Analytical and NumericalMethods to Study the MFPT and SR of a Stochastic Tumor-Immune Model

The Mean First-Passage Time (MFPT) and Stochastic Resonance (SR) of a stochastic tumor-immune model withnoise perturbation are discussed in this paper. Firstly, considering environmental perturbation, Gaussian whitenoise and Gaussian colored noise are introduced into a tumor growth model under immune surveillance. Asfollows, the long-time evolution of the tumor characterized by the Stationary Probability Density (SPD) and MFPTis obtained in theory on the basis of the Approximated Fokker-Planck Equation (AFPE). Herein the recurrenceof the tumor from the extinction state to the tumor-present state is more concerned in this paper. A moreefficient algorithmof Back-Propagation Neural Network (BPNN) is utilized in order to testify the correction of thetheoretical SPDandMFPT.With the existence of aweak signal, the functional relationship between Signal-to-NoiseRatio (SNR), noise intensities and correlation time is also studied. Numerical results show that both multiplicativeGaussian colored noise and additive Gaussian white noise can promote the extinction of the tumors, and themultiplicative Gaussian colored noise can lead to the resonance-like peak on MFPT curves, while the increasingintensity of the additiveGaussian white noise results in theminimum of MFPT. In addition, the correlation timesare negatively correlated with MFPT. As for the SNR, we find the intensities of both the Gaussian white noise andthe Gaussian colored noise, as well as their correlation intensity can induce SR. Especially, SNR is monotonouslyincreased in the case ofGaussian white noisewith the change of the correlation time.At last, the optimal parametersin BPNN structure are analyzed for MFPT from three aspects: the penalty factors, the number of neural networklayers and the number of nodes in each layer.

A modified stochastic model for LS+AR hybrid method and its application in polar motion short-term prediction

Short-term(up to 30 days)predictions of Earth Rotation Parameters(ERPs)such as Polar Motion(PM:PMX and PMY)play an essential role in real-time applications related to high-precision reference frame conversion.Currently,least squares(LS)+auto-regressive(AR)hybrid method is one of the main techniques of PM prediction.Besides,the weighted LS+AR hybrid method performs well for PM short-term prediction.However,the corresponding covariance information of LS fitting residuals deserves further exploration in the AR model.In this study,we have derived a modified stochastic model for the LS+AR hybrid method,namely the weighted LS+weighted AR hybrid method.By using the PM data products of IERS EOP 14 C04,the numerical results indicate that for PM short-term forecasting,the proposed weighted LS+weighted AR hybrid method shows an advantage over both the LS+AR hybrid method and the weighted LS+AR hybrid method.Compared to the mean absolute errors(MAEs)of PMX/PMY sho rt-term prediction of the LS+AR hybrid method and the weighted LS+AR hybrid method,the weighted LS+weighted AR hybrid method shows average improvements of 6.61%/12.08%and 0.24%/11.65%,respectively.Besides,for the slopes of the linear regression lines fitted to the errors of each method,the growth of the prediction error of the proposed method is slower than that of the other two methods.

A Stochastic Model to Assess the Epidemiological Impact of Vaccine Booster Doses on COVID-19 and Viral Hepatitis B Co-Dynamics with Real Data

A patient co-infected with COVID-19 and viral hepatitis B can be atmore risk of severe complications than the one infected with a single infection.This study develops a comprehensive stochastic model to assess the epidemiological impact of vaccine booster doses on the co-dynamics of viral hepatitis B and COVID-19.The model is fitted to real COVID-19 data from Pakistan.The proposed model incorporates logistic growth and saturated incidence functions.Rigorous analyses using the tools of stochastic calculus,are performed to study appropriate conditions for the existence of unique global solutions,stationary distribution in the sense of ergodicity and disease extinction.The stochastic threshold estimated from the data fitting is given by:R_(0)^(S)=3.0651.Numerical assessments are implemented to illustrate the impact of double-dose vaccination and saturated incidence functions on the dynamics of both diseases.The effects of stochastic white noise intensities are also highlighted.

Mixed D-vine copula-based conditional quantile model for stochastic monthly streamflow simulation

Copula functions have been widely used in stochastic simulation and prediction of streamflow.However,existing models are usually limited to single two-dimensional or three-dimensional copulas with the same bivariate block for all months.To address this limitation,this study developed a mixed D-vine copula-based conditional quantile model that can capture temporal correlations.This model can generate streamflow by selecting different historical streamflow variables as the conditions for different months and by exploiting the conditional quantile functions of streamflows in different months with mixed D-vine copulas.The up-to-down sequential method,which couples the maximum weight approach with the Akaike information criteria and the maximum likelihood approach,was used to determine the structures of multivariate Dvine copulas.The developed model was used in a case study to synthesize the monthly streamflow at the Tangnaihai hydrological station,the inflow control station of the Longyangxia Reservoir in the Yellow River Basin.The results showed that the developed model outperformed the commonly used bivariate copula model in terms of the performance in simulating the seasonality and interannual variability of streamflow.This model provides useful information for water-related natural hazard risk assessment and integrated water resources management and utilization.

