Large Initial Population Research Articles

A Markov Model for the Spread of Viruses in an Open Population

Inspired by methods of queueing theory, we propose a Markov model for the spread of viruses in an open population with an exogenous flow of infectives. We apply it to the diffusion of AIDS and hepatitis C diseases among drug users. From a mathematical point of view, the difference between the two viruses is shown in two parameters: the probability of curing the disease (which is 0 for AIDS but positive for hepatitis C) and the infection probability, which seems to be much higher for hepatitis. This model bears some resemblance to the M/M/∞ queueing system and is thus rather different from the models based on branching processes commonly used in the epidemiological literature. We carry out an asymptotic analysis (large initial population) and show that the Markov process is close to the solution of a nonlinear autonomous differential system. We prove both a law of large numbers and a functional central limit theorem to determine the speed of convergence towards the limiting system. The deterministic system itself converges, as time tends to ∞, to an equilibrium point. We then show that the sequence of stationary probabilities of the stochastic models shrinks to a Dirac measure at this point. This means that in a large population and for long-term analysis, we may replace the individual-based microscopic stochastic model with the macroscopic deterministic system without loss of precision. Moreover, we show how to compute the sensitivity of any functional of the Markov process with respect to a slight variation of any parameter of the model. This approach is applied to the spread of diseases among drug users, but could be applied to many other case studies in epidemiology.

A Markov Model for the Spread of Viruses in an Open Population

Inspired by methods of queueing theory, we propose a Markov model for the spread of viruses in an open population with an exogenous flow of infectives. We apply it to the diffusion of AIDS and hepatitis C diseases among drug users. From a mathematical point of view, the difference between the two viruses is shown in two parameters: the probability of curing the disease (which is 0 for AIDS but positive for hepatitis C) and the infection probability, which seems to be much higher for hepatitis. This model bears some resemblance to the M/M/∞ queueing system and is thus rather different from the models based on branching processes commonly used in the epidemiological literature. We carry out an asymptotic analysis (large initial population) and show that the Markov process is close to the solution of a nonlinear autonomous differential system. We prove both a law of large numbers and a functional central limit theorem to determine the speed of convergence towards the limiting system. The deterministic system itself converges, as time tends to ∞, to an equilibrium point. We then show that the sequence of stationary probabilities of the stochastic models shrinks to a Dirac measure at this point. This means that in a large population and for long-term analysis, we may replace the individual-based microscopic stochastic model with the macroscopic deterministic system without loss of precision. Moreover, we show how to compute the sensitivity of any functional of the Markov process with respect to a slight variation of any parameter of the model. This approach is applied to the spread of diseases among drug users, but could be applied to many other case studies in epidemiology.

Dynamics of long-term genomic selection

BackgroundSimulation and empirical studies of genomic selection (GS) show accuracies sufficient to generate rapid gains in early selection cycles. Beyond those cycles, allele frequency changes, recombination, and inbreeding make analytical prediction of gain impossible. The impacts of GS on long-term gain should be studied prior to its implementation.MethodsA simulation case-study of this issue was done for barley, an inbred crop. On the basis of marker data on 192 breeding lines from an elite six-row spring barley program, stochastic simulation was used to explore the effects of large or small initial training populations with heritabilities of 0.2 or 0.5, applying GS before or after phenotyping, and applying additional weight on low-frequency favorable marker alleles. Genomic predictions were from ridge regression or a Bayesian analysis.ResultsAssuming that applying GS prior to phenotyping shortened breeding cycle time by 50%, this practice strongly increased early selection gains but also caused the loss of many favorable QTL alleles, leading to loss of genetic variance, loss of GS accuracy, and a low selection plateau. Placing additional weight on low-frequency favorable marker alleles, however, allowed GS to increase their frequency earlier on, causing an initial increase in genetic variance. This dynamic led to higher long-term gain while mitigating losses in short-term gain. Weighted GS also increased the maintenance of marker polymorphism, ensuring that QTL-marker linkage disequilibrium was higher than in unweighted GS.ConclusionsLosing favorable alleles that are in weak linkage disequilibrium with markers is perhaps inevitable when using GS. Placing additional weight on low-frequency favorable alleles, however, may reduce the rate of loss of such alleles to below that of phenotypic selection. Applying such weights at the beginning of GS implementation is important.

