Gompertz Diffusion Research Articles

Review of: "A connection between Gompertz diffusion model and Vasicek Interest Rate model"

Review of: "A connection between Gompertz diffusion model and Vasicek Interest Rate model"

Review of: "A connection between Gompertz diffusion model and Vasicek interest rate model"

Review of: "A connection between Gompertz diffusion model and Vasicek interest rate model"

A connection between Gompertz diffusion model and Vasicek Interest Rate model

The main goal of this paper is to establish a new connection between the Gompertz diffusion model and the Vasicek Interest Rate model. These connection focus on elementary stochastic calculus and Itô's calculus. Firstly, we prove that the exponential of the Vasicek Interest Rate model is a Gompertz diffusion process. Secondly, we prove that the logarithm of the Gompertz diffusion process is a Vasicek Interest Rate model. New computations of the probability transition density function and the mean functions of the processes have quite simple formulations.

Some remarks on the links between the VIR and Gompertz diffusion models and the links between the CIR and Rayleigh diffusion models

The main purpose of this work is to establish new links between the stochastic Cox-Ingersoll-Ross (CIR) model and the stochastic Rayleigh diffusion process (SRDP) and the links between the Vasicek Interest Rate (VIR) process and the stochastic Gompertz diffusion process (SGDP). These links focus on elementary stochastic calculus and Itô’s calculus. Firstly, we prove that the square root of the CIR model is a SRDP. Secondly, we prove that the square of the SRDP is a CIR model. Thirdly, we prove that the exponential of the VIR model is a SGDP. Finally, we prove that the logarithm of the SGDP is a VIR model. New computations of the probability transition density function (PTDF) and the trend functions of the processes have quite simple formulations.

Caseinate vs bovine serum albumin on stabilizing zein nanoparticles for co-entrapping quercetin and curcumin

This work emphasized on the comparison of the particle characteristics and physicochemical properties of zein-caseinate and zein-BSA nanoparticles (NPs). Quercetin (Que) and curcumin (Cur) were used as the model bioactive to be co-encapsulated in the zein-caseinate and zein-BSA NPs by a combination of pH-driven and anti-solvent methods. The results demonstrated that compared with zein-caseinate, the zein-BSA exhibited smaller particle size, but stronger ability to encapsulate and retain Que and Cur during long-time storage. However, the zein-caseinate NPs showed higher thermal and ionic strength stability than zein-BSA NPs. In terms of the in-vitro bioactive release from zein-based NPs, zein-caseinate NPs appeared to show a higher Cur release rate but a lower Que release rate in comparison with zein-BSA NPs. Kinetic models revealed the Gompertz was the best fit model, indicating that the Gompertz diffusion contributed to the Que and Cur release from zein-based NPs. However, the two types of zein-based NPs showed very similar Que and Cur release profiles. These results indicated that zein-BSA and zein-caseinate composite NPs exhibited different particle characteristics and physicochemical properties in terms of encapsulation, release, and harsh environmental stability.

Preparation, characterization, and in vitro release kinetics of doxorubicin-loaded magnetosomes.

The doxorubicin (DOX) was successfully coupled to the magnetosomes from Acidithiobacillus ferrooxidans (At. ferrooxidans) by genipin bridging. The parameters (magnetosome concentration, DOX concentration, genipin concentration-, and cross-link time) expected for temperature significantly influenced the coupling rate. Bacterial magnetosome-doxorubicin complexes (BMDCs) were characterized by transmission electron microscope (TEM), particle size analyzer and Fourier transform infrared spectroscopy. Results indicated that BMDCs exhibited a mean particle size of 83.98mm and displayed a negative charge. The chemical reaction occurring between CO and NH group and the physical adsorption predominated by electrostatic interaction were found to involve in coupling. BMDCs can release 40% of DOX in simulated gastrointestinal conditions within 38h. Kinetic models including Higuchi, Korsmeyer-Peppas, Zero order, First order, Hixon-Crowell, Baker-Lonsdale, and Weibull and Gompertz were utilized to explore the release mechanism of DOX from BMDCs. All models were found to fit well (r2 ≥ 0.8144) with the release data and the Gompertz was the best fit model (r2 = 0.9742), implying that the complex mechanisms involving Fickian and Gompertz diffusion contributed to the release. These findings suggested that magnetosomes from At. ferrooxidans have great potential applications in biomedical and clinical fields as the carrier of target drug delivery systems in the future.

