Stochastic Growth Model Research Articles

A stochastic model for the bacterial growth exhibiting staged growth, desynchronization, saturation and persistence

We consider a model of population growth based on the stochastic variation of the population size-controlled duplication of bacterial cells. It is shown that the proper choice of the control function allows for reproducing a variety of regimes: a logistic growth with saturation, a hindered growth typical for persistent bacterial systems, and a linear population growth detected for some mycobacterial populations. When supplied with the rectangular function having the width equal to the generation time, this approach represents the solution generalizing Rubinow’s age-maturity model reproducing systems with desynchronisation and saturation. The model’s plausibility is confirmed by the direct comparison with real data for the growth of M. tuberculosis populations obtained with the BACTEC MGIT system under different conditions of growth synchronisation.

Scaling Limit of Multi-Type Invariant Measures via the Directed Landscape

Abstract This paper studies the large scale limits of multi-type invariant distributions and Busemann functions of planar stochastic growth models in the Kardar–Parisi–Zhang (KPZ) class. We identify a set of sufficient hypotheses for convergence of multi-type invariant measures of last-passage percolation (LPP) models to the stationary horizon (SH), which is the unique multi-type stationary measure of the KPZ fixed point. Our limit theorem utilizes conditions that are expected to hold broadly in the KPZ class, including convergence of the scaled last-passage process to the directed landscape. We verify these conditions for the six exactly solvable models whose scaled bulk versions converge to the directed landscape, as shown by Dauvergne and Virág. We also present a second, more general, convergence theorem with future applications to polymer models and particle systems. Our paper is the first to show convergence to the SH without relying on information about the structure of the multi-type invariant measures of the prelimit models. These results are consistent with the conjecture that the SH is the universal scaling limit of multi-type invariant measures in the KPZ class.

Growth, poverty trap and escape

The well-known Solow growth model is the workhorse model of the theory of economic growth, which studies capital accumulation in a model economy as a function of time with capital stock, labour and technology-based production as the basic ingredients. The capital is assumed to be in the form of manufacturing equipment and materials. Two important parameters of the model are: the saving fraction s of the output of a production function and the technology efficiency parameter A , appearing in the production function. The saved fraction of the output is fully invested in the generation of new capital and the rest is consumed. The capital stock also depreciates as a function of time due to the wearing out of old capital and the increase in the size of the labour population. We propose a stochastic Solow growth model assuming the saving fraction to be a sigmoidal function of the per capita capital kp . We derive analytically the steady state probability distribution P(kp) and demonstrate the existence of a poverty trap, of central concern in development economics. In a parameter regime, P(kp) is bimodal with the twin peaks corresponding to states of poverty and well-being, respectively. The associated potential landscape has two valleys with fluctuation-driven transitions between them. The mean exit times from the valleys are computed and one finds that the escape from a poverty trap is more favourable at higher values of A. We identify a critical value of Ac below (above) which the state of poverty (well-being) dominates and propose two early signatures of the regime shift occurring at Ac . The economic model, with conceptual foundations in nonlinear dynamics and statistical mechanics, shares universal features with dynamical models from diverse disciplines like ecology and cell biology.

Analysis of a stochastic hybrid Gompertz tumor growth model driven by Lévy noise

This article investigates a stochastic hybrid Gompertz tumor growth model driven by Lévy noise. The global existence of a unique positive solution, extinction, non-persistence in mean, and persistence in mean are well explored. Then, stochastic permanence is obtained by using stochastically ultimately upper bounded and lower bounded. An interesting fact is that the white noise, Markovian noise and Lévy noise can all affect the persistence and extinction of the Gompertz tumor growth model.

Social Capital, Government Expenditures, and Growth

Abstract This paper shows that social capital increases economic growth by raising government investment in human capital through better political incentives and selection. We provide empirical evidence that a greater share of output is spent on public education where social capital is higher, both across countries and across U.S. states. We develop a theoretical model of stochastic endogenous growth with imperfect political agency. Only some people correctly anticipate the future returns to current spending on public education. Greater social diffusion of information makes this knowledge more widespread among voters. As a result, social capital alleviates myopic political incentives to underinvest in human capital. It also helps voters select politicians who ensure high productivity in public education. Through this mechanism, we show that social capital raises the equilibrium growth rate of output and reduces its volatility.

Mean first-passage time for a stochastic tumor growth model with two different time delays

Mean first-passage time for a stochastic tumor growth model with two different time delays

Final nanoparticle size distribution under unusual parameter regimes.

