer around it, thereby a pathway for outward diffusion of CaO into the slag and subsequent formation of dicalcium silicate. In the present work, the lime dissolution behavior is described in terms of the chemical potential gradient of CaO between the lime particle and slag. The chemical potential gradient is more fundamental for calculating fluxes in concentrated melts in which the activities are far from ideal, which has been ignored by previous researchers. The model developed in the present work is used to compare the mass transfer coefficients for a variety of experimental conditions. In addition, other factors affecting the dissolution are lime particle morphology, activity of silica in slag, slag viscosity, temperature and turbulence in slag. [ABSTRACT FROM AUTHOR]

The functional integration of biological switches with synthetic building blocks enables the design of modular, stimulus‐responsive biohybrid materials. By connecting the individual modules via diffusible signals, information‐processing circuits can be designed. Such systems are, however, mostly limited to respond to either small molecules, proteins, or optical input thus limiting the sensing and application scope of the material circuits. Here, a highly modular biohybrid material is design based on CRISPR/Cas13a to translate arbitrary single‐stranded RNAs into a biomolecular material response. This system exemplified by the development of a cascade of communicating materials that can detect the tumor biomarker microRNA miR19b in patient samples or sequences specific for SARS‐CoV. Specificity of the system is further demonstrated by discriminating between input miRNA sequences with single‐nucleotide differences. To quantitatively understand information processing in the materials cascade, a mathematical model is developed. The model is used to guide systems design for enhancing signal amplification functionality of the overall materials system. The newly designed modular materials can be used to interface desired RNA input with stimulus‐responsive and information‐processing materials for building point‐of‐care suitable sensors as well as multi‐input diagnostic systems with integrated data processing and interpretation. [ABSTRACT FROM AUTHOR]

Malaria remains a global health problem despite the many attempts to control and eradicate it. There is an urgent need to understand the current transmission dynamics of malaria and to determine the interventions necessary to control malaria. In this paper, we seek to develop a fit-for-purpose mathematical model to assess the interventions needed to control malaria in an endemic setting. To achieve this, we formulate a malaria transmission model to analyse the spread of malaria in the presence of interventions. A sensitivity analysis of the model is performed to determine the relative impact of the model parameters on disease transmission. We explore how existing variations in the recruitment and management of intervention strategies affect malaria transmission. Results obtained from the study imply that the discontinuation of existing interventions has a significant effect on malaria prevalence. Thus, the maintenance of interventions is imperative for malaria elimination and eradication. In a scenario study aimed at assessing the impact of long-lasting insecticidal nets (LLINs), indoor residual spraying (IRS), and localized individual measures, our findings indicate that increased LLINs utilization and extended IRS coverage (with longer-lasting insecticides) cause a more pronounced reduction in symptomatic malaria prevalence compared to a reduced LLINs utilization and shorter IRS coverage. Additionally, our study demonstrates the impact of localized preventive measures in mitigating the spread of malaria when compared to the absence of interventions. [ABSTRACT FROM AUTHOR]

Communication is pivotal for the emergence of coordination and cooperation within teams. As a result, communication plays an important role in team dynamics as better communication between teammates can lead to more efficient and successful teams. To better understand communication dynamics and their impact on team performance, we develop a modelling framework of two-agent interaction dynamics in a discrete-time fashion. Our proposed model considers the communications between two agents based on each agent's personality and their individual impact on any received communication. Combined with data, we perform mathematical analysis and bifurcation diagrams to study how agents' personality and training may impact the quality of the team's communications and therefore their performance in task completion. The validations and parameter estimations of our model from data could potentially help us to select team members that could work together efficiently, and train members in the established team to collaborate better. [ABSTRACT FROM AUTHOR]

This paper is concerned with a numerical approach for grain drying process in a packed moving grain dryer, where the grain drying process is regarded as steady process based on a newly proposed steady state equation for moisture content of grain. The present method is compared with antecedent results suggested by Zare (Zare D. Computer simulation of rough rice drying in a batch dryer. Energy Convers Manag 2006;47:3241–54) conducted experiments for moisture content and temperature of grain dried in a fixed bed dryer, and shows good agreement. The present model is then extended to the case of the cylindrical dryer with a packed falling bed. The effects of operating parameters of the dryer (i.e. temperature and flow rate of inlet hot air and moving velocity of packed bed) on moisture content of grain are considered. The present approach can be applied to predict the positional distribution of physical quantities such as temperatures of air and grains as well as the relative humidity of air and the moisture content of grains in an arbitrarily-shaped packed moving grain dryer. [ABSTRACT FROM AUTHOR]

In this article, the solution of a mathematical model of childhood diseases with the Jumarie fractional derivative is discussed by coupling the Laplace variational iteration method (LVIM). The solution of the proposed model not only exists but is also unique. The proposed model is dimensionally balanced as well as stable. This article aims to present an analytical approximation solution for non-linear fractional-order differential equations (FDEs) with the Jumarie derivative. The model has also demonstrated relevant numerical and graphical descriptions. [ABSTRACT FROM AUTHOR]

