A spatially stochastic epidemic model with partial immunization shows in mean field approximation the reinfection threshold

Recently, the notion of a reinfection threshold in epidemiological models of only partial immunity has been debated in the literature. We present a rigorous analysis of a model of reinfection which shows a clear threshold behaviour at the parameter point where the reinfection threshold was originally described. Furthermore, we demonstrate that this threshold is the mean field version of a transition in corresponding spatial models of immunization. The reinfection threshold corresponds to the transition between annular growth of an epidemics spreading into a susceptible area leaving recovered behind and compact growth of a susceptible-infected-susceptible region growing into a susceptible area. This transition between annular growth and compact growth was described in the physics literature long before the reinfection threshold debate broke out in the theoretical biology literature.     

Resonance of the epidemic threshold in a periodic environment

Resonance between some natural period of an endemic disease and a seasonal periodic contact rate has been the subject of intensive study. This paper does not focus on resonance for endemic diseases but on resonance for emerging diseases. Periodicity can have an important impact on the initial growth rate and therefore on the epidemic threshold. Resonance occurs when the Euler-Lotka equation has a complex root with an imaginary part (i.e., a natural frequency) close to the angular frequency of the contact rate and a real part not too far from the Malthusian parameter. This is a kind of continuous-time analogue of work by Tuljapurkar on discrete-time population models, which in turn was motivated by the work by Coale on continuous-time demographic models with a periodic birth. We illustrate this resonance phenomenon on several simple epidemic models with contacts varying periodically on a weekly basis, and explain some surprising differences, e.g., between a periodic SEIR model with an exponentially distributed latency and the same model but with a fixed latency.     

A spatially extended stochastic model of the bacterial chemotaxis signalling pathway

We have combined two distinct but related stochastic approaches to model the Escherichia coli chemotaxis pathway. Reactions involving cytosolic components of the pathway were assumed to obey the laws of conventional stochastic chemical kinetics, while the clustered membrane receptors were represented in two-dimensional arrays similar to the Ising model. Receptors were assumed to flip between an active and an inactive state with probabilities dependent upon three energy inputs: ligand binding, methylation level due to adaptation, and the activity of neighbouring receptors. Examination of models with different lattice size and geometry showed that the sensitivity to stimuli increases with lattice size and the nearest-neighbour coupling strength up to a critical point, but this amplification was also accompanied by a proportional increase in steady-state noise. Multiple methylation of receptors resulted in diminished signal-to-noise ratio, but showed improved stability to variation in the coupling strength and increased gain. Under the best conditions the simulated output of a coupled lattice of receptors closely matched the time-course and amplitude found experimentally in living bacteria. The model also has some of the properties of a cellular automaton and shows an unexpected emergence of spatial patterns of methylation within the receptor lattice.     

Examining the hierarchy threshold model in a no-choice feeding assay

The hierarchy threshold model of individual insect diet predicts the acceptance or rejection of individual hosts when encountered by insects. One assumption of the hierarchy threshold model is that post-discrimination phase insects which accept lower ranked hosts will also accept hosts that are ranked higher. This assumption does not however suggest whether or not such insects behave differently when encountering these two hosts. This question is explored using Zygogramma bicolorata Pallister (Coleoptera: Chrysomelidae). Adult beetles that had just fed, or were one inter-meal interval after feeding, or were deprived for three or six days after feeding, were individually observed on a plant that either ranked high, Parthenium hysterophorus L., or low Xanthium occidentale Bertol. (Heliantheae: Ambrosiinae) in choice tests. Just-fed beetles were generally unresponsive to either host plant and did not feed. Beetles held for one inter-meal interval showed more feeding responses (shorter lag time between sampling and feeding, higher proportions of individuals feeding, and more time feeding) towards the higher ranked plant. As the beetles approached six days deprivation, behaviours of sampling, feeding and locomotion differed less between the two plants, but were still apparent. I conclude that food deprived Z. bicolorata beetles do still discriminate between the two plants. This result partially supports the hierarchy threshold model.     

