System Of Stochastic Differential Equations Research Articles

Artificial intelligence for COVID-19 spread modeling

Abstract This paper presents classification and analysis of the mathematical models of the spread of COVID-19 in different groups of population such as family, school, office (3–100 people), town (100–5000 people), city, region (0.5–15 million people), country, continent, and the world. The classification covers major types of models (time-series, differential, imitation ones, neural networks models and their combinations). The time-series models are based on analysis of time series using filtration, regression and network methods. The differential models are those derived from systems of ordinary and stochastic differential equations as well as partial differential equations. The imitation models include cellular automata and agent-based models. The fourth group in the classification consists of combinations of nonlinear Markov chains and optimal control theory, derived by methods of the mean-field game theory. COVID-19 is a novel and complicated disease, and the parameters of most models are, as a rule, unknown and estimated by solving inverse problems. The paper contains an analysis of major algorithms of solving inverse problems: stochastic optimization, nature-inspired algorithms (genetic, differential evolution, particle swarm, etc.), assimilation methods, big-data analysis, and machine learning.

Complex global dynamics of conditionally stable slopes: effect of initial conditions

In the present paper, we investigate the effect of the initial conditions on the dynamics of the spring-block landslide model. The time evolution of the studied model, which is governed by a system of stochastic delay differential equations, is analyzed in the mean-field approximation, which qualitatively exhibits the same dynamics as the initial model. The results of the numerical analysis show that changing the initial conditions has different effects in different parts of the parameter space of the model. Namely, moving away from the fixed-point initial conditions has a stabilizing effect on the dynamics when the noise, the friction parameters a (higher values) and c as well as the spring stiffness k are taken into account. The stabilization manifests itself in a complete suppression of the unstable dynamics or a partial limitation of the effect of some friction parameters. On the other hand, the destabilizing effect of changing the initial conditions occurs for the lower values of the friction parameters a and for b. The main feature of destabilization is the complete suppression of the sliding regime or a larger parameter range with a transient oscillatory regime. Our approach underlines the importance of analyzing the influence of initial conditions on landslide dynamics.

Lower and upper bounds for the explosion times of a system of semilinear SPDEs

In this paper, we obtain lower and upper bounds for the blow-up times for a system of semilinear stochastic partial differential equations. Under suitable assumptions, lower and upper bounds of the explosive times are obtained by using explicit solutions of an associated system of random partial differential equations and a formula due to Yor. We provide lower and upper bounds for the probability of finite-time blow-up solution as well. The above obtained results are also extended for semilinear SPDEs forced by two-dimensional Brownian motions.

Homogenization of a Multivariate Diffusion with Semipermeable Interfaces

AbstractWe study the homogenization problem for a system of stochastic differential equations with local time terms that models a multivariate diffusion in the presence of semipermeable hyperplane interfaces with oblique penetration. We show that this system has a unique weak solution and determine its weak limit as the distances between the interfaces converge to zero. In the limit, the singular local times terms vanish and give rise to an additional regular interface-induced drift.

Exponential Euler method for stiff stochastic differential equations with additive fractional Brownian noise

We discuss a system of stochastic differential equations with a stiff linear term and additive noise driven by fractional Brownian motions (fBms) with Hurst parameter H > 1 2 , which arise e.g. from spatial approximations of stochastic partial differential equations. For their numerical approximation, we present an exponential Euler scheme and show that it converges in the strong sense with an exact rate close to the Hurst parameter H. Further, based on E. Buckwar et al. [The numerical stability of stochastic ordinary differential equations with additive noise, Stoch. Dyn. 11 (2011), pp. 265–281], we conclude the existence of a unique stationary solution of the exponential Euler scheme that is pathwise asymptotically stable.

Solar extreme ultraviolet variability as a proxy for nanoflare heating diagnostics

Aims. We aim to improve the existing techniques to probe the nanoflare hypothesis for the coronal heating problem. For this purpose, we propose using the solar extreme ultraviolet (EUV) emission variability registered with modern space-based imagers. Methods. We followed a novel model-based approach. As a starting point, we used the EBTEL 0d hydrodynamic model. We integrated the arising system of stochastic differential equations to calculate the covariance matrix for plasma parameters. We then employed a Taylor expansion technique to relate model parameters with observable EUV intensity variation statistics. Results. We found that in the high-frequency approximation, the variability of the EUV emission is defined by the dimensionless factor ϖ, which is inversely proportional to the frequency. We calculated the factor ϖ throughout the solar disk and found that it does not exceed 0.01, except for the finite number of compact regions. The distribution of ϖ follows the power law with an index of ≈ − 2.6. To validate our approach, we used it to probe the temperature of the coronal plasma. We show that the line-of-sight temperature distribution is close to homogeneous with a mode of ≈1.25 MK, which is in perfect agreement with the results of the spectroscopic diagnostics.

