STOCHASTIC PROGNOSTICS FOR ROLLING ELEMENT BEARINGS

When can a deterministic model of a population system reveal what will happen on average?

This work presents a deterministic and a stochastic model for the simulation of industrial-size deionized water and water for injection (DI/WFI) systems. The objective of the simulations is to determine if additional DI/WFI demand from future production processes can be supported by an existing DI/WFI system. The models utilize discrete event simulation to compute the demand profile from the distribution system; they also use a continuous simulation to calculate the variation of the water level in the storage tank. Whereas the deterministic model ignores uncertainties, the stochastic model allows for both volume and schedule uncertainties. The Monte Carlo method is applied to solve the stochastic method. This paper compares the deterministic and stochastic models and shows that the deterministic model may be suitable for most applications and that the stochastic model should only be used if found necessary by the deterministic simulation. The models are programmed within Excel 2003 and are available for download as open public domain software (1), allowing for public modifications and improvements of the model. The proposed models may also be utilized to determine size or analyze the performance of other utilities, such as heat transfer media, drinking water, etc. Water for injection (WFI) and other pharmaceutical water distribution systems are notoriously difficult to analyze analytically due to the highly dynamic variable demand that is drawn from these systems. Discrete event simulation may provide an answer where the typical engineering approach of utilizing a diversity factor fails. This paper develops an Excel based deterministic and stochastic model for a WFI system with the latter allowing for the modeling of offtake volume and schedule uncertainty. The paper also compares the deterministic and stochastic models and shows that the deterministic model may be suitable for most applications while the stochastic model should only be used if found necessary. The models are available for download as open public domain software allowing for modifications and improvements of the model.

This study compares the newly developed stochastic agent-based model (ABM) and deterministic system within the context of COVID-19 transmission dynamics, using the Susceptible-Exposed-Infectious-Removed-Perception (SEIRP) compartments. The primary aim is to analyse the similarities and differences between these modelling approaches, providing insights into the emergent behaviours of the epidemiological systems. We also investigate how social phenomena like public perception affect the two epidemiological models’ outcomes, focusing on scenarios with varying awareness levels and proportions of severe cases. Our results show that while deterministic model outputs align well with the ABM for large populations, discrepancies emerge for small populations in the ABM due to the impacts of stochastic extinction and discreteness of individuals. In scenarios with high proportions of severe cases and for large population sizes, the deterministic model exhibits oscillatory behaviour. In this situation, the averaged ABM densities initially capture the fluctuation dynamics when a substantial number of realisations is used in simulation; however, this stochastic system exhibits diminishing fluctuations across different realisations, contributing to a consistent average akin to an endemic steady state over a longer period. Interestingly, as the number of realisations is reduced, the agreement between stochastic and deterministic systems in depicting recurrent outbreaks is evident (i.e., realisations-dependent dynamical behaviour). In the next case study, the joint effect of recovery, latency period, disease severity and public perception is explored, highlighting how different factors can combine to influence the systems’ outcomes. Notably, our case study finds that even when the deterministic model demonstrates the persistence of sustained oscillations, the ABM can depict an extinction state; this stochastic ABM observation is caused by the realisations in this epidemiological system fluctuating to a very low population density and being excluded over a long run. Overall, our findings suggest the importance of considering both deterministic and stochastic models in infectious disease modelling, highlighting the need for comprehensive analyses to guide evidence-based decision-making in public health and epidemiology.

We look at the behavior of biological oscillators, specifically analyzing a genetic circuit that has oscillatory behavior. Implementing the system in deterministic and stochastic models, we compare the two models using various tests and analyze the effect of stochastic noise on these oscillations. We first investigate the effect of volume and find that at small system volumes, stochastic events cause the system to exhibit more sporadic oscillations and a longer period of oscillation. Next, we develop classification systems for discerning the boundary in the parameter space where the model begins to show oscillations. This is more challenging in the stochastic model, due to persistent fluctuations. We compare different methods of analysis of our deterministic and stochastic models and visualize the conditions which lead to oscillation in each model. We find that the parameter range of oscillation is larger in the stochastic model than in the deterministic model. Finally, we find an increased rate of mRNA production can create greater noise in a system and amplify the difference between a stochastic and deterministic system.

