Four Applications of a Bivariate Pareto Distribution

The following model, of “latent structure” type, is considered: in each subpopulation, X and Y are random variables drawn independently from the same exponential distribution, and the parameter of the exponential distribution varies between subpopulations with a Gamma density. Over the whole population, X and Y are then positively correlated, and jointly have a bivariate PARETO distribution. Four examples show how this distribution is useful in analysing ordered contingency tables in which the two dimensions can be regarded as alternative measures of the same thing: the injuries to the two drivers in a road accident, or the severity of a lesion present in a patient as assessed by two physicians, for instance. Two extensions are considered: (a) allowing X and Y to have Gamma distributions, with each subpopulation having the same shape parameter but different scale parameters; (b) allowing the scale parameter for Y to be correlated with the scale parameter for X, rather than being identical to it. A new bivariate distribution with three shape parameters is derived, expressed in terms of a generalised hypergeometric function.     

The convergence in distribution of some simple epidemics

A group of n susceptible individuals exposed to a contagious disease isconsidered. It is assumed that at each point in time one or more susceptible individuals can contract the disease. The progress of this simple batch epidemic is modeled by a stochastic process X_n(t), t[0, ∞), representing the number of infectiveindividuals at time t. In this paper our analysis is restricted to simple batch epidemics with transition rates given by [α²X_n(t){n −X_n(t) +X_n(0)}]^1/2, t[0, ∞), α(0, ∞). This class of simple batch epidemics generalizes a model used and motivated by McNeil (1972) to describe simple epidemic situations. It is shown for this class of simple batch epidemics, that X_n(t), with suitable standardization, converges in distribution as n→∞ to a normal random variable for all t(0, t₀), and t₀ is evaluated.     

lea (likelihood‐based estimation of admixture): a program to estimate simultaneously admixture and time since the admixture event

We consider an admixture event, T generations in the past, where two ‘parental’ populations, P₁ and P₂, of size N₁ and N₂, contribute different proportions into the gene pool of an admixed population, H of size N_h. lea (likelihood‐based estimator of admixture) is a program which allows the user to obtain the posterior distribution of the parameters of the model. This includes p₁, the contribution of P₁, and t₁, t₂ and t_h, the time since the admixture event (scaled by the population size) for the three populations. lea allows the user to stop and restart the analyses at any time.     

Analysis of an SEIRS epidemic model with two delays

 A disease transmission model of SEIRS type with exponential demographic structure is formulated. All newborns are assumed susceptible, there is a natural death rate constant, and an excess death rate constant for infective individuals. Latent and immune periods are assumed to be constants, and the force of infection is assumed to be of the standard form, namely proportional to I(t)/N(t) where N(t) is the total (variable) population size and I(t) is the size of the infective population. The model consists of a set of integro-differential equations. Stability of the disease free proportion equilibrium, and existence, uniqueness, and stability of an endemic proportion equilibrium, are investigated. The stability results are stated in terms of a key threshold parameter. More detailed analyses are given for two cases, the SEIS model (with no immune period), and the SIRS model (with no latent period). Several threshold parameters quantify the two ways that the disease can be controlled, by forcing the number or the proportion of infectives to zero. Received 8 May 1995; received in revised form 7 November 1995     

Time series — information and prediction

A time series Y _t can be transformed into another time series V _t by means of a linear transformation. Should the matrix of that transformation have an inverse, the pair (Y _t, V _t) is called invertible. Based on the decomposition procedure for stationary time series introduced in a previous paper it is shown that a sufficient condition for the invertibility of the pair (Y _t, V _t) is that V _t be the first component of Y _t, i.e. V _t = V _t ¹ . By the invertibility property V _t ¹ can be used for forecasting, that is, predictions are made on V _t ¹ which is then transformed into Y _t. This is accomplished by means of a special kind of predictor permitting to make one-step-ahead forecasts in a straightforward way. Since the first component depends on a parameter i.e. V _t ¹ = V _t ¹ (), a procedure is proposed that allows us to find the optimal parameter value, = ₀. Thus, it is shown that better forecasting accuracy may result by fitting a simple autoregression to the first component V _t ¹ (₀) than if the process Y _t were described by a more elaborate model. Model building is therefore no longer a prerequisite in forecasting. The forecasting procedure is then extended so as to cope with the homogeneous nonstationary case, and examples are given to illustrate the forecasting accuracy as compared to customary model-based approaches. In the light of these results the problem of the information conveyed by the values of the series is discussed in terms of the spreading rate concept, thus highlighting the role of the current time value, as well as that of the remote values of the series, in forecasting stationary and nonstationary time series.     