Moving bed bioreactor is a promising technology for removing a wide range of contaminants from both municipal and industrial wastewater. For process optimization, it is crucial to have a detailed design of a reactor involving the development of a model that can predict the outcome of the operation. The present study attempts to develop a simplistic mathematical model for substrate removal in biofilms considering Fick’s Law of diffusion to illustrate the simultaneous substrate transport and its utilization within the biofilm and the microbial kinetics as explained by Monod’s theory. The mathematical solution is derived using second order differential equation with relevant boundary conditions. To investigate the applicability of the model for substrates with moderate biodegradability, experimental studies are conducted using synthetic oily wastewater, where oil and grease (O&G) primarily contribute to chemical oxygen demand (COD). The ease of practical implications of the model without any sophisticated software and tools would make it an economically viable option for process design in MBBR in treatment plants. The model is developed using FORTRAN language, implying that the same can be developed in other advanced languages. Furthermore, the model is experimentally validated at different hydraulic retention times, initial O&G concentration and biofilm density using the data obtained from laboratory-scale studies of an MBBR treating synthetic oily wastewater. The results indicate that a maximum relative error of 7.14% occurs with respect to the data obtained from the experimental analysis, implying the applicability of the model for treating moderately degradable pollutants in MBBR. Moreover, the performance precision of the model is carried out based on Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and Coefficient of Determination (R2), where the respective values of 0.005, 0.004 and 0.97 suggest the accuracy of the model. Discussion regarding the limitations and subsequent areas for future research clearly highlights the potential for further development of process design model for MBBR treating wastewater with moderate biodegradability.Highlights: A simplistic mathematical model is derived for substrate utilization in MBBR The model is based on Monod’s kinetics and Fick’s second law of diffusion It predicts effluent substrate concentration, biofilm thickness and substrate flux Model can be executed in any language including FORTRAN to get relevant parameters Experimental validation shows results within 10% deviation of model predicted data [ABSTRACT FROM AUTHOR]

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The problem of calculation prediction of the overall distortions of a large vessel made of aluminium alloy during friction stir welding (FSW) is considered. A mathematical model was developed using numerical methods of thermoplasticity analysis for determining the stress-strain state during FSW, by means of which it is possible to obtain residual plastic strains (the inherent strain function parameters) for both types of welded vessel joints (longitudinal and circumferential). This makes it possible to predict the overall distortions of a large cylindrical vessel with a great number of welded joints by the approximated method of inherent strains within the limits of the theory of elasticity. The reliability of the mathematical model for determination of the residual stresses and strains at FSW of aluminium alloy is confirmed by the agreement of the calculated distribution of the residual longitudinal stresses with the data of experimental measurements. This can contribute to ensuring the necessary accuracy of predicting the overall distortions of large vessels. The developed mathematical models and calculation algorithms can be effectively used for in-process prediction of the stress-strain state during assembly welding of large cylindrical vessels made of aluminium alloys. [ABSTRACT FROM AUTHOR]

The studied solar pot is a recent invention, which is made for environmentally friendly cooking or heating (by utilizing solar energy) of foods and liquids. Its structure is similar to a double pipe heat exchanger, it has an outer mantle and an inner cooking tank. The goals of the paper are proposing a new physically-based mathematical model describing the solar pot and carrying out computer experiments with it, assembling an experimental system of the pot connected with a solar collector and performing measurements on it. Based on the results, the solar pot can successfully be used for cooking or sterilizing foods or liquids during the studied time period, in Hungary. In particular, based on measured data, the temperature level needed for heat treatment (75 °C) can be maintained in the cooking tank for several hours (~5 h, on the average) in a typical day in May. [ABSTRACT FROM AUTHOR]

Clinical pharmacology is often the nexus in any cross‐disciplinary team that is seeking solutions for human healthcare issues. The use and application of real‐world data and artificial intelligence to better understand our ecosystem has influenced our view at the world and how we do things. This has resulted in remarkable advancements in the healthcare space and development of personalized medicines with great attributes from the application of models and simulations, contributing to a more efficient healthcare development process. A cross‐disciplinary symposium was held in December 2023, whereby experts from different scientific disciplines engaged in a high‐level discussion on the opportunities and challenges of mathematical models in different fields, possible future developments and decision making. A strong interlink amongst the disciplines represented was apparent, with clinical pharmacology identified as the one which integrates various scientific disciplines. Deliberate and strategic cross‐disciplinary dialogues are required to break out of the silos and implement integration for efficiency and cost‐effectiveness of novel interventions. [ABSTRACT FROM AUTHOR]

This research aims to investigate the relationships between the mathematical modelling process and the levels of covariational reasoning through a multiple instrumental case study with six college students solving a task in the context of science. We designed a modelling task based on a problem faced by Newton: obtaining higher temperature values than a thermometer could register; and analysed the participants’ modelling processes and covariational reasoning as they solved the task. The results show that the modelling task promoted covariational reasoning. The results also show that participants used covariational reasoning to solve the task, and the participants were unaware of nonlinear covariational reasoning which prevented them from making mathematical models closer to reality. Furthermore, participants required specific levels of covariational reasoning at specific steps in the mathematical modelling process, and as they advanced in the mathematical modelling process they exhibited more sophisticated levels of covariational reasoning. [ABSTRACT FROM AUTHOR]

