解析一类SIS和SIR传染病模型的稳定性

借助微分方程建立传染病SIS模型和SIR模型,进一步研究了一类SIS和SIR传染病模型,得出了决定SIS传染病是否发生的阈值;解析了SIR模型无病平衡点和地方平衡点的稳定性.     

具有阶段结构的SIS传染病模型

讨论了具有阶段结构的SIS传染病模型,给出了传染病最终消除和成为地方病的阈值。     

一类具有饱和反应率的脉冲免疫接种的SIS模型

研究了具饱和传染率的脉冲免疫接种SIS模型,得到了无病周期解全局渐近稳定的充分条件和系统持续生存的充分条件．     

媒体报道影响下具有Markov状态切换的随机SIS传染病模型阈值动力学行为研究

本文发展和研究了一类媒体报道影响下具有Markov切换的随机SIS传染病模型,探讨了利用Markov切换描述的天气变化和媒体报道对随机SIS传染病动力学模型的渐近性质的影响.利用Khasminskii方法、Markov链的遍历性和随机比较原理得到了在简单假设β_(ij)-β_(2jf)(I)≥σ~2_j下的闽值条件:即当R~S_O=△Σmj=1π_jC_j0时,疾病依概率1绝灭;当R~S_O0时,疾病随机持续,最终形成地方病.研究结果表明:色噪声和媒体报道对随机SIS传染病模型的阚值动力学行为都有较为显著的影响.     

一类病毒自身发生变异的传染病模型的全局分析

针对病毒变异前和变异后传染病患者具有不同的传染率情形,建立了一类分阶段传播的SIS模型,通过构造Liapunov函数和定性分析,得到病毒变异前和变异后传染病患者平衡点的存在条件以及它们的全局渐近稳定性。     

一类具有标准发生率的SIS型传染病模型的全局稳定性

研究一类具有标准发生率的SIS传染病模型,讨论了各类平衡点存在的条件;运用微分方程的定性理论,得到了无病平衡点E_1和地方病平衡点E_2的全局渐近稳定的充分条件.     

一类具有时滞和阶段结构的SIS传染病模型的稳定性与持久性

研究一类具有时滞和阶段结构的SIS传染病模型.通过分析特征方程,讨论了系统平衡点的局部稳定性,根据比较定理讨论了无病平衡点的全局稳定性,并证明了当地方病平衡点存在时系统是一致持续生存的.     

出生率密度依赖的肺结核传染病模型

考虑一类具有垂直传染和出生率密度依赖的肺结核传染病模型.在较弱的条件下,得到了疾病持久和灭绝的充分条件.同时也得到了相应的自治系统疾病持久和灭绝的充分条件.     

人口流动性对感染性疾病扩散与传播的影响

研究人口流动性对具有斑快结构的感染性疾病传播与扩散的影响,讨论了具有斑块结构感染性疾病SIS模型的全局稳定性,得到了该模型基本再生数的倍增效应．     

一类具有脉冲接种时滞的SEIRS传染病模型

建立了一类具有脉冲接种时滞的SEIRS传染病模型.利用频闪映射和脉冲微分方程比较定理对模型的等价系统进行分析,得到了模型无病周期解具有全局吸引性的存在条件,并且给出了疾病持久性的存在条件.     

Leslie matrix models as “stroboscopic snapshots” of McKendrick PDE models

 High dimensional Leslie matrix models have long been viewed as discretizations of McKendrick PDE models. However, these two fundamental classes of models can be linked in a completely different way. For populations with periodic birth pulses, Leslie models of any dimension can be viewed as “stroboscopic snapshots” (in time) of an associated impulsive McKendrick model; that is, the solution of the discrete model matches the solution of the corresponding continuous model at every discrete time step. In application, McKendrick models of populations with birth pulses can be used to identify the state of the population between the discrete census times of the associated Leslie model. Furthermore, McKendrick models describing populations with near-synchronous birth pulses can be viewed as realistic perturbations of the associated Leslie model. Received: 7 August 1997 / Revised version: 15 January 1998     

Density-dependent birth rate, birth pulses and their population dynamic consequences

 In most models of population dynamics, increases in population due to birth are assumed to be time-independent, but many species reproduce only during a single period of the year. We propose a single-species model with stage structure for the dynamics in a wild animal population for which births occur in a single pulse once per time period. Using the discrete dynamical system determined by the stroboscopic map, we obtain an exact periodic solution of systems which are with Ricker functions or Beverton-Holt functions, and obtain the threshold conditions for their stability. Above this threshold, there is a characteristic sequence of bifurcations, leading to chaotic dynamics, which implies that the dynamical behaviors of the single species model with birth pulses are very complex, including small-amplitude annual oscillations, large-amplitude multi-annual cycles, and chaos. This suggests that birth pulse, in effect, provides a natural period or cyclicity that allows for a period-doubling route to chaos. Received: 13 June 2001 / Revised version: 7 September 2001 / Published online: 8 February 2002     

