Structure-Preserving Dynamics of Stochastic Epidemic Model with the Saturated Incidence Rate

The structure-preserving features of the nonlinear stochastic models are positivity, dynamical consistency and boundedness. These features have a significant role in different fields of computational biology and many more. Unfortunately, the existing stochastic approaches in literature do not restore aforesaid structure-preserving features, particularly for the stochastic models. Therefore, these gaps should be occupied up in literature, by constructing the structure-preserving features preserving numerical approach. This writing aims to describe the structure-preserving dynamics of the stochastic model. We have analysed the effect of reproduction number in stochastic modelling the same as described in the literature for deterministic modelling. The usual explicit stochastic numerical approaches are time-dependent. We have developed the implicitly driven explicit approach for the stochastic epidemic model. We have proved that the newly developed approach is preserving the structural, dynamical properties as positivity, boundedness and dynamical consistency. Finally, convergence analysis of a newly developed approach and graphically illustration is also presented.     

Stochastic Numerical Analysis for Impact of Heavy Alcohol Consumption on Transmission Dynamics of Gonorrhoea Epidemic

This paper aims to perform a comparison of deterministic and stochastic models. The stochastic modelling is a more realistic way to study the dynamics of gonorrhoea infection as compared to its corresponding deterministic model. Also, the deterministic solution is itself mean of the stochastic solution of the model. For numerical analysis, first, we developed some explicit stochastic methods, but unfortunately, they do not remain consistent in certain situations. Then we proposed an implicitly driven explicit method for stochastic heavy alcohol epidemic model. The proposed method is independent of the choice of parameters and behaves well in all scenarios. So, some theorems and simulations are presented in support of the article.     

A new approach to solving stochastic programming problems with recourse

A new numerical approach to the solution of two-stage stochastic linear programming problems is described and evaluated. The approach avoids the solution of the first-stage problem and uses the underlying deterministic problem to generate a sequence of values of the first-stage variables which lead to successive improvements of the objective function towards the optimal policy. The model is evaluated using an example in which randomness is described by two correlated factors. The dynamics of these factors are described by stochastic processes simulated using lattice techniques. In this way, discrete distributions of the random parameters are assembled. The solutions obtained with the new iterative procedure are compared with solutions obtained with a deterministic equivalent linear programming problem. It is concluded that they are almost identical. However, the computational effort required for the new approach is negligible compared with that needed for the deterministic equivalent problem.     

迟滞的磁流变阻尼器的随机最优控制力

用Bouc-Wen迟滞模型描述磁流变阻尼器的动力学特性,分离阻尼器控制力的半主动部分和被动部分,被动部分结合到受控系统.先将该系统变换成等价的非迟滞的非线性随机控制系统,再运用随机平均法导出关于能量的It随机微分方程.根据随机动态规划原理,建立控制总能量的动态规划方程,并由此确定非clip的最优控制力.最后通过数值结果表明该控制力的有效性.     

随机最优控制下矩形薄板受面内随机参数激励的首次穿越研究

根据建立了四边简支受控矩形薄板受面内高斯白噪声激励的振动模型,并用Galerkin变分法将其化简为二自由度常微分非线性动力学方程。又利用拟不可积Hamilton系统平均理论将方程等价为一个一维的Ito随机过程,随后结合随机动态规划方法,得到了使系统可靠性最大的随机最优控制策略。最后建立了受控系统的条件可靠性函数所满足的Backward Kolmogorov(BK)方程,根据初始条件和边界条件得出数值结果。数值结果表明,随机最优控制对系统的可靠性提升有明显作用     

Stochastic projection schemes for deterministic linear elliptic partial differential equations on random domains

We present stochastic projection schemes for approximating the solution of a class of deterministic linear elliptic partial differential equations defined on random domains. The key idea is to carry out spatial discretization using a combination of finite element methods and stochastic mesh representations. We prove a result to establish the conditions that the input uncertainty model must satisfy to ensure the validity of the stochastic mesh representation and hence the well posedness of the problem. Finite element spatial discretization of the governing equations using a stochastic mesh representation results in a linear random algebraic system of equations in a polynomial chaos basis whose coefficients of expansion can be non‐intrusively computed either at the element or the global level. The resulting randomly parametrized algebraic equations are solved using stochastic projection schemes to approximate the response statistics. The proposed approach is demonstrated for modeling diffusion in a square domain with a rough wall and heat transfer analysis of a three‐dimensional gas turbine blade model with uncertainty in the cooling core geometry. The numerical results are compared against Monte–Carlo simulations, and it is shown that the proposed approach provides high‐quality approximations for the first two statistical moments at modest computational effort. Copyright © 2010 John Wiley & Sons, Ltd.     