In South Korea, the National Immunization Program has included one-dose varicella vaccination for 1-year-olds since 2005. This study examines the potential impact of introducing a two-dose varicella vaccination for children, along with zoster vaccination for adults, using either the zoster vaccine live (ZVL) or recombinant zoster vaccine (RZV). The investigation considered four strategies in a base case scenario. The first involved introducing zoster vaccination for 60-year-olds, with a 60 % coverage. The second strategy combined zoster vaccination with a second-dose varicella vaccination for 4-year-olds, with a 90 % coverage. An age-structured model spanning 50 years was employed, assuming a zoster vaccine catch-up campaign over the initial 5 years. Cost-effectiveness analyses were conducted, assessing incremental cost-effectiveness ratios (ICERs), incremental net monetary benefits (INMBs), and net loss under different ages at zoster vaccination (50, 60, 65, and 70 years) and varying willingness-to-pay (WTP) levels from ₩40 million ($34,998) to ₩84 million ($74,000). All strategies were cost-effective and significantly reduced herpes zoster (HZ) incidence, preventing approximately 3,077,000 to 7,609,000 cases, depending on the chosen strategy. The combined strategy prevented around 4,950,000 varicella and 653,000 HZ cases additionally. RZV outperformed ZVL by preventing twice as many HZ cases and offering greater QALY gains. However, ZVL was more cost-effective due to its lower cost. Probabilistic sensitivity analyses revealed that RZV became more cost-effective at higher WTP thresholds, exceeding ₩60.9 million ($53,193) in terms of ICER and ₩62.5 million ($54,591) for INMBs and net loss. The optimal age for zoster vaccination was 60 years concerning ICER but 50 years regarding INMB. Combining RZV with a two-dose varicella vaccination strategy reduced the disease burden and improved QALY more effectively, though ZVL remained more cost-effective at lower WTP levels. Decisions regarding vaccination policies should be balanced between the public health needs and WTP levels. [ABSTRACT FROM AUTHOR]

Quantitative Systems Pharmacology (QSP) has become a powerful tool in the drug development landscape. To facilitate its continued implementation and to further enhance its applicability, a symbiotic approach in which QSP is combined with artificial intelligence (AI) and machine learning (ML) seems key. This manuscript presents four case examples where the application of a symbiotic approach could unlock new insights from multidimensional data, including realworld data, potentially leading to breakthroughs in drug development. Besides the remarkable benefits (gAIns) that the symbiosis can offer, it does also carry potential challenges (pAIns) such as how to assess and quantify uncertainty, bias and error. Hence, to ensure a successful implementation, arising pAIns need to be acknowledged and carefully addressed. Successful implementation of the symbiotic QSP and ML/AI approach has the potential to serve as a catalyst, paving the way for a paradigm shift in drug development. [ABSTRACT FROM AUTHOR]

This study aims to characterise and understand the manifestation of critical mathematics consciousness (CMC) in the modelling process and the relationships that are established between mathematical modelling as a teaching methodology and the development of critical consciousness. Based on a contextualised topic of interest, the students produced mathematical models, and an improved analytical framework based on education for critical mathematical consciousness was used to analyse the interpretations and validations produced. In addition, a questionnaire was applied to ascertain consciousness at an individual level. The results show that the application of the analytical framework generated insights into the development of critical mathematics consciousness. Furthermore, it was noted that the use of questions seems to help in the development of mathematical modelling by deepening discussions and benefiting the development of CMC. This suggestion contributes to the evolution of CMC, through the reflection on social issues in the production of a model. It appears that mathematical modelling is favourable to the manifestation of CMC. [ABSTRACT FROM AUTHOR]

Background: By March 2023, the COVID-19 illness had caused over 6.8 million deaths globally. Countries restricted disease spread through nonpharmaceutical interventions (NPIs; e.g. social distancing). More severe "lockdowns" were also required to manage disease spread. Although lockdowns effectively reduce virus transmission, they substantially disrupt economies and individual well-being. Fortunately, the availability of vaccines provides alternative approaches to manage disease spread. Yet, vaccination programs take several months to implement fully, require further time for individuals to develop immunity following inoculation, may not have complete coverage and/or may be imperfectly efficacious against the virus. Given these aspects of a vaccination programme, it is important to understand how NPIs (such as lockdowns) can be used in conjunction with vaccination to achieve public health goals. Methods: We use mathematical methods to, investigate optimal approaches for vaccination under varying lockdown lengths and/or severities to prevent COVID-19-related deaths exceeding critical thresholds. Results: We find that increases in vaccination rate cause a disproportionate decrease in the length and severity lockdowns to keep mortality levels below a critical threshold. With vaccination, severe lockdowns can further reduce infections by up to 89%. Notably, we include simple demographics, modelling three groups: vulnerable, front-line workers, and non-vulnerable. We investigate the sequence of vaccination. One counter-intuitive finding is that even though the vulnerable group is high risk, demographically, this is a small group and critically, per person, vaccination therefore occurs more slowly. Hence vaccinating this group first achieves limited gains in overall disease control. Discussion: Importantly, we conclude that improved disease control may be best achieved by vaccinating the non-vulnerable group coupled with longer and/or more severe NPIs. [ABSTRACT FROM AUTHOR]