Unraveling pine wilt disease: Comparative study of stochastic and deterministic model using spectral method

Comparison of deterministic and stochastic SIS and SIR models in discrete time

Deterministic models have played an important role in biology over the past few centuries, and continue to play. However, there is a need to introduce a random factor due to the nature of biological processes, which led to the emergence of stochastic models. To bridge the gap between deterministic and stochastic models, and in appreciation of the huge efforts made in deterministic models, which have played a fundamental role in understanding, predicting, and controlling the transmission dynamics of infectious diseases, this study aims to clarify a method for creating stochastic model for one of most vital and recent deterministic, that is Covid-19 model with two doses, which showed the effect of receiving both doses in containing the disease. To achieve this goal, the deterministic model was studied extensively before converting to corresponding stochastic. The deterministic and stochastic systems were solved numerically and represented using MATAB tools. Keywords: Covid-19 deterministic model, stochastic model, next generation matrix method (NGMM), The diffusion matrix G.

The presence of a pathogen among multiple competing species has important ecological implications. For example, a pathogen may change the competitive outcome, resulting in replacement of a native species by a non-native species. Alternately, if a pathogen becomes established, there may be a drastic reduction in species numbers. Stochastic variability in the birth, death and pathogen transmission processes plays an important role in determining the success of species or pathogen invasion. We investigate these phenomena while studying the dynamics of deterministic and stochastic models for n competing species with a shared pathogen. The deterministic model is a system of ordinary differential equations for n competing species in which a single shared pathogen is transmitted among the n species. There is no immunity from infection, individuals either die or recover and become immediately susceptible, an SIS disease model. Analytical results about pathogen persistence or extinction are summarized for the deterministic model for two and three species and new results about stability of the infection-free state and invasion by one species of a system of n-1 species are obtained. New stochastic models are derived in the form of continuous-time Markov chains and stochastic differential equations. Branching process theory is applied to the continuous-time Markov chain model to estimate probabilities for pathogen extinction or species invasion. Finally, numerical simulations are conducted to explore the effect of disease on two-species competition, to illustrate some of the analytical results and to highlight some of the differences in the stochastic and deterministic models.