The Nature of Conformational Correlations in α- and β-Anomers of Nucleic Acid Components in Aqueous Solution by Nuclear Magnetic Resonance

Abstract

The solution distribution of combinations of the sugar ring puckering domains, C2′endo(S), C3′endo(N), and C4′-C5′ rotamers, +sc(g⁺), ap(t), -sc(g^?), in α and β-anomers in ribo- and deoxyribo- pyrimidine nucleic acid components can be determined from vicinal coupling constants (M. Remin, J. Biomol. Str. Dyn. 2, 211 (1984). A general correlation pattern with a conformational constant λ, reflecting an intrinsic physical property of the sugar - side chain ensemble, is developed and expressed in terms of four principles:

I) The +sc rotamer contributes to the C3′endo population to a higher extent (1 - Y_t) than to C2′endo,(l-Y_t-Y_g-/X_s).

II) The ap rotamer contributes to both C2′endo and C3′endo populations to the same extent (Y_t).

III) The—sc rotamer contributes only to the C2′endo population, (Y_g-/X_s).

IV) The molar fractions X_s, Y_t and Y_g- of conformations C2′endo, ap and—sc, respectively, are strongly correlated, λ = (Y_g-/X_s)/Y_t ≈ 0.5, and therefore Y_t is a basic variable parameter which determines all others in the correlation pattern.

In α-anomers, regardless of the type and conformation of the sugar ring and base, the molar fraction Y_t = 0.37 ± 0.02. This finding means that different α-anomers show one correlation pattern free of the influence of the base. In β-anomers, structure and conformation of the base are important factors which modulate (through Y_t) the correlation pattern, conserving its fundamental features. Y_t is considerably increased by a syn-oriented pyrimidine base, but decreases when the base is anti. The transition from anti to syn orientation of the base is followed by destabilization of (C2′endo, +sc) in favor of (C3′endo, ap). The principles of conformational correlations rationalize a variety of correlations observed in the past.     

Age and growth of Carcharhinus leucas in the northern Gulf of Mexico: incorporating variability in size at birth

Age and growth rates of bull shark Carcharhinus leucas[n = 255; 555–2230 mm fork length (L_F)] from the northern Gulf of Mexico were estimated from ring counts on vertebral sections collected from fishery‐dependent and ‐independent surveys. Two growth models were fitted to observed data: the von Bertalanffy growth model (VBGM) with t₀ as the third parameter and a modified version of the VBGM using a fixed size‐at‐birth intercept as the third parameter. To address the variability in size‐at‐birth, a Monte Carlo simulation was incorporated into the size‐at‐birth intercept. The sex‐specific growth models were not significantly different, allowing a sexes combined model to be generated. The traditional VBGM predicted a theoretical maximum size (L_∞) of 3007·1 mm L_F, a growth coefficient (K) of 0·042 year^?1 and a theoretical age at zero length (t₀) of –6·844 years. The modified VBGM with a fixed size‐at‐birth intercept of 565 mm L_F predicted an L_∞ of 2289·2 mm L_F and a K value of 0·089 year^?1. When comparing model estimates to previously published information, the traditional VBGM predicted a significantly lower theoretical maximum size and a higher growth coefficient than those produced using data collected during the 1980s. Overall, results obtained using the VBGM with a fixed size‐at‐birth produced more biologically realistic parameters than that of the VBGM with t₀. The Monte‐Carlo simulation incorporating variability in size‐at‐birth produced similar results to the VBGM using a fixed size‐at‐birth. This study provides the first attempt to incorporate variability at size‐at‐birth and provide measurements of variability around the individual parameter estimates for an elasmobranch.     

Molecular-Dynamics Simulations for the Density Autocorrelation Function in a Supercooled Fluid Phase

Abstract

We have re-calculated the self part of the density autocorrelation function F_s(k, t) (incoherent scattering function) for the binary soft-sphere fluid with a much longer molecular-dynamics (MD) simulation than our previous MD calculations, and with a larger system size (N = 4000) to a longer time window as well as to study a system-size dependence, if it exists. The full density autocorrelation function F(k, t) was also computed. It is found that all F(k, t)'s that we have computed in this work can be fitted over a wide range of time steps (at least over three figures of the decay) by a Williams-Watts stretched exponential function F_s(k, t) = A exp [— (t/t ₀)^β], where A, β and t ₀ are adjustable parameters. Other significant dynamical behaviours were also presented in mean square displacements and non-Gaussian parameters for highly supercooled fluids with N = 4000. The present results are compatible to our previous computations with N = 500, but a significant size dependence is suggested.     

