Stochastic Differential Equation Model Research Articles

Positivity and boundedness preserving numerical scheme for the stochastic epidemic model with square-root diffusion term

This work concerns about the numerical solution to the stochastic epidemic model proposed by Cai et al. [2]. The typical features of the model including the positivity and boundedness of the solution and the presence of the square-root diffusion term make this an interesting and challenging work. By modifying the classical Euler-Maruyama (EM) scheme, we generate a positivity and boundedness preserving numerical scheme, which is proved to have a strong convergence to the true solution over finite time intervals. We also demonstrate that the principle of this method is applicable to a bunch of popular stochastic differential equation (SDE) models, e.g. the mean-reverting square-root process, an important financial model, and the multi-dimensional SDE SIR epidemic model.

A Hybrid Epidemic Model to Explore Stochasticity in COVID-19 Dynamics

The dynamic nature of the COVID-19 pandemic has demanded a public health response that is constantly evolving due to the novelty of the virus. Many jurisdictions in the USA, Canada, and across the world have adopted social distancing and recommended the use of face masks. Considering these measures, it is prudent to understand the contributions of subpopulations—such as “silent spreaders”—to disease transmission dynamics in order to inform public health strategies in a jurisdiction-dependent manner. Additionally, we and others have shown that demographic and environmental stochasticity in transmission rates can play an important role in shaping disease dynamics. Here, we create a model for the COVID-19 pandemic by including two classes of individuals: silent spreaders, who either never experience a symptomatic phase or remain undetected throughout their disease course; and symptomatic spreaders, who experience symptoms and are detected. We fit the model to real-time COVID-19 confirmed cases and deaths to derive the transmission rates, death rates, and other relevant parameters for multiple phases of outbreaks in British Columbia (BC), Canada. We determine the extent to which SilS contributed to BC’s early wave of disease transmission as well as the impact of public health interventions on reducing transmission from both SilS and SymS. To do this, we validate our model against an existing COVID-19 parameterized framework and then fit our model to clinical data to estimate key parameter values for different stages of BC’s disease dynamics. We then use these parameters to construct a hybrid stochastic model that leverages the strengths of both a time-nonhomogeneous discrete process and a stochastic differential equation model. By combining these previously established approaches, we explore the impact of demographic and environmental variability on disease dynamics by simulating various scenarios in which a COVID-19 outbreak is initiated. Our results demonstrate that variability in disease transmission rate impacts the probability and severity of COVID-19 outbreaks differently in high- versus low-transmission scenarios.

Generative Stochastic Modeling of Strongly Nonlinear Flows with Non-Gaussian Statistics

Strongly nonlinear flows, which commonly arise in geophysical and engineering turbulence, are characterized by persistent and intermittent energy transfer between various spatial and temporal scales. These systems are difficult to model and analyze due to combination of high dimensionality and uncertainty, and there has been much interest in obtaining reduced models, in the form of stochastic closures, that can replicate their non-Gaussian statistics in many dimensions. Here, we propose a data-driven framework to model stationary chaotic dynamical systems through nonlinear transformations and a set of decoupled stochastic differential equations (SDEs). Specifically, we use optimal transport to find a transformation from the distribution of time-series data to a multiplicative reference probability measure such as the standard normal distribution. Then we find the set of decoupled SDEs that admit the reference measure as the invariant measure, and also closely match the spectrum of the transformed data. As such, this framework represents the chaotic time series as the evolution of a stochastic system observed through the lens of a nonlinear map. We demonstrate the application of this framework in Lorenz-96 system, a 10-dimensional model of high-Reynolds cavity flow, and reanalysis climate data. These examples show that SDE models generated by this framework can reproduce the non-Gaussian statistics of systems with moderate dimensions (e.g. 10 and more), and predict super-Gaussian tails that are not readily available from little training data. These findings suggest that this class of models provide an efficient hypothesis space for learning strongly nonlinear flows from small amounts of data.