The Discrete Beverton-Holt Model with Periodic Harvesting in a Periodically Fluctuating Environment

We investigate the effect of constant and periodic harvesting on the Beverton-Holt model in a periodically fluctuating environment. We show that in a periodically fluctuating environment, periodic harvesting gives a better maximum sustainable yield compared to constant harvesting. However, if one can also fix the environment, then constant harvesting in a constant environment can be a better option, especially for sufficiently large initial populations. Also, we investigate the combinatorial structure of the periodic sequence of carrying capacities and its effect on the maximum sustainable yield. Finally, we leave some questions worth further investigations.

The Beverton–Holt model with periodic and conditional harvesting

In this theoretical study, we investigate the effect of different harvesting strategies on the discrete Beverton–Holt model in a deterministic environment. In particular, we make a comparison between the constant, periodic and conditional harvesting strategies. We find that for large initial populations, constant harvest is more beneficial to both the population and the maximum sustainable yield. However, periodic harvest has a short-term advantage when the initial population is low, and conditional harvest has the advantage of lowering the risk of depletion or extinction. Also, we investigate the periodic character under each strategy and show that periodic harvesting drives population cycles to be multiples (period-wise) of the harvesting period.

Economic transformation, population growth and the long-run world income distribution

We present and calibrate a model where trade with advanced economies spurs development, and trade opportunities depend on the relative population in advanced and developing countries. As developing countries become advanced, prospects improve for the remaining developing countries. If population growth differentials between developing and advanced economies are small, economic development accelerates over time. Otherwise, long-run global prosperity requires a sufficiently large initial population in advanced countries. More open countries develop faster, but more openness by all developing countries may only modestly increase their aggregate growth. China's development may hurt developing countries in the short-run, but improves their long-run prospects.

Emergence and resilience of cooperation in the spatial prisoner's dilemma via a reward mechanism

We study the problem of the emergence of cooperation in the spatial Prisoner's Dilemma. The pioneering work by Nowak and May [1992. Evolutionary games and spatial chaos. Nature 415, 424–426] showed that large initial populations of cooperators can survive and sustain cooperation in a square lattice with imitate-the-best evolutionary dynamics. We revisit this problem in a cost–benefit formulation suitable for a number of biological applications. We show that if a fixed-amount reward is established for cooperators to share, a single cooperator can invade a population of defectors and form structures that are resilient to re-invasion even if the reward mechanism is turned off. We discuss analytically the case of the invasion by a single cooperator and present agent-based simulations for small initial fractions of cooperators. Large cooperation levels, in the sustainability range, are found. In the conclusions we discuss possible applications of this model as well as its connections with other mechanisms proposed to promote the emergence of cooperation.

Estimação de componentes de variância utilizando-se inferência Bayesiana e freqüentista em dados simulados sob heterogeneidade de variâncias

Foi simulado um genoma de 3.000 centimorgans de comprimento considerando uma única característica quantitativa, governada por 800 locos com dois alelos por loco. Segundo a estrutura genômica proposta, foram simulados 1.500 machos e 1.500 fêmeas que formaram a população-base. A partir da população-base foram formadas duas populações iniciais, uma grande e outra pequena. Dois tipos de estruturas de heterogeneidade de variâncias foram inseridos nas populações iniciais: heterogeneidade de variância genética aditiva e heterogeneidade de variâncias genética aditiva e ambiental. Para obtenção destas estruturas, foram feitos descartes estratégicos dos valores genéticos aditivos e ambientais de acordo com o tipo de heterogeneidade e o nível de variabilidade desejada: alta, média ou baixa. Os componentes de variância foram estimados por meio da metodologia Bayesiana via Amostragem de Gibbs e pelo método REML. Para a metodologia Bayesiana, foram utilizados três níveis de informação a priori: não-informativo, pouco informativo e informativo. Os métodos comparados apresentaram resultados semelhantes quando priors não-informativos foram utilizados e as populações de tamanho grande, de modo geral, apresentaram melhores estimativas. Para as populações pequenas, as análises realizadas considerando os níveis de variabilidade separadamente apresentaram maiores problemas, em virtude do pequeno tamanho das subpopulações formadas. Observou-se, para a metodologia Bayesiana, que o aumento no nível de informação a priori influenciou positivamente as estimativas dos componentes de variância, principalmente para as populações pequenas. Portanto, na presença de heterogeneidade de variâncias, as metodologias se comportam de forma semelhante. Entretanto, para populações pequenas a metodologia Bayesiana conduz a melhores estimativas quando informações adicionais estão disponíveis.

