Dissection of a mathematical model

Interdisciplinary models of broad systems with environmental, economic and sociopolitical interacting components are progressively developed to accept the challenge of dramatic global changes affecting the planet Earth and its ever increasing population. This paper, based on the author's experience in the interdisciplinary—physical, chemical, biological, economic and sociopolitical—modelling of the marine system, and using specific references to such models to provide concrete illustrations of the basic concepts, is an attempt to bring out the general substructural features of mathematical models.     

Optimal inspection of a complex system subject to periodic and opportunistic inspections and preventive replacements

This paper proposes two optimization models for the periodic inspection of a system with “hard-type” and “soft-type” components. Given that the failures of hard-type components are self-announcing, the component is instantly repaired or replaced, but the failures of soft-type components can only be detected at inspections. A system can operate with a soft failure, but its performance may be reduced. Although a system may be periodically inspected, a hard failure creates an opportunity for additional inspection (opportunistic inspection) of all soft-type components. Two optimization models are discussed in the paper. In the first, soft-type components undergo both periodic and opportunistic inspections to detect possible failures. In the second, hard-type components undergo periodic inspections and are preventively replaced depending on their condition at inspection. Soft-type and hard-type components are either minimally repaired or replaced when they fail. Minimal repair or replacement depends on the state of a component at failure; this, in turn, depends on its age. The paper formulates objective functions for the two models and derives recursive equations for their required expected values. It develops a simulation algorithm to calculate these expected values for a complex model. Several examples are used to illustrate the models and the calculations. The data used in the examples are adapted from a real case study of a hospital’s maintenance data for a general infusion pump.     

ESTIMATION OF SALMON ESCAPEMENT: MODELS WITH ENTRY,MORTALITY AND STOCHASTICITY

Understanding the dynamics of Prince William Sound pink salmon requires knowledge of the size of the spawning population in a stream over time. Periodic aerial surveys provide observations on the number of spawners, but the lack of daily observations requires a model to fill in the gaps. We develop a differential equation framework to represent the dynamics of escapement during the season. An exponential population growth model with a time-varying rate of growth is used for the number of spawners. The rate of growth consists of two primary components: the entry of salmon to the stream (escapement) and the mortality of spawners in the stream. The models for entry and mortality are also functions of time. The stochastic element of the model is based on a nonhomogeneous birth-and-death process which leads to a least squares estimation approach with either additive measurement or process errors. We illustrate the approach for a stream in Prince William Sound by fitting various models to observed spawner abundance, mortality counts from ground surveys and weir counts of the entry to the stream. We believe this approach could improve salmon escapement estimation, because the processes governing entry and mortality are explicitly considered.     

A novel hybrid decomposition-ensemble model based on VMD and HGWO for container throughput forecasting

This paper built a hybrid decomposition-ensemble model named VMD-ARIMA-HGWO-SVR for the purpose of improving the stability and accuracy of container throughput prediction. The latest variational mode decomposition (VMD) algorithm is employed to decompose the original series into several modes (components), then ARIMA models are built to forecast the low-frequency components, and the high-frequency components are predicted by SVR models which are optimized with a recently proposed swarm intelligence algorithm called hybridizing grey wolf optimization (HGWO), following this, the prediction results of all modes are ensembled as the final forecasting result. The error analysis and model comparison results show that the VMD is more effective than other decomposition methods such as CEEMD and WD, moreover, adopting ARIMA models for prediction of low-frequency components can yield better results than predicting all components by SVR models. Based on the results of empirical study, the proposed model has good prediction performance on container throughput data, which can be used in practical work to provide reference for the operation and management of ports to improve the overall efficiency and reduce the operation costs.     

On analytical and numerical approaches to division and label structured population models

Even among cells in the same population, the concentration of a protein or cellular constituent can vary considerably. This heterogeneity can arise from several sources, including differences in kinetic rates between cells and distribution of cellular constituents through cell division. Compartmental models have been used to describe the distribution of the number of divisions undergone by cells in a population. More recently, such models have been coupled with the dynamics of intracellular labels and analytical solutions to the division and label structured population equations have been found. However, such approaches have thus far focused on simple models of intracellular dynamics such as the decay of an intracellular label. In this work, we demonstrate that analytical solutions are possible for more general forms of intracellular dynamics offering the promise to lend mathematical insight into population dynamics in more realistic biological settings.     

