Stochastic Differential Equation Model Research Articles

Stability of a dengue epidemic model with independent stochastic perturbations

Dengue is a vector-borne disease mainly affecting tropical and subtropical regions. The transmission of dengue virus is affected by many stochastic factors. Hence, we incorporated stochastic perturbations into the framework of a deterministic compartmental model. For this stochastic differential equation model, we introduced an approximate analogue of the basic reproduction number to explore the existence of the almost sure exponential stability of disease free equilibrium. We proved the existence and uniqueness of a positive solution for the stochastic differential equation model, and derived stability conditions by using suitable Lyapunov functions. Numerical simulations for the stochastic differential equation model of dengue virus transmission were presented to illustrate our mathematical findings. The analysis for determining whether the disease free equilibrium is almost surely exponentially stable in stochastic settings can be applied to the analysis of many other vector-borne transmissions.

The effect of delay in viral production in within-host models during early infection

ABSTRACTDelay in viral production may have a significant impact on the early stages of infection. During the eclipse phase, the time from viral entry until active production of viral particles, no viruses are produced. This delay affects the probability that a viral infection becomes established and timing of the peak viral load. Deterministic and stochastic models are formulated with either multiple latent stages or a fixed delay for the eclipse phase. The deterministic model with multiple latent stages approaches in the limit the model with a fixed delay as the number of stages approaches infinity. The deterministic model framework is used to formulate continuous-time Markov chain and stochastic differential equation models. The probability of a minor infection with rapid viral clearance as opposed to a major full-blown infection with a high viral load is estimated from a branching process approximation of the Markov chain model and the results are confirmed through numerical simulations. In addition, parameter values for influenza A are used to numerically estimate the time to peak viral infection and peak viral load for the deterministic and stochastic models. Although the average length of the eclipse phase is the same in each of the models, as the number of latent stages increases, the numerical results show that the time to viral peak and the peak viral load increase.

Stochastic lattice-based modelling of malaria dynamics

BackgroundThe transmission of malaria is highly variable and depends on a range of climatic and anthropogenic factors. In addition, the dispersal of Anopheles mosquitoes is a key determinant that affects the persistence and dynamics of malaria. Simple, lumped-population models of malaria prevalence have been insufficient for predicting the complex responses of malaria to environmental changes.Methods and resultsA stochastic lattice-based model that couples a mosquito dispersal and a susceptible-exposed-infected-recovered epidemics model was developed for predicting the dynamics of malaria in heterogeneous environments. The Ithat{o} approximation of stochastic integrals with respect to Brownian motion was used to derive a model of stochastic differential equations. The results show that stochastic equations that capture uncertainties in the life cycle of mosquitoes and interactions among vectors, parasites, and hosts provide a mechanism for the disruptions of malaria. Finally, model simulations for a case study in the rural area of Kilifi county, Kenya are presented.ConclusionsA stochastic lattice-based integrated malaria model has been developed. The applicability of the model for capturing the climate-driven hydrologic factors and demographic variability on malaria transmission has been demonstrated.

Holistic water quality dynamics in rural artificial shallow water bodies

The water environment in diversely used rural artificial water bodies is generally varied with seasonal and diurnal changes, stability of which is significant in water resources management. Understanding of interaction among different water quality parameters that depend on their diurnal variations is the concern of this study. A rural homestead pond used for aquaculture in Bangladesh and a micro-dam used as an irrigation tank for paddy farming in Japan are chosen for contrasting the analysis of data. The observed data series of four typical water quality parameters exhibits the diurnal variations, which are primarily inferred to be driven by solar radiation and complex bio-chemical interactions. The study proposes a stochastic differential equation model to represent holistic water quality dynamics based on continuous measurements. The water quality parameters are considered as temporally continuous Markov process, where their individual effects on each parameter are evaluated in a specific time step and immediately reflected to the next observation. The model parameters are calibrated and the stability is discussed based on the eigenvalues of model parameters. The result mostly shows the mean reverting properties for dissolved oxygen and water temperature, while pH and oxidation reduction potential are rather depend on other parameters or external disturbance.

Perturbation-based inference for diffusion processes: Obtaining effective models from multiscale data

We consider the inference problem for parameters in stochastic differential equation (SDE) models from discrete time observations (e.g. experimental or simulation data). Specifically, we study the case where one does not have access to observations of the model itself, but only to a perturbed version that converges weakly to the solution of the model. Motivated by this perturbation argument, we study the convergence of estimation procedures from a numerical analysis point of view. More precisely, we introduce appropriate consistency, stability, and convergence concepts and study their connection. It turns out that standard statistical techniques, such as the maximum likelihood estimator, are not convergent methodologies in this setting, since they fail to be stable. Due to this shortcoming, we introduce and analyse a novel inference procedure for parameters in SDE models which turns out to be convergent. As such, the method is particularly suited for the estimation of parameters in effective (i.e. coarse-grained) models from observations of the corresponding multiscale process. We illustrate these theoretical findings via several numerical examples.