We explore the large-scale behavior of a stochastic model for nanoparticle growth in an unusual parameter regime. This model encompasses two types of reactions: nucleation, where n monomers aggregate to form a nanoparticle, and growth, where a nanoparticle increases its size by consuming a monomer. Reverse reactions are disregarded. We delve into a previously unexplored parameter regime. Specifically, we consider a scenario where the growth rate of the first newly formed particle is of the same order of magnitude as the nucleation rate, in contrast to the classical scenario where, in the initial stage, nucleation dominates over growth. In this regime, we investigate the final size distribution as the initial number of monomers tends to infinity through extensive simulation studies utilizing state-of-the-art stochastic simulation methods with an efficient implementation and supported by high-performance computing infrastructure. We observe the emergence of a deterministic limit for the particle's final size density. To scale up the initial number of monomers to approximate the magnitudes encountered in real experiments, we introduce a novel approximation process aimed at faster simulation. Remarkably, this approximating process yields a final size distribution that becomes very close to that of the original process when the available monomers approach infinity. Simulations of the approximating process further support the conjecture of the emergence of a deterministic limit.

Modeling of ice-matrix composite fracture

This article discusses the results of a study that used the structural approach to conduct multiscale modeling of composite materials. These materials are composed of an ice matrix that has been enhanced with high-strength basalt fibers. The research involved numerical modeling to analyze the fracture process of composite materials made by freezing fresh water with added basalt fibers. The analysis included experimental data from samples of pure ice and composites with different amounts of filler. The effective modulus of elasticity of the composites was determined by considering the quantity of basalt fiber reinforcement. The discrepancy between the strength obtained by finite element complex ANSYS numerical calculation and the experimental data was revealed by a series of macroscopic test calculations of the composite material sample, according to which the effective modulus of elasticity of the composite was recalculated and corrections were introduced into the Voight and Reuss models, taking into account the non-uniform distribution of the fibers and the non-ideal adhesion between the fibers and water ice. A stochastic model of crack growth at the micro level was also used with the data on the diameter and length of basalt fibers and their random distribution in the layers of the composite material. Using the newly acquired effective elastic moduli and the SmartCrackGrowth algorithm, a satisfactory agreement of the crack growth rate with the obtained by stochastic calculation was achieved. The refined method of effective elastic moduli and multilevel model of stress-strain state calculation of composite materials are recommended for assessing the strength of ice cover and designing winter roads with high load-bearing capacity and long operational duration in the Arctic and Subarctic environments.

An analytical approach to evaluate life-cycle cost of deteriorating pipelines

Optimal inspection and maintenance planning for deteriorating pipelines is important for cost-effective risk management decisions. The objective of this study is to develop an analytical approach for determining the expected value and standard deviation of life-cycle cost (LCC) for deteriorating pipelines, which can incorporate imperfect repair and inspection actions without making independent assumptions of failure and repair events. The framework is developed based on a decision tree model by using analytical methods to evaluate events and considers the impact of the inspection schedules and possible repair actions on the probability distributions of failure times. To illustrate the proposed framework, a steel transmission pipeline with a corrosion defect is used, where a stochastic model of corrosion growth is assumed and burst failure is considered as the pipeline failure mode. The case study indicates that the failure cost due to two or more failure occurrences can be ignored; and the inspection interval, repair threshold and type, the length of lifetime considered, and unit failure cost all play important roles in the expected value of the total cost. The variation in the LCC estimation is found to be significant, and it has a great impact on determination of optimal decision variables.

Ergodic periodic measure of a stochastic budworm growth model with periodic coefficients

A stochastic budworm growth model with periodic coefficients is formulated and explored. Sufficient conditions for the existence of a unique ergodic periodic measure are given. An example based on exemplary data is also provided to illustrate the main results.

Modeling the Effect of Spatial Structure on Solid Tumor Evolution and Circulating Tumor DNA Composition.

Circulating tumor DNA (ctDNA) monitoring, while sufficiently advanced to reflect tumor evolution in real time and inform cancer diagnosis, treatment, and prognosis, mainly relies on DNA that originates from cell death via apoptosis or necrosis. In solid tumors, chemotherapy and immune infiltration can induce spatially variable rates of cell death, with the potential to bias and distort the clonal composition of ctDNA. Using a stochastic evolutionary model of boundary-driven growth, we study how elevated cell death on the edge of a tumor can simultaneously impact driver mutation accumulation and the representation of tumor clones and mutation detectability in ctDNA. We describe conditions in which invasive clones are over-represented in ctDNA, clonal diversity can appear elevated in the blood, and spatial bias in shedding can inflate subclonal variant allele frequencies (VAFs). Additionally, we find that tumors that are mostly quiescent can display similar biases but are far less detectable, and the extent of perceptible spatial bias strongly depends on sequence detection limits. Overall, we show that spatially structured shedding might cause liquid biopsies to provide highly biased profiles of tumor state. While this may enable more sensitive detection of expanding clones, it could also increase the risk of targeting a subclonal variant for treatment. Our results indicate that the effects and clinical consequences of spatially variable cell death on ctDNA composition present an important area for future work.