Stochastic wave models play a central role in our present-day wave modelling capabilities. They are frequently used to compute wave statistics, to generate boundary conditions and to include wave effects in coupled model systems. Historically, such models were developed to predict the wave field evolution in deep water where the conditions of Gaussianity generally hold. However, in recent decades, such models have been applied to the shallower coastal environment where the stochastic representation of the dominant wave physics becomes questionable. This is primarily due to the increased influence of wave nonlinearity and the additional depth-induced wave processes that are dominant in this region.<br/><br/>Unfortunately, the two most dominant wave processes in the surf zone: depth-induced wave breaking and nonlinear triad wave-wave interactions are also the least well represented and understood. This is due to both their complexity and the scarcity of analytical solutions for realistic wave fields. As such, they represent a significant obstacle in the accurate modelling of the wave dynamics in the coastal region. Providing accurate representations of these wave processes is essential to answering the questions demanded from stochastic wave models from coastal engineers for coastal management and design. Such advancements are necessary to improve our understanding of wave-induced processes, to reduce costs in managing the coastal environment and to tackle contemporary issues such as uncertainties with respect to increased sea level rise.<br/><br/>Due to the complexity of depth-induced wave breaking, a complete representation of this wave process does not exist for both stochastic and deterministic modelling frameworks. Although there is extensive literature on the subject of parameterizing depth-induced wave breaking in a stochastic sense, these parameterizations are inconsistent with theory, observations and (deterministic) model predictions. In particular, present-day modelling defaults perform poorly over (near-)horizontal bathymetries with over-enhanced wave dissipation of locally-generated waves and insufficient dissipation of swell waves. Equally, nonlinear triad wave-wave interactions are poorly represented in stochastic wave models due to the problem of closure and the impractical computational expense of more accurate representations. In particular, the most commonly applied parameterization in the wave literature incorrectly predicts the evolution of the spectral shape, and the convergence to an equilibrium high-frequency tail deep in the surf zone. Correctly resolving these issues is essential for the management of many of the activities occurring at the coast; from the design of coastal defenses to feasibility studies for wave energy converters, from port operation and availability to vessel navigation, from understanding the ecology at the coast to the fisheries, and from managing leisure and tourism to safety at the coast.<br/><br/> In this work, we investigate the process of depth-induced wave breaking through a comprehensive analysis of the literature and a comparison of modelling performance. Here, we use an extensive set of wave observations representing a large range of wave conditions and bathymetric profiles. The analysis demonstrates that no currently available depth-induced breaking source term is capable of sufficiently representing the process of depth-induced wave breaking. This is shown to be in agreement with the wave literature with parameterizations either over-predicting wave dissipation for locally generated waves or under-predicting wave dissipation for non-locally generated waves over (near-)horizontal bathymetries. To address this issue, a new joint scaling using both local wave and bathymetric conditions is proposed. Using both the normalized characteristic wave number and local bottom slope unifies two approaches prevalent in the wave literature. This is shown to improve the model performance for the dissipation of both locally and non-locally generated waves over (near-)horizontal bathymetries. <br/><br/>Furthermore, the validity of the assumption that wave dissipation can be modelled as analogous to a 1D dissipative bore is explored. Subsequently, a heuristic directional modification is introduced for depth-induced wave breaking dissipation models. This directionally partitions the 2D spectrum into several directional partitions that are assumed to be unidirectional. Model results demonstrate that the effect of the directional partitioning is to reduce the dissipation of wave energy and to enhance the significant wave height; in agreement with field measurements. Not only is this modification shown to be applicable to the joint wave breaking parameterization proposed in this study, but also for well-established parameterizations. <br/><br/>The effects of both the proposed scaling and directional modification are then reviewed from an operational context and are compared to state-of-the-art source terms, field observations and a hypothetical storm representative of Dutch design conditions. Such design conditions are expected to be representative of design conditions found globally. In an environment where storm intensities may be increasing, for example due to global warming, the results of wave breaking models near the coast under such extreme conditions become of greater relevance. The influence of wave breaking models in coupled model systems is anticipated to provide important new insights in understanding the various wave-driven processes along our coasts. <br/><br/>Next, the representation of the nonlinear triad wave-wave interactions in stochastic wave models is reviewed. In particular, the collinear approximation used to transform 1D triad source terms for implementation in 2D stochastic wave models is revisited. These approximations are necessitated by considerations of computational efficiency. The conventional collinear approximation is shown to be inconsistent at the unidirectional limit and to be a primary source of modelling error. Instead of converging to the values predicted by the 1D triad source terms at the unidirectional limit, the energy transfers as computed by stochastic wave models are shown to become unbounded. This results in a dimensional calibration coefficient which is at least an order of magnitude smaller than that found in the wave literature. Consequently, for directional wave conditions, 1D triad source terms implemented with the conventional collinear approximation insufficiently capture the wave evolution. To address this problem, a new collinear approximation is presented which accounts for the wave energy contained within a finite directional bandwidth. This collinear approximation is shown to converge correctly at the unidirectional limit and to agree well with predictions from a second-order accurate deterministic wave model. In particular, better agreement is shown in the modelling prediction of the spectral shape and related integral parameters, e.g. wave period, under idealized wave conditions. Under certain conditions, these error reductions are shown to be more significant than differences between the underlying triad models.<br/><br/>The contribution of this work demonstrates that while the underlying theory underpinning stochastic wave modelling in the coastal environment still remains questionable, the accurate determination of wave statistics in the coastal zone is tenable. With the advancements presented in this study, the new source terms correspond better with the current wave literature and are shown to provide significant steps forward over existing default source terms. The developments presented here are anticipated to form the foundation for future source term research, and to be used for the representation of the dominant wave physics in the coastal environment in operational wave models.