Identifiability analysis for models of the translation kinetics after mRNA transfection

Mechanistic models are a powerful tool to gain insights into biological processes. The parameters of such models, e.g. kinetic rate constants, usually cannot be measured directly but need to be inferred from experimental data. In this article, we study dynamical models of the translation kinetics after mRNA transfection and analyze their parameter identifiability. That is, whether parameters can be uniquely determined from perfect or realistic data in theory and practice. Previous studies have considered ordinary differential equation (ODE) models of the process, and here we formulate a stochastic differential equation (SDE) model. For both model types, we consider structural identifiability based on the model equations and practical identifiability based on simulated as well as experimental data and find that the SDE model provides better parameter identifiability than the ODE model. Moreover, our analysis shows that even for those parameters of the ODE model that are considered to be identifiable, the obtained estimates are sometimes unreliable. Overall, our study clearly demonstrates the relevance of considering different modeling approaches and that stochastic models can provide more reliable and informative results.

Parameter and uncertainty estimation in stochastic differential equation models with multi-rate data and nonstationary disturbances

Updated methods are proposed for estimating model parameters and error-covariance matrices in stochastic differential equation (SDE) models. New expressions, based on the LAMLE (Laplace Approximation Maximum Likelihood Estimation) and LAB (Laplace Approximation Bayesian) SDE estimation methods are derived so that LAMLE and LAB can be used for systems with multi-rate data and nonstationary disturbances. The updated LAMLE method is tested using simulated data from a two-state continuous stirred tank reactor (CSTR) model. Comparisons are made with the results obtained using the continuous-time stochastic method (CTSM), confirming that the updated LAMLE algorithm provides reliable estimates for unknown model parameters, measurement noise variances, and process-error variances. In all situations tested, the updated LAMLE algorithm converged to reliable parameter and uncertainty estimates, while avoiding convergence difficulties encountered by CTSM in some situations. The updated LAMLE and LAB algorithms will be useful for parameter estimation in online models used in advanced process monitoring and control applications where multi-rate data and nonstationary disturbances are often encountered. Estimates of measurement noise variances and process-error variances will be helpful for state-estimator tuning.

A classification of the dynamics of three-dimensional stochastic ecological systems

The classification of the long-term behavior of dynamical systems is a fundamental problem in mathematics. For both deterministic and stochastic dynamics specific classes of models verify Palis’ conjecture: the long-term behavior is determined by a finite number of stationary distributions. In this paper we consider the classification problem for stochastic models of interacting species. For a large class of three-species, stochastic differential equation models, we prove a variant of Palis’ conjecture: the long-term statistical behavior is determined by a finite number of stationary distributions and, generically, three general types of behavior are possible: 1) convergence to a unique stationary distribution that supports all species, 2) convergence to one of a finite number of stationary distributions supporting two or fewer species, 3) convergence to convex combinations of single species, stationary distributions due to a rock-paper-scissors type of dynamic. Moreover, we prove that the classification reduces to computing Lyapunov exponents (external Lyapunov exponents) that correspond to the average per-capita growth rate of species when rare. Our results stand in contrast to the deterministic setting where the classification is incomplete even for three-dimensional, competitive Lotka–Volterra systems. For these SDE models, our results also provide a rigorous foundation for ecology’s modern coexistence theory (MCT) which assumes the external Lyapunov exponents determine long-term ecological outcomes.

Deterministic and Stochastic Prey–Predator Model for Three Predators and a Single Prey

In this paper, a deterministic prey–predator model is proposed and analyzed. The interaction between three predators and a single prey was investigated. The impact of harvesting on the three predators was studied, and we concluded that the dynamics of the population can be controlled by harvesting. Some sufficient conditions were obtained to ensure the local and global stability of equilibrium points. The transcritical bifurcation was investigated using Sotomayor’s theorem. We performed a stochastic extension of the deterministic model to study the fluctuation environmental factors. The existence of a unique global positive solution for the stochastic model was investigated. The exponential–mean–squared stability of the resulting stochastic differential equation model was examined, and it was found to be dependent on the harvesting effort. Theoretical results are illustrated using numerical simulations.

Analysis of a new stochastic Gompertz diffusion model for untreated human glioblastomas

In this paper, we analyze a new Ito stochastic differential equation model for untreated human glioblastomas. The model was the best fit of the average growth and variance of 94 pairs of a data set. We show the existence and uniqueness of solutions in the positive spatial domain. When the model is restricted in the finite domain [Formula: see text], we show that the boundary point 0 is unattainable while the point [Formula: see text] is reflecting attainable. We prove there is a unique ergodic stationary distribution for any non-zero noise intensity, and obtain the explicit probability density function for the stationary distribution. By using Brownian bridge, we give a representation of the probability density function of the first passage time when the diffusion process defined by a solution passes the point [Formula: see text] firstly. We carry out numerical studies to illustrate our analysis. Our mathematical and numerical analysis confirms the soundness of our randomization of the deterministic model in that the stochastic model will set down to the deterministic model when the noise intensity approaches zero. We also give physical interpretation of our stochastic model and analysis.

