Search

Background: Malaria is the world's most fatal and challenging parasitic disease, caused by the Plasmodium parasite, which is transmitted to humans by the bites of infected female mosquitoes. Bangladesh is the most vulnerable region to spread malaria because of its geographic position. In this paper, we have considered the dynamics of vector-host models and observed the stochastic behavior. This study elaborates on the seasonal variability and calculates the probability of disease outbreaks., Methods: We present a model for malaria disease transmission and develop its corresponding continuous-time Markov chain (CTMC) representation. The proposed vector-host models illustrate the malaria transmission model along with sensitivity analysis. The deterministic model with CTMC curves is depicted to show the randomness in real scenarios. Sequentially, we expand these studies to a time-varying stochastic vector-host model that incorporates seasonal variability. Phase plane analysis is conducted to explore the characteristics of the disease, examine interactions among various compartments, and evaluate the impact of key parameters. The branching process approximation is developed for the corresponding vector-host model to calculate the probability outbreak. Numerous numerical results are accomplished to observe the analytical investigation., Results: Seasonality and contact patterns affect the dynamics of disease outbreaks. The numerical illustration provides that the probability of a disease outbreak depends on the infected host or vector. Additionally, periodic transmission rates have a great influence on the probability outbreak. The basic reproduction number (R 0 ) is derived, which is the main justification for studying the dynamical behavior of epidemic models., Conclusions: Seasonal variability significantly impacts malaria transmission, and the probability of disease outbreaks is influenced by time and the initial number of infected individuals. Moreover, the branching process approximation is applicable when the population size is large enough and the basic reproduction number is less than 1. In the future, such analysis can help decision-makers understand the impact of various parameters and their stochastic behavior in the vector-host model to prevent such types of disease outbreaks., Competing Interests: Declaration of competing interest The authors declare no conflict of interest., (Copyright © 2024 Elsevier B.V. All rights reserved.)

First-passage properties of continuous stochastic processes confined in a one-dimensional interval are well described. However, for jump processes (discrete random walks), the characterization of the corresponding observables remains elusive, despite their relevance in various contexts. Here we derive exact asymptotic expressions for the leftward, rightward, and complete exit-time distributions from the interval [0,x] for symmetric jump processes starting from x_{0}=0, in the large x and large time limit. We show that both the leftward probability F_{[under 0]̲,x}(n) to exit through 0 at step n and rightward probability F_{0,[under x]̲}(n) to exit through x at step n exhibit a universal behavior dictated by the large-distance decay of the jump distribution parametrized by the Levy exponent μ. In particular, we exhaustively describe the n≪(x/a_{μ})^{μ} and n≫(x/a_{μ})^{μ} limits and obtain explicit results in both regimes. Our results finally provide exact asymptotics for exit-time distributions of jump processes in regimes where continuous limits do not apply.

The Ornstein-Uhlenbeck process is interpreted as Brownian motion in a harmonic potential. This Gaussian Markov process has a bounded variance and admits a stationary probability distribution, in contrast to the standard Brownian motion. It also tends to a drift towards its mean function, and such a process is called mean reverting. Two examples of the generalized Ornstein-Uhlenbeck process are considered. In the first one, we study the Ornstein-Uhlenbeck process on a comb model, as an example of the harmonically bounded random motion in the topologically constrained geometry. The main dynamical characteristics (as the first and the second moments) and the probability density function are studied in the framework of both the Langevin stochastic equation and the Fokker-Planck equation. The second example is devoted to the study of the effects of stochastic resetting on the Ornstein-Uhlenbeck process, including stochastic resetting in the comb geometry. Here the nonequilibrium stationary state is the main question in task, where the two divergent forces, namely, the resetting and the drift towards the mean, lead to compelling results in the cases of both the Ornstein-Uhlenbeck process with resetting and its generalization on the two-dimensional comb structure.

Cell fate transitions have been studied from various perspectives, such as the transition between stable states, or the transition between stable states and oscillation states. However, there is a lack of study on the stochastic transition between different oscillation states. Here, we aim to explore the stochastic transition mechanism for the switching between oscillations. By employing a landscape and flux theory for a simplified two-dimensional model, we quantify the stochastic dynamics and the global stability of the double oscillation system, and find that the system will escape the starting limit cycle at the position where the flux is large, and cross the barrier between oscillations at the position where the barrier is lower. We also calculate the transition path between limit cycle states based on the minimum action path (MAP) theory. So, the barrier height based on landscape topography and probability flux govern the stochastic transition process between limit cycles, which is further supported by the analysis of mean first passage time (MFPT). We provide a way to calculate the critical points where the switching behavior most likely occurs along a cycle. We validate these conclusions in a realistic biological system; the NF- κ B gene regulatory system. The results for the potential landscape, flux and transition path further our understanding of the underlying mechanism of stochastic transitions between different oscillation states.

We study the probability of fixation in a stochastic two-species competition model. By identifying a naturally occurring fast timescale, we derive an approximation to the associated backward Kolmogorov equation that allows us to obtain an explicit closed form solution for the probability of fixation of either species. We use our result to study fitness tradeoff strategies and show that, despite some tradeoffs having nearly negligible effects on the corresponding deterministic dynamics, they can have large implications for the outcome of the stochastic system., (© 2022. The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.)

The COVID-19 pandemic has drastically shifted the way people work. While many businesses can operate remotely, a large number of jobs can only be performed on-site. Moreover as businesses create plans for bringing workers back on-site, they are in need of tools to assess the risk of COVID-19 for their employees in the workplaces. This study aims to fill the gap in risk modeling of COVID-19 outbreaks in facilities like offices and warehouses. We propose a simulation-based stochastic contact network model to assess the cumulative incidence in workplaces. First-generation cases are introduced as a Bernoulli random variable using the local daily new case rate as the success rate. Contact networks are established through randomly sampled daily contacts for each of the first-generation cases and successful transmissions are established based on a randomized secondary attack rate (SAR). Modification factors are provided for SAR based on changes in airflow, speaking volume, and speaking activity within a facility. Control measures such as mask wearing are incorporated through modifications in SAR. We validated the model by comparing the distribution of cumulative incidence in model simulations against real-world outbreaks in workplaces and nursing homes. The comparisons support the model's validity for estimating cumulative incidences for short forecasting periods of up to 15 days. We believe that the current study presents an effective tool for providing short-term forecasts of COVID-19 cases for workplaces and for quantifying the effectiveness of various control measures. The open source model code is made available at github.com/abhineetgupta/covid-workplace-risk., Competing Interests: This does not alter our adherence to PLOS ONE policies on sharing data and materials.