For an engineered genetic oscillator, deterministic analysis indicates sustained oscillations and stochastic simulations show irregular or absent oscillations. Since the major difference is in the modeling of the promoters, we have performed a detailed analysis of a generic repressible promoter system. Deterministic, stochastic, thermodynamic, and hybrid models were developed for the promoter. The average behavior of the stochastic model converged to the thermodynamic model. The deterministic model is a special case of the thermodynamic model. The hybrid model could lock into the off state. Adding an unrelated background reaction allowed it to exit that state. Increasing the background rate allowed the hybrid model to converge to thermodynamic and stochastic model. Adding a background reaction to the stochastic oscillator simulation noticeably improved its performance.

Ergodic stationary distribution and extinction of a hybrid stochastic SEQIHR epidemic model with media coverage, quarantine strategies and pre-existing immunity under discrete Markov switching

This article is an introduction to capacity planning and basic models used to support capacity planning and expansion decisions. The importance of making good capacity decisions by firms is described followed by definitions of various measures of capacity. The main elements of capacity decisions such as size, timing, and location of capacity additions are discussed. The impact of flexibility of capacity and the level of automation on a firm's productivity, the nature of capacity decisions as a function of the production processes, and implications of interactions between capacity levels at firms in a supply chain are discussed. Some standard deterministic and stochastic models in the literature for making capacity decisions are presented. A key insight from continuous time deterministic models is that it is optimal to add capacity at equal time intervals under several demand and capacity cost scenarios. A basic stochastic capacity expansion model is provided that can be transformed to a deterministic model with a different discount rate under fairly general conditions on the demand process. Finally, a stochastic programming model and heuristic solution approaches are described—this is a flexible approach to model capacity expansion decisions when there is uncertainty in both demand and other parameters.

Stochastic constitutive modeling of elastic-plastic materials with uncertain properties

Introduction: This study aims to provide a mathematical description of the production planning model (method). There will occasionally be restrictions on the raw material availability for production activities due to this element of uncertainty. Objectives: The researcher will outline and analyze the various perspectives, arguments, and opinions that exist in the literature, and look for relationships between them Methods: This research was compiled using qualitative research methods and descriptive and analytical journal literature review approaches. Most of the data used are secondary data. Result: Deterministic and stochastic mathematical models are the two broad categories into which production planning mathematical models fall. In deterministic models, the data is assumed to be confident at this point. In the meantime, the stochastic model uses data that reflects the "best prediction" of unknown values. These deterministic methods finally address average or worst-case value issues even though many human judgment and quantitative models have been developed to anticipate variables with this uncertainty. Even though they have been studied extensively in the literature, deterministic production planning models are generally not helpful when uncertainty is considered. There are two types of manufacturing-related uncertainties: system uncertainty and manufacturing environment uncertainty. The second category consists of supply and demand uncertainties and other uncertainties outside the purview of production process management. Conclusion: Even though they have been studied extensively in the literature, deterministic production planning models are generally not helpful when uncertainty is considered. There are two types of manufacturing-related uncertainties: system uncertainty and manufacturing environment uncertainty. Uncertainties like supply and demand issues outside the purview of industrial process management are included in the second category.

For the last few years, annual honeybee colony losses have been center of key interest for many researchers throughout the world. The spread of the parasitic mite and its interaction with specific honeybee viruses carried by Varroa mites has been linked to the decline of honeybee colonies. In this investigation, we consider honeybee-virus and honeybee-infected mite-virus models. We perform sensitivity analysis locally and globally to see the effect of the parameters on the basic reproduction number for both models and to understand the disease dynamics in detail. We use the continuous-time Markov chain model to develop and analyze stochastic epidemic models corresponding to both deterministic models. By using the disease extinction process, we compare both deterministic and stochastic models. We have observed that the numerically approximated probability of disease extinction based on 30 000 sample paths agrees well with the calculated probability using multitype branching process approximation. In particular, it is observed that the disease extinction probability is higher when infected honeybees spread the disease instead of infected mites. We conduct a sensitivity analysis for the stochastic model also to examine how the system parameters affect the probability of disease extinction. We have also derived the equation for the expected time required to reach disease-free equilibrium for stochastic models. Finally, the effect of the parameters on the expected time is represented graphically.