Competitive numerical analysis for stochastic HIV/AIDS epidemic model in a two‐sex population

This study is an attempt to explain a reliable numerical analysis of a stochastic HIV/AIDS model in a two‐sex population considering counselling and antiretroviral therapy (ART). The authors are comparing the solutions of the stochastic and deterministic HIV/AIDS epidemic model. Here, an endeavour has been made to explain the stochastic HIV/AIDS epidemic model is comparatively more pragmatic in contrast with the deterministic HIV/AIDS epidemic model. The effect of threshold number H ^* holds on the stochastic HIV/AIDS epidemic model. If H ^*  < 1 then condition helps us to control disease in a two‐sex human population while H ^*  > 1 explains the persistence of disease in the two‐sex human population. Lamentably, numerical methods such as Euler–Maruyama, stochastic Euler, and stochastic Runge–Kutta do not work for large time step sizes. The recommended structure preserving framework of the stochastic non‐standard finite difference (SNSFD) scheme conserve all vital characteristics such as positivity, boundedness, and dynamical consistency defined by Mickens. The effectiveness of counselling and ART may control HIV/AIDS in a two‐sex population.Inspec keywords: diseases, stochastic processes, epidemics, patient treatment, finite difference methodsOther keywords: two‐sex human population, antiretroviral therapy, competitive numerical analysis, stochastic HIV‐AIDS epidemic model, structure preserving framework, stochastic nonstandard finite difference scheme, SNSFD scheme, deterministic HIV‐AIDS epidemic model     

随机结构弹性屈曲的递推求解方法

采用随机收敛的非正交的多项式展式表示未知的随机屈曲特征值和屈曲模态,利用摄动技巧,建立了随机结构弹性屈曲的递推求解方法。算例表明,和基于泰勒展开的摄动随机有限元方法相比,方法的结果能在较宽的随机涨落范围内更好地逼近蒙特卡洛模拟结果,即使只采用前四阶非正交多项式展式,逼近的结果仍然较好。     

Dynamic response and reliability analysis of structures with uncertain parameters

An original approach for dynamic response and reliability analysis of stochastic structures is proposed. The probability density evolution equation is established which implies that incremental rate of the probability density function is related to the structural response velocity. Therefore, the response analysis of stochastic structures becomes an initial‐value partial differential equation problem. For the dynamic reliability problem, the solution can be derived through solving the probability density evolution equation with an initial value condition and an absorbing boundary condition corresponding to specified failure criterion. The numerical algorithm for the proposed method is suggested by combining the precise time integration method and the finite difference method with TVD scheme. To verify and validate the proposed method, a SDOF system and an 8‐storey frame with random parameters are investigated in detail. In the SDOF system, the response obtained by the proposed method is compared with the counterparts by the exact solution. The responses and the reliabilities of a frame with random stiffness, subject to deterministic excitation or random excitation, are evaluated by the proposed method as well. The mean, the standard deviation and the reliabilities are compared, respectively, with the Monte Carlo simulation. The numerical examples verify that the proposed method is of high accuracy and efficiency. Moreover, it is found that the probability transition of structural responses is like water flowing in a river with many whirlpools, showing complexity of probability transition process of the stochastic dynamic responses. Copyright © 2004 John Wiley & Sons, Ltd.     

弹性限位浮筏系统的随机振动建模与算法

对具有弹性限位的浮筏隔振系统,采用单边约束动力学方程,引入接触力,建立系统的确定性方程。再设系统的激励受随机因素的影响,导出系统的随机离散动力学模型,给出离散的计算格式。由于浮筏系统中限位器的个数很多,系统受冲击后,限位器接触状态的可能性非常多,因此对于限位器的接触状态的判别采用了人工神经网络,使得接触计算得以实现。最后通过实例分析对系统的随机动力学特性进行分析,给出噪声对系统振动周期性的影响,以及限位器间隙与位移方差之间的关系。     

A family of weak stochastic Newmark methods for simplified and efficient Monte Carlo simulations of oscillators

Engineers are often more concerned with the computation of statistical moments (or mathematical expectations) of the response of stochastically driven dynamical systems than with the determination of path‐wise response histories. With this in perspective, weak stochastic solutions of dynamical systems, modelled as n degrees‐of‐freedom (DOF) mechanical oscillators and driven by additive and/or multiplicative white noise (or, filtered white noise) processes, are considered in this study. While weak stochastic solutions are simpler and quicker to compute than strong (sample path‐wise) solutions, it must be emphasized that sample realizations of weak solutions have no path‐wise similarity with strong solutions. However, the statistical moment of any continuous and sufficiently differentiable deterministic function of the weak stochastic response is ‘close’ to that of the true response (if it exists) within a certain order of a given time step size. Computation of weak response therefore assumes great significance in the context of simulations of stochastically driven dynamical systems (oscillators) of engineering interest. To efficiently generate such weak responses, a novel class of weak stochastic Newmark methods (WSNMs), based on implicit Ito–Taylor expansions of displacement and velocity fields, is proposed. The resulting multiple stochastic integrals (MSIs) in these expansions are replaced by a set of random variables with considerably simpler and discrete probability densities. In fact, yet another simplifying feature of the present strategy is that there is no need to model and compute some of the higher‐level MSIs. Estimates of error orders of these methods in terms of a given time step size are derived and a proof of global convergence provided. Numerical illustrations are provided and compared with exact solutions whenever available to demonstrate the accuracy, simplicity and higher computational speed of WSNMs vis‐à‐vis a few other popularly used stochastic integration schemes as well as the path‐wise versions of the stochastic Newmark scheme. Copyright © 2006 John Wiley & Sons, Ltd.     