A stochastic model of chemorepulsion with additive noise and nonlinear sensitivity

We consider a stochastic partial differential equation (SPDE) model for chemorepulsion, with non-linear sensitivity on the one-dimensional torus. By establishing an a priori estimate independent of the initial data, we show that there exists a pathwise unique, global solution to the SPDE. Furthermore, we show that the associated semi-group is Markov and possesses a unique invariant measure, supported on a Hölder–Besov space of positive regularity, which the solution law converges to exponentially fast. The a priori bound also allows us to establish tail estimates on the L^p norm of the invariant measure which are heavier than Gaussian.

Exponential Growth Model and Stochastic Population Models: A Comparison via Population Data

A population dynamic model explains the changes of a population in the near future, given its current status and the environmental conditions that the population is exposed to. In modelling a population dynamic, deterministic model and stochastic models are used to describe and predict the observed population. For modelling population size deterministic model may provide sufficient biological understanding about the system, but if the population numbers do become small, then a stochastic model is necessary with certain conditions. In this study, both types of models such exponential, discrete-time Markov chain (DTMC), continuous-time Markov chain (CTMC) and stochastic differential equation (SDE) are applied to goat population data. Results from the simulations of stochastic realisations as well as deterministic counterparts are shown and tested by root mean square error (RMSE). The SDE model gives the smallest RMSE value which indicate the best model in fitting the data.

Optimal index and averaging principle for Itô–Doob stochastic fractional differential equations

In this paper, a class of Itô–Doob stochastic fractional differential equations (Itô–Doob SFDEs) models are discussed. Using the time scale transformation method, we consider the averaging principle of the transformed equations and establish the relevant results. At the same time, we find that the optimal index for the original Itô–Doob SFDEs can be determined, the selection of such index is similar to the classical stochastic differential equations model.

Topological approximate Bayesian computation for parameter inference of an angiogenesis model.

MotivationInferring the parameters of models describing biological systems is an important problem in the reverse engineering of the mechanisms underlying these systems. Much work has focused on parameter inference of stochastic and ordinary differential equation models using Approximate Bayesian Computation (ABC). While there is some recent work on inference in spatial models, this remains an open problem. Simultaneously, advances in topological data analysis (TDA), a field of computational mathematics, have enabled spatial patterns in data to be characterized.ResultsHere, we focus on recent work using TDA to study different regimes of parameter space for a well-studied model of angiogenesis. We propose a method for combining TDA with ABC to infer parameters in the Anderson–Chaplain model of angiogenesis. We demonstrate that this topological approach outperforms ABC approaches that use simpler statistics based on spatial features of the data. This is a first step toward a general framework of spatial parameter inference for biological systems, for which there may be a variety of filtrations, vectorizations and summary statistics to be considered. Availability and implementationAll code used to produce our results is available as a Snakemake workflow from github.com/tt104/tabc_angio.

An optimization‐based stochastic model of the two‐compartment pharmacokinetics

The original pharmacokinetics (PK) two‐compartment model illustrates the change law of the drug in the central and peripheral chambers after administration, respectively. Although this deterministic model has derived many practical conclusions, it is based on simplifications and neglects noise, which is inherent to pharmacological processes. The actual pharmacological processes are always influenced by factors that cannot be entirely understood or modeled explicitly. Modeling without considering these phenomena may impact the accuracy and the related conclusions. A stochastic type of model can capture such noise. We proposed a novel two‐compartment PK model concerning drug administration through intravenous route by combining optimal control theory and stochastic analysis. The selection of the objective function was based on the goal of obtaining the best possible therapeutic effect with the least possible drug dosage. Firstly, we extended the original PK two‐compartment model based on optimal control and proved the existence and uniqueness of the switch in the control. Moreover, considering the possible uncertain factors, we added disturbances to the distance between the drug concentration and equilibrium point of the dynamic system and extended the model to a stochastic differential equation model. Qualitative and quantitative analyses showed that optimal controls were bang‐bang, that is, alternating the drug dosages at a full dose with rest‐periods in‐between. Our analysis provided a schedule for optimal dosage and timing. The solutions of the model provided estimates of the drug concentration at any given time. Finally, we simulated the model using R and showed that the numerical method is stable.

