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.     

Dynamics of games and genes: Discrete versus continuous time

It is shown that in the classical model of population genetics (Fisher-Wright-Haldane, discrete or continuous version) every solution p(t) converges to equilibrium for t → ∞. For related models of evolutionary games (with non-symmetric matrices) it is shown that the transformation that describes the dynamics is a diffeomorphism (in particular one-to-one).     

Practical considerations for measuring the effective reproductive number,Rt

Estimation of the effective reproductive number R_t is important for detecting changes in disease transmission over time. During the Coronavirus Disease 2019 (COVID-19) pandemic, policy makers and public health officials are using R_t to assess the effectiveness of interventions and to inform policy. However, estimation of R_t from available data presents several challenges, with critical implications for the interpretation of the course of the pandemic. The purpose of this document is to summarize these challenges, illustrate them with examples from synthetic data, and, where possible, make recommendations. For near real-time estimation of R_t, we recommend the approach of Cori and colleagues, which uses data from before time t and empirical estimates of the distribution of time between infections. Methods that require data from after time t, such as Wallinga and Teunis, are conceptually and methodologically less suited for near real-time estimation, but may be appropriate for retrospective analyses of how individuals infected at different time points contributed to the spread. We advise caution when using methods derived from the approach of Bettencourt and Ribeiro, as the resulting R_t estimates may be biased if the underlying structural assumptions are not met. Two key challenges common to all approaches are accurate specification of the generation interval and reconstruction of the time series of new infections from observations occurring long after the moment of transmission. Naive approaches for dealing with observation delays, such as subtracting delays sampled from a distribution, can introduce bias. We provide suggestions for how to mitigate this and other technical challenges and highlight open problems in R_t estimation.     

The decay of genetic variability in geographically structured populations. II

The ultimate rate of approach to equilibrium in the infinite stepping-stone model is calculated. The analysis is restricted to a single locus in the absence of selection, and every mutant is assumed to be new to the population. Let f(t, x) be the probability that two homologous genes separated by the vector x in generation t are the same allele. It is supposed that $\text{f(}\text{0, x}\text{) = O(x}{}^{\mathrm{?2?\eta }}\text{), η > 0,}\text{as}\text{x ≡ ¦}\text{x}\text{¦ → ∞}$. In the absence of mutation, f(t, x) tends to unity at the rate $\text{t}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}$ in one dimension and (ln t)^?1 in two dimensions. Thus, the loss of genetic variability in two dimensions is so slow that evolutionary forces not considered in this model would supervene long before a two-dimensional natural population became completely homogeneous. If the mutation rate, u, is not zero f(t, x) asymptotically approaches equilibrium at the rate $\text{(1 ? u)}{}^{\mathrm{2t}}\text{t}{}^{\mathrm{<mtext>?3</mtext><mtext>2</mtext>}}$ in one dimension and (1 ? u)^2tt^?1(lnt)^?2 in two dimensions. Integral formulas are presented for the spatial dependence of the deviation of f(t, x) from its stationary value as t → ∞, and for large separations this dependence is shown to be (const + x) in one dimension and (const + ln x) in two dimensions. All the results are the same for the Malécot model of a continuously distributed population provided the number of individuals per colony is replaced by the population density. The relatively slow algebraic and logarithmic rates of convergence for the infinite habitat contrast sharply with the exponential one for a finite habitat.     

Asymptotic Distributions of Coalescence Times and Ancestral Lineage Numbers for Populations with Temporally Varying Size