Bayesian deconvolution analysis of pulsatile hormone concentration profiles

Many hormones are secreted into the circulatory system in a pulsatile manner and are cleared exponentially. The most common method of analyzing these systems is to deconvolve the hormone concentration into a secretion function and a clearance function. Accurate estimation of the model parameters depends on the number and location of the secretion pulses. To date, deconvolution analysis assumes the number and approximate location of these pulses are known a priori. In this article, we present a novel Bayesian approach to deconvolution that jointly models the number of pulses along with all other model parameters. Our method stochastically searches for the secretion pulses. This is accomplished by viewing the set of parameters that define the pulses as a point process. Pulses are determined by a birth-death process which is embedded in Markov chain Monte Carlo algorithm. This idea originated with Stephens (2000, Annals of Statistics 28, 40-74) in the context of finite mixture model density estimation, where the number of mixture components is unknown. There are several advantages that our model enjoys over the traditional frequentist approaches. These advantages are highlighted with four datasets consisting of serum concentration levels of luteinizing hormone obtained from ovariectomized ewes.     

Comparison of statistical models to analyse the genetic effect on within-litter variance in pigs

Genetics affects not only the weight of piglets at birth but also the variability of birth weight within litter. Previous studies on this topic assigned the sample standard deviation of piglet birth weights within litter as an observation to the sow. However, the contribution of the difference in mean birth weight per sex on the within-litter variance has been neglected so far. This work deals with the genetic effect on within-litter variance when different statistical models with different distributional assumptions are used and considers the sex effect and appropriate weights per trait. Traits were formed from the pooled sample variance of male and female birth weights within litter. A linear model approach was fitted to the logarithmized within-litter variance and the sample standard deviation. A generalized linear model with gamma-distributed residuals and log-link function was applied to the untransformed sample variance. Models were compared by analysing data from 9439 litters from Landrace and Large White of a commercial breeding programme. The estimates of heritability for different traits ranged from 7% to 11%. Although the generalized linear mixed model is preferred from a mathematical view, the rank correlations between breeding values of the linear mixed models and the generalized linear mixed model were relatively high, i.e. 94% to 98%. By residual diagnostics, a linear mixed model using the weighted and pooled within-litter standard deviation was identified as most suitable.     

Wavelet analysis of nonequilibrium ionic currents in human heart sodium channel (hH1a)

Nonequilibrium response spectroscopy (NRS), the technique of using rapidly fluctuating voltage pulses in the study of ion channels, is applied here. NRS is known to drive an ensemble of ion channels far from equilibrium where, it has been argued, new details of ion channel kinetics can be studied under nonequilibrium conditions. In this paper, a single-pulse NRS technique with custom-designed waveforms built from wavelets is used. The pulses are designed to produce different responses from two competing models of a human heart isoform of the sodium channel (hH1a). Experimental data using this new type of pulses are obtained through whole-cell recordings from mammalian cells (HEK 293). Wavelet analysis of the model response and the experimental data is introduced to show how these NRS pulses can aid in distinguishing the better of the two models and thus introduces another important application of this new technique.     

Simple mathematical models for interacting wild and transgenic mosquito populations

Two discrete-time models for interacting populations of wild and genetically altered mosquito are presented, where the genetically altered mosquitoes are grouped into a single population without distinguishing their zygosity. The birth and death rates for both populations are density-dependent, and the mating rates between the mosquitoes are assumed to be either constant or proportional to the total populations for the two models, respectively. The existence and stability of the boundary and positive equilibria are investigated. In particular, it is shown that bifurcations from both boundary and positive equilibria can appear for the model with proportional mating rates. Stable equilibria, periodic-doubling bifurcations, aperiodic oscillations, and chaotic behavior are all illustrated by numerical simulations.     

Competition and evolutionary stability of plants in a spatially structured habitat

We modelled the population dynamics of two types of plants with limited dispersal living in a lattice structured habitat. Each site of the square lattice model was either occupied by an individual or vacant. Each individual reproduced to its neighbors. We derived a criterion for the invasion of a rare type into a population composed of a resident type based on a pair-approximation method, in which the dynamics of both average densities and the nearest neighbor correlations were considered. Based on this invasibility criterion, we showed that, when there is a tradeoff between birth and death rates, the evolutionarily stable type is the one that has the highest ratio of birth rate to mortality. If these types are different species, they form segregated spatial patterns in the lattice model in which intraspecific competitive interactions occur more frequently than interspecific interactions. However, stable coexistence is not possible in the lattice model contrary to results from completely mixed population models. This clearly shows that the casual conclusion, based on traditional well mixed population models, that different species can coexist if intraspecific competition is stronger than interspecific competition, does not hold for spatially structured population models.     