Likelihood Function through the Delta Approximation in Mixed SDE Models

Stochastic differential equations (SDE) appropriately describe a variety of phenomena occurring in random environments, such as the growth dynamics of individual animals. Using appropriate weight transformations and a variant of the Ornstein–Uhlenbeck model, one obtains a general model for the evolution of cattle weight. The model parameters are α, the average transformed weight at maturity, β, a growth parameter, and σ, a measure of environmental fluctuations intensity. We briefly review our previous work on estimation and prediction issues for this model and some generalizations, considering fixed parameters. In order to incorporate individual characteristics of the animals, we now consider that the parameters α and β are Gaussian random variables varying from animal to animal, which results in SDE mixed models. We estimate parameters by maximum likelihood, but, since a closed-form expression for the likelihood function is usually not possible, we approximate it using our proposed delta approximation method. Using simulated data, we estimate the model parameters and compare them with existing methodologies, showing that the proposed method is a good alternative. It also overcomes the existing methodologies requirement of having all animals weighed at the same ages; thus, we apply it to real data, where such a requirement fails.

Stochastic differential equation model of Covid-19: Case study of Pakistan

In recent years, the world lived a horrible nightmare named the Covid-19 pandemic. It’s changing lifestyle caused the closure of critical establishments like schools, sports halls, companies, etc., as well as causing damage and menace for humanity. Mathematical models are helpful tools for modeling and analysing the infectious disease transmission This article presents a stochastic model of the Covid-19 epidemic for a population with five compartments. We give a numerical analysis of the proposed stochastic model. Also, we compare it with results of the corresponding deterministic model.

Noise inference for ergodic Lévy driven SDE

We study inference for the driving Lévy noise of an ergodic stochastic differential equation (SDE) model, when the process is observed at high-frequency and long time and when the drift and scale coefficients contain finite-dimensional unknown parameters. By making use of the Gaussian quasi-likelihood function for the coefficients, we derive a stochastic expansion for functionals of the unit-time residuals, which clarifies some quantitative effect of plugging in the estimators of the coefficients, thereby enabling us to take several inference procedures for the driving-noise characteristics into account. We also present new classes and methods available in YUIMA for the simulation and the estimation of a Lévy SDE model. We highlight the flexibility of these new advances in YUIMA using simulated and real data.

Simulation of Noise Experiments Using a Stochastic Differential Equation Model

Simulation of Noise Experiments Using a Stochastic Differential Equation Model

A stochastic hybrid differential equation model: Analysis and application

Infectious diseases represent a real challenge to humanity and a true challenge for researchers to propose relevant solutions in order to reduce the number of infected individuals. Mathematical modeling of infectious diseases represents a better way to understand and control the spread of epidemics. In this work, we propose a stochastic SIQS epidemic model with a nonlinear incidence function and Markov switching. Firstly, we present our proposed stochastic model and its parameters. Secondly, we show the global existence and uniqueness of the positive solution. Then, we show a sufficient condition for the extinction of disease. Finally, we give numerical simulation to enrich our analytical results.

Data-Driven Continuous-Time Modeling of Power System Uncertainty Using a Multi-Dimensional Stochastic Differential Equation

This paper proposes a generalized continuous random process modeling framework for imposing any desired probability distribution, autocorrelation and power spectral density of measured data of power system uncertainty by developing a multi-dimension stochastic differential equation (SDE) together with a quadratic output equation. An algorithm for determining appropriate parameters of the proposed SDE model is presented. The effectiveness of derived random modeling scheme is verified by carrying out comparative simulations using actual time series data of wind speed, solar radiation, PJM dynamic regulation, and wind power. The proposed SDE model can be readily integrated in various electrical component models for reliable operation, stability assessment and control of renewable power systems.

Modelling and Simulating Dynamics Efficiency of Rural-Community Banks (RCBs) in Ghana

The present work seeks to develop a model for measuring efficiency of RCBs in Ghana by means of financial key performance indicators pairing macroeconomic indicators. A stochastic differential equation model for predicting the efficiency of RCBs in the future is developed and simulated using gaussian jumps to evaluate the models’ performance in unpredicted situations with four distinct phases of efficiency. Unique solution Exit multiple 4-dimensional stochastic differential equations and Macroeconomic model is proven to be the best-fitting model for the data with the lowest information criterion.