Stochastic Analysis and Modeling of Velocity Observations in Turbulent Flows

Highly turbulent water flows,often encountered near human constructions like bridge piers,spillways,and weirs,display intricate dynamics characterized by the formation of eddies and vortices.These formations,varying in sizes and lifespans,significantly influence the distribution of fluid velocities within the flow.Subsequently,the rapid velocity fluctuations in highly turbulent flows lead to elevated shear and normal stress levels.For this reason,to meticulously study these dynamics,more often than not,physical modeling is employed for studying the impact of turbulent flows on the stability and longevity of nearby structures.Despite the effectiveness of physical modeling,various monitoring challenges arise,including flow disruption,the necessity for concurrent gauging at multiple locations,and the duration of measurements.Addressing these challenges,image velocimetry emerges as an ideal method in fluid mechanics,particularly for studying turbulent flows.To account for measurement duration,a probabilistic approach utilizing a probability density function(PDF)is suggested to mitigate uncertainty in estimated average and maximum values.However,it becomes evident that deriving the PDF is not straightforward for all turbulence-induced stresses.In response,this study proposes a novel approach by combining image velocimetry with a stochastic model to provide a generic yet accurate description of flow dynamics in such applications.This integration enables an approach based on the probability of failure,facilitating a more comprehensive analysis of turbulent flows.Such an approach is essential for estimating both short-and long-term stresses on hydraulic constructions under assessment.

Asymptotic Analysis of a Stochastic Model of Mosquito-Borne Disease with the Use of Insecticides and Bet Nets

Ross’ epidemic model describing the transmission of malaria uses two classes of infection, one for humans and one for mosquitoes. This paper presents a stochastic extension of a deterministic vector-borne epidemic model based only on the class of human infectious. The consistency of the model is established by proving that the stochastic delay differential equation describing the model has a unique positive global solution. The extinction of the disease is studied through the analysis of the stability of the disease-free equilibrium state and the persistence of the model. Finally, we introduce some numerical simulations to illustrate the obtained results.

Stochastic Models to Mitigate Sparse Sensor Attacks in Continuous-Time Non-Linear Cyber-Physical Systems

Cyber-Physical Systems are very vulnerable to sparse sensor attacks.But current protection mechanisms employ linear and deterministic models which cannot detect attacks precisely.Therefore,in this paper,we propose a new non-linear generalized model to describe Cyber-Physical Systems.This model includes unknown multivariable discrete and continuous-time functions and different multiplicative noises to represent the evolution of physical processes and randomeffects in the physical and computationalworlds.Besides,the digitalization stage in hardware devices is represented too.Attackers and most critical sparse sensor attacks are described through a stochastic process.The reconstruction and protectionmechanisms are based on aweighted stochasticmodel.Error probability in data samples is estimated through different indicators commonly employed in non-linear dynamics(such as the Fourier transform,first-return maps,or the probability density function).A decision algorithm calculates the final reconstructed value considering the previous error probability.An experimental validation based on simulation tools and real deployments is also carried out.Both,the new technology performance and scalability are studied.Results prove that the proposed solution protects Cyber-Physical Systems against up to 92%of attacks and perturbations,with a computational delay below 2.5 s.The proposed model shows a linear complexity,as recursive or iterative structures are not employed,just algebraic and probabilistic functions.In conclusion,the new model and reconstructionmechanism can protect successfully Cyber-Physical Systems against sparse sensor attacks,even in dense or pervasive deployments and scenarios.

Dynamics of Advantageous Mutant Spread in Spatial Death-Birth and Birth-Death Moran Models

The spread of an advantageous mutation through a population is of fundamental interest in population genetics. While the classical Moran model is formulated for a well-mixed population, it has long been recognized that in real-world applications, the population usually has an explicit spatial structure which can significantly influence the dynamics. In the context of cancer initiation in epithelial tissue, several recent works have analyzed the dynamics of advantageous mutant spread on integer lattices, using the biased voter model from particle systems theory. In this spatial version of the Moran model, individuals first reproduce according to their fitness and then replace a neighboring individual. From a biological standpoint, the opposite dynamics, where individuals first die and are then replaced by a neighboring individual according to its fitness, are equally relevant. Here, we investigate this death-birth analogue of the biased voter model. We construct the process mathematically, derive the associated dual process, establish bounds on the survival probability of a single mutant, and prove that the process has an asymptotic shape. We also briefly discuss alternative birth-death and death-birth dynamics, depending on how the mutant fitness advantage affects the dynamics. We show that birth-death and death-birth formulations of the biased voter model are equivalent when fitness affects the former event of each update of the model, whereas the birth-death model is fundamentally different from the death-birth model when fitness affects the latter event.