Association between GSTM1 polymorphism and DNA adduct concentration in the occupational workers exposed to PAHs: A meta-analysis

Increasing investigations have been conducted on the association between DNA adducts and glutathione S-transferase Mu 1 (GSTM1) null genotype in occupationally exposed population. However, the results were controversial. The objective of the present study was to perform a meta-analysis to better understand the possible association between DNA adduct levels and GSTM1 genotype in occupational exposure population. Among a total of 167 literature searched from frequently-used databases, 7 articles corresponding to the specific criteria were enrolled into the meta-analysis. There was a significant increase of DNA adduct levels in occupationally exposed workers compared with control groups (p = 0.003). Additionally, DNA adduct levels among the carriers of null GSTM1 were significantly higher than those of active GSTM1 carriers in exposure workers (p = 0.017). Egger's test (p = 0.056) and Begg's test (p = 0.368) indicated that there was no evidence of publication bias. In conclusion, workers exposed to polycyclic aromatic hydrocarbons (PAHs) were at high risk to form DNA adducts, and the occupationally exposed workers who carried null GSTM1 were more susceptible to damage from PAHs.     

Life history and initial assessment of fishing impacts on the by‐catch species Dules auriga (Teleostei: Serranidae) in southern Brazil

The life history of Dules auriga, a small hermaphrodite serranid species inhabiting deep waters and a frequent component of the discarded catch of bottom trawling in southern Brazil, was studied to assess the fishery effects on the stock through the estimation of the remaining spawning‐potential ratio. Sampling was conducted throughout a year and included specimens to determine sex, maturity and age. Age was validated by the edge type and marginal‐increment analysis. The oldest and the largest individuals were 9 years and 195 mm total length. Growth parameters fitted to the von Bertalanffy equation were L_∞ = 178·34 mm, k = 0·641 year^?1 and t₀ = ?0·341 years. Length and age at first maturity were 140·72 mm and 2 years, respectively. The reproductive season was throughout the austral spring and summer. The assessment of the effects of fishing showed that it may have resulted in a loss of 50% of the spawning potential. This loss may be higher when taking into account the uncertainty in the life‐history parameters and could be considered of concern for the population. Fast growth, moderate longevity, long spawning season, small size and age at maturity make D. auriga relatively resilient to the removal of biomass by fishing. When considering the uncertainty, however, the losses of the spawning potential have been severely reducing the population resilience in the face of ecosystem changes.     

Approximation of the Basic Reproduction Number <Emphasis Type="Italic">R</Emphasis><Subscript>0</Subscript> for Vector-Borne Diseases with a Periodic Vector Population

The main purpose of this paper is to give an approximate formula involving two terms for the basic reproduction number R ₀ of a vector-borne disease when the vector population has small seasonal fluctuations of the form p(t) = p ₀ (1+ε cos (ωt − φ)) with ε ≪ 1. The first term is similar to the case of a constant vector population p but with p replaced by the average vector population p ₀. The maximum correction due to the second term is (ε²/8)% and always tends to decrease R ₀. The basic reproduction number R ₀ is defined through the spectral radius of a linear integral operator. Four numerical methods for the computation of R ₀ are compared using as example a model for the 2005/2006 chikungunya epidemic in La Réunion. The approximate formula and the numerical methods can be used for many other epidemic models with seasonality. MSC 92D30 ⋅ 45C05 ⋅ 47A55     

The Estimation of a Multivariate Fitness Function from Several Samples Taken from a Population

The situation is considered where the multivariate distribution of certain variables X₁, X₂, …, X_p is changing with time in a population because natural selection related to the X's is taking place. It is assumed that random samples taken from the population at times t₁, t₂, …, t_s are available and it is desirable to estimate the fitness function w_t(x₁, x₂,…,x_p) which shows how the number of individuals with X_i = x_i, i = 1, 2, …, p at time t is related to the number of individuals with the same X values at time zero. Tests for population changes are discussed and indices of the selection on the population dispersion and the population mean are proposed. The situation with a multivariate normal distribution is considered as a special case. A maximum likelihood method that can be applied with any form of population distribution is proposed for estimating w_t. The methods discussed in the paper are illustrated with data on four dimensions of male Egyptian skulls covering a time span from about 4500 B.C. to about 300 A.D. In this case there seems to have been very little selection on the population dispersion but considerable selection on means.     