The study of diffusion in biological materials is crucial for fields like food science, engineering, and pharmaceuticals. Research that combines numerical and analytical methods is needed to better understand diffusive phenomena across various dimensions and under variable boundary conditions within food matrices. This study aims to bridge this gap by examining the diffusion of substances through biological materials analytically and numerically, calculating diffusivity and conducting surface analysis. The research proposes a process for sweetening Bing-type cherries (Prunus avium) using sucrose/xylitol solutions and a staining technique utilising erythrosine and red gardenia at varying concentrations (119, 238 and 357 ppm) and temperatures (40, 50 and 60 °C). Given the fruit's epidermis resistance, the effective diffusivities of skin were inferior to those in flesh. Temperature and concentration synergise in enhancing diffusion coefficients and dye penetration within the food matrix (357 ppm and 60 °C). Red gardenia displayed significant temperature-dependent variation (p = 0.001), whereas erythrosine dye remained stable by temperature changes (p > 0.05). Gardenia's effective diffusivities in cherry flesh and skin, at 357 ppm and 60 °C, 3.89E−08 and 6.61E−09 m2/s, respectively, significantly differed from those obtained at lower temperatures and concentrations. The results highlight the temperature-concentration impacts on mass transfer calculations for food colouring processes and preservation methodologies. [ABSTRACT FROM AUTHOR]

Efficient steam energy production was essential for reducing energy consumption and operational costs while enhancing productivity, particularly in industrial settings prone to explosions due to boiler parameter control issues. This challenge was especially acute in the food and beverage industry amid rising energy costs and stricter environmental regulations, highlighting the importance of optimizing steam energy production. This study focused on refining operational parameters in a steam production plant to maximize steam energy output. It utilized mathematical models and optimization tools to identify ideal operational conditions and investigate extreme scenarios. Design-Expert version 13.0 statistical software and Response Surface Methodology (RSM) via Centre Composite Design (CCD) were employed to create a comprehensive design matrix encompassing key variables like time, pressure levels, temperature, mass flow rate, and steam energy production across three experimental levels. The research revealed that increased pressure and time significantly boosted steam energy production by leveraging water's energy content rise under initial conditions, thus improving efficiency by reducing required water mass circulation. Moreover, elevated temperature and extended operation enhanced economizer efficiency, leading to increased heat recovery and reduced steam generation. Steam generation also increased with temperature and time due to the pressure rise during boiling, necessitating more energy for steam conversion. An optimum yield of steam energy of 620 Cal was attained at a time, pressure, temperature, and mass flow rate of 1 h, 16.97 MPa, 249.5°C, and 59.85 kg/s, respectively. The mathematical model developed is accurate, reliable, responsive, and can replicate the experimental data due to the high F-value (24.48), low CV (0.94) lowp-value (< 0.005), and high R2 (0.9821) value close to 1. This research promises to enhance the efficiency of steam energy production in the food and beverage industry by reducing the need for resource-intensive experimental procedures, thus lowering costs and resource consumption. [ABSTRACT FROM AUTHOR]

This paper proposes a mathematical model for the shading profiles of a PV module with thin, long linear elements. The model includes the brightness distribution over the entire shading region (umbra, penumbra, and antumbra). A corresponding calculation code in the form of m-files has been prepared for the MATLAB environment. The input data for the calculations are the coordinates of the Sun's position in the sky, the dimensions and spatial orientation of the shading element, and the spatial orientation of the shaded PV module. The correctness of the model was verified by a measurement experiment carried out under actual outdoor weather conditions. Statistical analysis of the comparison between the measurement data from the experiment and the model showed its high accuracy. As part of this research work, it was also checked how shading with thin linear elements affects the current–voltage characteristics of the module. It turned out that even a small linear shading could reduce the power output of the module by more than 6%, with the distribution of this shading across the individual cells of the module being extremely important. [ABSTRACT FROM AUTHOR]

An upcoming trial may provide further evidence that adolescent/adult-targeted BCG revaccination prevents sustained Mycobacterium tuberculosis infection, but its public health value depends on its impact on overall tuberculosis morbidity and mortality, which will remain unknown. Using previously calibrated models for India and South Africa, we simulated BCG revaccination assuming 45% prevention-of-infection efficacy, and we evaluated scenarios varying additional prevention-of-disease efficacy between +50% (reducing risk) and −50% (increasing risk). Given the assumed prevention-of-infection efficacy and range in prevention-of-disease efficacy, BCG revaccination may have a positive health impact and be cost-effective. This may be useful when considering future evaluations and implementation of adolescent/adult BCG revaccination. [ABSTRACT FROM AUTHOR]

Pruning residues, which occur every year in orchards and have many different utilization potentials, are an important issue for fruit producers. The shredding process is indispensable and critical for the utilization of these residues. The performance of the shredding process is affected by the operating parameters of the shredding machine as well as the moisture content of the residues to be shredded. In this study, olive pruning residues with three different moisture contents were shredded at three different rotor speeds in the developed shredding system. We determined how the power requirement of the shredder changed under different conditions, and empirical models were developed. The experiments showed that the average power requirement of the shredder ranged from 7.32 to 10.81 kW, and it was found that residues with low moisture content decreased the power values, while higher rotor speeds increased the power requirement. The developed final model has a mean absolute error (MAE) of 0.376, a root mean square error (RMSE) of 0.441, and a correlation coefficient (R2) of 0.859. The model serves as a reliable tool for estimating power requirements in the shredding of olive pruning residues, enabling the selection of the optimal rotor speed based on moisture content. [ABSTRACT FROM AUTHOR]