A mean field Ising model for cortical rotation in amphibian one-cell stage embryos

We propose a new physical mechanism of cortical rotation generation in one-cell embryos of amphibians based on a phase transition in the ensemble of microtubules localized to the cortical region of the cell interior. Microtubules, protein polymers formed from tubulin heterodimers, are highly negatively charged, which results in strong electrostatic interactions over tens of nanometers, even in the presence of counterions that partially screen electrostatic interactions. A simplified model that offers a plausible representation of these effects is based on the Ising Hamiltonian, which has been robustly applied to explain a wide range of order-disorder transitions in physics, chemistry and other sciences. An Ising model phase transition, especially with the supercooperative flow alignment effect of global rotation of the cortex, provides an alternative to models of cortical rotation based on microtubule polymerization or motor molecules. Insofar as there is any reality to the concept that microtubules are involved in consciousness, we propose that cortical rotation in the one-cell embryo is a better place to look for the purported microtubule entanglement or coherence properties than the adult brain.     

武陵源风景名胜区旅游生态环境演变趋势与阈值分析

武陵源旅游业迅速发展的同时。景区生态环境质量逐年下降引起联合国教科文组织世界遗产委员会的关注：武陵源风景名胜区核心景区较突出的生态环境问题,主要表现在大气环境质量逐年降低;金鞭溪水质明显恶化;生物多样性受到威胁;景区城镇化、工商业比速度加快等。通过实地监测并全面分析武陵源风景名胜区环境演变趋势。发现住宿设施对环境的影响。比其他游乐设施更为明显,是武陵源风景名胜区旅游生态环境的关键影响因子。景区水质监测值的变化最大,表明水体是景区生态环境中最为脆弱的环境因子：随着游客的逐年增多。旅游生态环境呈现出加速恶化的趋势。通过建立基于环境脆弱因子的阈值模型,计算得出在不超出武陵源风景名胜区景区最为脆弱的环境因子——景区水质标准：总磷≤0．02(mg/L)前提下。景区上游接待区旅游生态阈值是春季宾馆可使用面积容限为29650m^2。夏季为76425m^2。秋季为13625m^2。冬季为8325m^2。不同类型的风景区。生态环境脆弱因子不同。宾馆建筑面积容限位也不同。沿着本文思路。可计算出其他类型风景区阈值。     

华南稻田农田害鼠复合防治指标的理论研究

依据对华南地区稻田3种主要害鼠黄毛鼠、板齿鼠和小家鼠的田间捕获率调查,以及对害鼠造成的相应水稻产量损失率（y）研究,建立了水稻产量损失率与鼠密度关系的多元线性回归方程。确定板齿鼠作为本项研究的标准鼠种,利用3种农田鼠害的捕获率在回归方程中对应的系数,计算出农田害鼠的标准当量损害指标。结果显示,每个标准鼠单位造成的水稻产量损失率分别相当于4.084个黄毛鼠当量单位,或31.593个小家鼠当量单位。进一步,对原始调查数据进行二次处理,建立了标准当量（x）与（y）的相互关系数学模型;文中配合EIL计算公式,求出了在不同水稻产量和不同灭鼠效果等条件下的华南农田害鼠的复合防治指标。     

Statistical inference in a stochastic epidemic SEIR model with control intervention: Ebola as a case study

A stochastic discrete-time susceptible-exposed-infectious-recovered (SEIR) model for infectious diseases is developed with the aim of estimating parameters from daily incidence and mortality time series for an outbreak of Ebola in the Democratic Republic of Congo in 1995. The incidence time series exhibit many low integers as well as zero counts requiring an intrinsically stochastic modeling approach. In order to capture the stochastic nature of the transitions between the compartmental populations in such a model we specify appropriate conditional binomial distributions. In addition, a relatively simple temporally varying transmission rate function is introduced that allows for the effect of control interventions. We develop Markov chain Monte Carlo methods for inference that are used to explore the posterior distribution of the parameters. The algorithm is further extended to integrate numerically over state variables of the model, which are unobserved. This provides a realistic stochastic model that can be used by epidemiologists to study the dynamics of the disease and the effect of control interventions.     