A Study of Electron Forbush Decreases with a 3D SDE Numerical Model

Abstract Because of the precise measurements of the cosmic ray electron flux by the PAMELA and AMS02, Electron Forbush decreases (Fds) have recently been observed for the first time. This serves as motivation to perform a numerical study of electron Forbush decreases with an advanced time-dependent, three-dimensional (3D) stochastic differential equation model, developed earlier to study proton Fds. The model includes a realistic interstellar electron spectrum reconstructed from Voyager observations, and diffusion and drift coefficients to reproduce the modulated spectrum observed by PAMELA in 2009. On the basis of this numerical model, electron Fd profiles for a range of rigidities are simulated. In addition, a systematic comparison between electron and proton Fds during different solar polarity epochs is performed. This approach gives insight into the rigidity dependence of the heliospheric diffusion coefficients and of drift effects over two magnetic field polarity cycles. We find that during an A > 0 epoch, the recovery time of a 1 GV proton Fd is remarkably shorter than the 1 GV electrons, whereas the electron Fd display a faster recovery during an A < 0 epoch. This model clear predicts a charge-sign dependent effect in the recovery time of Fds but less so for their magnitude.

Bayesian Objective Functions for Estimating Parameters in Nonlinear Stochastic Differential Equation Models with Limited Data

An Approximate Bayesian Expectation Maximization (ABEM) methodology and a Laplace Approximation Bayesian (LAB) methodology are developed for estimating parameters in nonlinear stochastic differential equation (SDE) models of chemical processes. These new methodologies are more powerful than previous maximum-likelihood methodologies for SDEs because they enable modelers to account for prior information about unknown parameters and initial conditions. The ABEM methodology is suitable for situations in which the modeler can assume that measurement noise variances are well-known, whereas LAB includes measurement noise variances among the parameters that require estimation. Both techniques estimate the magnitude of stochastic terms included in the differential equations to account for model mismatch and unknown process disturbances. The proposed ABEM and LAB methodologies are illustrated using a nonlinear continuous stirred tank reactor (CSTR) case study, with simulated data sets generated using a variety of s...

A spatially varying stochastic differential equation model for animal movement

Animal movement exhibits complex behavior which can be influenced by unobserved environmental conditions. We propose a model which allows for a spatially varying movement rate and spatially varying drift through a semiparametric potential surface and a separate motility surface. These surfaces are embedded in a stochastic differential equation framework which allows for complex animal movement patterns in space. The resulting model is used to analyze the spatially varying behavior of ants to provide insight into the spatial structure of ant movement in the nest.

A new approach for fitting growth models in random environment

Nonlinear growth models are widely employed in Animal sciences for describing growth of various species ofanimals. Nonlinear estimation procedures are generally employed for estimation of parameters. However, onelimitation of these models is that they are applicable only when the data are available at equidistant epochs. Another limitation is that the fluctuations in the system cannot be satisfactorily explained simply by adding an error term to the deterministic formulation. The purpose of this article is to bring to the notice of Animal scientists the new approach of Stochastic differential equation modelling, which is capable of incorporating both the above aspects. The methodology is discussed by considering Gompertz growth model. Relevant SAS codes for fitting the model are developed. Finally, the methodology is illustrated on secondary monthly pig weight data, collected at thepiggery farm of Indian Veterinary Research Institute, Izatnagar, Bareilly, India.

Non-Gaussian quasi-likelihood estimation of SDE driven by locally stable Lévy process

We address estimation of parametric coefficients of a pure-jump Lévy driven univariate stochastic differential equation (SDE) model, which is observed at high frequency over a fixed time period. It is known from the previous study (Masuda, 2013) that adopting the conventional Gaussian quasi-maximum likelihood estimator then leads to an inconsistent estimator. In this paper, under the assumption that the driving Lévy process is locally stable, we extend the Gaussian framework into a non-Gaussian counterpart, by introducing a novel quasi-likelihood function formally based on the small-time stable approximation of the unknown transition density. The resulting estimator turns out to be asymptotically mixed normally distributed without ergodicity and finite moments for a wide range of the driving pure-jump Lévy processes, showing much better theoretical performance compared with the Gaussian quasi-maximum likelihood estimator. Extensive simulations are carried out to show good estimation accuracy. The case of large-time asymptotics under ergodicity is briefly mentioned as well, where we can deduce an analogous asymptotic normality result.