Realistic models of biological processes typically involve interacting components on multiple scales, driven by changing environment and inherent stochasticity. Such models are often analytically and numerically intractable. We revisit a dynamic maximum entropy method that combines a static maximum entropy with a quasi-stationary approximation. This allows us to reduce stochastic non-equilibrium dynamics expressed by the Fokker-Planck equation to a simpler low-dimensional deterministic dynamics, without the need to track microscopic details. Although the method has been previously applied to a few (rather complicated) applications in population genetics, our main goal here is to explain and to better understand how the method works. We demonstrate the usefulness of the method for two widely studied stochastic problems, highlighting its accuracy in capturing important macroscopic quantities even in rapidly changing non-stationary conditions. For the Ornstein-Uhlenbeck process, the method recovers the exact dynamics whilst for a stochastic island model with migration from other habitats, the approximation retains high macroscopic accuracy under a wide range of scenarios in a dynamic environment., Competing Interests: The authors have declared that no competing interests exist.

This paper presents new theoretical results on the multi-periodicity of recurrent neural networks with time delays evoked by periodic inputs under stochastic disturbances and state-dependent switching. Based on the geometric properties of activation function and switching threshold, the neuronal state space is partitioned into 5 n regions in which 3 n ones are shown to be positively invariant with probability one. Furthermore, by using Itô's formula, Lyapunov functional method, and the contraction mapping theorem, two criteria are proposed to ascertain the existence and mean-square exponential stability of a periodic orbit in every positive invariant set. As a result, the number of mean-square exponentially stable periodic orbits increases to 3 n from 2 n in a neural network without switching. Two illustrative examples are elaborated to substantiate the efficacy and characteristics of the theoretical results., Competing Interests: Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (Copyright © 2021 Elsevier Ltd. All rights reserved.)

People's intuitions about mathematical and statistical concepts often include features that are not a part of the formal definitions. We argue that randomness and related concepts (events happening "accidentally", "coincidentally" or "by chance") are typically assumed to occur in a context of small rather than large events. Five experiments were designed to test the hypothesis of an association between perceived randomness and size. In Experiment 1 and 2, statements describing small outcomes as due to chance were judged to be more natural and to make better sense than corresponding statements about large outcomes (or about small outcomes not due to chance). Experiment 3 showed that people imagine that stories about randomness in daily life should preferably start with small events, even when they eventually turn out to be consequential (e.g., stories about an apparently random meeting ending with marriage). Experiment 4 demonstrated that small changes in a graph of a random walk were seen as random, whereas large changes were perceived as potentially nonrandom. Finally, Experiment 5 showed that small animals are believed to display more random behavior than larger ones. This applied also to fictional creatures with nonsense names, where size was implicitly suggested by the names' phonetic qualities. Analogical instances can be found in the history of science, all the way back to Lucretius' doctrine of the tiny "swerves" of atoms. The pervasive association between smallness and randomness might be partly due to real-world observations and partly to cognitive and motivational constraints.

Humans are notoriously bad at understanding probabilities, exhibiting a host of biases and distortions that are context dependent. This has serious consequences on how we assess risks and make decisions. Several theories have been developed to replace the normative rational expectation theory at the foundation of economics. These approaches essentially assume that (subjective) probabilities weight multiplicatively the utilities of the alternatives offered to the decision maker, although evidence suggest that probability weights and utilities are often not separable in the mind of the decision maker. In this context, we introduce a simple and efficient framework on how to describe the inherently probabilistic human decision-making process, based on a representation of the deliberation activity leading to a choice through stochastic processes, the simplest of which is a random walk. Our model leads naturally to the hypothesis that probabilities and utilities are entangled dual characteristics of the real human decision making process. It predicts the famous fourfold pattern of risk preferences. Through the analysis of choice probabilities, it is possible to identify two previously postulated features of prospect theory: the inverse S-shaped subjective probability as a function of the objective probability and risk-seeking behavior in the loss domain. It also predicts observed violations of stochastic dominance, while it does not when the dominance is "evident". Extending the model to account for human finite deliberation time and the effect of time pressure on choice, it provides other sound predictions: inverse relation between choice probability and response time, preference reversal with time pressure, and an inverse double-S-shaped probability weighting function. Our theory, which offers many more predictions for future tests, has strong implications for psychology, economics and artificial intelligence., Competing Interests: The authors have declared that no competing interests exist.

In decision-making situations individuals rarely have complete information available to select the best option and often show decisional randomness, i.e. given the same amount of knowledge individuals choose different options at different times. Dysfunctional processes resulting in altered decisional randomness can be considered a target process for psychiatric disorders, yet these processes remain poorly understood. Advances in computational modeling of decision-making offer a potential explanation for decisional randomness by positing that decisions are implemented in the brain through accumulation of noisy evidence, causing a generally less preferred option to be chosen at times by chance. One such model, the linear ballistic accumulator (LBA), assumes that individuals accumulate information for each option independently over time and that the first option to reach a threshold will be selected. To investigate the mechanisms of decisional randomness, we applied the LBA to a decision-making task in which risk and expected value (EV) were explicitly signaled prior to making a choice, and estimated separate drift rates for each of the four task stimuli (representing high and low EV and high and low risk). We then used the fitted LBA parameters to predict subject response rates on held-out trials for each of the 6 possible stimulus pairs. We found that choices predicted by LBA were correlated with actual choices across subjects for all stimulus pairs. Taken together, these findings suggest that sequential sampling models can account for decisional randomness on an explicit probabilistic task, which may have implications for understanding decision-making in healthy individuals and in psychiatric populations., Competing Interests: The authors have declared that no competing interests exist.

A stochastic simulation model allows to study and represents in a simplified manner the behavior of complex variables in terms of probability. In this context, the objective of this work is to present, through the use of information and communication technology tools, the applicability of simulation models and methods, in studies of indicators within the health sector. Through the development of a case study, this work demonstrates the potential of the @Risk and Excel technological tools in the construction of stochastic models that allow health professionals to predict, monitor and support decision making in the treatment and monitoring of indicators and indices of a population.

