Statistical Insights into Stochastic Glucose-InsulinDynamics: Modeling Insulin Degradation with the Michaelis-Menten Function

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.

Event Abstract Back to Event Stochastic modeling of neuronal signaling Jukka Intosalmi1*, Tiina Manninen1, Keijo Ruohonen1 and Marja-Leena Linne1 1 Tampere University of Technology, Finland The time evolution of biochemical systems in neurons is traditionally modeled using deterministic ordinary differential equations (ODEs). Chemical reactions, however, are random in nature, and the deterministic approach is valid only for a restricted class of systems. Stochastic models take random fluctuations into account and are thus more realistic. Biochemical reactions can be modeled stochastically using numerous different methods. An ideal model would have the following three important properties. First, the model should be as realistic as possible, second, the mathematical method should be easily implementable as a computer algorithm, and third, the algorithm should be computationally effective. Naturally, all these three conditions cannot be fulfilled at the same time. Some realistic modeling approaches can be derived directly from chemical kinetics without making any approximations. Such approaches are called exact. A good example of an exact modeling approach is the stochastic simulation algorithm (SSA) developed by Gillespie. The SSA is applicable when the molecular populations in the system are small, but it becomes computationally inefficient when the numbers of molecules increase. In order to construct stochastic models that can be effectively simulated, new mathematical approaches have to be explored. As an approximate method also stochastic differential equations (SDEs) have been considered a promising way to model biochemical reactions stochastically. The SDE approach is attractive especially if we consider a system for which the SSA is computationally inefficient and the traditional deterministic ODE approach cannot be used as a good approximation. SDE models treat the chemical populations as real numbers and the construction of the model is based on the law of mass action, similarly as the construction of the traditional deterministic modeling approach. The SDE model, however, takes the random fluctuations into account by describing the time evolution of the system with the stochastic Itô process instead of the deterministic set of ODEs. The Itô process is basically a set of coupled stochastic differential equations. Although SDE modeling usually leads to equations that cannot be solved analytically, the solutions can be approximated numerically using different numerical integration methods. Thus, SDEs provide a mathematically rigorous way to construct models that can be simulated effectively. In order to investigate how well the SDE approach models biochemical systems, the results have to be compared with experimental data or with the simulation results from some exact simulation procedure. In this study, we use the methodology of spectral analysis to compare the simulation results of the SDE model and the SSA. As case studies, we consider e.g. calcium binding and protein kinase C (PKC) signal transduction pathway. We investigate the nature of noise in different modeling approaches and try to interpret the meaning of the noise in real biological systems. The main goal of our study is to find the most essential components and parameters in the SDE model and to adjust them so that the model is capable of giving similar results as the SSA. The simulations are carried out using the distributed computing resources (GRID) provided by Techila Technologies Ltd. Conference: Neuroinformatics 2008, Stockholm, Sweden, 7 Sep - 9 Sep, 2008. Presentation Type: Poster Presentation Topic: Computational Neuroscience Citation: Intosalmi J, Manninen T, Ruohonen K and Linne M (2008). Stochastic modeling of neuronal signaling. Front. Neuroinform. Conference Abstract: Neuroinformatics 2008. doi: 10.3389/conf.neuro.11.2008.01.023 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 28 Jul 2008; Published Online: 28 Jul 2008. * Correspondence: Jukka Intosalmi, Tampere University of Technology, Tampere, Finland, jukka.intosalmi@tut.fi Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne Google Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne Google Scholar Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne PubMed Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

This article presents an analysis of deterministic models of computer virus epidemic propagation (SIR models) and their classification. The main areas of research into these models are highlighted. An analysis of existing stochastic models based on the SIR model and their diversity are presented. A method for constructing a stochastic SIR model based on the classical SIR model, represented by a system of Ito stochastic differential equations with a Wiener process, is proposed. The possibility of constructing a stochastic and deterministic model of an information system protected against computer virus infection with probability 1 is demonstrated: a stochastic model, in which infection by viruses occurs continuously, and a deterministic model, in which the virus is present in the information system. A mathematical stochastic model of an information system protected from computer virus outbreaks is constructed as a system of stochastic differential equations whose first integrals are invariants preserved with probability 1. A certain functional relationship between model variables, maintaining a constant value, is considered as the system's security indicator. Introducing a compensator (program control with probability 1 (PCP1)) into the model allows the specified security indicator, described by the model variables, to be maintained with probability 1. Introducing a compensator (program control with probability 1 (PCP1)) into the model allows the specified security indicator, described by the model variables, to be maintained with probability 1. Similarly, based on the proposed algorithm, a deterministic model of an information system protected from computer virus infection is constructed. A control similar to programmed control with probability 1 (PCP1) is introduced into the constructed model, which allows the invariants to be maintained. A distinctive feature of the proposed models is that they preserve invariants associated with the properties that ensure the security of the information system. The behavior of the constructed models is studied using numerical simulation in the MathCad environment. Based on the research results, conclusions were drawn on the possibility of using the proposed method in constructing stochastic models based on other models of epidemic spread, as well as for models of protecting an information system from the spread of a computer virus epidemic.