Derivation of Continuum Models from An Agent-based Cancer Model: Optimization and Sensitivity Analysis.

Agent-based models provide a formidable tool for exploring complex and emergent behaviour of biological systems as well as accurate results but with the drawback of needing a lot of computational power and time for subsequent analysis. On the other hand, equation-based models can more easily be used for complex analysis in a much shorter timescale. This paper formulates an ordinary differential equations and stochastic differential equations model to capture the behaviour of an existing agent-based model of tumour cell reprogramming and applies it to optimization of possible treatment as well as dosage sensitivity analysis. For certain values of the parameter space a close match between the equation-based and agent-based models is achieved. The need for division of labour between the two approaches is explored.

Multi-Dimensional Software Tool for OSS Project Management Considering Cloud with Big Data

We discuss a method of software dependability assessment based on stochastic differential equation modeling in order to consider the interesting factors of the numbers of components, cloud applications and users. Then, we consider the determination of an optimum software maintenance time, minimizing the total expected software cost. In particular, we develop the three-dimensional AIR application for reliability and cost optimization analyses based on the proposed method. Moreover, the three-dimensional application based on NW.js, known as the latest technology, is developed in this paper. Furthermore, we show several numerical examples of the developed application software to evaluate the performance of software optimization and reliability assessment tools for the big data on cloud computing.

Surrogate modelling for the prediction of spatial fields based on simultaneous dimensionality reduction of high-dimensional input/output spaces.

Time-consuming numerical simulators for solving groundwater flow and dissolution models of physico-chemical processes in deep aquifers normally require some of the model inputs to be defined in high-dimensional spaces in order to return realistic results. Sometimes, the outputs of interest are spatial fields leading to high-dimensional output spaces. Although Gaussian process emulation has been satisfactorily used for computing faithful and inexpensive approximations of complex simulators, these have been mostly applied to problems defined in low-dimensional input spaces. In this paper, we propose a method for simultaneously reducing the dimensionality of very high-dimensional input and output spaces in Gaussian process emulators for stochastic partial differential equation models while retaining the qualitative features of the original models. This allows us to build a surrogate model for the prediction of spatial fields in such time-consuming simulators. We apply the methodology to a model of convection and dissolution processes occurring during carbon capture and storage.

A deterministic and stochastic model for the system dynamics of tumor–immune responses to chemotherapy

Modern medical studies show that chemotherapy can help most cancer patients, especially for those diagnosed early, to stabilize their disease conditions from months to years, which means the population of tumor cells remained nearly unchanged in quite a long time after fighting against immune system and drugs. In order to better understand the dynamics of tumor–immune responses under chemotherapy, deterministic and stochastic differential equation models are constructed to characterize the dynamical change of tumor cells and immune cells in this paper. The basic dynamical properties, such as boundedness, existence and stability of equilibrium points, are investigated in the deterministic model. Extended stochastic models include stochastic differential equations (SDEs) model and continuous-time Markov chain (CTMC) model, which accounts for the variability in cellular reproduction, growth and death, interspecific competitions, and immune response to chemotherapy. The CTMC model is harnessed to estimate the extinction probability of tumor cells. Numerical simulations are performed, which confirms the obtained theoretical results.

A stochastic differential equation model for the foraging behavior of fish schools

Constructing models of living organisms locating food sources has important implications for understanding animal behavior and for the development of distribution technologies. This paper presents a novel simple model of stochastic differential equations for the foraging behavior of fish schools in a space including obstacles. The model is studied numerically. Three configurations of space with various food locations are considered. In the first configuration, fish swim in free but limited space. All individuals can find food with large probability while keeping their school structure. In the second and third configurations, they move in limited space with one and two obstacles, respectively. Our results reveal that the probability of foraging success is highest in the first configuration, and smallest in the third one. Furthermore, when school size increases up to an optimal value, the probability of foraging success tends to increase. When it exceeds an optimal value, the probability tends to decrease. The results agree with experimental observations.

Extremely rare collapse and build-up of turbulence in stochastic models of transitional wall flows.