Transcription is a pivotal step in gene expression yet a complex biochemical process. It occurs often in a bursty manner, leading to cell-to-cell variability important for the cells surviving in complex environments. Quantitative experiments of such stochastic transcription call for model analysis in a systematic rather than case-by-case fashion. Here we analyze two general yet biologically-reasonable classes of stochastic transcription models: the first class called promoter models considers that promoter structure is general, and the second class called queuing models considers that waiting-time distributions are general. For the former, we show that there are conjugate relationships between models with different transcription exits. This property well reveals the mechanic principle of stochastic transcription in complex cases of transcription factor regulation. For the latter, we establish an integral equation for the mRNA moment-generating function, through which stationary mRNA moments of any orders can be analytically derived. Finally, we analyze parametric regions for robust unimodality and bimodality. The overall analysis not only lays a foundation for quantitative analysis of stochastic transcription but also reveals how the upstream promoter kinetics impact the downstream expression dynamics.

Post-transcriptional regulation of gene expression is often a critical component of cellular processes involved in cell-fate decisions. Correspondingly, considerable efforts have focused on modeling post-transcriptional regulation of stochastic gene expression and on quantifying its impact on the mean and variance of protein levels. However, the impact of post-transcriptional regulation on rare events corresponding to large deviations in gene expression is less well understood. Here, we study a simple model involving post-transcriptional control of gene expression and characterize the impact of regulation on large deviations in activity fluctuations. We derive analytical results for the large deviation function for protein production rate and for the corresponding driven process which characterizes system fluctuations conditional on the rare event. Our results suggest that fluctuations in burst size rather than frequency play a dominant role in controlling rare events. The results derived also provide insight into how post-transcriptional regulation can be used to fine-tune the probability of rare fluctuations and to thereby control phenotypic variation in a population of cells.

In this paper, we analyze the effect of environment noise on the transmission dynamics of a stochastic hepatitis B virus (HBV) infection model with intervention strategies. By using the Markov semigroups theory, we define the stochastic basic reproduction number and find it can be used to govern disease extinction or persistence. When it is less than one, under a mild extra condition, the stochastic system has a disease-free equilibrium and the disease is predicted to die out with probability one. When it is greater than one, under mild extra conditions, the model admits a stationary distribution which means the persistence of the disease. Thus, we observe that larger intensity of noise (resulting in a smaller stochastic basic reproduction number) can suppress the emergence of hepatitis B outbreak. Numerical simulations are also carried out to investigate the influence of information intervention strategies that may change individual behavior and protect the susceptible from infection. Our analysis shows that the environmental noise can greatly a ect the long-term behavior of the system, highlighting the importance of the role of intervention strategies in the control of hepatitis B.

In this paper, we proposed a frequency dependent fitness-based process, which is an extension of both the standard Moran process and the Wright-Fisher process. Some individuals are selected into a parent's pool and reproduce. Then the offspring is selected to replace individuals in the entire parent generation. We explored the influence of the size of parent pool and the number of offspring on a single cooperator's fixation. The less offspring leads to higher fixation probability of s single cooperator. Meanwhile, the fixation probability decreases with the growth of the local level. In other words, the direction of the number of offspring's impact on fixation probability is in accordance with that of the local level's impact. The less offspring in one generation or the smaller parent's pool contributes to promoting cooperation with the fitness-based updating rule., (Copyright © 2018 Elsevier Ltd. All rights reserved.)

Background and Objectives: Biological systems with intertwined feedback loops pose a challenge to mathematical modeling efforts. Moreover, rare events, such as mutation and extinction, complicate system dynamics. Stochastic simulation algorithms are useful in generating time-evolution trajectories for these systems because they can adequately capture the influence of random fluctuations and quantify rare events. We present a simple and flexible package, BioSimulator.jl, for implementing the Gillespie algorithm, τ-leaping, and related stochastic simulation algorithms. The objective of this work is to provide scientists across domains with fast, user-friendly simulation tools., Methods: We used the high-performance programming language Julia because of its emphasis on scientific computing. Our software package implements a suite of stochastic simulation algorithms based on Markov chain theory. We provide the ability to (a) diagram Petri Nets describing interactions, (b) plot average trajectories and attached standard deviations of each participating species over time, and (c) generate frequency distributions of each species at a specified time., Results: BioSimulator.jl's interface allows users to build models programmatically within Julia. A model is then passed to the simulate routine to generate simulation data. The built-in tools allow one to visualize results and compute summary statistics. Our examples highlight the broad applicability of our software to systems of varying complexity from ecology, systems biology, chemistry, and genetics., Conclusion: The user-friendly nature of BioSimulator.jl encourages the use of stochastic simulation, minimizes tedious programming efforts, and reduces errors during model specification., (Copyright © 2018. Published by Elsevier B.V.)

The state-of-art research in the field of life's organization confronts the need to investigate a number of interacting components, their properties and conditions of sustainable behaviour within a natural system. In biology, ecology and life sciences, the performance of such stable system is usually related to homeostasis, a property of the system to actively regulate its state within a certain allowable limits. In our previous work, we proposed a deterministic model for systems' homeostasis. The model was based on dynamical system's theory and pairwise relationships of competition, amensalism and antagonism taken from theoretical biology and ecology. However, the present paper proposes a different dimension to our previous results based on the same model. In this paper, we introduce the influence of inter-component relationships in a system, wherein the impact is characterized by direction (neutral, positive, or negative) as well as its (absolute) value, or strength. This makes the model stochastic which, in our opinion, is more consistent with real-world elements affected by various random factors. The case study includes two examples from areas of hydrobiology and medicine. The models acquired for these cases enabled us to propose a convincing explanation for corresponding phenomena identified by different types of natural systems.

Human randomness perception is commonly described as biased. This is because when generating random sequences humans tend to systematically under- and overrepresent certain subsequences relative to the number expected from an unbiased random process. In a purely theoretical analysis we have previously suggested that common misperceptions of randomness may actually reflect genuine aspects of the statistical environment, once cognitive constraints are taken into account which impact on how that environment is actually experienced (Hahn & Warren, Psychological Review , 2009). In the present study we undertake an empirical test of this account, comparing human-generated against unbiased process-generated binary sequences in two experiments. We suggest that comparing human and theoretically unbiased sequences using metrics reflecting the constraints imposed on human experience provides a more meaningful picture of lay people's ability to perceive randomness. Finally, we propose a simple generative model of human random sequence generation inspired by the Hahn and Warren account. Taken together our results question the notion of bias in human randomness perception. (PsycINFO Database Record, ((c) 2018 APA, all rights reserved).)