Inference on an heteroscedastic Gompertz tumor growth model

We consider a non homogeneous Gompertz diffusion process whose parameters are modified by generally time-dependent exogenous factors included in the infinitesimal moments. The proposed model is able to describe tumor dynamics under the effect of anti-proliferative and/or cell death-induced therapies. We assume that such therapies can modify also the infinitesimal variance of the diffusion process. An estimation procedure, based on a control group and two treated groups, is proposed to infer the model by estimating the constant parameters and the time-dependent terms. Moreover, several concatenated hypothesis tests are considered in order to confirm or reject the need to include time-dependent functions in the infinitesimal moments. Simulations are provided to evaluate the efficiency of the suggested procedures and to validate the testing hypothesis. Finally, an application to real data is considered.

Stochastic growth pattern of untreated human glioblastomas predicts the survival time for patients

Glioblastomas are highly malignant brain tumors. Knowledge of growth rates and growth patterns is useful for understanding tumor biology and planning treatment logistics. Based on untreated human glioblastoma data collected in Trondheim, Norway, we first fit the average growth to a Gompertz curve, then find a best fitted white noise term for the growth rate variance. Combining these two fits, we obtain a new type of Gompertz diffusion dynamics, which is a stochastic differential equation (SDE). Newly collected untreated human glioblastoma data in Seattle, US, re-verify our model. Instead of growth curves predicted by deterministic models, our SDE model predicts a band with a center curve as the tumor size average and its width as the tumor size variance over time. Given the glioblastoma size in a patient, our model can predict the patient survival time with a prescribed probability. The survival time is approximately a normal random variable with simple formulas for its mean and variance in terms of tumor sizes. Our model can be applied to studies of tumor treatments. As a demonstration, we numerically investigate different protocols of surgical resection using our model and provide possible theoretical strategies.

Powers of the Stochastic Gompertz and Lognormal Diffusion Processes, Statistical Inference and Simulation

In this paper, we study a new family of Gompertz processes, defined by the power of the homogeneous Gompertz diffusion process, which we term the powers of the stochastic Gompertz diffusion process. First, we show that this homogenous Gompertz diffusion process is stable, by power transformation, and determine the probabilistic characteristics of the process, i.e., its analytic expression, the transition probability density function and the trend functions. We then study the statistical inference in this process. The parameters present in the model are studied by using the maximum likelihood estimation method, based on discrete sampling, thus obtaining the expression of the likelihood estimators and their ergodic properties. We then obtain the power process of the stochastic lognormal diffusion as the limit of the Gompertz process being studied and go on to obtain all the probabilistic characteristics and the statistical inference. Finally, the proposed model is applied to simulated data.

Analysis and Prediction on Vehicle Ownership Based on an Improved Stochastic Gompertz Diffusion Process

This paper aims at introducing a new improved stochastic differential equation related to Gompertz curve for the projection of vehicle ownership growth. This diffusion model explains the relationship between vehicle ownership and GDP per capita, which has been studied as a Gompertz-like function before. The main innovations of the process lie in two parts: by modifying the deterministic part of the original Gompertz equation, the model can present the remaining slow increase when the S-shaped curve has reached its saturation level; by introducing the stochastic differential equation, the model can better fit the real data when there are fluctuations. Such comparisons are carried out based on data from US, UK, Japan, and Korea with a time span of 1960–2008. It turns out that the new process behaves better in fitting curves and predicting short term growth. Finally, a prediction of Chinese vehicle ownership up to 2025 is presented with the new model, as China is on the initial stage of motorization with much fluctuations in growth.

On comparison of the estimators of the Hurst index and the diffusion coefficient of the fractional Gompertz diffusion process

We study some estimators of the Hurst index and the diffusion coefficient of the fractional Gompertz diffusion process and prove that they are strongly consistent and most of them are asymptotically normal. Moreover, we compare the asymptotic behavior of these estimators with the aid of computer simulations.

Modeling tumor growth in the presence of a therapy with an effect on rate growth and variability by means of a modified Gompertz diffusion process

In experimental studies on tumor growth, whenever the time evolution of the relative volume of a tumor in an untreated (control) group can be fitted by a Gompertz diffusion process there is a possibility that an antiproliferative therapy, which modifies the growth rate of the process that fits the treated group, may also affect its variability. The present paper proposes several procedures for the estimation of the time function included in the infinitesimal variance of the new process, as well as the time function affecting the growth rate (which is included in the infinitesimal mean). Also, a hypothesis testing is designed to confirm or refute the need for including such a time-dependent function in the infinitesimal variance. In order to validate and compare the proposed procedures a simulation study has been carried out. In addition, a proposal is made for a strategy aimed at finding the optimal combination of procedures for each case. A real data application concerning the effects of cisplatin on a patient-derived xenograft (PDX) tumor model showcases the advantages of this model over others that have been used in the past.