This paper presents a numerical and theoretical study of multistability in two stochastic models of transitional wall flows. An algorithm dedicated to the computation of rare events is adapted on these two stochastic models. The main focus is placed on a stochastic partial differential equation model proposed by Barkley. Three types of events are computed in a systematic and reproducible manner: (i) the collapse of isolated puffs and domains initially containing their steady turbulent fraction; (ii) the puff splitting; (iii) the build-up of turbulence from the laminar base flow under a noise perturbation of vanishing variance. For build-up events, an extreme realization of the vanishing variance noise pushes the state from the laminar base flow to the most probable germ of turbulence which in turn develops into a full blown puff. For collapse events, the Reynolds number and length ranges of the two regimes of collapse of laminar-turbulent pipes, independent collapse or global collapse of puffs, is determined. The mean first passage time before each event is then systematically computed as a function of the Reynolds number r and pipe length L in the laminar-turbulent coexistence range of Reynolds number. In the case of isolated puffs, the faster-than-linear growth with Reynolds number of the logarithm of mean first passage time T before collapse is separated in two. One finds that ln(T)=A_{p}r-B_{p}, with A_{p} and B_{p} positive. Moreover, A_{p} and B_{p} are affine in the spatial integral of turbulence intensity of the puff, with the same slope. In the case of pipes initially containing the steady turbulent fraction, the length L and Reynolds number r dependence of the mean first passage time T before collapse is also separated. The author finds that T≍exp[L(Ar-B)] with A and B positive. The length and Reynolds number dependence of T are then discussed in view of the large deviations theoretical approaches of the study of mean first passage times and multistability, where ln(T) in the limit of small variance noise is studied. Two points of view, local noise of small variance and large length, can be used to discuss the exponential dependence in L of T. In particular, it is shown how a T≍exp[L(A^{'}R-B^{'})] can be derived in a conceptual two degrees of freedom model of a transitional wall flow proposed by Dauchot and Manneville. This is done by identifying a quasipotential in low variance noise, large length limit. This pinpoints the physical effects controlling collapse and build-up trajectories and corresponding passage times with an emphasis on the saddle points between laminar and turbulent states. This analytical analysis also shows that these effects lead to the asymmetric probability density function of kinetic energy of turbulence.

Empirical evaluated SDE modelling for dimensionality-reduced systems and its predictability estimates

This paper develops and validates a method of empirical modelling for a dimensionality-reduced system of a nonlinear dynamical system based on the framework of the stochastic differential equation (SDE). Following the mathematical theorem corresponding to some inverse problem of the probability theory, we derive the empirically evaluating formulae for the drift vector and diffusion matrix. Focusing on a low-dimensional dynamical system of the Lorenz system, we empirically reconstruct an SDE that approximates the original time-series on the projected 2-dimensional plane. The distribution of the ensemble variance of solutions generated by the numerical SDE well agrees with that of the trajectories of the projected time-series, which indicates the ability of the SDE modelling to represent local predictability. Moreover, we also compare our SDE constructing method with the conventional Mori–Zwanzig projected operator method, which is used to derive a generalised Langevin equation for dimensionality-reduced systems, to assess the applicability of the obtained SDE model derived by the presented method.

Stochastic Differential Equation Model for Spontaneous Emission and Carrier Noise in Semiconductor Lasers

We present a new stochastic differential equation model for the spontaneous emission noise and carrier noise in semiconductor lasers. The correlations between these two types of noise have often been neglected in recent studies of the effects of the noise on the laser dynamics. However, the classic results of Henry show that the intensity noise and the carrier noise are strongly negatively correlated. Our model demonstrates how to properly account for these correlations since the corresponding diffusion coefficients agree exactly with those derived by Henry. We show that in fact in the correct model the spontaneous emission noise and the carrier noise are driven by the same Wiener processes. Furthermore, we demonstrate that the nonzero correlation time of the physical noise affects the mean dynamics of both the electric field amplitude and the carrier number. We show that these are systematic corrections that can be described by additional drift terms in the model.

A stochastic partial differential equation model for the pricing of mortgage-backed securities

We develop a dynamic structural model for the wealth of individual mortgagors in a mortgage pool. We model the process of default and prepayment and, by taking a limit as the pool size goes to infinity, derive a stochastic partial differential equation (SPDE) which can be used to describe the evolution of the loss process from the pool. We prove existence and uniqueness of solutions to this SPDE and show how our model is able to capture, in a flexible way, the prices of credit risky tranches of mortgage-backed securities under different market conditions.

A probabilistic model for the numerical solution of initial value problems

We study connections between ordinary differential equation (ODE) solvers and probabilistic regression methods in statistics. We provide a new view of probabilistic ODE solvers as active inference agents operating on stochastic differential equation models that estimate the unknown initial value problem (IVP) solution from approximate observations of the solution derivative, as provided by the ODE dynamics. Adding to this picture, we show that several multistep methods of Nordsieck form can be recasted as Kalman filtering on q-times integrated Wiener processes. Doing so provides a family of IVP solvers that return a Gaussian posterior measure, rather than a point estimate. We show that some such methods have low computational overhead, nontrivial convergence order, and that the posterior has a calibrated concentration rate. Additionally, we suggest a step size adaptation algorithm which completes the proposed method to a practically useful implementation, which we experimentally evaluate using a representative set of standard codes in the DETEST benchmark set.