Thermal conductivity is an important property for many iron cast components, and the lack of widely accepted thermal conductivity model for cast iron, especially grey cast iron, motivates the efforts in this research area. The present study contributes to understanding the effects alloy microstructure has on thermal conductivity. A thermal conductivity model for a pearlitic cast iron has been proposed, based on the as-cast alloy composition and microstructural parameters obtained at different solidification rates. According to the model, available parallel heat transfer paths formed by connected graphite flakes across eutectic cells are determined by the space between dendrite arms. The uncertainties both for model inputs and for validation measurements have been estimated. Sensitivity analysis has been conducted to result in better understanding of the model behaviour. The agreement between modelled and measured thermal conductivities has been achieved within 5% on the average for the investigated samples. [ABSTRACT FROM AUTHOR]

Mathematical modelling of physical phenomena is based on the laws of physics, but for complicated processes, phenomenological models could enhance the descriptive and prescriptive power of the analysis. This paper describes some hybrid models, where in addition to the physics-driven part, some phenomenological variables (based on observations) are added. The internal variables widely used in continuum mechanics for modelling dissipative processes and the phenomenological variables used in modelling neural impulses are described and compared. The appendices describe two models of neural impulses and test problems for two classical cases: the wave equation and the diffusion equation. These test problems demonstrate the usage of phenomenological variables for describing dissipation as well as amplification. [ABSTRACT FROM AUTHOR]

Many neurodegenerative diseases (NDs) are characterized by the slow spatial spread of toxic protein species in the brain. The toxic proteins can induce neuronal stress, triggering the Unfolded Protein Response (UPR), which slows or stops protein translation and can indirectly reduce the toxic load. However, the UPR may also trigger processes leading to apoptotic cell death and the UPR is implicated in the progression of several NDs. In this paper, we develop a novel mathematical model to describe the spatiotemporal dynamics of the UPR mechanism for prion diseases. Our model is centered around a single neuron, with representative proteins P (healthy) and S (toxic) interacting with heterodimer dynamics (S interacts with P to form two S’s). The model takes the form of a coupled system of nonlinear reaction–diffusion equations with a delayed, nonlinear flux for P (delay from the UPR). Through the delay, we find parameter regimes that exhibit oscillations in the P- and S-protein levels. We find that oscillations are more pronounced when the S-clearance rate and S-diffusivity are small in comparison to the P-clearance rate and P-diffusivity, respectively. The oscillations become more pronounced as delays in initiating the UPR increase. We also consider quasi-realistic clinical parameters to understand how possible drug therapies can alter the course of a prion disease. We find that decreasing the production of P, decreasing the recruitment rate, increasing the diffusivity of S, increasing the UPR S-threshold, and increasing the S clearance rate appear to be the most powerful modifications to reduce the mean UPR intensity and potentially moderate the disease progression. [ABSTRACT FROM AUTHOR]

Focusing on improving the performance of the hydrogen (H2)-based direct reduction shaft furnace (HSF), the current work was undertaken to evaluate the potential benefits of an operation featuring nitrogen (N2) mixing in feed gas using a computational fluid dynamics (CFD) model that describes the in-furnace gas-solid countercurrent reactive flows. A set of simulation cases was carried out under different N2 flow rates and top pressures. Variation in the latter operating parameter was conducted with the intent to mitigate the issue of H2 dilution caused by N2 mixing. The results showed that the in-furnace thermochemical state deteriorates if the N2 flow rate is inadequate. The state is gradually improved by increasing the N2 flow rate as more sensible heat is delivered into the process, thereby resulting in better degrees of solid reduction and H2 utilization. An increase in the top pressure gives rise to higher gas density that enhances the driving force and thus facilitates the reduction reaction. A higher solid reduction degree is consequently achieved by elevating the top pressure. When the top pressure exceeds 5.0 atm, however, the increase in solid reduction degree becomes marginal, while the energy required for compressing the feed gas continues to rise linearly. [ABSTRACT FROM AUTHOR]

The goal of this paper is to utilize the homotopy perturbation method (HPM) and Laplace transform to provide an approximate analytical expression to the non-linear time-dependent reaction diffusion equation arising in a mathematical model of an immobilized enzyme system with external mass transfer resistance. This mathematical model is a non-steady, non-linear reaction diffusion equation based on Michaelis–Menten kinetics. Approximate analytical expressions are also provided for various geometries of the enzyme catalytic pellets, namely, planar, cylindrical, and spherical. Obtained semi-analytical expressions are proven to fit for all the parameters appearing in the system and for all the geometries of enzyme catalytic pellets. When comparing the numerical and approximate analytical solutions, satisfactory results are obtained. Also, approximate analytical expressions of the effectiveness factor (EF) of the immobilized system are presented, and the effect of parameters on the EF is also analyzed. [ABSTRACT FROM AUTHOR]

The knowledge‐based prerequisite framework (KPF) is an alternative to the course‐based prerequisite framework (CPF), which is widely used for curriculum design. The KPF is more flexible because it only requires essential prerequisite knowledge, while the CPF is more rigid and requires students to take all prerequisite courses. Since the number of prerequisite knowledge terms is, in general, much greater than the number of prerequisite courses, flexibility can cause additional complexity. Furthermore, the KPF inevitably requires handling semantics of defined knowledge terms. This work presents a novel Artificial Intelligence (AI) Planning mathematical model that enables the KPF by automatically verifying prerequisite knowledge and incorporating hierarchical semantic relationships among knowledge terms into the model. The proposed model significantly improves the quality of course planning solutions by finding hidden or better solutions that could not be obtained without semantics consideration. The results of the comprehensive experiments show the optimality of the solutions obtained by the mathematical model and demonstrate the outperformance of incorporation of the semantics into the mathematical model, in terms of the quality of solutions. Finally, the experimental results on scalability show the necessity of the development of efficient heuristic algorithms. [ABSTRACT FROM AUTHOR]