Harmonic moments of inhomogeneous branching processes

We study the harmonic moments of Galton-Watson processes that are possibly inhomogeneous and have positive values. Good estimates of these are needed to compute unbiased estimators for noncanonical branching Markov processes, which occur, for instance, in the modelling of the polymerase chain reaction. By convexity, the ratio of the harmonic mean to the mean is at most 1. We prove that, for every square-integrable branching mechanism, this ratio lies between 1-A/k and 1-A/k for every initial population of size k>A. The positive constants A and Aͤ are such that A≥Aͤ, are explicit, and depend only on the generation-by-generation branching mechanisms. In particular, we do not use the distribution of the limit of the classical martingale associated with the Galton-Watson process. Thus, emphasis is put on nonasymptotic bounds and on the dependence of the harmonic mean upon the size of the initial population. In the Bernoulli case, which is relevant for the modelling of the polymerase chain reaction, we prove essentially optimal bounds that are valid for every initial population size k≥1. Finally, in the general case and for sufficiently large initial populations, similar techniques yield sharp estimates of the harmonic moments of higher degree.

Harmonic moments of inhomogeneous branching processes

We study the harmonic moments of Galton-Watson processes that are possibly inhomogeneous and have positive values. Good estimates of these are needed to compute unbiased estimators for noncanonical branching Markov processes, which occur, for instance, in the modelling of the polymerase chain reaction. By convexity, the ratio of the harmonic mean to the mean is at most 1. We prove that, for every square-integrable branching mechanism, this ratio lies between 1-A/k and 1-A/k for every initial population of size k>A. The positive constants A and Aͤ are such that A≥Aͤ, are explicit, and depend only on the generation-by-generation branching mechanisms. In particular, we do not use the distribution of the limit of the classical martingale associated with the Galton-Watson process. Thus, emphasis is put on nonasymptotic bounds and on the dependence of the harmonic mean upon the size of the initial population. In the Bernoulli case, which is relevant for the modelling of the polymerase chain reaction, we prove essentially optimal bounds that are valid for every initial population size k≥1. Finally, in the general case and for sufficiently large initial populations, similar techniques yield sharp estimates of the harmonic moments of higher degree.

Likely path to extinction in simple branching models with large initial population

We give explicit formulae for most likely paths to extinction in simple branching models when initial population is large. In discrete time, we study the Galton-Watson process, and in continuous time, the branching diffusion. The most likely paths are found with the help of the large deviation principle (LDP). We also find asymptotics for the extinction probability, which gives a new expression in continuous time and recovers the known formula in discrete time. Due to the nonnegativity of the processes, the proof of LDP at the point of extinction uses a nonstandard argument of independent interest.

NEUTRINO SPECTRUM DISTORTION DUE TO OSCILLATIONS AND ITS BBN EFFECT

We study the distortion of electron neutrino energy spectrum due to oscillations with the sterile neutrino νe↔νs, for different initial populations of the sterile state δNs at the onset of oscillations. The influence of this spectrum distortion on Big Bang Nucleosynthesis is analyzed. Only the case of an initially empty sterile state was studied in previous publications. The primordial abundance of 4He is calculated for all possible δNs:0≤δNs≤1 in the model of oscillations, effective after electron neutrino decoupling, for which the spectrum distortion effects on the neutron–proton transitions are the strongest. It is found that the spectrum distortion effect may be dominant, not only in the case of small δNs, but also in the case of large initial population of νs. For example, in the resonant case it may play a considerable role even for very large δNs~0.8. Cosmological constraints on neutrino mixing for small δNs are discussed.