Confidence intervals for demographic projections based on products of random matrices

This work is concerned with the growth of age-structured populations whose vital rates vary stochastically in time and with the provision of confidence intervals. In this paper a model Y_{t + 1}(ω) = X_{t + 1}(ω)Y_t(ω) is considered, where Y_t is the (column) vector of the numbers of individuals in each age class at time t, X is a matrix of vital rates, and ω refers to a particular realization of the process that produces the vital rates. It is assumed that {X_i} is a stationary sequence of random matrices with nonnegative elements and that there is an integer n₀ such that any product X_{j + n0} ··· X_{j + 1}X_j has all its elements positive with probability one. Then, under mild additional conditions, strong laws of large numbers and central limit results are obtained for the logarithms of the components of Y_t. Large-sample estimators of the parameters in these limit results are derived. From these, confidence intervals on population growth and growth rates can be constructed. Various finite-sample estimators are studied numerically. The estimators are then used to study the growth of the striped bass population breeding in the Potomac River of the eastern United States.     

Asymptotic theory for analyzing dose-response survival data with application to the low-dose extrapolation problem

A model of the form P(t, d)=1−;exp{-(t)σ^K_i=0q_idⁱ}, q_i 0, is proposed for analyzing dose-response survival data with right censoring. The q_i's in the dose polynomial are estimated by maximizing the Cox partial likelihood, and given these estimates. Λ(t) is estimated nonparametrically by an estimator proposed by Breslow. Large-sample properties of these estimators are established. Estimates and related large-sample properties are provided for the “virtually safe dose” and other parameters for assessing low-dose carcinogenic risk as a function of age, using data from animal carcinogenesis experiments.     

A simulation program based on a structured population model for biotechnological yeast processes

Summary A segregated population model for budding yeasts and a simulation program based on it are presented. They enable the study of bioprocesses utilizing yeasts in steady and perturbed conditions and in particular the comparison between the model predictions and the experimental results obtained by flow cytometry, which allows the measurement of segregated parameters of cell populations.Nomenclature a genealogical age - A parameter of the budding law - CV coefficient of variation - F _in(t) volumetric input flow - F _out(t) volumetric output flow - h parameter of the division law - K _s parameter of the Monod's law - m cell mass - M _i discretized cell mass - m _b (a,s) critical mass level for budding - m _p cell mass at the time of budding - n(t) cell number per unit volume - n _p number of sub-populations - n _c number of channels - p (a, i, j, k) discrete density function - Q parameter of the budding law - s(t) substrate concentration - S _in(t) substrate concentration in the input flow - t time - T _m minimal length of the budded phase - V(t) culture volume - x(t) biomass concentration - Y yield coefficient - channel width - (s) specific growth rate - _max parameter of the Monod's law     

A trait‐based approach to predict population genetic structure in bees

Understanding population genetic structure is key to developing predictions about species susceptibility to environmental change, such as habitat fragmentation and climate change. It has been theorized that life‐history traits may constrain some species in their dispersal and lead to greater signatures of population genetic structure. In this study, we use a quantitative comparative approach to assess if patterns of population genetic structure in bees are driven by three key species‐level life‐history traits: body size, sociality, and diet breadth. Specifically, we reviewed the current literature on bee population genetic structure, as measured by the differentiation indices Nei's G_ST, Hedrick's G′_ST, and Jost's D. We then used phylogenetic generalised linear models to estimate the correlation between the evolution of these traits and patterns of genetic differentiation. Our analyses revealed a negative and significant effect of body size on genetic structure, regardless of differentiation index utilized. For Hedrick's G′_ST and Jost's D, we also found a significant impact of sociality, where social species exhibited lower levels of differentiation than solitary species. We did not find an effect of diet specialization on population genetic structure. Overall, our results suggest that physical dispersal or other functions related to body size are among the most critical for mediating population structure for bees. We further highlight the importance of standardizing population genetic measures to more easily compare studies and to identify the most susceptible species to landscape and climatic changes.     