Impact of the HITECH Act on physicians' adoption of electronic health records.

The Health Information Technology for Economic and Clinical Health (HITECH) Act has distributed billions of dollars to physicians as incentives for adopting certified electronic health records (EHRs) through the meaningful use (MU) program ultimately aimed at improving healthcare outcomes. The authors examine the extent to which the MU program impacted the EHR adoption curve that existed prior to the Act. Bass and Gamma Shifted Gompertz (G/SG) diffusion models of the adoption of "Any" and "Basic" EHR systems in physicians' offices using consistent data series covering 2001-2013 and 2006-2013, respectively, are estimated to determine if adoption was stimulated during either a PrePay (2009-2010) period of subsidy anticipation or a PostPay (2011-2013) period when payments were actually made. Adoption of Any EHR system may have increased by as much as 7 percentage points above the level predicted in the absence of the MU subsidies. This estimate, however, lacks statistical significance and becomes smaller or negative under alternative model specifications. No substantial effects are found for Basic systems. The models suggest that adoption was largely driven by "imitation" effects (q-coefficient) as physicians mimic their peers' technology use or respond to mandates. Small and often insignificant "innovation" effects (p-coefficient) are found suggesting little enthusiasm by physicians who are leaders in technology adoption. The authors find weak evidence of the impact of the MU program on EHR uptake. This is consistent with reports that many current EHR systems reduce physician productivity, lack data sharing capabilities, and need to incorporate other key interoperability features (e.g., application program interfaces).

Estimating and determining the effect of a therapy on tumor dynamics by means of a modified Gompertz diffusion process

A modified Gompertz diffusion process is considered to model tumor dynamics. The infinitesimal mean of this process includes non-homogeneous terms describing the effect of therapy treatments able to modify the natural growth rate of the process. Specifically, therapies with an effect on cell growth and/or cell death are assumed to modify the birth and death parameters of the process. This paper proposes a methodology to estimate the time-dependent functions representing the effect of a therapy when one of the functions is known or can be previously estimated. This is the case of therapies that are jointly applied, when experimental data are available from either an untreated control group or from groups treated with single and combined therapies. Moreover, this procedure allows us to establish the nature (or, at least, the prevalent effect) of a single therapy in vivo. To accomplish this, we suggest a criterion based on the Kullback–Leibler divergence (or relative entropy). Some simulation studies are performed and an application to real data is presented.

EU regional unemployment as a transnational matter: An analysis via the Gompertz diffusion processs

AbstractAt the end of 1990s, Danny Quah devoted several papers to the analysis of polarization and stratification in the convergence processes of economies, creating the image of the ‘convergence clubs’ and suggesting the importance of studying the distribution dynamics of the macroeconomic variables. As for the labour markets, Overman and Puga (2002) showed that a progressive polarization of unemployment was in fact occurring among the European regions in 1986–1996, causing a phenomenon of cross‐border clusterization. Here we propose to analyse the evolution of the unemployment rates of the EU 27 regions in the last two decades assuming that the unemployment rates evolve according to a Gompertz stochastic process. The estimated parameters of the process – intrinsic growth rate, deceleration factor, volatility – represent the evolutionary path of the unemployment rate and allow for estimating the steady state of the process. A cluster analysis is performed on the steady state values of the unemployment rates. The analysis confirms the emergence of several ‘convergence clubs’ among the European regional labour markets, which are compared to the clusters resulting from the more traditional clusterization on the current unemployment rates.

Inferring the effect of therapies on tumor growth by using diffusion processes

Modeling the effect of therapies in cancer animal models remains a challenge. This point may be addressed by considering a diffusion process that models the tumor growth and a modified process that includes, in its infinitesimal mean, a time function modeling the effect of the therapy. In the case of a Gompertz diffusion process, where a control group and one or more treated groups are examined, a methodology to estimate this function has been proposed by Albano et al. (2011). This method has been applied to infer the effect of cisplatin and doxorubicin+cyclophosphamide on breast cancer xenografts. Although this methodology can be extended to other diffusion processes, it has an important restriction: it is necessary that a known diffusion process adequately fits the control group. Here, we propose the use of a stochastic process for a hypothetical control group, in such a way that both the control and the treated groups can be modeled by modified processes of the former. Thus, the comparison between models would allow estimating the real effect of the therapy. The new methodology has been validated by inferring the effects in breast cancer models, and we have checked the robustness of the procedure against the choice of stochastic model for the hypothetical control group. Finally, we have also applied the methodology to infer the effect of a therapeutic peptide and ovariectomy on the growth of a breast cancer xenograft, and its efficiency in modeling the effect of different treatments in the absence of control group data is shown.