Controlled synthesis of 2-D and 3-D dendritic platinum nanostructures.

Seeding and autocatalytic reduction of platinum salts in aqueous surfactant solution using ascorbic acid as the reductant leads to remarkable dendritic metal nanostructures. In micellar surfactant solutions, spherical dendritic metal nanostructures are obtained, and the smallest of these nanodendrites resemble assemblies of joined nanoparticles and the nanodendrites are single crystals. With liposomes as the template, dendritic platinum sheets in the form of thin circular disks or solid foamlike nanomaterials can be made. Synthetic control over the morphology of these nanodendrites, nanosheets, and nanostructured foams is realized by using a tin-porphyrin photocatalyst to conveniently and effectively produce a large initial population of catalytic growth centers. The concentration of seed particles determines the ultimate average size and uniformity of these novel two- and three-dimensional platinum nanostructures.

Life-history models of extinction: a test with island spiders.

This study analyzes extinction patterns for two species of orb spiders monitored annually on 77 islands over a continuous 20-yr period. One species, Argiope argentata, has large populations sometimes crashing quickly to extinction and a much weaker relation of extinction likelihood to population size than does the other species, Metepeira datona. Demographic models were built for both species and matched against observations. Differences between the species in life-history traits-estimated with measurements from the field-together with incorporation of demographic stochasticity, a population ceiling, and environmental stochasticity, were necessary to fit the observed extinction curves. As predicted from life-history patterns, long-term population growth rates (and hence predicted extinction probabilities) are relatively very sensitive to values of juvenile survivorship. Models are also sensitive to variation in the population ceiling and environmental noise, which tend to act in a complementary manner. A simple model with no age structure was able to fit the data on large initial population sizes but not on small initial population sizes, showing that life cycle characteristics interact with the various sources of stochasticity and hence have to be taken into account to produce a precise model of the extinction process.

Factors affecting the success of translocations of the black-faced impala in Namibia

This study analysed 21 translocations of the vulnerable black-faced impala ( Aepyceros melampus petersi) to 20 Namibian game farms that occurred between 1970 and 2001, seeking characteristics of the translocated populations and the release sites that significantly correlated with the success of the translocations. Characteristics considered were: initial population size; presence of cheetah and leopard; area; habitat type; occurrence within the historical range of the subspecies and occurrence of trophy hunting. Success of translocations was described by whether the population had a positive growth rate. The success rate of translocations of black-faced impala (62%) was higher than shown in other studies of vertebrate translocations. Initial population size was paramount to the success of translocations. Releases of larger populations were more likely to lead to positive population growth rates than were releases of small populations. The presence of cheetah also influenced the success of translocated populations. In the presence of cheetah, small populations translocated to game farms were significantly less likely to be viable than larger populations. Recommendations for the management of this vulnerable antelope include introducing large initial populations, ideally more than 15 animals, rather than attempting to eliminate cheetah following translocations of impala.

Branching Systems with Long-Living Particles at the Critical Dimension

A spatial branching process is considered in which particles have a lifetime law with a tail index smaller than one. It is shown that at the critical dimension, unlike classical branching particle systems the population does not suffer local extinction when started from a spatially homogeneous Poissonian initial population. In fact, persistent convergence to a mixed Poissonian particle system is shown. The random intensity of the limiting process is characterized in law by the random density in a space point of a related age-dependent superprocess at a fixed time. The proof relies on a refined study of the system starting from asymptotically large but finite initial populations.

Long term behavior of solutions of the Lotka-Volterra system under small random perturbations

A stochastic analogue of the Lotka –Volterra model for predator–prey relationshipis obtained when the birth rate of the prey and the death rate of the predator are perturbed by independent white noises with intensities of order $\varepsilon^2$, where $\varepsilon>0$ is a small parameter.The evolution of this system is studied on large time intervals of $O(1/\varepsilon^2)$. It is shown that for small initial population sizes the stochastic model is adequate, whereas for large initial population sizes it is not as suitable, because it leads to ever-increasing fluctuations in population sizes, although it still precludes extinction. New results for the classical deterministic Lotka–Volterra model are obtained by a probabilistic method; we show in particular that large population sizes of predator and prey coexist only for a very short time, and most of the time one of the populations is exponentially small.