A stochastic model for predicting the stage emergence of Plutella xylostella under field conditions

The diamondback moth, Plutella xylostella, is a worldwide pest of brassicas, and its biology and ecology have been extensively studied over recent years. Despite the importance of mathematical models to the management of insect pests, no stochastic model has been developed to date for P. xylostella. In this context, the study aimed to develop a stochastic model capable of describing the stage emergence of P. xylostella under field conditions. The stochastic model was developed using simple nonlinear functions based on the laboratory data on development times under constant temperatures. Comparison between estimated and observed cumulative proportions of egg hatch, pupation and adult emergence recorded in the field in Southern Brazil shows that the model accurately describes the stage emergence of P. xylostella. The developed model shows potential to estimate the stage emergence of P. xylostella under field conditions, and can add significant advances to the management of this pest.     

A simple stochastic model for complex coextinctions in mutualistic networks: robustness decreases with connectance

Understanding and predicting species extinctions and coextinctions is a major goal of ecological research in the face of a biodiversity crisis. Typically, models based on network topology are used to simulate coextinctions in mutualistic networks. However, such topological models neglect two key biological features of species interactions: variation in the intrinsic dependence of species on the mutualism, and variation in the relative importance of each interacting partner. By incorporating both types of variation, we developed a stochastic coextinction model capable of simulating extinction cascades far more complex than those observed in previous topological models. Using a set of empirical mutualistic networks, we show that the traditional topological model may either underestimate or overestimate the number and likelihood of coextinctions, depending on the intrinsic dependence of species on the mutualism. More importantly, contrary to topological models, our stochastic model predicts extinction cascades to be more likely in highly connected mutualistic communities.     

A possible mechanism of stochastic resonance in the light of an extra-classical receptive field model of retinal ganglion cells

Traditionally the intensity discontinuities in an image are detected as zero-crossings of the second derivative with the help of a Laplacian of Gaussian (LOG) operator that models the receptive field of retinal Ganglion cells. Such zero-crossings supposedly form a raw primal sketch edge map of the external world in the primary visual cortex of the brain. Based on a new operator which is a linear combination of the LOG and a Dirac-delta function that models the extra-classical receptive field of the ganglion cells, we find that zero-crossing points thus generated, store in presence of noise, apart from the edge information, the shading information of the image in the form of density variation of these points. We have also shown that an optimal image contrast produces best mapping of the shading information to such zero-crossing density variation for a given amount of noise contamination. Furthermore, we have observed that an optimal amount of noise contamination reproduces the minimum optimal contrast and hence gives rise to the best representation of the original image. We show that this phenomenon is similar in nature to that of stochastic resonance phenomenon observed in psychophysical experiments.     

A discrete,stochastic model of colonial phytoplankton population size structure: Development and application to in situ imaging-in-flow cytometer observations of Dinobryon

The scientific community lacks models for the dynamic changes in population size structure that occur in colonial phytoplankton. This is surprising, as size is a key trait affecting many aspects of phytoplankton ecology, and colonial forms are very common. We aim to fill this gap with a new discrete, stochastic model of dynamic changes in phytoplankton colonies' population size structure. We use the colonial phytoplankton Dinobryon as a proof-of-concept organism. The model includes four stochastic functions—division, stomatocyst production, colony breakage, and colony loss—to determine Dinobryon population size structure and populations counts. Although the functions presented here are tailored to Dinobryon, the model is readily adaptable to represent other colonial taxa. We demonstrate how fitting our model to in situ observations of colony population size structure can provide a powerful approach to explore colony size dynamics. Here, we have (1) collected high-frequency in situ observations of Dinobryon in Lac (Lake) Montjoie (Quebec, Canada) in 2013 with a moored Imaging FlowCytobot (IFCB) and (2) fit the model to those observations with a genetic algorithm solver that extracts parameter estimates for each of the four stochastic functions. As an example of the power of this model-data integration, we also highlight ecological insights into Dinobryon colony size and stomatocyst production. The Dinobryon population was enriched in larger, flagellate-rich colonies near bloom initiation and shifted to smaller and emptier colonies toward bloom decline.     

A new SEIR epidemic model with applications to the theory of eradication and control of diseases, and to the calculation of R0

We present a novel SEIR (susceptible-exposure-infective-recovered) model that is suitable for modeling the eradication of diseases by mass vaccination or control of diseases by case isolation combined with contact tracing, incorporating the vaccine efficacy or the control efficacy into the model. Moreover, relying on this novel SEIR model and some probabilistic arguments, we have found four formulas that are suitable for estimating the basic reproductive numbers R(0) in terms of the ratio of the mean infectious period to the mean latent period of a disease. The ranges of R(0) for most known diseases, that are calculated by our formulas, coincide very well with the values of R(0) estimated by the usual method of fitting the models to observed data.     