DETERMINISTIC AND STOCHASTIC MODEL FOR THE TRANSMISSION OF LASSA FEVER

Many models on the transmission dynamics of Lasser fever were based on purely deterministic approach. This approach does not put into cognizance randomness which is inherent in disease transmission resulting from differences in immunity levels, contact patterns, hygienic practices and mutation rates among so many other possibilities.  In this work, we attempt to demonstrate the impact of uncertainties in the mode of transmission of Lassa fever by subjecting the dynamics to some white noise modeled by the Brownian motion as a Wiener process. An existing deterministic model involving the Susceptible, Exposed, Infected and Recovered (SEIR) individuals was transformed into a stochastic differential equation model by applying the procedure proposed by Allen et al (2008). The resulting system of Stochastic Differential Equations (SDE) was solved numerically using the Milstein scheme for SDEs. An algorithm for the method was developed and implemented in Python programming language. Numerical simulations of the model was done using four sets of parameters;  , representing the natural birth rate, the natural death rate , the recovering rate from infected to recovered, transmission rate from exposed to infected ,transmission rate from susceptible to exposed  are carried out to investigate the transmission dynamic of Lassa fever. The results of the simulations indicate that randomness does affect transmission of Lassa fever. We therefore recommend that factors such as social behavior, hygienic practices, contact patters, mutation rate should be considered while formulating mathematics models of disease transmission.

Application of Non-Linear Evolution Stochastic Equations with Asymptotic Null Controllability Analysis

This paper investigated system of stochastic differential equations with prominence on disparities of drift parameters. These problems were solved analytical by adopting the Ito’s method of solution and three different investment solutions were obtained consequently. The necessary conditions were achieved which govern various drift parameters in assessing financial markets. Therefore, the impressions on each solution of investors in financial markets were analyzed graphically. Secondly, stock price data of Transco, LTD were analyzed which covariance matrix were considered and analysis were logically extended to stochastic vector differential equation where control measures were incorporated that would help in predicting different stock price processes, and the result obtained by exploring the properties of the fundamental matrix solution where asymptotic null controllability results were obtained by the singularity of the controllability matrix a function of the drift. Finally, the effects of the significant parameters of stochastic variables were successfully discussed.

Balanced implicit Patankar-Euler methods for positive solutions of stochastic differential equations of biological regulatory systems.

Stochastic differential equations (SDEs) are a powerful tool to model fluctuations and uncertainty in complex systems. Although numerical methods have been designed to simulate SDEs effectively, it is still problematic when numerical solutions may be negative, but application problems require positive simulations. To address this issue, we propose balanced implicit Patankar-Euler methods to ensure positive simulations of SDEs. Instead of considering the addition of balanced terms to explicit methods in existing balanced methods, we attempt the deletion of possible negative terms from the explicit methods to maintain positivity of numerical simulations. The designed balanced terms include negative-valued drift terms and potential negative diffusion terms. The proposed method successfully addresses the issue of divisions with very small denominators in our recently designed stochastic Patankar method. Stability analysis shows that the balanced implicit Patankar-Euler method has much better stability properties than our recently designed composite Patankar-Euler method. Four SDE systems are used to examine the effectiveness, accuracy, and convergence properties of balanced implicit Patankar-Euler methods. Numerical results suggest that the proposed balanced implicit Patankar-Euler method is an effective and efficient approach to ensure positive simulations when any appropriate stepsize is used in simulating SDEs of biological regulatory systems.

Nonlinear stochastic model for epidemic disease prediction by optimal filtering perspective

Understanding and predicting novel diseases have become very important owing to the huge global health burden. The organization and study of mathematical models are critical in predicting disease behavior of the disease. In this paper, a new stochastic Susceptible‐Infected‐Recovered‐Death (SIRD) model for spreading epidemic disease is investigated. First, the deterministic SIRD model is considered, and then, by allowing randomness in the recovery and death rates that are not deterministic, the system of nonlinear stochastic differential equations is derived. For the suggested model, the existence and uniqueness of a positive global solution are demonstrated. The parameter estimation is done with the conditional least square estimator for deterministic models and the maximum likelihood estimator for stochastic ones. After that, we investigate a nonadditive state‐space model for spreading epidemic disease by considering infected as the hidden process variable. The problem of the hidden process variable from noisy observations is filtered, predicted, and smoothed using a recursive Bayesian technique. For estimating the hidden number of infected variables, closed‐form solutions are obtained. Finally, numerical simulations with both simulated and real data are performed to demonstrate the efficiency and accuracy of the current work.