High-order harmonic generation of ZnO crystals in chirped and static electric fields

High harmonic generation in ZnO crystals under chirped single-color field and static electric field are investigated by solving the semiconductor Bloch equation(SBE). It is found that when the chirp pulse is introduced, the interference structure becomes obvious while the harmonic cutoff is not extended. Furthermore, the harmonic efficiency is improved when the static electric field is included. These phenomena are demonstrated by the classical recollision model in real space affected by the waveform of laser field and inversion symmetry. Specifically, the electron motion in k-space shows that the change of waveform and the destruction of the symmetry of the laser field lead to the incomplete X-structure of the crystal-momentum-resolved(k-resolved) inter-band harmonic spectrum. Furthermore, a pre-acceleration process in the solid four-step model is confirmed.

Research and application of composite stochastic resonance in enhancement detection

Aiming at the problem that the intermediate potential part of the traditional bistable stochastic resonance model cannot be adjusted independently, a new composite stochastic resonance(NCSR) model is proposed by combining the Woods–Saxon(WS) model and the improved piecewise bistable model. The model retains the characteristics of the independent parameters of WS model and the improved piecewise model has no output saturation, all the parameters in the new model have no coupling characteristics. Under α stable noise environment, the new model is used to detect periodic signal and aperiodic signal, the detection results indicate that the new model has higher noise utilization and better detection effect.Finally, the new model is applied to image denoising, the results showed that under the same conditions, the output peak signal-to-noise ratio(PSNR) and the correlation number of NCSR method is higher than that of other commonly used linear denoising methods and improved piecewise SR methods, the effectiveness of the new model is verified.

A stochastic two-dimensional intelligent driver car-following model with vehicular dynamics

The law of vehicle movement has long been studied under the umbrella of microscopic traffic flow models,especially the car-following(CF)models.These models of the movement of vehicles serve as the backbone of traffic flow analysis,simulation,autonomous vehicle development,etc.Two-dimensional(2D)vehicular movement is basically stochastic and is the result of interactions between a driver's behavior and a vehicle's characteristics.Current microscopic models either neglect 2D noise,or overlook vehicle dynamics.The modeling capabilities,thus,are limited,so that stochastic lateral movement cannot be reproduced.The present research extends an intelligent driver model(IDM)by explicitly considering both vehicle dynamics and 2D noises to formulate a stochastic 2D IDM model,with vehicle dynamics based on the stochastic differential equation(SDE)theory.Control inputs from the vehicle include the steer rate and longitudinal acceleration,both of which are developed based on an idea from a traditional intelligent driver model.The stochastic stability condition is analyzed on the basis of Lyapunov theory.Numerical analysis is used to assess the two cases:(i)when a vehicle accelerates from a standstill and(ii)when a platoon of vehicles follow a leader with a stop-and-go speed profile,the formation of congestion and subsequent dispersion are simulated.The results show that the model can reproduce the stochastic 2D trajectories of the vehicle and the marginal distribution of lateral movement.The proposed model can be used in both a simulation platform and a behavioral analysis of a human driver in traffic flow.

Exit problem of stochastic SIR model with limited medical resource

Nonlinearity and randomness are both the essential attributes for the real world,and the case is the same for the models of infectious diseases,for which the deterministic models can not give a complete picture of the evolution.However,although there has been a lot of work on stochastic epidemic models,most of them focus mainly on qualitative properties,which makes us somewhat ignore the original meaning of the parameter value.In this paper we extend the classic susceptible-infectious-removed(SIR)epidemic model by adding a white noise excitation and then we utilize the large deviation theory to quantitatively study the long-term coexistence exit problem with epidemic.Finally,in order to extend the meaning of parameters in the corresponding deterministic system,we tentatively introduce two new thresholds which then prove rational.

High-order targeted essentially non-oscillatory scheme for two-fluid plasma model

The weakly ionized plasma flows in aerospace are commonly simulated by the single-fluid model,which cannot describe certain nonequilibrium phenomena by finite collisions of particles,decreasing the fidelity of the solution.Based on an alternative formulation of the targeted essentially non-oscillatory(TENO)scheme,a novel high-order numerical scheme is proposed to simulate the two-fluid plasmas problems.The numerical flux is constructed by the TENO interpolation of the solution and its derivatives,instead of being reconstructed from the physical flux.The present scheme is used to solve the two sets of Euler equations coupled with Maxwell's equations.The numerical methods are verified by several classical plasma problems.The results show that compared with the original TENO scheme,the present scheme can suppress the non-physical oscillations and reduce the numerical dissipation.