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

Neurons in the brain express intrinsic dynamic behavior which is known to be stochastic in nature. A crucial question in building models of neuronal excitability is how to be able to mimic the dynamic behavior of the biological counterpart accurately and how to perform simulations in the fastest possible way. The well-established Hodgkin-Huxley formalism has formed to a large extent the basis for building biophysically and anatomically detailed models of neurons. However, the deterministic Hodgkin-Huxley formalism does not take into account the stochastic behavior of voltage-dependent ion channels. Ion channel stochasticity is shown to be important in adjusting the transmembrane voltage dynamics at or close to the threshold of action potential firing, at the very least in small neurons. In order to achieve a better understanding of the dynamic behavior of a neuron, a new modeling and simulation approach based on stochastic differential equations and Brownian motion is developed. The basis of the work is a deterministic one-compartmental multi-conductance model of the cerebellar granule cell. This model includes six different types of voltage-dependent conductances described by Hodgkin-Huxley formalism and simple calcium dynamics. A new model for the granule cell is developed by incorporating stochasticity inherently present in the ion channel function into the gating variables of conductances. With the new stochastic model, the irregular electrophysiological activity of an in vitro granule cell is reproduced accurately, with the same parameter values for which the membrane potential of the original deterministic model exhibits regular behavior. The irregular electrophysiological activity includes experimentally observed random subthreshold oscillations, occasional spontaneous spikes, and clusters of action potentials. As a conclusion, the new stochastic differential equation model of the cerebellar granule cell excitability is found to expand the range of dynamics in comparison to the original deterministic model. Inclusion of stochastic elements in the operation of voltage-dependent conductances should thus be emphasized more in modeling the dynamic behavior of small neurons. Furthermore, the presented approach is valuable in providing faster computation times compared to the Markov chain type of modeling approaches and more sophisticated theoretical analysis tools compared to previously presented stochastic modeling approaches.

Epidemic modeling is a key component in the assessment of infectious diseases transmission and guiding the public health response. This report explores two primary approaches to epidemic modeling: It has categorized the models into deterministic model where the disease spread is depicted using differential equations and stochastic models where probability forms the basis of modeling the spread of the diseases. These approaches offer essential information about the nature of epidemic behaviors, for example, the ability to forecast the trends of epidemic curves, and calculation of the basic reproduction number (R0), assessment of child and caregiver, identification of protective/environmental factors, and assessment of intervention approaches. The SIR and SEIR models based on ordinary differential equations that trace the shifting between Susceptible, Infected and Recovered populations. They are especially good in large data scenarios where randomness, or the lack of it, has negligible or no bearing and are fast in providing long-term trends. But they do not target small or more heterogenic groups and do not account for stochastic fluctuations. On the other hand, stochastic models take into consideration the variability of the disease’s transmission as well as the recovery time for disease, they are used for small population size or variable conditions. These models, frequently calibrated through computations involving Monte Carlo simulation or stochastic differential equations, allow for a wider envisaged set of outcomes and shed light on such scenarios as disease wiping out or super spread events. However, they present higher computational costs and are sensitive with parameter estimates as well. It highlights how problematic mathematical modeling is for evidence-informed decision making in public health, as evidenced by COVID-19 pandemic. This highlights the need for constant refinement of the models – data streaming, population variability, and adaptation of the model to real-world emerging issues such as climatic change and zoonotic diseases. Applying the respective strengths of deterministic and stochastic models, further research is encouraged as well as policy implications associated with the interaction between these models and infectious disease outbreaks

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

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.

Two new stochastic epidemic models, a continuous-time Markov chain model and a stochastic differential equation model, are formulated. These are based on a deterministic model that includes vaccination and is applicable to pertussis. For some parameter values, the deterministic model exhibits a backward bifurcation if the vaccine is imperfect. Thus a region of bistability exists in a subset of parameter space. The dynamics of the stochastic epidemic models are investigated in this region of bistability, and compared with those of the deterministic model. In this region the probability distribution associated with the infective population exhibits bimodality with one mode at the disease-free equilibrium and the other at the larger endemic equilibrium. For population sizes N=/>1000, the deterministic and stochastic models agree, but for small population sizes the stochastic models indicate that the backward bifurcation may have little effect on the disease dynamics.

Traffic congestion in urban areas is posing many challenges, and a traffic flow model that accurately predicts traffic conditions can be useful in responding to them. With the limitation of infrastructure and the difficulty for real world experiment, traffic simulation is a good tool for the validation and future potential applications of the traffic flow model. This research presents a stochastic traffic flow model which uses a stochastic partial differential equation to describe the evolution of traffic flow on the highway. The stochastic model is calibrated and validated by real traffic data on a highway and is proved to have better predictive power than the deterministic model. Also, a microscopic traffic simulation is built and calibrated by the same highway data, and the validation result shows that the simulation is capable of representing the real traffic. By using traffic simulation to further explore the capability of the stochastic model, the validation demonstrated that the prediction works well at most locations, and the interpolation error may be improved by considering the influence of ramps and the change in the number of lanes.

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.