小鼠低菌模型的构建及多形拟杆菌对其补偿生长作用的研究

目的探讨高剂量抗生素构建低菌小鼠模型的可行性,并研究小鼠在不同生理状态下肠内拟杆菌的演替规律及对小鼠生长性能的影响。方法将120只SPF级的昆明小鼠分成3组：A组为空白对照组;B组为模型组,用22g/（kg·BW）的氨苄青霉素连续灌胃小鼠3d构建低菌动物模型;C组在构建低菌小鼠模型的基础上每天早晚2次各灌胃1ml的多形拟杆菌发酵液（10^9／m1）,连续3d。用注射器稀释法测定小鼠肠内拟杆菌的活菌数,并测定各组小鼠体重和日增重。结果高剂量的抗生素处理可以使小鼠保持16d的低菌状态,但是给小鼠带来了一定的应激;C组小鼠在灌胃多形拟杆菌发酵液后,肠内拟杆菌在22～34d迅速生长并形成优势定植,且体重和日增重在实验后期显著增加即补偿生长。结论低菌模型较常规动物更能敏感地反映添加拟杆菌的生物学效应,但对小鼠有一定应激;多形拟杆菌对低菌小鼠具有明显的补偿生长效应,它可以缓解应激并使小鼠更快的恢复到正常的生理状态。     

A mathematical model for pH patterns in the rhizospheres of growth zones

In the classical model by Nye (1981), the main process for the change in pH across the rhizosphere is assumed to be diffusion. The classical model focuses on the non-growing part of the root and assumes that the distribution of ion fluxes along the root is spatially uniform. We consider the rhizosphere of the growth zone and take into account the root growth rate and spatially varying flux along the root surface. We present both analytical (dimensional analysis) and experimental (computational) evidence of the importance of taking into account the root growth rate. We describe a conceptual and mathematical model to analyse the pH field around the root tip over time. The model is used with published data to show that, for typical growth rates in sandy soil, the pH field becomes steady (independent of time) after 6 h. Dimensional analysis reveals that a version of the Péclet number, related to the quotient of root elongation rate and proton diffusivity, can be used to predict the extent of the rhizosphere and the time required for it to become steady. For Péclet numbers much greater than 1 (soils), the root influences soil pH for distances on the millimetre scale. In contrast, for Péclet numbers much less than one (agar, aqueous solution), the root influences substrate pH for radial distances on the scale of centimetres. We also present some evidence that agar-contact techniques to measure the soil pH may not be appropriate for measuring the millimetre-scale gradients in soil pH.     

A stochastic formulation of the gompertzian growth model for in vitro bactericidal kinetics: parameter estimation and extinction probability

Time-kill curves have frequently been employed to study the antimicrobial effects of antibiotics. The relevance of pharmacodynamic modeling to these investigations has been emphasized in many studies of bactericidal kinetics. Stochastic models are needed that take into account the randomness of the mechanisms of both bacterial growth and bacteria-drug interactions. However, most of the models currently used to describe antibiotic activity against microorganisms are deterministic. In this paper we examine a stochastic differential equation representing a stochastic version of a pharmacodynamic model of bacterial growth undergoing random fluctuations, and derive its solution, mean value and covariance structure. An explicit likelihood function is obtained both when the process is observed continuously over a period of time and when data is sampled at time points, as is the custom in these experimental conditions. Some asymptotic properties of the maximum likelihood estimators for the model parameters are discussed. The model is applied to analyze in vitro time-kill data and to estimate model parameters; the probability of the bacterial population size dropping below some critical threshold is also evaluated. Finally, the relationship between bacterial extinction probability and the pharmacodynamic parameters estimated is discussed.     