The costs and benefits of commonality in assemble-to-order systems with a (Q,r)-policy for component replenishment

In assemble-to-order systems, it has been shown that replacing a number of specific components by a smaller number of general-purpose, common components can reduce required safety stock levels due to the benefits of risk pooling. Previous research using single-period models has shown that even if the common components are somewhat more expensive than the unique components they replace, the benefits of risk pooling often outweigh the added purchasing costs. However, this has been shown often not to be the case with multiple-period models – in the long run, the added purchasing costs dominate the benefits of risk pooling. However, both single- and multiple-period models ignore one of the benefits of commonality – order pooling. With commonality, demand is pooled into a smaller number of components, reducing the required number of orders (or setups). After re-optimizing the order quantities and order intervals, both the ordering costs and cyclic carrying costs are reduced. This paper develops a model to consider the assemble-to-order environment where components are replenished according to a (Q,r)-policy. Results show that order pooling is a significant benefit; in many cases it is much more important than the risk-pooling benefit. In contrast to multiple-period models, there is often a total cost benefit with commonality, even when the common component is several percent more expensive than the unique components it replaces.     

From behavioural to population level: Growth and competition

The aim of this work is to build models of population dynamics for growth and competition interaction by starting with detailed models at the individual level. At the individual level, we start with detailed models where the growth is described by linear terms. By considering individual interferences and by using aggregation methods, we show that the population level, different growth equation can emerge. We present an example of the emergence of logistic growth and an example of the emergence of logistic growth with Allee effect. Furthermore, in the case of two populations, we show that individual interferences can lead at the population level, to a model which has the same qualitative dynamics behaviour as the Lotka-Volterra competition model. Finally, we show that our model brings to light the effects of spatial heterogeneity on competition models. First, we find the stabilizing effects but also we show that destabilizing effects can occur.     

Multivariate sensitivity analysis for dynamic models with both random and random process inputs

Dynamic models with both random and random process inputs are frequently used in engineering. However, sensitivity analysis (SA) for such models is still a challenging problem. This paper, therefore, proposes a new multivariate SA technique to aid the safety design of these models. The new method can decompose the SA of dynamic models into a series of SA of their principle components based on singular value decomposition, which will make the SA of dynamic models much more efficient. It is shown that the effect of both random and random process inputs on the uncertainty of dynamic output can be measured from their effects on both the distributions and directions of the principle components, based on which the individual sensitivities are defined. The generalized sensitivities are then proposed to synthesize the information that is spread between the principal components to assess the influence of each input on the entire uncertainty of dynamic output. The properties of the new sensitivities are derived and an efficient estimation algorithm is proposed based on unscented transformation. Numerical results are discussed with application to a hydrokinetic turbine blade model, where the new method is compared with the existing variance-based method.     

Variable selection for generalized varying coefficient partially linear models with diverging number of parameters

Semiparametric models with diverging number of predictors arise in many contemporary scientific areas.Variable selection for these models consists of two components:model selection for non-parametric components and selection of significant variables for the parametric portion.In this paper,we consider a variable selection procedure by combining basis function approximation with SCAD penalty.The proposed procedure simultaneously selects significant variables in the parametric components and the nonparametric components.With appropriate selection of tuning parameters,we establish the consistency and sparseness of this procedure.     

Periodic solutions in periodic state-dependent delay equations and population models

Sufficient and realistic conditions are obtained for the existence of positive periodic solutions in periodic equations with state-dependent delay. The method involves the application of the coincidence degree theorem and estimations of uniform upper bounds on solutions. Applications of these results to some population models are presented. These application results indicate that seasonal effects on population models often lead to synchronous solutions. In addition, we may conclude that when both seasonality and time delay are present and deserve consideration, the seasonality is often the generating force for the often observed oscillatory behavior in population densities.

     

一类线性混合模型的谱分解估计

谱分解估计(SDE)是新近提出的关于线性混合模型参数的一种新的估计方法,此方法的一个突出特点是同时给出固定效应参数和方差分量的显式解估计．本文就含两个方差分量的线性混合模型,对谱分解估计的性质做了进一步的研究,获得了方差分量的SDE和方差分析估计相等的充分必要条件,证明了在一定的条件下方差分量的SDE为一致最小方差无偏估计．     

Mathematical modelling of drug resistant malaria parasites and vector populations

In this paper, we describe the phenomenon of drug resistant malaria infection in a hyperendemic region by a system of ordinary differential equations models. Different situations on both vectors and humans are described. The first presentations deal with single population each for both humans and vectors (with the human population fixed). These models are without resistant strains. The second group of models deal with both resistant and sensitive strains. Our focus in analysing the models is on establishing a unique positive asymptotic equilibrium for each of them. It is shown that (under suitable conditions in most cases) the equilibrium points are (locally) asymptotically stable. The biological significance of each of these points emerges as a by‐product. Copyright © 2002 John Wiley & Sons, Ltd.     