The distributions of coalescence times and ancestral lineage numbers play an essential role in coalescent modeling and ancestral inference. Both exact distributions of coalescence times and ancestral lineage numbers are expressed as the sum of alternating series, and the terms in the series become numerically intractable for large samples. More computationally attractive are their asymptotic distributions, which were derived in Griffiths (1984) for populations with constant size. In this article, we derive the asymptotic distributions of coalescence times and ancestral lineage numbers for populations with temporally varying size. For a sample of size n, denote by T_m the mth coalescent time, when m + 1 lineages coalesce into m lineages, and A_n(t) the number of ancestral lineages at time t back from the current generation. Similar to the results in Griffiths (1984), the number of ancestral lineages, A_n(t), and the coalescence times, T_m, are asymptotically normal, with the mean and variance of these distributions depending on the population size function, N(t). At the very early stage of the coalescent, when t → 0, the number of coalesced lineages n − A_n(t) follows a Poisson distribution, and as m → n, n(n ? 1)T_m/2N(0) follows a gamma distribution. We demonstrate the accuracy of the asymptotic approximations by comparing to both exact distributions and coalescent simulations. Several applications of the theoretical results are also shown: deriving statistics related to the properties of gene genealogies, such as the time to the most recent common ancestor (TMRCA) and the total branch length (TBL) of the genealogy, and deriving the allele frequency spectrum for large genealogies. With the advent of genomic-level sequencing data for large samples, the asymptotic distributions are expected to have wide applications in theoretical and methodological development for population genetic inference.     

Response of population size to changing vital rates in random environments

Population size and population growth rate respond to changes in vital rates like survival and fertility. In deterministic environments change in population growth rate alone determines change in population size. In random environments, population size at any time t is a random variable so that change in population size obeys a probability distribution. We analytically show that, in a density-independent population, the proportional change in population size with respect to a small proportional change in a vital rate has an asymptotic normal distribution. Its mean grows linearly at a rate equal to the elasticity of the long-term stochastic growth rate λ _S while the standard deviation scales as $\sqrt t$ . Consequently, a vital rate with a larger elasticity of λ _S may produce a larger mean change in population size compared to one with a smaller elasticity of λ _S. But a given percentage change in population size may be more likely when the vital rate with smaller elasticity is perturbed. Hence, the response of population size to perturbation of a vital rate depends not only on the elasticity of the population growth rate but also on the variance in change in population size. Our results provide a formula to calculate the probability that population size changes by a given percentage that works well even for short time periods.     

Characterization of the physiological state of fungi by dynamics of colony emergence on solid media

Poisson distribution was shown to be applicable to the dynamics of emergence of fungal colonies on plates inoculated with pure cultures or environmental samples, indicating the possibility for application of Hattori approach for assessment of the physiological state of fungi. The differences in physiological activity of different fungal species and genera, between spores and mycelia, or between the fungal populations from different environments, were revealed using the t _r (delay time for colony emergence) and λ (potential capacity for growth) parameters.     

The incorporation of cytochrome oxidase into newly-forming yeast mitochondrial membranes

Cytochrome c oxidase extracted from a yeast auxotroph grown on different fatty acid supplements, and assayed at various temperatures (0→37°), gives transition points (T_t) when the data are expressed in an Arrhenius plot: T_t for linoleic, oleic and elaidic acids were 7.7°, 10.2° and 21.8° respectively. Lipid anlaysis of isolated mitochondria established that there is a change in lipid profile when yeast is first grown aerobically on linoleic acid and then transferred to elaidic acid under anaerobic consitions. These two observations bave been used as a basis for the investigation of the assembly of cytochrome c oxidase when cells are oxygen induced for mitochondriogenesisvia a linoleic, N₂ elaidic, O₂ transfer experiment. A three-stage assembly process is proposed consisting of (i) 0.0→0.25 h, auto-assembly of oxidase from preformed, chloramphenicol-sensitive precursors and newly synthesized cycloheximide-sensitive precursors, (ii) 0.25 h→0.5h, a continuation of (i) plusde novo synthesis of new precursors of both kinds and (iii) 0.5 h onwards; normal, synchronisedde novo synthesis. This model predicts one population of oxidase molecules at 0.0 h (linoleic T_t) and two populations of oxidase at 0.5 h (end of stage (ii)); one from the anaerobic precursors (linoleic T_t) and one for thede nove, aerobic oxidase (elaidic T_t): these predictions are confirmed by Arrhenius profiles constructed from samples taken at 0.0, 0.5, 1.0 and 3.0 h after oxygen challenge.     