Investigating macroecological patterns in coarse-grained microbial communities using the stochastic logistic model of growth

The structure and diversity of microbial communities are intrinsically hierarchical due to the shared evolutionary history of their constituents. This history is typically captured through taxonomic assignment and phylogenetic reconstruction, sources of information that are frequently used to group microbes into higher levels of organization in experimental and natural communities. Connecting community diversity to the joint ecological dynamics of the abundances of these groups is a central problem of community ecology. However, how microbial diversity depends on the scale of observation at which groups are defined has never been systematically examined. Here, we used a macroecological approach to quantitatively characterize the structure and diversity of microbial communities among disparate environments across taxonomic and phylogenetic scales. We found that measures of biodiversity at a given scale can be consistently predicted using a minimal model of ecology, the Stochastic Logistic Model of growth (SLM). This result suggests that the SLM is a more appropriate null-model for microbial biodiversity than alternatives such as the Unified Neutral Theory of Biodiversity. Extending these within-scale results, we examined the relationship between measures of biodiversity calculated at different scales (e.g. genus vs. family), an empirical pattern previously evaluated in the context of the Diversity Begets Diversity (DBD) hypothesis (Madi et al., 2020). We found that the relationship between richness estimates at different scales can be quantitatively predicted assuming independence among community members, demonstrating that the DBD can be sufficiently explained using the SLM as a null model of ecology. Contrastingly, only by including correlations between the abundances of community members (e.g. as the consequence of interactions) can we predict the relationship between estimates of diversity at different scales. The results of this study characterize novel microbial patterns across scales of organization and establish a sharp demarcation between recently proposed macroecological patterns that are not and are affected by ecological interactions.

Investigating macroecological patterns in coarse-grained microbial communities using the stochastic logistic model of growth.

The structure and diversity of microbial communities are intrinsically hierarchical due to the shared evolutionary history of their constituents. This history is typically captured through taxonomic assignment and phylogenetic reconstruction, sources of information that are frequently used to group microbes into higher levels of organization in experimental and natural communities. Connecting community diversity to the joint ecological dynamics of the abundances of these groups is a central problem of community ecology. However, how microbial diversity depends on the scale of observation at which groups are defined has never been systematically examined. Here, we used a macroecological approach to quantitatively characterize the structure and diversity of microbial communities among disparate environments across taxonomic and phylogenetic scales. We found that measures of biodiversity at a given scale can be consistently predicted using a minimal model of ecology, the Stochastic Logistic Model of growth (SLM). This result suggests that the SLM is a more appropriate null-model for microbial biodiversity than alternatives such as the Unified Neutral Theory of Biodiversity. Extending these within-scale results, we examined the relationship between measures of biodiversity calculated at different scales (e.g. genus vs. family), an empirical pattern previously evaluated in the context of the Diversity Begets Diversity (DBD) hypothesis (Madi et al., 2020). We found that the relationship between richness estimates at different scales can be quantitatively predicted assuming independence among community members, demonstrating that the DBD can be sufficiently explained using the SLM as a null model of ecology. Contrastingly, only by including correlations between the abundances of community members (e.g. as the consequence of interactions) can we predict the relationship between estimates of diversity at different scales. The results of this study characterize novel microbial patterns across scales of organization and establish a sharp demarcation between recently proposed macroecological patterns that are not and are affected by ecological interactions.

Stability and mean growth rate of stochastic Solow model driven by jump–diffusion process

This work focuses on the stability and mean growth rate of stochastic Solow growth model. The uncertainty in our model is generated by the technology part, which includes fluctuations on the technological accumulation and jump growth from new inventions and ideas. We introduce the criteria of stability, including stochastic stability and exponential stability, and the sufficient condition for the existence of balanced growth path. Furthermore, we precisely calculate the higher-order moment of effective capital–labor ratio and the mean growth rate of major economic quantities on the balanced growth path. Our analysis sheds light on the impact of various stochastic effects on economic quantities and provides insight into the stability of the stochastic Solow model with technological uncertainty.

Локальная предельная теорема для ветвящихся случайных систем Гальтона–Ватсона с иммиграцией и бесконечной дисперсией

The paper considers a stochastic population growth model, which is the critical Galton-Watson stochastic branching system allowing immigration. We consider the case when the mathematical expectation of the law of influx of immigrant particles and the variance of the law of reproduction of aboriginal particles have infinite values. The asymptotic properties of transition probabilities are studied in the case when the states of the system are irreversible. For this case, a local limit theorem has been established.