基于Markov链的重组细胞恒化培养的随机建模分析

本文研究了一类重组细胞恒化培养的连续时间Markov链模型.首先利用累积母函数表示出数字特征所满足的矩方程,然后通过对数正态分布近似的矩封闭技术得到了封闭后的矩方程,最后运用Euler-Maruyama方法构建了时间和状态都是连续的It随机微分方程.为了验证矩封闭的合理性,利用数值模拟给出了确定模型、随机模型和矩封闭后的方程的比较,并分析了重组细胞的变化趋势,结果表明其随机游走趋势与相应确定性模型是一致的.     

Stochastic Newmark scheme

In this paper, we are concerned with numerical resolution of the dynamical equation of linear oscillators. An answer is given to the following question: can one use widely implemented Newmark's algorithm when the acceleration is a white noise instead of a regular deterministic one? After showing drawbacks of such a practice, a modification of the usual Newmark schemes is introduced, to be used for the numerical resolution of such a stochastic dynamical equation. The modification is only in the stochastic part, so that already implemented softwares can be used, the input only having to be carefully adapted. After that, mathematical analysis of this new scheme is fulfilled. First and second moments, as well as the power spectral density resulting from the use of this scheme are compared with the corresponding quantities resulting from the theoretical process, and also from usual Newmark scheme and Euler schemes. Almost sure convergence of this new scheme is proved.     

First-passage reliability of high-dimensional nonlinear systems under additive excitation by the ensemble-evolving-based generalized density evolution equation

The reliability analysis of high-dimensional stochastic dynamical systems subjected to random excitations has long been one of the major challenges in civil and various engineering fields. Despite great efforts, no satisfactory method with high efficiency and accuracy has been available as yet for high-dimensional systems even when they are linear systems, not to mention generic nonlinear systems. In the present paper, a novel method by imposing appropriate absorbing boundary condition on the newly developed ensemble-evolving-based generalized density evolution equation (EV-GDEE) combined with a feasible numerical method is proposed to capture the time-variant first-passage reliability of high-dimensional systems enforced by additive white noise excitation. In the proposed method, the equivalent drift coefficients in EV-GDEE can be estimated by analytical expression or captured by some representative deterministic dynamic analyses. Further, imposing the absorbing boundary condition and then solving the EV-GDEE, a one-or two-dimensional partial differential equation (PDE), yield the remaining probability density of the response of interest. Consequently, by integrating the remaining probability density, the numerical solution of time-variant first-passage reliability can be obtained. Several numerical examples are illustrated to verify the efficiency and accuracy of the proposed method. Compared to the Monte Carlo simulation, the proposed method is of much higher efficiency. Problems to be further studied are finally discussed.     

邻接高耸结构的随机最优控制

文基于随机动态规划原理与随机平均法，提出耦合相邻高耸结构的随机最优控制方法。先建立任意层数并在任意层高处控制联接的耦合结构的缩聚模型，再运用随机平均法导出关于模态能量的It6随机微分方程，应用随机动态规划原理建立动态规划方程，由此可确定最优控制律。将结构的响应控制化为模态能量控制，缩减控制系统的维数。用高斯随机过程模拟地震激励，可计及其功率谱特性。数值结果表明该耦合结构控制方法的有效性。     

Task assignment under uncertainty: stochastic programming and robust optimisation approaches

The assignment of tasks to teams is a challenging combinatorial optimisation problem. The uncertainty in the tasks’ execution processes further complicates the assignment decisions. This study investigates a variant of the typical assignment problem, in which each task can be divided into two parts, one is deterministic and the other is uncertain with respect to their workloads. From the stochastic perspective, this paper proposes both a stochastic programming model that can cope with arbitrary probability distributions of tasks’ random workload requirements, and a robust optimisation model that is applicable to situations in which limited information about probability distributions is available. An example of its application in the software project management is given. Some numerical experiments are also performed to validate the effectiveness of the proposed models and the relationships between the two models.