We propose a method to obtain phase portraits for stochastic systems. Starting from the Fokker-Planck equation, we separate the dynamics into a convective and a diffusive part. We show that stable and unstable fixed points of the convective field correspond to maxima and minima of the stationary probability distribution if the probability current vanishes at these points. Stochastic phase portraits, which are vector plots of the convective field, therefore indicate the extrema of the stationary distribution and can be used to identify stochastic bifurcations that change the number and stability of these extrema. We show that limit cycles in stochastic phase portraits can indicate ridges of the probability distribution, and we identify a novel type of stochastic bifurcation, where the probability maximum moves to the edge of the system through a gap between the two nullclines of the convective field.

In order to maintain homeostasis, mature cells removed from the top compartment of hierarchical tissues have to be replenished by means of differentiation and self-renewal events happening in the more primitive compartments. As each cell division is associated with a risk of mutation, cell division patterns have to be optimized, in order to minimize or delay the risk of malignancy generation. Here we study this optimization problem, focusing on the role of division tree length, that is, the number of layers of cells activated in response to the loss of terminally differentiated cells, which is related to the balance between differentiation and self-renewal events in the compartments. Using both analytical methods and stochastic simulations in a metapopulation-style model, we find that shorter division trees are advantageous if the objective is to minimize the total number of one-hit mutants in the cell population. Longer division trees on the other hand minimize the accumulation of two-hit mutants, which is a more likely evolutionary goal given the key role played by tumor suppressor genes in cancer initiation. While division tree length is the most important property determining mutant accumulation, we also find that increasing the size of primitive compartments helps to delay two-hit mutant generation.

Although the detection of metastases radically changes prognosis of and treatment decisions for a cancer patient, clinically undetectable micrometastases hamper a consistent classification into localized or metastatic disease. This chapter discusses mathematical modeling efforts that could help to estimate the metastatic risk in such a situation. We focus on two approaches: (1) a stochastic framework describing metastatic emission events at random times, formalized via Poisson processes, and (2) a deterministic framework describing the micrometastatic state through a size-structured density function in a partial differential equation model. Three aspects are addressed in this chapter. First, a motivation for the Poisson process framework is presented and modeling hypotheses and mechanisms are introduced. Second, we extend the Poisson model to account for secondary metastatic emission. Third, we highlight an inherent crosslink between the stochastic and deterministic frameworks and discuss its implications. For increased accessibility the chapter is split into an informal presentation of the results using a minimum of mathematical formalism and a rigorous mathematical treatment for more theoretically interested readers.

In this work we consider fixation of an allele in a population. Fixation is key to understanding the way long-term evolutionary change occurs at the gene and molecular levels. Two basic aspects of fixation are: (i) the chance it occurs and (ii) the way the gene frequency progresses to fixation. We present exact results for both aspects of fixation for the Wright-Fisher model. We give the exact fixation probability for some different schemes of frequency-dependent selection. We also give the corresponding exact stochastic difference equation that generates frequency trajectories which ultimately fix. Exactness of the results means selection need not be weak. There are possible applications of this work to data analysis, modelling, and tests of approximations. The methodology employed illustrates that knowledge of the fixation probability, for all initial frequencies, fully characterises the dynamics of the Wright-Fisher model. The stochastic equations for fixing trajectories allow insight into the way fixation occurs. They provide the alternative picture that fixation is driven by the injection of one carrier of the fixing allele into the population each generation. The stochastic equations allow explicit calculation of some properties of fixing trajectories and their efficient simulation. The results are illustrated and tested with simulations., (Copyright © 2017 Elsevier Ltd. All rights reserved.)

This paper suggests a probability-dominant user equilibrium (PdUE) model to describe the selfish routing equilibrium in a stochastic traffic network. At PdUE, travel demands are only assigned to the most dominant routes in the same origin-destination pair. A probability-dominant rerouting dynamic model is proposed to explain the behavioral mechanism of PdUE. To facilitate applications, the logit formula of PdUE is developed, of which a well-designed route set is not indispensable and the equivalent varitional inequality formation is simple. Two routing strategies, i.e., the probability-dominant strategy (PDS) and the dominant probability strategy (DPS), are discussed through a hypothetical experiment. It is found that, whether out of insurance or striving for perfection, PDS is a better choice than DPS. For more general cases, the conducted numerical tests lead to the same conclusion. These imply that PdUE (rather than the conventional stochastic user equilibrium) is a desirable selfish routing equilibrium for a stochastic network, given that the probability distributions of travel time are available to travelers.

Mathematical models of the cardiac cell have started to include markovian representations of the ionic channels instead of the traditional Hodgkin & Huxley formulations. There are many reasons for this: Markov models are not restricted to the idea of independent gates defining the channel, they allow more complex description with specific transitions between open, closed or inactivated states, and more importantly those states can be closely related to the underlying channel structure and conformational changes., Methods: We used the LabVIEW ® and MATLAB ® programs to implement the simulator MarkoLAB that allow a dynamical 3D representation of the markovian model of the channel. The Monte Carlo simulation was used to implement the stochastic transitions among states. The user can specify the voltage protocol by setting the holding potential, the step-to voltage and the duration of the stimuli., Results: The most studied feature of a channel is the current flowing through it. This happens when the channel stays in the open state, but most of the time, as revealed by the low open probability values, the channel remains on the inactive or closed states. By focusing only when the channel enters or leaves the open state we are missing most of its activity. MarkoLAB proved to be quite useful to visualize the whole behavior of the channel and not only when the channel produces a current. Such dynamic representation provides more complete information about channel kinetics and will be a powerful tool to demonstrate the effect of gene mutations or drugs on the channel function., Conclusions: MarkoLAB provides an original way of visualizing the stochastic behavior of a channel. It clarifies concepts, such as recovery from inactivation, calcium- versus voltage-dependent inactivation, and tail currents. It is not restricted to ionic channels only but it can be extended to other transporters, such as exchangers and pumps. This program is intended as a didactical tool to illustrate the dynamical behavior of a channel. It has been implemented in two platforms MATLAB ® and LabVIEW ® to enhance the target users of this new didactical tool. The computational cost of implementing a stochastic simulation is within the range of a personal computer performance; making MarkoLAB suitable to be run during a lecture or presentation., (Copyright © 2017 Elsevier Ltd. All rights reserved.)