This work explores the mathematical modelling capabilities of various iterations of ChatGPT, focusing on their performance across tasks of differing complexity and openness. The study examines the abilities of GPT-3.5, GPT-4.0, and a more instructed version, GPT-MM, in multiple scenarios. It is observed that all versions demonstrate basic mathematical problem-solving skills. However, their effectiveness varies with increasing task complexity. While GPT-4.0 and GPT-MM show marginal improvements in providing detailed solutions, significant challenges persist, especially in moderate to complex modelling contexts where comprehending the nuances of tasks becomes challenging. Additionally, the study suggests that the openness of modelling tasks has a limited impact on performance, highlighting that mathematical and contextual complexities play more critical roles. The implications of these observations are discussed in terms of potential enhancements to teaching methodologies and the integration of AI tools like GPT in educational settings. This reiterates the importance of further research to fully understand the capabilities and limitations of AI tools and ensure their effective use in education. [ABSTRACT FROM AUTHOR]

This study describes, characterises and analyses the optimisation of truck entry and exit time windows, which includes a cross-docking centre planning strategy, using the simulation environment of the AnyLogic software environment. The article describes the use of software integers (IP*) and heuristics (IP1 and IP2) for the time windows of a cross-docking centre. Based on a mathematical model using an integer number program (IP*) and heuristics (IP1 and IP2), a scheduling problem is investigated. This program consists in minimizing the penalties for violation of the time windows for inbound and outbound trucks. It also enables to calculate the operation time of time windows on an ongoing basis. The cross-docking-center model is a layout, which includes logistic elements of technological processes, created using the AnyLogic software. It is found out by using order diagrams that after unloading arriving trucks, the goods are collected in four stages, the first two of which (queuing and waiting for collection) determine the loading rate of goods in the trucks departing from the cross-dock. After determining the main stages of order collecting using the cross-docking strategy, the dependences of the average time of unloading and loading goods are studied. The average unloading and loading time are found to be 30 and 25 minutes, respectively. Based on the time dependences, a diagram of the average time of a truck in the dock, being about an hour - an hour and a half, is created. [ABSTRACT FROM AUTHOR]

The development and refinement of air ventilation systems aboard ships are paramount for ensuring safety and operational efficiency. This comprehensive review systematically evaluates the existing literature on shipboard air ventilation and associated fire incidents, employing both bibliometric analysis and mathematical modeling methodologies. It has been observed that ventilation systems are predominantly classified into three categories: mechanical ventilation, natural ventilation, and hybrid systems combining mechanical ventilation with air conditioning (AC). These systems are imperative for regulating temperature, particularly in critical areas such as engine rooms, and for mitigating risks related to leaks and fires. International Maritime Organization (IMO) standards are consistently adhered to, reinforcing the effectiveness of these systems in controlling onboard environments. Furthermore, the application of mathematical models offers significant insights, facilitating the calculation and prediction of outcomes pertinent to ventilation-related research. These models prove crucial for addressing prevalent issues such as leaks and fire hazards within the maritime industry. Conversely, a bibliometric analysis highlights emerging research trends, identifying "ventilation," "ship," "Computational Fluid Dynamics (CFD)," "ship engine," "leakage," and "fire onboard" as focal keywords. This analysis not only underscores the current research focus but also guides future investigations, encouraging ongoing advancement and innovation in maritime research. This review thus serves as a vital resource for researchers aiming to explore and expand upon the thematic areas identified, potentially leading to breakthroughs in the design and implementation of ship ventilation systems. [ABSTRACT FROM AUTHOR]

Background: Throughout the SARS-CoV-2 pandemic, policymakers have had to navigate between recommending voluntary behaviour change and policy-driven behaviour change to mitigate the impact of the virus. While individuals will voluntarily engage in self-protective behaviour when there is an increasing infectious disease risk, the extent to which this occurs and its impact on an epidemic is not known. Methods: This paper describes a deterministic disease transmission model exploring the impact of individual avoidance behaviour and policy-mediated avoidance behaviour on epidemic outcomes during the second wave of SARS-CoV-2 infections in Ontario, Canada (September 1, 2020 to February 28, 2021). The model incorporates an information feedback function based on empirically derived behaviour data describing the degree to which avoidance behaviour changed in response to the number of new daily cases COVID-19. Results: Voluntary avoidance behaviour alone was estimated to reduce the final attack rate by 23.1%, the total number of hospitalizations by 26.2%, and cumulative deaths by 27.5% over 6 months compared to a counterfactual scenario in which there were no interventions or avoidance behaviour. A provincial shutdown order issued on December 26, 2020 was estimated to reduce the final attack rate by 66.7%, the total number of hospitalizations by 66.8%, and the total number of deaths by 67.2% compared to the counterfactual scenario. Conclusion: Given the dynamics of SARS-CoV-2 in a pre-vaccine era, individual avoidance behaviour in the absence of government action would have resulted in a moderate reduction in disease however, it would not have been sufficient to entirely mitigate transmission and the associated risk to the population in Ontario. Government action during the second wave of the COVID-19 pandemic in Ontario reduced infections, protected hospital capacity, and saved lives.