Forecasting of peak electricity demand in Mauritius using the non-homogeneous Gompertz diffusion process

In this study, the non-homogeneous Gompertz diffusion process (NHGDP) is used to model the monthly peak electricity demand in Mauritius in order to predict the future values on the basis of a Genetic Algorithm (GA) approach. Our model is developed based a key economic indicator which is the gross domestic product (GDP) and the weather factors such as temperature, hours of sunshine and humidity. Genetic Algorithm then searches for the best coefficients by minimizing the root mean square error. Monthly data from January 2005 to December 2008 are considered to test the model. Finally, the Artificial Neural Network (ANN) is used to forecast each independent variable for the year 2009 and the NHGDP model is validated for that year. Our results show that the model provides an accurate and reliable prediction for the monthly peak electricity demand in Mauritius.

Evolutionary model of an anonymous consumer durable market

An analytic model is presented that considers the evolution of a market of durable goods. The model suggests that after introduction goods spread always according to a Bass diffusion. However, this phase will be followed by a diffusion process for durable consumer goods governed by a variation-selection-reproduction mechanism and the growth dynamics can be described by a replicator equation.The theory suggests that products play the role of species in biological evolutionary models. It implies that the evolution of man-made products can be arranged into an evolutionary tree. The model suggests that each product can be characterized by its product fitness. The fitness space contains elements of both sites of the market, supply and demand. The unit sales of products with a higher product fitness compared to the mean fitness increase. Durables with a constant fitness advantage replace other goods according to a logistic law. The model predicts in particular that the mean price exhibits an exponential decrease over a long time period for durable goods. The evolutionary diffusion process is directly related to this price decline and is governed by Gompertz equation. Therefore it is denoted as Gompertz diffusion.Describing the aggregate sales as the sum of first, multiple and replacement purchase the product life cycle can be derived. Replacement purchase causes periodic variations of the sales determined by the finite lifetime of the good (Juglar cycles). The model suggests that both, Bass- and Gompertz diffusion may contribute to the product life cycle of a consumer durable.The theory contains the standard equilibrium view of a market as a special case. It depends on the time scale, whether an equilibrium or evolutionary description is more appropriate. The evolutionary framework is used to derive also the size, growth rate and price distribution of manufacturing business units. It predicts that the size distribution of the business units (products) is lognormal, while the growth rates exhibit a Laplace distribution. Large price deviations from the mean price are also governed by a Laplace distribution (fat tails). These results are in agreement with empirical findings. The explicit comparison of the time evolution of consumer durables with empirical investigations confirms the close relationship between price decline and Gompertz diffusion, while the product life cycle can be described qualitatively for a long time period.

The Bivariate Gompertz Diffusion Model for Tree Diameter and Height Distribution

Abstract With this research we present a new method for describing the bivariate diameter and height distribution of trees growing in a pure, uneven-aged forest. We use a stochastic differential equation framework to derive a bivariate age-dependent probability density function of tree diameter and height when the tree diameter and height follow a bivariate stochastic Gompertz shape growth process. We also adopt the two-dimensional transition probability function methodology for growth modeling of forest stands. The bivariate stochastic Gompertz model is fit to diameter and height observations for 1,575 pine trees in the Dubrava district of Lithuania. A considerable advantage of the bivariate stochastic Gompertz growth model is that the model parameters are easily interpretable. All results are implemented in the symbolic algebra system MAPLE.

The effect of nested grid sampling on the parameter estimation of a spatial Gompertz diffusion

This paper evaluates the effects of using data observed on regular nested grids on the parameter estimates of a two-parameter Gompertz diffusion model. This new spatial diffusion process represents a technically more complex stage of Gompertz modeling. Firstly, the diffusion model is introduced through an appropriate transformation of a two-parameter Gaussian diffusion process. Probabilistic characteristics of this model, such as the transition densities and the trend functions, are obtained. Secondly, statistical estimation is considered using data obtained on a regular or irregular grid; the explicit expression of the likelihood equations and the parameter estimators are given for regular grids. Finally, a simulation experiment illustrates the results of this paper.