Gene expression is inherently stochastic. Advanced single-cell microscopy techniques together with mathematical models for single gene expression led to important insights in elucidating the sources of intrinsic noise in prokaryotic and eukaryotic cells. In addition to the finite size effects due to low copy numbers, translational bursting is a dominant source of stochasticity in cell scenarios involving few short lived mRNA transcripts with high translational efficiency (as is typically the case for prokaryotes), causing protein synthesis to occur in random bursts. In the context of gene regulation cascades, the Chemical Master Equation (CME) governing gene expression has in general no closed form solution, and the accurate stochastic simulation of the dynamics of complex gene regulatory networks is a major computational challenge. The CME associated to a single gene self regulatory motif has been previously approximated by a one dimensional time dependent partial integral differential equation (PIDE). However, to the best of our knowledge, multidimensional versions for such PIDE have not been developed yet. Here we propose a multidimensional PIDE model for regulatory networks involving multiple genes with self and cross regulations (in which genes can be regulated by different transcription factors) derived as the continuous counterpart of a CME with jump process. The model offers a reliable description of systems with translational bursting. In order to provide an efficient numerical solution, we develop a semilagrangian method to discretize the differential part of the PIDE, combined with a composed trapezoidal quadrature formula to approximate the integral term. We apply the model and numerical method to study sustained stochastic oscillations and the development of competence, a particular case of transient differentiation attained by certain bacterial cells under stress conditions. We found that the resulting probability distributions are distinguishable from those characteristic of other transient differentiation processes. In this way, they can be employed as markers or signatures that identify such phenomena from bacterial population experimental data, for instance. The computational efficiency of the semilagrangian method makes it suitable for purposes like model identification and parameter estimation from experimental data or, in combination with optimization routines, the design of gene regulatory networks under molecular noise., (Copyright © 2017 Elsevier Ltd. All rights reserved.)

The stochastic simulation algorithm has been used to generate exact trajectories of biochemical reaction networks. For each simulation step, the simulation selects a reaction and its firing time according to a probability that is proportional to the reaction propensity. We investigate in this paper new efficient formulations of the stochastic simulation algorithm to improve its computational efficiency. We examine the selection of the next reaction firing and reduce its computational cost by reusing the computation in the previous step. For biochemical reactions with delays, we present a new method for computing the firing time of the next reaction. The principle for computing the firing time of our approach is based on recycling of random numbers. Our new approach for generating the firing time of the next reaction is not only computationally efficient but also easy to implement. We further analyze and reduce the number of propensity updates when a delayed reaction occurred. We demonstrate the applicability of our improvements by experimenting with concrete biological models.

Stochastic evolution of Chemical Reactions Networks (CRNs) over time is usually analyzed through solving the Chemical Master Equation (CME) or performing extensive simulations. Analysing stochasticity is often needed, particularly when some molecules occur in low numbers. Unfortunately, both approaches become infeasible if the system is complex and/or it cannot be ensured that initial populations are small. We develop a probabilistic logic for CRNs that enables stochastic analysis of the evolution of populations of molecular species. We present an approximate model checking algorithm based on the Linear Noise Approximation (LNA) of the CME, whose computational complexity is independent of the population size of each species and polynomial in the number of different species. The algorithm requires the solution of first order polynomial differential equations. We prove that our approach is valid for any CRN close enough to the thermodynamical limit. However, we show on four case studies that it can still provide good approximation even for low molecule counts. Our approach enables rigorous analysis of CRNs that are not analyzable by solving the CME, but are far from the deterministic limit. Moreover, it can be used for a fast approximate stochastic characterization of a CRN., (Copyright © 2016 The Authors. Published by Elsevier Ireland Ltd.. All rights reserved.)

We propose a modelling framework to analyse the stochastic behaviour of heterogeneous, multi-scale cellular populations. We illustrate our methodology with a particular example in which we study a population with an oxygen-regulated proliferation rate. Our formulation is based on an age-dependent stochastic process. Cells within the population are characterised by their age (i.e. time elapsed since they were born). The age-dependent (oxygen-regulated) birth rate is given by a stochastic model of oxygen-dependent cell cycle progression. Once the birth rate is determined, we formulate an age-dependent birth-and-death process, which dictates the time evolution of the cell population. The population is under a feedback loop which controls its steady state size (carrying capacity): cells consume oxygen which in turn fuels cell proliferation. We show that our stochastic model of cell cycle progression allows for heterogeneity within the cell population induced by stochastic effects. Such heterogeneous behaviour is reflected in variations in the proliferation rate. Within this set-up, we have established three main results. First, we have shown that the age to the G1/S transition, which essentially determines the birth rate, exhibits a remarkably simple scaling behaviour. Besides the fact that this simple behaviour emerges from a rather complex model, this allows for a huge simplification of our numerical methodology. A further result is the observation that heterogeneous populations undergo an internal process of quasi-neutral competition. Finally, we investigated the effects of cell-cycle-phase dependent therapies (such as radiation therapy) on heterogeneous populations. In particular, we have studied the case in which the population contains a quiescent sub-population. Our mean-field analysis and numerical simulations confirm that, if the survival fraction of the therapy is too high, rescue of the quiescent population occurs. This gives rise to emergence of resistance to therapy since the rescued population is less sensitive to therapy., (Copyright © 2016 The Authors. Published by Elsevier Ltd.. All rights reserved.)

In spot-scanning particle therapy, inverse treatment planning is usually limited to finding the optimal beam fluences given the beam trajectories and energies. We address the much more challenging problem of jointly optimizing the beam fluences, trajectories and energies. For this purpose, we design a simulated annealing algorithm with an exploration mechanism that balances the conflicting demands of a small mixing time at high temperatures and a reasonable acceptance rate at low temperatures. Numerical experiments substantiate the relevance of our approach and open new horizons to spot-scanning particle therapy.