A stochastic lipid structured model for macrophage dynamics in atherosclerotic plaques.

We propose to model certain aspects of the dynamics of a macrophage that moves randomly in a one dimensional space in arterial wall tissue and grows by accumulating localized lipid particles, thus reducing its motility. This phenomenon has been observed in the context of atherosclerotic plaque formation. For this purpose, we use a system of stochastic differential equations satisfied by the position and diffusion coefficient of a Brownian particle whose diffusion coefficient is modified at each visit to the origin and with a dumping coefficient. The novelty of the model, with respect to Bénichou et al. (Phys Rev E 85(2):021137, 2012), Meunier et al. (Acta Appl Math 161:107-126, 2019), is to include offloading of lipids through the dumping term. We find explicit necessary and sufficient conditions for macrophage trapping in the locally enriched region.

A brief review on stability investigations of numerical methods for systems of stochastic differential equations

<abstract><p>A modest review on stability investigations of numerical methods for systems of Itô-interpreted stochastic differential equations (SDEs) driven by $ m $-dimensional Wiener processes $ W = (W^1, W^2, ..., W^m) $ is presented in $ \mathbb{R}^d $. Since the problem of relevance of 1D test equations for multidimensional numerical methods has not completely been solved so far, we suggest to use the Krein-Perron-Frobenius theory of positive operators on positive cones of $ \mathbb{R}^{d \times d} $, instead of classic stability functions with values in $ \mathbb{C}^1 $, which is only relevant for the very restricted case of "simultaneously diagonalizable" SDEs. Our main focus is put on the concept of asymptotic mean square and almost sure (a.s.) stability for systems with state-dependent noise (multiplicative case), and the concept of exact preservation of asymptotic probabilistic quantities for systems with state-independent noise (additive case). The asymptotic exactness of midpoint methods with any equidistant step size $ h $ is worked out in order to underline their superiority within the class of all drift-implicit, classic theta methods for multidimensional, bilinear systems of SDEs. Balanced implicit methods with appropriate weights can also provide a.s. exact, asymptotically stable numerical methods for pure diffusions. The review on numerical stability is based on "major breakthroughs" of research for systems of SDEs over the last 35 years, with emphasis on applicability to all dimensions $ d \ge 1 $.</p></abstract>

Stochastic Lotka–Volterra mutualism model with jumps

The existence and uniqueness of the global positive solution are proved for the system of stochastic differential equations describing a two-species Lotka–Volterra mutualism model disturbed by white noise, centered and noncentered Poisson noises. For the considered system, sufficient conditions of stochastic ultimate boundedness, stochastic permanence, nonpersistence and strong persistence in the mean are obtained.

A study of stochastic epidemic model driven by liouville fractional brownian motion coupled with seasonal air pollution

Air pollution can cause and provoke respiratory diseases. It is an important topic to the public, particularly in developing countries. Since there are many uncertain factors in the environment, stochastic differential equation model is a powerful tool to study the changes of air pollution and the transmission of infectious diseases. The removal of air pollutants as well as the transmission of diseases can be influenced by random perturbations with memories. In this research, we develop a mathematical model in the form of a system of stochastic differential equations driven by fractional Brownian motion of Liouville-type, coupled with seasonal air pollution, to study the dynamics of infectious respiratory disease spread. As a result , by using stochastic calculus techniques, we derive the equation for the level of air pollution.

Representing a second-order Ito equation as an equation with a given force structure

The problem of constructing equivalent equations with a given structure of forces by the given system of stochastic equations is considered. The equivalence of equations in the sense of almost surely is investigated. The paper determines the conditions under which a given system of second-order Ito stochastic differential equations is represented in the form of stochastic Lagrange equations with non-potential forces of a certain structure. Necessary and sufficient conditions for the representability of stochastic equations in the form of stochastic equations with non-potential forces admitting the Rayleigh function are obtained. The obtained results are illustrated by an example of motion of a symmetric satellite in a circular orbit, assuming a change in pitch under the action of gravitational and aerodynamic forces.