Probabilistic Model of Microbial Cell Growth,Division, and Mortality

After a short time interval of length δt during microbial growth, an individual cell can be found to be divided with probability P_d(t)δt, dead with probability P_m(t)δt, or alive but undivided with the probability 1 − [P_d(t) + P_m(t)]δt, where t is time, P_d(t) expresses the probability of division for an individual cell per unit of time, and P_m(t) expresses the probability of mortality per unit of time. These probabilities may change with the state of the population and the habitat''s properties and are therefore functions of time. This scenario translates into a model that is presented in stochastic and deterministic versions. The first, a stochastic process model, monitors the fates of individual cells and determines cell numbers. It is particularly suitable for small populations such as those that may exist in the case of casual contamination of a food by a pathogen. The second, which can be regarded as a large-population limit of the stochastic model, is a continuous mathematical expression that describes the population''s size as a function of time. It is suitable for large microbial populations such as those present in unprocessed foods. Exponential or logistic growth with or without lag, inactivation with or without a “shoulder,” and transitions between growth and inactivation are all manifestations of the underlying probability structure of the model. With temperature-dependent parameters, the model can be used to simulate nonisothermal growth and inactivation patterns. The same concept applies to other factors that promote or inhibit microorganisms, such as pH and the presence of antimicrobials, etc. With P_d(t) and P_m(t) in the form of logistic functions, the model can simulate all commonly observed growth/mortality patterns. Estimates of the changing probability parameters can be obtained with both the stochastic and deterministic versions of the model, as demonstrated with simulated data.     

Identity Coefficients in Finite Populations. I. Evolution of Identity Coefficients in a Random Mating Diploid Dioecious Population

Properties of identity relation between genes are discussed, and a derivation of recurrent equations of identity coefficients in a random mating, diploid dioecious population is presented. Computations are run by repeated matrix multiplication. Results show that for effective population size (N_e) larger than 16 and no mutation, a given identity coefficient at any time t can be expressed approximately as a function of (1—f), (1—f)³ and (1— f)⁶, where f is the mean inbreeding coefficient at time t. Tables are presented, for small N_e values and extreme sex ratios, showing the pattern of change in the identity coefficients over time. The pattern of evolution of identity coefficients is also presented and discussed with respect to N _eu, where u is the mutation rate. Applications of these results to the evolution of genetic variability within and between inbred lines are discussed.     

Rheology and Confocal Reflectance Microscopy as Probes of Mechanical Properties and Structure during Collagen and Collagen/Hyaluronan Self-Assembly

In this work, the gelation of three-dimensional collagen and collagen/hyaluronan (HA) composites is studied by time sweep rheology and time lapse confocal reflectance microscopy (CRM). To investigate the complementary nature of these techniques, first collagen gel formation is investigated at concentrations of 0.5, 1.0, and 1.5 mg/mL at 37°C and 32°C. The following parameters are used to describe the self-assembly process in all gels: the crossover time (t_c), the slope of the growth phase (k_g), and the arrest time (t_a). The first two measures are determined by rheology, and the third by CRM. A frequency-independent rheological measure of gelation, t_g, is also measured at 37°C. However, this quantity cannot be straightforwardly determined for gels formed at 32°C, indicating that percolation theory does not fully capture the dynamics of collagen network formation. The effects of collagen concentration and gelation temperature on k_g, t_c, and t_a as well as on the mechanical properties and structure of these gels both during gelation and at equilibrium are elucidated. Composite collagen/HA gels are also prepared, and their properties are monitored at equilibrium and during gelation at 37°C and 32°C. We show that addition of HA subtly alters mechanical properties and structure of these systems both during the gelation process and at equilibrium. This occurs in a temperature-dependent manner, with the ratio of HA deposited on collagen fibers versus that distributed homogeneously between fibers increasing with decreasing gelation temperature. In addition to providing information on collagen and collagen/HA structure and mechanical properties during gelation, this work shows new ways in which rheology and microscopy can be used complementarily to reveal details of gelation processes.     