Stochasticity plays an important role in the evolutionary dynamic of cyclic dominance within a finite population. To investigate the stochastic evolution process of the behaviour of bounded rational individuals, we model the Rock-Scissors-Paper (RSP) game as a finite, state dependent Quasi Birth and Death (QBD) process. We assume that bounded rational players can adjust their strategies by imitating the successful strategy according to the payoffs of the last round of the game, and then analyse the limiting distribution of the QBD process for the game stochastic evolutionary dynamic. The numerical experiments results are exhibited as pseudo colour ternary heat maps. Comparisons of these diagrams shows that the convergence property of long run equilibrium of the RSP game in populations depends on population size and the parameter of the payoff matrix and noise factor. The long run equilibrium is asymptotically stable, neutrally stable and unstable respectively according to the normalised parameters in the payoff matrix. Moreover, the results show that the distribution probability becomes more concentrated with a larger population size. This indicates that increasing the population size also increases the convergence speed of the stochastic evolution process while simultaneously reducing the influence of the noise factor.

Traditional moment-closure methods need to assume that high-order cumulants of a probability distribution approximate to zero. However, this strong assumption is not satisfied for many biochemical reaction networks. Here, we introduce convergent moments (defined in mathematics as the coefficients in the Taylor expansion of the probability-generating function at some point) to overcome this drawback of the moment-closure methods. As such, we develop a new analysis method for stochastic chemical kinetics. This method provides an accurate approximation for the master probability equation (MPE). In particular, the connection between low-order convergent moments and rate constants can be more easily derived in terms of explicit and analytical forms, allowing insights that would be difficult to obtain through direct simulation or manipulation of the MPE. In addition, it provides an accurate and efficient way to compute steady-state or transient probability distribution, avoiding the algorithmic difficulty associated with stiffness of the MPE due to large differences in sizes of rate constants. Applications of the method to several systems reveal nontrivial stochastic mechanisms of gene expression dynamics, e.g., intrinsic fluctuations can induce transient bimodality and amplify transient signals, and slow switching between promoter states can increase fluctuations in spatially heterogeneous signals. The overall approach has broad applications in modeling, analysis, and computation of complex biochemical networks with intrinsic noise.

The problem of finding an optimum within a set of possibilities that represent the varying successfulness of numerous subjects competing with one another is highly relevant in the field of ecosystem interactions. We propose a method for solving this problem by the application of the Nash equilibrium concept, which is frequently used in ecology. The proposed model is based on the transformation of the initial payoff vectors of subjects that interact in different situations into a statistical set of symmetrical game matrices that consist of permutations of payoff values. The equilibrium solution is expressed as values of the probability of Nash equilibrium occurrence with uniform distribution over all possible permutations based on uncertainty of positions of payoff values in the matrix. We assume that this equilibrium solution provides information on the distribution of the degree of stability among individual situations and interacting subjects. In this paper, we validate this assumption and demonstrate its application to a dataset that represents interspecies interactions in plant ecology. We propose that the use of the Nash equilibrium in the analysis of datasets formalized according to the Pareto optimality scheme is applicable in numerous other contexts.

The structure and function of microbial communities is deeply influenced by the physical and chemical architecture of the local microenvironment and the abundance of its community members. The complexity of this natural parameter space has made characterization of the key drivers of community development difficult. In order to facilitate these characterizations, we have developed a microwell platform designed to screen microbial growth and interactions across a wide variety of physical and initial conditions. Assembly of microbial communities into microwells was achieved using a novel biofabrication method that exploits well feature sizes for control of innoculum levels. Wells with incrementally smaller size features created populations with increasingly larger variations in inoculum levels. This allowed for reproducible growth measurement in large (20 μm diameter) wells, and screening for favorable growth conditions in small (5, 10 μm diameter) wells. We demonstrate the utility of this approach for screening and discovery using 5 μm wells to assemble P. aeruginosa colonies across a broad distribution of innoculum levels, and identify those conditions that promote the highest probability of survivial and growth under spatial confinement. Multi-member community assembly was also characterized to demonstrate the broad potential of this platform for studying the role of member abundance on microbial competition, mutualism and community succession.

The Trojan Y-Chromosome (TYC) strategy, an autocidal genetic biocontrol method, has been proposed to eliminate invasive alien species. In this work, we develop a Markov jump process model for this strategy, and we verify that there is a positive probability for wild-type females going extinct within a finite time. Moreover, when sex-reversed Trojan females are introduced at a constant population size, we formulate a stochastic differential equation (SDE) model as an approximation to the proposed Markov jump process model. Using the SDE model, we investigate the probability distribution and expectation of the extinction time of wild-type females by solving Kolmogorov equations associated with these statistics. The results indicate how the probability distribution and expectation of the extinction time are shaped by the initial conditions and the model parameters.

In stem cell differentiation, a pluripotent stem cell becomes progressively specialized and generates specific cell types through a series of epigenetic processes. How cells can precisely determine their fate in a fluctuating environment is a currently unsolved problem. In this paper, we suggest an abstract gene regulatory network to describe mathematically the differentiation phenomenon featuring stochasticity, divergent cell fates, and robustness. The network consists of three functional motifs: an upstream chaotic motif, a buffering motif of incoherent feed forward loop capable of generating a pulse, and a downstream motif which is bistable. The dynamic behavior is typically a transient chaos with fractal basin boundaries. The trajectories take transiently chaotic journeys before divergently settling down to the bistable states. The ratio of the probability that the high state is achieved to the probability that the low state is reached can maintain a constant in a population of cells with varied molecular fluctuations. The ratio can be turned up or down when proper parameters are adjusted. The model suggests a possible mechanism for the robustness against fluctuations that is prominently featured in pluripotent cell differentiations and developmental phenomena.