108 years of change in spatial pattern following selective harvest of a Pinus ponderosa stand in northern Arizona,USA

Questions: How did an initial tree harvest in 1894 influence the spatial and temporal patterns of Pinus ponderosa recruitment? How do these patterns compare to our understanding of P. ponderosa stand dynamics prior to Euro‐American settlement? How might spatial pattern information, particularly with respect to patch characteristics, inform current restoration and management practices? Location: A 2.59‐ha permanent sample plot in the Fort Valley Experimental Forest, Flagstaff, Arizona. The plot was selectively harvested in 1894 and measured in 1909 and 2002. Methods: We used historical stem‐map and ledger data, contemporary data, and dendrochronological techniques to reconstruct stand structure (tree size, age, location) in three scenarios: (1) unharvested (1909), (2) harvested (1909), and (3) contemporary (2002). We used Clark and Evans' R, Ripley's K(t) univariate analysis, and correlogram analysis to assess the spatial pattern in each scenario. We also used Ripley's K₁₂(t) bivariate analysis and tree age data to examine spatial and temporal recruitment patterns as observed in the contemporary scenario. Results and Conclusions: The unharvested stand was aggregated at scales up to 28 m. The selective harvest accentuated the spatial patchiness of the stand in 1909 and changed spatial patterns by homogenizing tree size within patches. By 2002, the stand was a single patch dominated by small trees. Post‐harvest recruitment patterns were not spatially random; Pinus seedlings initially established in natural grass openings and then proceeded to fill‐in stump patches created by harvesting. Knowledge of spatial pattern should be explicitly incorporated into restoration activities in these forests.     

A mathematical model for malaria involving differential susceptibility,exposedness and infectivity of human host

The main purpose of this article is to formulate a deterministic mathematical model for the transmission of malaria that considers two host types in the human population. The first type is called “non-immune” comprising all humans who have never acquired immunity against malaria and the second type is called “semi-immune”. Non-immune are divided into susceptible, exposed and infectious and semi-immune are divided into susceptible, exposed, infectious and immune. We obtain an explicit formula for the reproductive number, R ₀ which is a function of the weight of the transmission semi-immune-mosquito-semi-immune, R _0a , and the weight of the transmission non-immune-mosquito-non-immune, R _0e . Then, we study the existence of endemic equilibria by using bifurcation analysis. We give a simple criterion when R ₀ crosses one for forward and backward bifurcation. We explore the possibility of a control for malaria through a specific sub-group such as non-immune or semi-immune or mosquitoes.     

Growth of E. coli in a stirred tank and in an air lift tower reactor with an outer loop

E. coli ATCC 11105 was cultivated in a 10-1 stirred tank reactor and in a 60-1 tower loop reactor in batch and continuous operation. By on-line measurements of O₂ and CO₂ concentrations in the outlet gas, pH, temperature, cell mass concentration X as well as dissolved O₂ concentration along the tower in the broth, gas holdup, broth recirculation rate through the loop and by offline measurements of substrate concentration DOC and cell mass concentration along the tower, the maximum specific growth rate _m , yield coefficients Y _X/S. Y _X/DOC and were evaluated in stirred tank and tower loop in batch and continuous cultures with and without motionless mixers in the tower and at different broth circulation rates through the loop. To control the accuracy of the measurements the C balance was calculated and 95% of the C content was covered.The biological parameters determined depend on the mode of operation as well as on the reactor used. Furthermore, they depend on the recirculation rate of the broth and built-ins in the tower. The unstructured cell and reactor models are unable to explain these differences. Obviously, structured cell and reactor models are needed. The cell mass concentration can be determined on line by NADH fluorescence in balanced growth, if the model parameters are determined under the same operational conditions in the same reactor.List of Symbols a, b empirical parameters in Eq. (1) - CPR kg/(m³ h) CO₂ production rate - C kg/m³ concentration - D l/h dilution rate - DOC kg/m³ dissolved organic carbon - I net. fluorescence intensity - K _S kg/m³ Monod constant - k _L a l/h volumetric mass transfer coefficient - OTR kg/(m³ h) oxygen transfer rate - OUR kg/(m³ h) oxygen utilization rate - RQ = CPR/OUR respiratory quotient - S kg/m³ substrate concentration - t h,min, s time - t _u min recirculation time - t _M min mixing time - v m³/h volumetric flow rate through the loop - X kg/m³ (dry) cell mass concentration - Y _X/S yield coefficient of cell mass with regard to the consumed substrate - Y _X/DOC yield coefficient of the cell mass with regard to the consumed DOC - Y _X/O yield coefficient of the cell mass with regard to the consumed oxygen - Z relative distance in the tower from the aerator with regard to the height of the aerated broth - l/h specific growth rate - _m l/h maximum specific growth rate Indices f feed - e outlet