Random Differential Equations Research Articles

Pathwise methods for the integration of a stochastic SVIR model

We propose an approach for the precise numerical integration of a stochastic SVIR model defined by a stochastic differential equation (SDE) with non‐globally Lipschitz continuous coefficients and multiplicative noise. This equation, based on a compartmental epidemic model, describes a continuous vaccination strategy with environmental noise effects. By means of an appropriate invertible continuous transformation, we link the solution to the stochastic SVIR model to the solution of an auxiliary random differential equation (RDE) that has an Ornstein–Uhlenbeck process as the only input parameter of the system. In this way, based on this explicit conjugacy between both equations, new pathwise numerical schemes are constructed for the SVIR model. In particular, we propose an exponential method that outperforms other integrators in the literature and is able to approximate, with high stability, meaningful probabilistic features of the continuous system, including its stationary distribution and ergodicity. A simulation study is presented to illustrate the practical performance of the introduced methods, and a comparative analysis with other integrators commonly used for the simulation of epidemiological models is performed.

On the solution of random linear difference equations with Laplace transform method

In this study, Laplace transformation, which is very important for solutions to initial value problems, is examined. To solve the initial value problem of a discrete‐time equation, Laplace implements the conversion method. Here, Laplace transformation is used to obtain an approach to the solutions of random difference equations formed by randomizing components of deterministic difference equations. For random behavior of linear difference equations under random effects, uniform, geometric, binomial, Poisson, and Bernouilli distributions are used, and approximate expected value, variance, standard deviation, and confidence interval of equations obtained by Laplace transformation are calculated. The results were obtained through the Maple package program.

Weak random periodic solutions of random dynamical systems

AbstractWe first introduce the concept of weak random periodic solutions of random dynamical systems. Then, we discuss the existence of such periodic solutions. Further, we introduce the definition of weak random periodic measures and study their relationship with weak random periodic solutions. In particular, we establish the existence of invariant measures of random dynamical systems by virtue of their weak random periodic solutions. We use concrete examples to illustrate the weak random periodic phenomena of dynamical systems induced by random and stochastic differential equations.

Statistical laws of a one-dimensional model of turbulent flows subject to an external random force

This paper provides mathematical and numerical analysis of a one-dimensional model of turbulent flow generating the anomalous cascade of the inviscid conserved quantity. The model is based on the generalized Constantin–Lax–Majda–DeGregorio (gCLMG) equation with viscous dissipation under a large-scale forcing. Suppose that the forcing function and the initial data are random variables defined on a certain probability space. Then, the equation is regarded as a random partial differential equation. We prove the global existence of a unique solution to the gCLMG equation, from which a stochastic process is defined. In addition, by approximating the solutions numerically by Galerkin approximation of random variables with generalized polynomial chaos, we confirm the existence of a steady distribution. We find that the steady distribution reproduces qualitatively the same cascades of the energy and the enstrophy spectra as those of a turbulent flow generated by randomly moving pulse (Matsumoto and Sakajo 2016 Phys. Rev. E 93 053101). We also investigate the structure functions, showing intermittency.

Finite-time control for USV path tracking under input saturation with random disturbances

In this paper, a finite-time path tracking control method for unmanned surface vessel (USVs) subject to random disturbances and input saturation is proposed. A class of noise may be non-white one is employed to describe wave disturbance more accurately for real sea conditions. The constraints are considered to describe the limited output force of USVs. A finite-time control strategy is introduced to ensure that the requirements of tracking the desired position and heading within finite time. The noise-to-state stability in probability (NSS-P) within finite time, under an asymptotic gain in probability (AG-P) of state signals, is proved via the theoretical framework of random differential equations (RDEs). Finally, simulation results are presented to show the effectiveness of the designed finite-time path tracking control scheme.

Functional delay random semilinear differential equations

Functional delay random semilinear differential equations

Estimation of Parameter Distributions for Reaction-Diffusion Equations with Competition using Aggregate Spatiotemporal Data.