Cell colonization during embryonic development involves cells migrating and proliferating over growing tissues. Unsuccessful colonization, resulting from genetic causes, can result in various birth defects. However not all individuals with the same mutation show the disease. This is termed incomplete penetrance, and it even extends to discordancy in monozygotic (identical) twins. A one-dimensional agent-based model of cell migration and proliferation within a growing tissue is presented, where the position of every cell is recorded at any time. We develop a new model that approximates this agent-based process - rather than requiring the precise configuration of cells within the tissue, the new model records the total number of cells, the position of the most advanced cell, and then invokes an approximation for how the cells are distributed. The probability mass function (PMF) for the most advanced cell is obtained for both the agent-based model and its approximation. The two PMFs compare extremely well, but using the approximation is computationally faster. Success or failure of colonization is probabilistic. For example for sufficiently high proliferation rate the colonization is assured. However, if the proliferation rate is sufficiently low, there will be a lower, say 50%, chance of success. These results provide insights into the puzzle of incomplete penetrance of a disease phenotype, especially in monozygotic twins. Indeed, stochastic cell behavior (amplified by disease-causing mutations) within the colonization process may play a key role in incomplete penetrance, rather than differences in genes, their expression or environmental conditions., (Copyright © 2015 Elsevier Ltd. All rights reserved.)

We present a novel model that describes the within-host evolutionary dynamics of parasites undergoing antigenic variation. The approach uses a multi-type branching process with two types of entities defined according to their relationship with the immune system: clans of resistant parasitic cells (i.e. groups of cells sharing the same antigen not yet recognized by the immune system) that may become sensitive, and individual sensitive cells that can acquire a new resistance thus giving rise to the emergence of a new clan. The simplicity of the model allows analytical treatment to determine the subcritical and supercritical regimes in the space of parameters. By incorporating a density-dependent mechanism the model is able to capture additional relevant features observed in experimental data, such as the characteristic parasitemia waves. In summary our approach provides a new general framework to address the dynamics of antigenic variation which can be easily adapted to cope with broader and more complex situations., (Copyright © 2015 Elsevier Ltd. All rights reserved.)

Antiviral prophylaxis forms a significant component of health management plans for many countries around the world. A number of studies have shown that the delays typically encountered in distributing these antivirals to households, following the first infectious case, can result in their efficacy being severely reduced. Here, we investigate the use of contact tracing as a method to reduce the delays and hence mitigate the reduction in efficacy of antivirals. We assess the usefulness of contact tracing in terms of the probability of a major outbreak. It is found, with parameter distributions appropriate to the 2009 H1N1 pandemic and distributions reflecting commonly experienced delays, that standard contact tracing renders an outbreak impossible approximately one in five times compared with approximately one in ten times in its absence. A contact-tracing efficiency of 50% would see further improvements with an outbreak being impossible approximately one in four times, and a reduction of the median probability of a major outbreak from 0.41 to below 0.27., (© The authors 2014. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.)

Minimal hardware implementations able to cope with the processing of large amounts of data in reasonable times are highly desired in our information-driven society. In this work we review the application of stochastic computing to probabilistic-based pattern-recognition analysis of huge database sets. The proposed technique consists in the hardware implementation of a parallel architecture implementing a similarity search of data with respect to different pre-stored categories. We design pulse-based stochastic-logic blocks to obtain an efficient pattern recognition system. The proposed architecture speeds up the screening process of huge databases by a factor of 7 when compared to a conventional digital implementation using the same hardware area.

Here we characterize the low-noise regime of a stochastic model for a negative self-regulating binary gene. The model has two stochastic variables, the protein number and the state of the gene. Each state of the gene behaves as a protein source governed by a Poisson process. The coupling between the two gene states depends on protein number. This fact has a very important implication: There exist protein production regimes characterized by sub-Poissonian noise because of negative covariance between the two stochastic variables of the model. Hence the protein numbers obey a probability distribution that has a peak that is sharper than those of the two coupled Poisson processes that are combined to produce it. Biochemically, the noise reduction in protein number occurs when the switching of the genetic state is more rapid than protein synthesis or degradation. We consider the chemical reaction rates necessary for Poisson and sub-Poisson processes in prokaryotes and eucaryotes. Our results suggest that the coupling of multiple stochastic processes in a negative covariance regime might be a widespread mechanism for noise reduction.

Efforts to reconstruct phylogenetic trees and understand evolutionary processes depend fundamentally on stochastic models of speciation and mutation. The simplest continuous-time model for speciation in phylogenetic trees is the Yule process, in which new species are "born" from existing lineages at a constant rate. Recent work has illuminated some of the structural properties of Yule trees, but it remains mostly unknown how these properties affect sequence and trait patterns observed at the tips of the phylogenetic tree. Understanding the interplay between speciation and mutation under simple models of evolution is essential for deriving valid phylogenetic inference methods and gives insight into the optimal design of phylogenetic studies. In this work, we derive the probability distribution of interspecies covariance under Brownian motion and Ornstein-Uhlenbeck models of phenotypic change on a Yule tree. We compute the probability distribution of the number of mutations shared between two randomly chosen taxa in a Yule tree under discrete Markov mutation models. Our results suggest summary measures of phylogenetic information content, illuminate the correlation between site patterns in sequences or traits of related organisms, and provide heuristics for experimental design and reconstruction of phylogenetic trees., (Copyright © 2014 Elsevier Ltd. All rights reserved.)

Background: Estimating the required dose in radiotherapy is of crucial importance since the administrated dose should be sufficient to eradicate the tumor and at the same time should inflict minimal damage on normal cells. The probability that a given dose and schedule of ionizing radiation eradicates all the tumor cells in a given tissue is called the tumor control probability (TCP), and is often used to compare various treatment strategies used in radiation therapy., Method: In this paper, we aim to investigate the effects of including cell-cycle phase on the TCP by analyzing a stochastic model of a tumor comprised of actively dividing cells and quiescent cells with different radiation sensitivities. Moreover, we use a novel numerical approach based on the method of characteristics for partial differential equations, validated by the Gillespie algorithm, to compute the TCP as a function of time., Results: We derive an exact phase-diagram for the steady-state TCP of the model and show that at high, clinically-relevant doses of radiation, the distinction between active and quiescent tumor cells (i.e. accounting for cell-cycle effects) becomes of negligible importance in terms of its effect on the TCP curve. However, for very low doses of radiation, these proportions become significant determinants of the TCP. We also present the results of TCP as a function of time for different values of asymmetric division factor., Conclusion: We observe that our results differ from the results in the literature using similar existing models, even though similar parameters values are used, and the reasons for this are discussed.