Modifying the ‘pulse–reserve’ paradigm for deserts of North America: precipitation pulses,soil water,and plant responses

The ‘pulse–reserve’ conceptual model—arguably one of the most-cited paradigms in aridland ecology—depicts a simple, direct relationship between rainfall, which triggers pulses of plant growth, and reserves of carbon and energy. While the heuristics of ‘pulses’, ‘triggers’ and ‘reserves’ are intuitive and thus appealing, the value of the paradigm is limited, both as a conceptual model of how pulsed water inputs are translated into primary production and as a framework for developing quantitative models. To overcome these limitations, we propose a revision of the pulse–reserve model that emphasizes the following: (1) what explicitly constitutes a biologically significant ‘rainfall pulse’, (2) how do rainfall pulses translate into usable ‘soil moisture pulses’, and (3) how are soil moisture pulses differentially utilized by various plant functional types (FTs) in terms of growth? We explore these questions using the patch arid lands simulation (PALS) model for sites in the Mojave, Sonoran, and Chihuahuan deserts of North America. Our analyses indicate that rainfall variability is best understood in terms of sequences of rainfall events that produce biologically-significant ‘pulses’ of soil moisture recharge, as opposed to individual rain events. In the desert regions investigated, biologically significant pulses of soil moisture occur in either winter (October–March) or summer (July–September), as determined by the period of activity of the plant FTs. Nevertheless, it is difficult to make generalizations regarding specific growth responses to moisture pulses, because of the strong effects of and interactions between precipitation, antecedent soil moisture, and plant FT responses, all of which vary among deserts and seasons. Our results further suggest that, in most soil types and in most seasons, there is little separation of soil water with depth. Thus, coexistence of plant FTs in a single patch as examined in this PALS study is likely to be fostered by factors that promote: (1) separation of water use over time (seasonal differences in growth), (2) relative differences in the utilization of water in the upper soil layers, or (3) separation in the responses of plant FTs as a function of preceding conditions, i.e., the physiological and morphological readiness of the plant for water-uptake and growth. Finally, the high seasonal and annual variability in soil water recharge and plant growth, which result from the complex interactions that occur as a result of rainfall variability, antecedent soil moisture conditions, nutrient availability, and plant FT composition and cover, call into question the use of simplified vegetation models in forecasting potential impacts of climate change in the arid zones in North America.     

Dynamics of a physiologically structured population in a time-varying environment

Physiologically structured population models have become a valuable tool to model the dynamics of populations. In a stationary environment such models can exhibit equilibrium solutions as well as periodic solutions. However, for many organisms the environment is not stationary, but varies more or less regularly. In order to understand the interaction between an external environmental forcing and the internal dynamics in a population, we examine the response of a physiologically structured population model to a periodic variation in the food resource. We explore the addition of forcing in two cases: (A) where the population dynamics is in equilibrium in a stationary environment, and (B) where the population dynamics exhibits a periodic solution in a stationary environment. When forcing is applied in case A, the solutions are mainly periodic. In case B the forcing signal interacts with the oscillations of the unforced system, and both periodic and irregular (quasi-periodic or chaotic) solutions occur. In both cases the periodic solutions include one and multiple period cycles, and each cycle can have several reproduction pulses.     

A unified approach to the Richards-model family for use in growth analyses: Why we need only two model forms

This paper advances a unified approach to the modeling of sigmoid organismal growth. There are numerous studies on growth, and there have been several proposals and applications of candidate models. Still, a lack of interpretation of the parameter values persists and, consequently, differences in growth patterns have riddled this field. A candidate regression model as a tool should be able to assess and compare growth-curve shapes, systematically and precisely. The Richards models constitute a useful family of growth models that amongst a multitude of parameterizations, re-parameterizations and special cases, include familiar models such as the negative exponential, the logistic, the Bertalanffy and the Gompertz. We have reviewed and systemized this family of models. We demonstrate that two specific parameterizations (or re-parameterizations) of the Richards model are able to substitute, and thus to unify all other forms and models. This unified-Richards model (with its two forms) constitutes a powerful tool for an interpretation of important characteristics of observed growth patterns, namely, [I] maximum (relative) growth rate (i.e., slope at inflection), [II] age at maximum growth rate (i.e., time at inflection), [III] relative mass or length at maximum growth rate (i.e., relative value at an inflection), [IV] value at age zero (i.e., birth, hatching or germination), and [V] asymptotic value (i.e., adult weight or length). These five parameters can characterize uniquely any sigmoid-growth data. To date most studies only compare what is referred to as the “growth-rate constant” or simply “growth rate” (k). This parameter can be interpreted as neither relative nor actual growth rate, but only as a parameter that affects the slope at inflection. We fitted the unified-Richards and five other candidate models to six artificial data sets, generated from the same models, and made a comparison based on the corrected Akaike’s Information Criterion (AICc). The outcome may in part be the result of the random generation of data points. Still, in conclusion, the unified-Richards model performed consistently well for all data sets, despite the penalty imposed by the AICc.