Reaction-diffusion equations have been used to model a wide range of biological phenomenon related to population spread and proliferation from ecology to cancer. It is commonly assumed that individuals in a population have homogeneous diffusion and growth rates; however, this assumption can be inaccurate when the population is intrinsically divided into many distinct subpopulations that compete with each other. In previous work, the task of inferring the degree of phenotypic heterogeneity between subpopulations from total population density has been performed within a framework that combines parameter distribution estimation with reaction-diffusion models. Here, we extend this approach so that it is compatible with reaction-diffusion models that include competition between subpopulations. We use a reaction-diffusion model of glioblastoma multiforme, an aggressive type of brain cancer, to test our approach on simulated data that are similar to measurements that could be collected in practice. We use Prokhorov metric framework and convert the reaction-diffusion model to a random differential equation model to estimate joint distributions of diffusion and growth rates among heterogeneous subpopulations. We then compare the new random differential equation model performance against other partial differential equation models' performance. We find that the random differential equation is more capable at predicting the cell density compared to other models while being more time efficient. Finally, we use k-means clustering to predict the number of subpopulations based on the recovered distributions.

Modeling the biological growth with a random logistic differential equation

We modeled biological growth using a random differential equation (RDE), where the initial condition is a random variable, and the growth rate is a suitable stochastic process. These assumptions let us obtain a model that represents well the random growth process observed in nature, where only a few individuals of the population reach the maximal size of the species, and the growth curve for every individual behaves randomly. Since we assumed that the initial condition is a random variable, we assigned a priori density, and we performed Bayesian inference to update the initial condition’s density of the RDE. The Karhunen–Loeve expansion was then used to approximate the random coefficient of the RDE. Then, using the RDE’s approximations, we estimated the density f(p, t). Finally, we fitted this model to the biological growth of the giant electric ray (or Cortez electric ray) Narcine entemedor. Simulations of the solution of the random logistic equation were performed to construct a curve that describes the solutions’ mean for each time. As a result, we estimated confidence intervals for the mean growth that described reasonably well the observed data. We fit the proposed model with a training dataset, and the model is tested with a different dataset. The model selection is performed with the square of the errors.

About the Existence and Uniqueness Theorem of Fuzzy Random Ordinary Differential Equations

Ordinary differential equations that includes stochastic processes in their vector fields are called random ordinary differential equations, that are considered through this work as a Weiner process or also called Brownian motion. In this paper, fuzzy random ordinary differential equations are considered, in which the fuzziness appears in the initial conditions in terms of triangular fuzzy numbers. Such equations are crucial in the theory of random dynamical systems and/or modern control theory and therefore the existence of a unique solution of such equations is of great importance. The statement and the proof of the existence and uniqueness theorem of fuzzy random ordinary differential equations is the main objective of this paper, which is proved using Banach contraction mapping theorem

Assessing the efficacy of health countermeasures on arrival time of infectious diseases.

Public health measures to control the international spread of infectious diseases include strengthening quarantines and sealing borders. Although these measures are effective in delaying the importation of infectious diseases, they also have a significant economic impact by stopping the flow of people and goods. The arrival time of infectious diseases is often used to assess quarantine effectiveness. Although the arrival time is highly dependent on the number of infected cases in the endemic country, direct comparisons have not yet been made. Therefore, this study derives an explicit relationship between the number of infected cases and arrival time. Transmission behavior is stochastic, and deterministic models are not always realistic. In this study, random differential equations, which are differential equations with stochastic processes, were used to describe the dynamics of infection in an endemic country. Furthermore, the flow of travelers from the endemic country was described in terms of survival time, and the arrival time in each country was calculated. A scenario in which PCR kits were distributed between endemic and disease-free countries was also considered, and the impact of different distribution rates on arrival time was evaluated. The simulation results showed that increasing the distribution of PCR kits in the endemic country was more effective in delaying arrival times than using PCR kits in quarantine in disease-free countries. It was also found that increasing the proportion of identified infected persons in the endemic country, leading to isolation, was more important and effective in delaying arrival times than increasing the number of PCR tests.

Some Results and Analysis of Nonlocal Special Random Impulsive Fractional Differential Equations

Some Results and Analysis of Nonlocal Special Random Impulsive Fractional Differential Equations

Existence and uniqueness of solutions for random impulsive differential equation

In this paper, we study the existence and uniqueness of the mild solutions for random impulsive differential equations through fixed point technique. An example is provided to illustrate the theory.

Sensitivity analysis and random dynamics for a mathematical model of tuberculosis transmission

In this study, a recent mathematical model for the spread of tuberculosis disease will be analyzed under random conditions. The sensitivity of the basic reproduction number to the changes in the parameters will be analyzed using the forward normalized sensitivity index. Additionally, random disease transmission dynamics will be analyzed by using simulations of the random differential equation system. Random behavior of disease transmission such as expected compartment sizes and coefficients of variations will be investigated to comment on the randomness of the spread.