Logical models have widely been used to gain insights into the biological behavior of gene regulatory networks (GRNs). Most logical models assume a synchronous update of the genes' states in a GRN. However, this may not be appropriate, because each gene may require a different period of time for changing its state. In this article, asynchronous stochastic Boolean networks (ASBNs) are proposed for investigating various asynchronous state-updating strategies in a GRN. As in stochastic computation, ASBNs use randomly permutated stochastic sequences to encode probability. Investigated by several stochasticity models, a GRN is considered to be subject to noise and external perturbation. Hence, both stochasticity and asynchronicity are considered in the state evolution of a GRN. As a case study, ASBNs are utilized to investigate the dynamic behavior of a T helper network. It is shown that ASBNs are efficient in evaluating the steady-state distributions (SSDs) of the network with random gene perturbation. The SSDs found by using ASBNs show the robustness of the attractors of the T helper network, when various stochasticity and asynchronicity models are considered to investigate its dynamic behavior.

Most chemical and biological processes can be viewed as reaction networks in which different pathways often compete kinetically for transformation of substrates into products. An enzymatic process is an example of such phenomena when biological catalysts create new routes for chemical reactions to proceed. It is typically assumed that the general process of product formation is governed by the pathway with the fastest kinetics at all time scales. In contrast to the expectation, here we show theoretically that at time scales sufficiently short, reactions are predominantly determined by the shortest pathway (in the number of intermediate states), regardless of the average turnover time associated with each pathway. This universal phenomenon is demonstrated by an explicit calculation for a system with two competing reversible (or irreversible) pathways. The time scales that characterize this regime and its relevance for single-molecule experimental studies are also discussed.

Approximations based on moment-closure (MA) are commonly used to obtain estimates of the mean molecule numbers and of the variance of fluctuations in the number of molecules of chemical systems. The advantage of this approach is that it can be far less computationally expensive than exact stochastic simulations of the chemical master equation. Here, we numerically study the conditions under which the MA equations yield results reflecting the true stochastic dynamics of the system. We show that for bistable and oscillatory chemical systems with deterministic initial conditions, the solution of the MA equations can be interpreted as a valid approximation to the true moments of the chemical master equation, only when the steady-state mean molecule numbers obtained from the chemical master equation fall within a certain finite range. The same validity criterion for monostable systems implies that the steady-state mean molecule numbers obtained from the chemical master equation must be above a certain threshold. For mean molecule numbers outside of this range of validity, the MA equations lead to either qualitatively wrong oscillatory dynamics or to unphysical predictions such as negative variances in the molecule numbers or multiple steady-state moments of the stationary distribution as the initial conditions are varied. Our results clarify the range of validity of the MA approach and show that pitfalls in the interpretation of the results can only be overcome through the systematic comparison of the solutions of the MA equations of a certain order with those of higher orders.

A reconstruction methodology based on threshold energy based energy minimization (TA) and different-phase-neighbor (DPN)-based pixel swapping is presented. The TA method uses an energy threshold rather than probabilities as an acceptance criteria for annealing steps. The DPN-based pixel selection method gives priority to pixels which are segregated from clusters instead of random selection. An in-house solver has been developed to obtain two-dimensional reconstructions of heterogeneous two-phase mediums. Compared to conventional simulated annealing with random pixel swapping, the proposed method was found to achieve an optimal structure with up to an order of magnitude reduction in energy. When selecting a threshold tolerance value, the proposed method showed a 50% improvement in convergence time compared to conventional simulated annealing with random pixel swapping. The improved algorithm is used to study the effect of multiple correlation functions during the reconstruction. It was found that a combination of two-point correlation function and lineal path function for both phases results in most accurate reconstructions.

We develop a mean-field approach for multicomponent stochastic spatially extended systems and use it to obtain a multivariate nonlinear self-consistent Fokker-Planck equation defining the probability density of the state of the system, which describes a well-known model of autocatalytic chemical reaction (brusselator) with spatially correlated multiplicative noise, and to study the evolution of probability density and statistical characteristics of the system in the process of spatial pattern formation. We propose the finite-difference method for the numerical solving of a general class of multivariate nonlinear self-consistent time-dependent Fokker-Planck equations. We illustrate the accuracy and reliability of the method by applying it to an exactly solvable nonlinear Fokker-Planck equation (NFPE) for the Shimizu-Yamada model [Prog. Theor. Phys. 47, 350 (1972)] and nonlinear Fokker-Planck equation [Desai and Zwanzig, J. Stat. Phys. 19, 1 (1978)] obtained for a nonlinear stochastic mean-field model introduced by Kometani and Shimizu [J. Stat. Phys. 13, 473 (1975)]. Taking the problems indicated above as an example, the accuracy of the method is compared with the accuracy of Hermite distributed approximating functional method [Zhang et al., Phys. Rev. E 56, 1197 (1997)]. Numerical study of the NFPE solutions for a stochastic brusselator shows that in the region of Turing bifurcation several types of solutions exist if noise intensity increases: unimodal solution, transient bimodality, and an interesting solution which involves multiple "repumping" of probability density through bimodality. Additionally, we study the behavior of the order parameter of the system under consideration and show that the second type of solution arises in the supercritical region if noise intensity values are close to the values appropriate for the transition from bimodal stationary probability density for the order parameter to the unimodal one.

In 2 × 2 prisoner's dilemma games, network reciprocity is one mechanism for adding social viscosity, which leads to cooperative equilibrium. This study introduced an intriguing framework for the strategy update rule that allows any combination of a purely deterministic method, imitation max (IM), and a purely probabilistic one, pairwise Fermi (Fermi-PW). A series of simulations covering the whole range from IM to Fermi-PW reveals that, as a general tendency, the larger fractions of stochastic updating reduce network reciprocity, so long as the underlying lattice contains no noise in the degree of distribution. However, a small amount of stochastic flavor added to an otherwise perfectly deterministic update rule was actually found to enhance network reciprocity. This occurs because a subtle stochastic effect in the update rule improves the evolutionary trail in games having more stag-hunt-type dilemmas, although the same stochastic effect degenerates evolutionary trails in games having more chicken-type dilemmas. We explain these effects by dividing evolutionary trails into the enduring and expanding periods defined by Shigaki et al. [Phys. Rev. E 86, 031141 (2012)].

We introduce multikangaroo Markov processes and provide a general procedure for evaluating a certain type of stochastic functional. We calculate analytically the large deviation properties. We apply our results to zero-crossing statistics and to stochastic thermodynamics, including the derivation of the fluctuation theorem and the large deviation properties for the stochastic entropy production in a typical solid state device.