A dynamic multibody model of the physiological knee to predict internal loads during movement in gravitational field

Obtaining tibio-femoral (TF) contact forces, ligament deformations and loads during daily life motor tasks would be useful to better understand the aetiopathogenesis of knee joint diseases or the effects of ligament reconstruction and knee arthroplasty. However, methods to obtain this information are either too simplified or too computationally demanding to be used for clinical application. A multibody dynamic model of the lower limb reproducing knee joint contact surfaces and ligaments was developed on the basis of magnetic resonance imaging. Several clinically relevant conditions were simulated, including resistance to hyperextension, varus–valgus stability, anterior–posterior drawer, loaded squat movement. Quadriceps force, ligament deformations and loads, and TF contact forces were computed. During anterior drawer test the anterior cruciate ligament (ACL) was maximally loaded when the knee was extended (392 N) while the posterior cruciate ligament (PCL) was much more stressed during posterior drawer when the knee was flexed (319 N). The simulated loaded squat revealed that the anterior fibres of ACL become inactive after 60° of flexion in conjunction with PCL anterior bundle activation, while most components of the collateral ligaments exhibit limited length changes. Maximum quadriceps and TF forces achieved 3.2 and 4.2 body weight, respectively. The possibility to easily manage model parameters and the low computational cost of each simulation represent key points of the present project. The obtained results are consistent with in vivo measurements, suggesting that the model can be used to simulate complex and clinically relevant exercises.     

A predictive model to locate ancient forests in the Cross Timbers of Osage County, Oklahoma

The Cross Timbers are a mosaic of upland deciduous forest, savanna, and glade that typifies the broad ecotone between the eastern deciduous forest and the grasslands of the southern Great Plains. The pre-settlement Cross Timbers may have covered some 7,909,700 ha from central Texas, across Oklahama into eastern Kansas, and today may represent the least disturbed forest ecosystem of comparable size still left in the eastern United States. Extensive tree-ring research indicates that ancient forests dominated by 200 to 400 year old post-oaks ( Quercus stellata Wang) survive throughout the Cross Timbers, particularly in Oklahoma. These ancient forests persist largely because the Cross Timbers formation is non-commercial for timber production, and has not experienced large-scale industrial logging. Because ancient forest relics are often found on stressful non-commercial sites in the Cross Timbers and elsewhere, it is possible to design predictive models to locate the specific terrain where undisturbed forests are likely to survive. A predictive model for southern Osage County, Oklahoma, was developed based on the steep, infertile soils of the Niotaze-Darnell complex. We tested the model with field inspection and tree-ring analysis of fifty randomly selected belt transects, and 74% of the sampled terrain is still old-growth Cross Timbers woodland. This translates into 8200 ha of ancient Cross Timbers on this single site type in southern Osage County. The abundance of ancient forest in the Cross Timbers is not widely appreciated. However, large contiguous tracts of ancient Cross Timbers up to 700 ha were identified with this predictive model, strongly supporting inferences concerning the relatively undisturbed nature of this ecosystem.     

A stochastic model for the mutation-selection balance in an infinite asexual population with a genome of fixed size

A stochastic model is presented which describes the evolution of a genome of a haploid species in an infinite population. The genome is a finite set of elements. The elements are divided into different classes according to their effect on the fitness of the organism. Repeated mutations of the genome elements are permitted, in particular positive mutations are introduced. The distribution of the deleterious elements in the genome with respect to the impact on the fitness is given after the replication step. The steady state is fully described including the distribution and the fitness.     

A partial linear model in the outcome-dependent sampling setting to evaluate the effect of prenatal PCB exposure on cognitive function in children

Outcome-dependent sampling (ODS) has been widely used in biomedical studies because it is a cost-effective way to improve study efficiency. However, in the setting of a continuous outcome, the representation of the exposure variable has been limited to the framework of linear models, due to the challenge in terms of both theory and computation. Partial linear models (PLM) are a powerful inference tool to nonparametrically model the relation between an outcome and the exposure variable. In this article, we consider a case study of a PLM for data from an ODS design. We propose a semiparametric maximum likelihood method to make inferences with a PLM. We develop the asymptotic properties and conduct simulation studies to show that the proposed ODS estimator can produce a more efficient estimate than that from a traditional simple random sampling design with the same sample size. Using this newly developed method, we were able to explore an open question in epidemiology: whether in utero exposure to background levels of polychlorinated biphenyls (PCBs) is associated with children's intellectual impairment. Our model provides further insights into the relation between low-level PCB exposure and children's cognitive function. The results shed new light on a body of inconsistent epidemiologic findings.