Recurrence and transience of random difference equations in the critical case

Étant donnés des vecteurs aléatoires i.i.d. (M1,Q1),(M2,Q2),… tels que M>0 et Q≥0 p.s., et P(Q=0)<1, nous étudions l’équation aux différences aléatoires Xn=MnXn−1+Qn, n=1,2,… dans le cas critique, lorsque la marche aléatoire avec incréments logM1, logM2,… est oscillante. Nous obtenons des conditions pour la récurrence nulle et la transience de la chaîne de Markov (Xn)n≥0, en utilisant notamment des techniques développées dans l’article lié (J. Appl. Probab. 54 (2017) 1089–1110), qui traite d’un autre cas présentant la dichotomie récurrence nulle/transience.

On One Approach to the Study of the Periodic Problem for Random Differential Equations

On One Approach to the Study of the Periodic Problem for Random Differential Equations

Accurate and Efficient Simulation of Very High-Dimensional Neural Mass Models with Distributed-Delay Connectome Tensors

This paper introduces methods and a novel toolbox that efficiently integrates high-dimensional Neural Mass Models (NMMs) specified by two essential components. The first is the set of nonlinear Random Differential Equations (RDEs) of the dynamics of each neural mass. The second is the highly sparse three-dimensional Connectome Tensor (CT) that encodes the strength of the connections and the delays of information transfer along the axons of each connection. To date, simplistic assumptions prevail about delays in the CT, often assumed to be Dirac-delta functions. In reality, delays are distributed due to heterogeneous conduction velocities of the axons connecting neural masses. These distributed-delay CTs are challenging to model. Our approach implements these models by leveraging several innovations. Semi-analytical integration of RDEs is done with the Local Linearization (LL) scheme for each neural mass, ensuring dynamical fidelity to the original continuous-time nonlinear dynamic. This semi-analytic LL integration is highly computationally-efficient. In addition, a tensor representation of the CT facilitates parallel computation. It also seamlessly allows modeling distributed delays CT with any level of complexity or realism. This ease of implementation includes models with distributed-delay CTs. Consequently, our algorithm scales linearly with the number of neural masses and the number of equations they are represented with, contrasting with more traditional methods that scale quadratically at best. To illustrate the toolbox's usefulness, we simulate a single Zetterberg-Jansen and Rit (ZJR) cortical column, a single thalmo-cortical unit, and a toy example comprising 1000 interconnected ZJR columns. These simulations demonstrate the consequences of modifying the CT, especially by introducing distributed delays. The examples illustrate the complexity of explaining EEG oscillations, e.g., split alpha peaks, since they only appear for distinct neural masses. We provide an open-source Script for the toolbox.

Robust a posteriori estimates for the stochastic Cahn-Hilliard equation

We derive a posteriori error estimates for a fully discrete finite element approximation of the stochastic Cahn-Hilliard equation. The a posteriori bound is obtained by a splitting of the equation into a linear stochastic partial differential equation and a nonlinear random partial differential equation. The resulting estimate is robust with respect to the interfacial width parameter and is computable since it involves the discrete principal eigenvalue of a linearized (stochastic) Cahn-Hilliard operator. Furthermore, the estimate is robust with respect to topological changes as well as the intensity of the stochastic noise. We provide numerical simulations to demonstrate the practicability of the proposed adaptive algorithm.

Constructing reliable approximations of the random fractional Hermite equation: solution, moments and density

We extend the study of the random Hermite second-order ordinary differential equation to the fractional setting. We first construct a random generalized power series that solves the equation in the mean square sense under mild hypotheses on the random inputs (coefficients and initial conditions). From this representation of the solution, which is a parametric stochastic process, reliable approximations of the mean and the variance are explicitly given. Then, we take advantage of the random variable transformation technique to go further and construct convergent approximations of the first probability density function of the solution. Finally, several numerically simulations are carried out to illustrate the broad applicability of our theoretical findings.

Quasimonotone random and stochastic functional differential equations with applications

Quasimonotone random and stochastic functional differential equations with applications

On Solutions Of Random Partial Differential Equations With Laplace Adomian Decomposition Method

In this study, random partial differential equations obtained by randomly choosing the coefficients or initial conditions of partial differential equations will be analyzed. With the help of Laplace Adomian Decomposition Method and Homotopy Analysis Method, approximate analytical solutions of random partial differential equations were obtained. Initial conditions and parameters are made into random variables with normal distribution and gamma distribution. Probability characteristics such as expected value, variance and confidence intervals of the obtained random partial differential equation are calculated. Obtained results will be plotted with the help of MATLAB (2013a), package program and random results will be interpreted.