The Stochastic Asymptotic Stability Analysis in Two Species Lotka-Volterra Model

The asymptotic stability of two species stochastic Lotka-Volterra model is explored in this paper. Firstly, the Lotka-Volterra model with random parameter is built and reduced into the equivalent deterministic system by orthogonal polynomial approximation. Then, the linear stability theory and Routh-Hurwitz criterion for nonlinear deterministic systems are applied to the equivalent one. At last, at the aid of Lyapunov second method, we obtain that as the random intensity or statistical parameter of random variable is changed, the stability about stochastic Lotka-Volterra model is different from the deterministic system.

Calibration of quantitative rescattering model for simulating vortex high-order harmonic generation driven by Laguerre–Gaussian beam with nonzero orbital angular momentum

We calibrate the macroscopic vortex high-order harmonic generation(HHG)obtained by the quantitative rescattering(QRS)model to compute single-atom induced dipoles against that by solving the time-dependent Schr?dinger equation(TDSE).We show that the QRS perfectly agrees with the TDSE under the favorable phase-matching condition,and the QRS can accurately predict the main features in the spatial profiles of vortex HHG if the phase-matching condition is not good.We uncover that harmonic emissions from short and long trajectories are adjusted by the phase-matching condition through the time-frequency analysis and the QRS can simulate the vortex HHG accurately only when the interference between two trajectories is absent.This work confirms that it is an efficient way to employ the QRS model in the single-atom response for precisely simulating the macroscopic vortex HHG.

Deterministic and Stochastic Analysis of a New Rumor Propagation Model with Nonlinear Propagation Rate in Social Network

This paper presents a study on a new rumor propagation model with nonlinear propagation rate and secondary propagation rate. We divide the total population into three groups, the ignorant, the spreader and the aware. The nonlinear incidence rate describes the psychological impact of certain serious rumors on social groups when the number of individuals spreading rumors becomes larger. The main contributions of this work are the development of a new rumor propagation model and some results of deterministic and stochastic analysis of the rumor propagation model. The results show the influence of nonlinear propagation rate and stochastic fluctuation on the dynamic behavior of the rumor propagation model by using Lyapunov function method and stochastic related knowledge. Numerical examples and simulation results are given to illustrate the results obtained.

Optimal Quota-Share and Excess-of-Loss Reinsurance and Investment with Heston’s Stochastic Volatility Model

An optimal quota-share and excess-of-loss reinsurance and investment problem is studied for an insurer who is allowed to invest in a risk-free asset and a risky asset.Especially the price process of the risky asset is governed by Heston's stochastic volatility(SV)model.With the objective of maximizing the expected index utility of the terminal wealth of the insurance company,by using the classical tools of stochastic optimal control,the explicit expressions for optimal strategies and optimal value functions are derived.An interesting conclusion is found that it is better to buy one reinsurance than two under the assumption of this paper.Moreover,some numerical simulations and sensitivity analysis are provided.

A CLASS OF STATIONARY MODELS OF SINGULAR STOCHASTIC CONTROL

A class of stationary models of singular stochastic control has been studied, in which the state is extended to solution of a class of S.D.E. from Wiener process. The existence of optimal control has been proved in all cases under some weaker conditions, and the structure of optimal control may be characterized.

Sampled-data Observer Design for a Class of Stochastic Nonlinear Systems Based on the Approximate Discrete-time Models

In this paper,we studied the approximate sampleddata observer design for a class of stochastic nonlinear systems.Euler-Maruyama approximation was investigated in this paper because it is the basis of other higher precision numerical methods,and it preserves important structures of the nonlinear systems.Also,the form of Euler-Maruyama model is simple and easy to be calculated.The results provide a reference for sampled-data observer design method for such stochastic nonlinear systems,and may be useful to many practical control applications,such as tracking control in mechanical systems.And the effectiveness of the approach is demonstrated by a simulation example.

Simplifying Stochastic Mathematical Models of Biochemical Systems

Stochastic modeling of biochemical reactions taking place at the cellular level has become the subject of intense research in recent years. Molecular interactions in a single cell exhibit random fluctuations. These fluctuations may be significant when small populations of some reacting species are present and then a stochastic description of the cellular dynamics is required. Often, the biochemically reacting systems encountered in applications consist of many species interacting through many reaction channels. Also, the dynamics of such systems is typically non-linear and presents multiple time-scales. Consequently, the stochastic mathematical models of biochemical systems can be quite complex and their analysis challenging. In this paper, we present a method to reduce a stochastic continuous model of well-stirred biochemical systems, the Chemical Langevin Equation, while preserving the overall behavior of the system. Several tests of our method on models of practical interest gave excellent results.