Heterogeneity of Times Required for Germination and Outgrowth from Single Spores of Nonproteolytic Clostridium botulinum

Knowledge of the distribution of growth times from individual spores and quantification of this biovariability are important if predictions of growth in food are to be improved, particularly when, as for Clostridium botulinum, growth is likely to initiate from low numbers of spores. In this study we made a novel attempt to determine the distributions of times associated with the various stages of germination and subsequent growth from spores and the relationships between these stages. The time to germination (t_germ), time to emergence (t_emerg), and times to reach the lengths of one (t_C1) and two (t_C2) mature cells were quantified for individual spores of nonproteolytic C. botulinum Eklund 17B using phase-contrast microscopy and image analysis. The times to detection for wells inoculated with individual spores were recorded using a Bioscreen C automated turbidity reader and were compatible with the data obtained microscopically. The distributions of times to events during germination and subsequent growth showed considerable variability, and all stages contributed to the overall variability in the lag time. The times for germination (t_germ), emergence (t_emerg − t_germ), cell maturation (t_C1 − t_emerg), and doubling (t_C2 − t_C1) were not found to be correlated. Consequently, it was not possible to predict the total duration of the lag phase from information for just one of the stages, such as germination. As the variability in postgermination stages is relatively large, the first spore to germinate will not necessarily be the first spore to produce actively dividing cells and start neurotoxin production. This information can make a substantial contribution to improved predictive modeling and better quantitative microbiological risk assessment.     

The helix-coil transformation for tropocollagen solutions and its relationship to transformations involving the crystalline form of the protein

Collagen molecules can exist, in the aggregate crystalline state (C) and in solutions, as tropocollagen helices (H) or random coils (RC). We have compared the role of temperature, salt type, and salt concentration, C₈, on the transformations: C → RC (I), H → RC (II), and H → C (III) under near-isoelectric conditions. Processes I and II occur in the direction from ordered to disordered form on increasing temperature, while process III occurs in the direction crystal → dissolved helices on lowering temperature. The order of both anions and cations for increasing the stability of the dissolved random coiled form is similar for processes I and II, but the order of the anions for increasing the stability of the dissolved helices, according to process III, is reversed with respect to the Hofmeister series. T_I and T_II are greatly and continuously depressed by some salts on increasing C₈ while, for other salts, they are at first slightly depressed and then raised. T_III, instead, is at first raised and then lowered on increasing C₈. For a given salt the field of stability of each form is determined, on a pseudo-phase diagram, by the relative positions of the T_I versus C₈, the T_II versus C₈, and the T_III versus C₈, curves. The similarity of the effect of temperature, salt type, and concentration on processes I and II supports the contention that both processes are controlled by the interaction of the random coiled form with the salt solution. The phenomenon of crystallization by increasing temperature is complicated by large time effects and, often by lack of reversibility. Assuming, nevertheless, that there is an underlying equilibrium process, two possible interpretations are proposed. The peculiar trend of the T_III versus C₈ curve is justified on the basis that it reflects the usual change from salting in to salting out on increasing C₈, allowing for the fact that the dissolved phase is stable in the low-temperature range. The reversal of the order of anions for process III can be most simply justified in terms of the slight cat ionic character of the tropocollagen units. Accordingly, the dissolved helices will be more; stable of the crystalline form depending upon the occurrence of a net charge different from zero on the protein, as well as upon low ionic strength and low temperature. Minute alterations of KCl concentrations are able to cause isothermal precipitation of fibers and probably constitute a controlling factor for the fibrogenesis in vivo.     