Stochastic differential equations mixed model for individual growth with the inclusion of genetic characteristics

In early work we have studied a class of stochastic differential equation (SDE) models, for which the Gompertz and the Bertalanffy-Richards stochastic models are particular cases, to describe individual growth in random environments, and applied it to model cattle weight evolution using real data. We have started to work on these type of models considering that the model parameters are fixed, i.e. the same for all the animals. Aiming to incorporate variability among individuals, we consider that the model parameters can be random variables, resulting in SDE mixed models. In additon, here we consider SDE mixed models, allowing the parameters to be random and propose to incorporate each animal's genetic characteristics considering the transformed animal's weight at maturity to be a function of its genetic values. The main objective is to extend the SDE mixed model to the more realistic case where the individual genetic value becomes an important component in the estimated growth curve. For the estimation of the model parameters we have used maximum likelihood estimation theory. Estimates and asymptotic confidence intervals of the parameters are presented. A comparison with SDE non-mixed model and SDE mixed model without the inclusion of genetic characteristics is shown with the conclusion that the incorporation of some genetic characteristics in the model parameters improves the estimation of the animal's growth curve.

First-passage-time statistics of growing microbial populations carry an imprint of initial conditions

In exponential population growth, variability in the timing of individual division events and environmental factors (including stochastic inoculation) compound to produce variable growth trajectories. In several stochastic models of exponential growth we show power-law relationships that relate variability in the time required to reach a threshold population size to growth rate and inoculum size. Population-growth experiments in E. coli and S. aureus with inoculum sizes ranging between 1 and 100 are consistent with these relationships. We quantify how noise accumulates over time, finding that it encodes—and can be used to deduce—information about the early growth rate of a population.

Development of a quantitative microbiological spoilage risk assessment (QMSRA) model for cooked ham sliced at retail

A quantitative microbiological spoilage risk assessment model (QMSRA) for cooked ham sliced at retail was developed based on a stochastic growth model for lactic acid bacteria (LAB), which are considered as the specific spoilage organisms (SSO), and a “spoilage-response” relationship characterizing the variability in consumer's perception of spoilage. In a simulation involving 10,000 cooked ham purchases, the QMSRA model predicted a median of zero spoilage events for up to 4.5 days of storage. After storage times of 5 and 6 days, the model predicted 1,790 and 8,570 spoilage events, respectively. A sensitivity analysis showed that domestic storage temperature was the most significant factor affecting LAB concentration in cooked ham, followed by the LAB contamination level at slicing. A scenario analysis was performed testing better temperature control of consumer's refrigerators, better hygiene conditions during slicing and a combination of the two strategies. Among the tested scenarios, a 2 log reduction in the LAB contamination at slicing combined with a 2 °C decrease in domestic storage temperature resulted in zero risk of spoilage for up to 12 days of storage. The QMSRA model developed in the present study can be a useful tool for quality management decisions.

Extinction time in growth models subject to binomial catastrophes *

Populations are often subject to catastrophes that lead to significant reductions in the number of individuals. Many stochastic growth models have been considered to explain such dynamics. Among the reported results, it has been considered whether dispersion strategies, at times of catastrophes, increase the survival probability of the population. In this paper, we contrast dispersion strategies by comparing the mean extinction times of a population under conditions of near-certain extinction. Specifically, we consider populations subject to binomial catastrophes, where the population size is reduced according to a binomial law when a catastrophe occurs. Our findings delineate the optimal strategy (dispersion or non-dispersion) based on variations in model parameter values.

Discrete-Time Gompertz Model for Adana Breed Pigeons

The mathematical animal growth models in the literature are not in the form of linear models. These growth models in the literature are not in linear form. There are different numerical analysis methods for the estimation of the parameters found in these functions and specific software have been produced to estimate the unknown parameters in these mathematical models and to apply these methods. In these nonlinear mathematical growth models, there may be more than one parameter. For these and other reasons, the number of mathematical numerical operations in estimating parameters is quite high. In this study, the discrete time stochastic Gompertz model (DTSGM) was considered to determine the growth of Adana pigeons. A model is used in which the parameter in the model is estimated by an adaptive Kalman filter (AKF). The aim of this research is to reveal the validity of both the model and the estimation method for Adana breed domestic pigeons. Daily weight measurements of 28 Adana pigeons were considered and estimated using DTSGM and AKF methods in this framework. DTSGM in conjunction with AKF has been shown to provide a convenient analysis tool for modeling daily weight estimates of Adana pigeons.