Mathematical modelling of physical phenomena is based on the laws of physics, but for complicated processes, phenomenological models could enhance the descriptive and prescriptive power of the analysis. This paper describes some hybrid models, where in addition to the physics-driven part, some phenomenological variables (based on observations) are added. The internal variables widely used in continuum mechanics for modelling dissipative processes and the phenomenological variables used in modelling neural impulses are described and compared. The appendices describe two models of neural impulses and test problems for two classical cases: the wave equation and the diffusion equation. These test problems demonstrate the usage of phenomenological variables for describing dissipation as well as amplification.

The focus of this study is to investigate the energy performance of a direct-expansion solar-assisted heat pump water heating system (DX-SAHPWH). The system consists of an unglazed solar collector-evaporator, which can absorb heat from solar energy and air ambient simultaneously, a condenser in the form of a coil immersed in a hot water storage tank, a thermostatic expansion valve and a hermetic reciprocating compressor. The performance of the heat pump system is evaluated using a developed mathematical model under Matlab code. The modelling method is based on lumped and distributed parameter approach of different system components. Numerical calculations were carried-out to study the influence of different parameters, such as ambient temperature, solar radiation intensity and polytropic index on the system performance. Additionally, in order to evaluate the long-term system performance, the system’s model was applied on a case study of a single-family building located in Djelfa (Algeria), which represents the coldest arid region of the country. The results showed that the solar radiation intensity and ambient temperature have significant effects on the heat pump performance. A COP of 5.9 and a collector-evaporator efficiency of 1.9 were obtained at high solar radiation of 850 W/m2 resulting in lower heating time (29 min). In addition, results revealed that the system can operate even at lower ambient temperature due to its ability to take advantage of heat from the ambient air. The results from the case study gave a COP ranging from 2.3 to 3.8, which enhance the promising adoption of this system in domestic hot water production to respond to people daily life needs with clean, abundant and renewable energy.

Cross-contamination of foodborne pathogenic bacteria is a widespread phenomenon in food processing and production, storage and transportation, and household consumption. It is regarded as one of the important causes of foodborne disease outbreaks. Using mathematical models to simulate the transfer of foodborne pathogenic bacteria in different food media enhances our understanding of the pattern of transmission of the pathogens through cross-contamination. Determining the key links in the microbial cross-contamination by combined use of cross-contamination modelling with microbiological risk assessment and designing novel antimicrobial coatings will effectively prevent and control the occurrence of microbial cross-contamination. Therefore, this article reviews the food safety problems caused by cross-contamination of pathogenic bacteria in the food industry and household settings, and provides a detailed overview of the latest progress in mathematical modelling of cross-contamination of foodborne pathogenic bacteria, and describes the application of microbial risk assessment combined with hazard analysis and critical control point (MRA-HACCP) and antimicrobial coatings in the prevention and control of cross-contamination of foodborne pathogenic bacteria from the perspectives of both global control and source control. It concludes with an outlook on future research and control strategies for foodborne pathogen cross-contamination.

Abstract The study of diffusion in biological materials is crucial for fields like food science, engineering, and pharmaceuticals. Research that combines numerical and analytical methods is needed to better understand diffusive phenomena across various dimensions and under variable boundary conditions within food matrices. This study aims to bridge this gap by examining the diffusion of substances through biological materials analytically and numerically, calculating diffusivity and conducting surface analysis. The research proposes a process for sweetening Bing-type cherries (Prunus avium) using sucrose/xylitol solutions and a staining technique utilising erythrosine and red gardenia at varying concentrations (119, 238 and 357 ppm) and temperatures (40, 50 and 60 °C). Given the fruit's epidermis resistance, the effective diffusivities of skin were inferior to those in flesh. Temperature and concentration synergise in enhancing diffusion coefficients and dye penetration within the food matrix (357 ppm and 60 °C). Red gardenia displayed significant temperature-dependent variation (p = 0.001), whereas erythrosine dye remained stable by temperature changes (p > 0.05). Gardenia's effective diffusivities in cherry flesh and skin, at 357 ppm and 60 °C, 3.89E−08 and 6.61E−09 m2/s, respectively, significantly differed from those obtained at lower temperatures and concentrations. The results highlight the temperature-concentration impacts on mass transfer calculations for food colouring processes and preservation methodologies.