人口与基础教育发展趋势预测模型研究及应用

以西藏人口与基础教育发展为研究对象,应用灰色系统理论建立了一个新的出生人口预测模型,还分别建立了小学和初中招生数、分年级在校生规模、在校生总量的预测模型,并结合近几年西藏自治区教育统计结果,给出了应用实例.     

Bifurcation analysis of a population model and the resulting SIS epidemic model with delay

This paper deals with the model for matured population growth proposed in Cooke et al. [Interaction of matiration delay and nonlinear birth in population and epidemic models, J. Math. Biol. 39 (1999) 332–352] and the resulting SIS epidemic model. The dynamics of these two models are still largely undetermined, and in this paper, we perform some bifurcation analysis to the models. By applying the global bifurcation theory for functional differential equations, we are able to show that the population model allows multiple periodic solutions. For the SIS model, we obtain some local bifurcation results and derive formulas for determining the bifurcation direction and the stability of the bifurcated periodic solution.     

Reviews on the Research of Dynamic Mortality Models

Mortality forecasting is the basis of population forecasting. In recent years, new progress has been made in mortality models. From the earliest static mortality models, mortality models have been developed into dynamic forecasting models including time terms, such as Lee-Carter model family, CBD model family and so on. This paper reviews and sorts out relevant literature on mortality forecasting models. With the development of dynamic models, some scholars have developed a series of mortality improvement models based on the level of mortality improvement. In addition, with the progress of mortality research, multi-population mortality modeling attracted the attention of researchers, and the multi-population forecasting models have been constantly developed and improved, which play an important role in the mortality forecasting. With the continuous enrichment and innovation of mortality model research methods, new statistical methods (such as machine learning) have been applied in mortality modeling, and the accuracy of fitting and prediction has been improved. In addition to the extension of classical modeling methods, issues such as small-area population or missing data of the population, the elderly population, the related population mortality modeling are still worth studying.     

Using the EPQ for coordinated planning of a product with partial backordering and its components

While there has been considerable work over the years on multistage lot-sizing models, particularly in an MRP environment, there has been relatively little work on systems recognizing the WIP effects when there is gradual conversion of the components into the final product, as in production planning using the EPQ model for planning the final product. Here we consider lot-sizing planning for a two-stage system in which the final product is planned using an EPQ model with partial backordering and the production of the components is controlled using basic EPQ models without backordering.     

Semi-varying coefficient models with a diverging number of components

Semiparametric models with both nonparametric and parametric components have become increasingly useful in many scientific fields, due to their appropriate representation of the trade-off between flexibility and efficiency of statistical models. In this paper we focus on semi-varying coefficient models (a.k.a. varying coefficient partially linear models) in a “large n, diverging p” situation, when both the number of parametric and nonparametric components diverges at appropriate rates, and we only consider the case p=o(n). Consistency of the estimator based on B-splines and asymptotic normality of the linear components are established under suitable assumptions. Interestingly (although not surprisingly) our analysis shows that the number of parametric components can diverge at a faster rate than the number of nonparametric components and the divergence rates of the number of the nonparametric components constrain the allowable divergence rates of the parametric components, which is a new phenomenon not established in the existing literature as far as we know. Finally, the finite sample behavior of the estimator is evaluated by some Monte Carlo studies.     

Convergence dynamics and pseudo almost periodicity of a class of nonautonomous RFDEs with applications

This paper studies the dynamics of a system of retarded functional differential equations (i.e., RFDEs), which generalize the Hopfield neural network models, the bidirectional associative memory neural networks, the hybrid network models of the cellular neural network type, and some population growth model. Sufficient criteria are established for the globally exponential stability and the existence and uniqueness of pseudo almost periodic solution. The approaches are based on constructing suitable Lyapunov functionals and the well-known Banach contraction mapping principle. The paper ends with some applications of the main results to some neural network models and population growth models and numerical simulations.