Plasticity of the thermal requirements of exotherms and adaptation to environmental conditions

In exothermal organisms, temperature is an important determinant of the rate of ecophysiological processes, which monotonically increase between the minimum (t_{d min}) and maximum (t_{d max}) temperatures typical for each species. In insects, t_{d min} and t_{d max} are correlated and there is a approximately 20°C interval (thermal window W_T = t_{d max} − t_{d min}) between them over which insects can develop. We assumed that other exotherms have similar thermal windows because the thermal kinetics of their physiological processes are similar. In this study, we determined the thermal requirements for germination in plants. Seeds of 125 species of Central European wild herbaceous and crop plants were germinated at nine constant temperatures between 5 and 37°C, and the time to germination of 50% of the seeds D and rate of germination R (=1/D) were determined for each temperature and the Lactin model used to determine t_{d min}, t_{d max}, and W_T. The average width of the thermal windows for seeds was significantly wider (mean 24°C, 95% CI 22.7–24.2°C), varied more (between 14.5 and 37.5°C) and development occurred at lower temperatures than recorded for insects. The limiting temperatures for germination, t_{d min} and t_{d max}, were not coupled, so the width of the thermal window increased with both a decrease in t_{d min} and/or increase in t_{d max}. Variation in W_T was not associated with taxonomic affiliation, adult longevity, or domestication of the different species, but tends to vary with seed size. Plants are poor at regulating their temperature and cannot move to a more suitable location and as a consequence have to cope with wider ranges in temperatures than insects and possibly do this by having wider thermal windows. Synthesis: The study indicated specificity of W_T in different exotherm taxa and/or their development stages.     

The asymptotic normality of a simple batch epidemic process

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 $\text{[α}{}^2\text{X}{}_{\mathrm{n}}\text{(t){n ?X}{}_{\mathrm{n}}\text{(t) +X}{}_{\mathrm{n}}\text{(0)}]}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{, t∈[0, ∞), α∈(0, ∞)}$. 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.     

<Emphasis Type="Italic">l</Emphasis>=1 diocotron instability of single charged plasmas

The linear stability analysis of the l=1 diocotron perturbations in a low density single charged plasma confined in a cylindrical Penning trap is critically revisited. Particular attention is devoted to the instability due to the presence of one or more stationary points in the radial profile of the azimuthal rotation frequency. The asymptotic analysis of Smith and Rosenbluth for the case of a single-bounded plasma column (algebraic instability proportional to t^1/2) is generalized in a few respects. In particular, the existence of unperturbed density profiles that give rise to l=1 algebraic instabilities growing with time proportionally to t^1?1/m, m≥ 3 being the order of a stationary point in the rotation frequency profile, and even proportionally to t, is pointed out. It is also shown that smoothing the density jumps of a multistep density profile can convert algebraically growing perturbations into exponentially decaying modes. The relevant damping rates are computed. The asymptotic analysis (t → ∞) of the fundamental diocotron perturbations is then generalized to the case of a cylindrical Penning trap with an additional coaxial inner conductor. It is shown that the algebraic instability found in the case of a single-bounded plasma column becomes exponential at longer times. The relevant linear growth rate is computed by a suitable inverse Laplace transform (contour integral in the complex plane). In the particular case of an uncharged inner conductor of radius a, the growth rate is shown to scale as a^4/3 for a → 0. The theoretical results are compared with the numerical solution of the linearized two-dimensional drift Poisson equations.     

Diminished Confidence Levels in the Interval Estimation of an Unknown Regressor in Model I of Linear Regression

For the model y=β₀ + β₁ x + e (model I of linear regression) in the literature confidence estimators of an unknown position x₀ are given at which either the expectation of y is given (see FIELLER, 1944; FINNEY, 1952), or realizations of y are given (see GRAYBILL, 1961). These confidence regions with level 1—α need not be intervals. The occurrence of interval shape is a random event. Its probability is equal to the power of the t test for the examination of the hypothesis H: β₁ = 0. The papers mentioned above claim to provide confidence intervals with level 1 ? α. But because of the restriction of (1 —α)—confidence regions to intervals the true confidence probability is the conditional probability W_c: W_c = P (the confidence region covers x₀| the region has interval shape). Here this conditional probability is shown to be less than 1 —α. Evidence on the possible deviations from 1 —α has been obtained by simulations.     