A cloud manufacturing (CMfg) system is presented as a novel service- and customer-oriented manufacturing paradigm that integrates the distributed manufacturing enterprises to share their manufacturing capabilities or resources and collaborate as an interconnected system in a dynamic environment. Since the high performance of this system depends on the formation of a suitable group of manufacturing service providers, this paper develops an integrated c onfiguration design and capacity planning problem for the CMfg system by considering the dynamic environment of this system. In this regard, dynamic service providers and dynamic demand are considered as two aspects of the dynamic nature of this system. A multi-period multi-objective mathematical model is proposed by maximising the utilities of all three stakeholders of the system. Moreover, three extensions of a discrete multi-objective grey wolf optimiser (DMOGWO) algorithm are devised to solve the medium- and large-scale instances. A comprehensive computational experiment is conducted to assess the performance of the developed meta-heuristic algorithms. Furthermore, by carrying out a sensitivity analysis, some managerial insight is suggested for the managers. [ABSTRACT FROM AUTHOR]

Harvesting energy from nonlinear systems has been at the centre of research in the energy harvesting community. Many such proposed systems are single nonlinear harvester. While these systems show an increase in bandwidth of harvesting frequency, overall, they are not effective enough in power generation. This article studies power harvesting and frequency bandwidth characteristics of an array of harvesters. Multiple harvesters are considered with linear and nonlinear coupling between the harvesters. The phenomena of internal resonance (IR) and stochastic resonance (SR) are reported. The IR in multiple coupled nonlinear harvesters is explored using multiple-scale analysis. A parametric study is conducted to demonstrate the effect of coupling strength, frequency mistuning, innate nonlinearity and other parameters. The parametric study helped establish effective ways to increase bandwidth. Moreover, a stochastically loaded linearly coupled bistable harvester array is numerically analysed to find the effect of coupling strength and array size on the phenomenon of SR and on the system's harvesting performance. Through these studies, the potential of multiple coupled nonlinear harvesters in enhanced energy harvesting is demonstrated under both harmonic and stochastic excitation. This article is part of the theme issue 'Celebrating the 15th anniversary of the Royal Society Newton International Fellowship'. [ABSTRACT FROM AUTHOR]

• Critical releasable charges (CRCs) are modeled by mechanism analysis. • Predicted CRCs are generally within 16 % compared with measured and simulated values. • Effect of air speed on concentration field under CRCs is studied by CFD. • Transition value of air speed is 0.1 and 0.2 m/s for 8 g and 14.2 g CRC respectively. Propane (R290) is a promising refrigerant for split-type household air conditioners with the advantages of energy saving, environmentally friendliness and low charge. Nevertheless, the large scale use of R290 has been hindered by its high flammability. The potential fire hazards can be relieved by reducing R290 releasable charge, and numerous methods have been put forward. In the authors' previous papers, the method of refrigerant pump-down was proposed and a mathematical model of critical releasable charge (CRC, ignition threshold of leakage) was developed by dimensional analysis, but it is difficult to reflect the actual physical process of R290 dispersion. To reveal the leakage and dispersion mechanism of R290, a new numerical model of CRC is developed based on mechanism analysis in this work, in which the turbulent jet model, Coanda effect and dimensional analysis are involved. The predicted CRCs agree well with experimental and simulated data. Besides, the influence of indoor air speed with its direction parallel to the IDU panel on R290 concentration field is investigated by computational fluid dynamics (CFD) under different releasable charges. It is found that there is a transition value of air speed, below which the fire hazards decrease greatly with the increase of air speed, on the contrary, when the air speed is above the transition value, raising air speed has little impact on fire hazards. [ABSTRACT FROM AUTHOR]

Manufacturing plays a vital role in fostering economic growth, yet it is one of the major consumers of energy. Considering energy consumption when devising an Aggregate Production Plan (APP) is a determinant of a more energy-efficient production schedule at the operational level. Manufacturers in emerging economies envisage planning challenges when energy reforms are introduced. A guide for analysis and decision-making in the transitional phase of phasing out energy subsidies is essential. This work tackles this problem by developing two mixed-integer linear programming models for energy-aware production planning. The first model targets maximising profit while explicitly using the energy cost as one of the cost elements. The second is an energy-driven model to minimise energy consumption while including profit as a constraint. The proposed models were validated via a real-life dataset. The models provide the decision-makers with the flexibility to improve their production plans' energy efficiency by either developing plans that reduce energy consumption without sacrificing profit or developing production plans that increase profit without increasing energy consumption. The results imply that adopting Energy-Efficient Production Planning (EEPP) is substantial with high subsidy levels. For low subsidy levels, strategic decision-making entailing the deployment of technological solutions is more efficient. [ABSTRACT FROM AUTHOR]

Most organisms have evolved specific mechanisms to respond to changes in environmental conditions such as light and temperature over the course of day. These periodic changes in the physiology and behaviour of organisms, referred to as circadian rhythms, are a consequence of intricate molecular mechanisms in the form of transcription and translational feedback loops. The plant circadian regulatory network is a complex web of interconnected feedback loops involving various transcription factors such as CCA1, LHY, PRRs, TOC1, LUX, ELF3, ELF4, RVE8, and more. This network enables plants to adapt and thrive in diverse environmental conditions. It responds to entrainment signals, including light, temperature, and nutrient concentrations and interacts with most of the physiological functions such as flowering, growth and stress response. Mathematical modelling of these gene regulatory networks enables a deeper understanding of not only the function but also the perturbations that may affect the plant growth and function with changing climate. Over the years, numerous mathematical models have been developed to understand the diverse aspects of plant circadian regulation. In this review, we have delved into the systematic development of these models, outlining the model components and refinements over time. We have also highlighted strengths and limitations of each of the models developed so far. Finally, we conclude the review by describing the prospects for investigation and advancement of these models for better understanding of plant circadian regulation.Key message: Plant circadian rhythms are regulated by a complex network of interactions. This review summarizes the mathematical models that have been developed until now to explore plant circadian rhythms. [ABSTRACT FROM AUTHOR]