Novel intelligent predictive networks for analysis of chaos in stochastic differential SIS epidemic model with vaccination impact

In this paper, stochastic predictive computing networks are exploited to investigate the dynamics of the SIS with vaccination impact based epidemic model (SISV-EM) represented by nonlinear systems of stochastic differential equations (SDEs) by exploitation of artificial neural networks (ANNs) with the backpropagated Levenberg-Marquardt technique (BLMT) i.e., (ANNs-BLMT) to approximate the solution behavior. The stochastic nonlinear SISV-EM is governed with three classes: susceptible, infectious, and vaccinated populations. The referenced or target datasets for ANNs-BLMT are constructed by employing Euler-Maruyama (EM) scheme for solving stochastic differential systems in case of sufficiently various nonlinear SISV-EM scenarios by varying the percentage of vaccination for newly born, the coefficient of transmission, the natural mortality rates, the infectious rates of recovery, the rate at which vaccinated people lose their immunity, the rate of death caused by disease, the proportion of vaccinated against susceptible and the white noise in the environment. Based on arbitrary training, testing, and validation samples from the referenced dataset, the ANNs-BLMT provides an approximate solution for the stochastic nonlinear SISV-EM, with significant correlations to the referenced results. Exhaustive simulation-based results using error histograms, mean square errors, and regression analyses further demonstrate that the proposed ANNs-BLMT is efficient, consistent, and accurate for solving SISV-EM.

Stochastic stability of solutions for a fourth-order stochastic differential equation with constant delay

In this paper, we present sufficient conditions to ensure the stochastic asymptotic stability of the zero solution for a specific type of fourth-order stochastic differential equation (SDE) with constant delay. By reducing the fourth-order SDE to a system of first-order SDEs, we utilize a fourth-order quadratic function to derive an appropriate Lyapunov functional. This functional is then employed to establish standard criteria for the nonlinear functions present in the SDE. The stability result obtained in this study is novel and extends the existing findings on stability in fourth-order differential equations. Additionally, we provide an illustrative example to demonstrate the significance and accuracy of our main result.

Construction of analytical solutions to systems of two stochastic differential equations

Abstract A scheme for the stochastization of systems of ordinary differential equations (ODEs) based on Itô calculus is presented in this article. Using the presented techniques, a system of stochastic differential equations (SDEs) can be constructed in such a way that eliminating the stochastic component yields the original system of ODEs. One of the main benefits of this scheme is the ability to construct analytical solutions to SDEs with the use of special vector-valued functions, which significantly differs from the randomization approach, which can only be applied via numerical integration. Moreover, using the presented techniques, a system of ODEs and SDEs can be constructed from a given diffusion function, which governs the uncertainty of a particular process.

Assessment of Stochastic Numerical Schemes for Stochastic Differential Equations with “White Noise” Using Itô’s Integral

Stochastic Differential Equations (SDEs) model physical phenomena dominated by stochastic processes. They represent a method for studying the dynamic evolution of a physical phenomenon, like ordinary or partial differential equations, but with an additional term called “noise” that represents a perturbing factor that cannot be attached to a classical mathematical model. In this paper, we study weak and strong convergence for six numerical schemes applied to a multiplicative noise, an additive, and a system of SDEs. The Efficient Runge–Kutta (ERK) technique, however, comes out as the top performer, displaying the best convergence features in all circumstances, including in the difficult setting of multiplicative noise. This result highlights the importance of researching cutting-edge numerical techniques built especially for stochastic systems and we consider to be of good help to the MATLAB function code for the ERK method.

Application of a Stochastic Extension of the Analytical Design of Aggregated Regulators to a Multidimensional Biomedical Object

The results of the application of the methods of the synergetic control theory to a high-dimensional immunology object with uncertainty in its descriptions are reported. The control here is the therapy treated as a problem for constructing an optimal cure program. The control object is presented in continuous and discrete forms, i.e., mathematical models given by a system of ordinary differential equations with a bounded disturbance and a system of stochastic difference equations, respectively. Two algorithms for deriving robust regulators applicable to a 10-dimensional nonlinear multi-loop system with unstable limit states, which models an immune response to the hepatitis B infection, are obtained. Analytical control design for a continuous model relies on the method of nonlinear adaptation on the target manifold. The second algorithm represents a stochastic extension of the method of analytical design of aggregated discrete regulators minimizing the variance of the target macro variable. The numerical simulation of the developed control systems indicates the performance of the designed control algorithms. The results of this study can be used as a component part of the mathematical tools of expert systems and decision support systems.