Simulations of tokamak discharges with fast L-H transitions

Results are presented from the simulations of discharges with fast L-H transitions in the JET tokamak. During a transition, electron temperature perturbations propagate into the plasma core over a time much shorter than the transport time characteristic of this device. It is shown that the experimentally observed variations in the electron temperature may be caused by the change in the particle source intensity in the plasma when the atomic flux decreases, which is detected from the drop in the intensity of the D_α hydrogen spectral line. Hence, the experiments under consideration can be explained without the assumption about the nonlocal character of transport processes in tokamaks, which was made in some papers devoted to JET experiments. The plasma component responsible for the apparent nonlocal character of transport processes is the neutral component, whose propagation time across the plasma column is sufficiently short (t<100 μs). __________ Translated from Fizika Plazmy, Vol. 28, No. 1, 2002, pp. 3–8. Original Russian Text Copyright ? 2002 by Leonov.     

Estimation of Noise-Free Variance to Measure Heterogeneity

Variance is a statistical parameter used to characterize heterogeneity or variability in data sets. However, measurements commonly include noise, as random errors superimposed to the actual value, which may substantially increase the variance compared to a noise-free data set. Our aim was to develop and validate a method to estimate noise-free spatial heterogeneity of pulmonary perfusion using dynamic positron emission tomography (PET) scans. On theoretical grounds, we demonstrate a linear relationship between the total variance of a data set derived from averages of n multiple measurements, and the reciprocal of n. Using multiple measurements with varying n yields estimates of the linear relationship including the noise-free variance as the constant parameter. In PET images, n is proportional to the number of registered decay events, and the variance of the image is typically normalized by the square of its mean value yielding a coefficient of variation squared (CV ²). The method was evaluated with a Jaszczak phantom as reference spatial heterogeneity (CV_r ²) for comparison with our estimate of noise-free or ‘true’ heterogeneity (CV _t ²). We found that CV _t ² was only 5.4% higher than CV _r ². Additional evaluations were conducted on 38 PET scans of pulmonary perfusion using ¹³NN-saline injection. The mean CV _t ² was 0.10 (range: 0.03–0.30), while the mean CV ² including noise was 0.24 (range: 0.10–0.59). CV _t ² was in average 41.5% of the CV ² measured including noise (range: 17.8–71.2%). The reproducibility of CV _t ² was evaluated using three repeated PET scans from five subjects. Individual CV _t ² were within 16% of each subject''s mean and paired t-tests revealed no difference among the results from the three consecutive PET scans. In conclusion, our method provides reliable noise-free estimates of CV _t ² in PET scans, and may be useful for similar statistical problems in experimental data.     

Water-soluble glycoproteins from Cannabis sativa (Thailand)

Glycoproteins were extracted with water from leaves of Cannabis sativa grown from seeds of Thailand origin. By ion exchange chromatography the material was separated into a neutral and an acidic fraction. Both glycoprotein fractions contained arabinose, galactose, glucose, mannose and xylose, and in addition rhamnose and galacturonic acid were present in the acidic fraction. The carbohydrate moieties were investigated by methylation analysis and Smith-degradation, whereas the glycopeptide linkage was studied by alkaline hydrolysis in the presence of NaBH₄ and Na₂SO₃, respectively. This linkage was shown to be of the serine-O-galactoside type. The carbohydrate structure is highly branched, the majority of branches terminating in arabinofuranose end groups. Arabinose is also present in the chain, predominantly (1 → 4)- and/or (1 → 5)-linked. Galactose makes up most of the main chain as (1 → 3)-linked residues but also constitutes end groups and branch points, as do mannose and/or glucose. Xylose and rhamnose are present as (1 → 4)- and (1 → 2)-linked units, respectively. Galacturonic acid is assumed to be (1 → 4)- linked with some branching at 3 position. The amino acid hydroxyproline, present in the glycoprotein of South African Cannabis leaves, was absent in the corresponding Thailand material.