ＦＴ‐ＩＣＲ質量分析計におけるイオン間クーロン相互作用によるピーク合体現象とイオン損失の３次元多粒子シミュレーション

A three dimensional many particle model is proposed for computer simulation of Fourier transform ion cyclotron resonance (FT-ICR) mass spectra. The model is relatively simple and includes Coulomb interactions between ions in an FT-ICR cell. FT-ICR mass spectra of CO+ and C2H4+ mixtures are simulated at CO+:C2H4+=1:1, 2:1, and 1:2 for low, medium, and high ion density. The Coulomb interaction at CO+:C2H4+=1:1 appears to be the most significant from the results on peak coalescence phenomenon. The simulation performance is estimated through comparison with Mitchell's simulation model. Ion loss due to the space charge effect during detection period is investigated for ions of the same (C2H4+), two different (C2H4+: m/z 50.0=1:1) and two closely spaced (CO+:C2H4+=1:1) mass-to-charge ratios as a function of initial ion population. In the case of different m/z ions, particularly for closely spaced m/z ions, the ion loss ratio increases considerably with initial ion population. On the other hand, for the same m/z ions no significant ion loss occurs even at a large initial ion population.

The effects of reduced agent-area insecticide treatments for rangeland grasshopper (Orthoptera: Acrididae) control on bird densities.

The effects of insecticidal control programs for grasshoppers on the population density of birds were assessed under operational conditions in two rangeland sites in Wyoming during an outbreak (19 to 56 grasshoppers m(-2) on >1,500 ha). Malathion, carbaryl, and fipronil were evaluated using large-scale (>/=260 ha) reduced agent-area treatments (RAAT), in which these chemicals were applied at 50% of the standard rate (except fipronil, for which there is no standard in North America) to 80, 50, and 33% of the areas (alternating treated and untreated swaths), respectively. Grasshopper population densities were reduced by 73 to 99%, at 14 and 28 days after treatment. Significant differences among treatments within the sites were apparent, and despite disparities in the vegetation, topography, and grasshopper populations between sites, some consistent patterns of response by bird populations were found. Relative to pretreatment densities and untreated plots, at 14 days after treatment, bird densities were enhanced at one site and suppressed at the other; at 28 days after treatment bird populations were generally suppressed at both sites. However, absolute densities increased in some plots, indicating that birds still had adequate food. There appeared to be no relationship between posttreatment grasshopper and bird densities, perhaps due to the large initial insect populations. Fipronil plots generally had higher avian population densities (nongregarious insectivores and total birds) than carbaryl plots, which had bird densities greater than or equal to malathion plots. These differences were probably a function of both the toxicological properties of the compounds and the proportion of the infested land that was treated. Based on these findings, we must consider the possibility that compounds with relatively high toxicity to birds and long residual effects applied to widely spaced swaths may cause no more (or perhaps less) environmental impact than less toxic and persistent insecticides that must be applied to large proportions of infested rangeland. http://link.springer-ny. com/link/service/journals/00244/bibs/37n4p519.html</++ +HEA

Response of selected tree species to culling of introduced Australian brushtail possums Trichosurus vulpecula at Waipoua Forest, Northland, New Zealand

Population trends of colonising brushtail possums Trichosurus vulpecula were monitored between 1981 and 1993 at Waipoua Forest in Northland, New Zealand. Canopy defoliation and possum-related damage to eight plant indicator species were assessed annually for four years after an aerial poison operation in 1990, to determine whether the operation had reduced possum numbers and halted possum-related damage to the vegetation. Trap catch estimates of possum density increased by over 70% during the 1980s. The poison operation killed 87% of possums, and ongoing leg-hold trapping (1991–1993) maintained population levels at 7–9% trap catch. Reduction of the possum population did not produce a significant positive short-term vegetation response, but did halt the continuing downward trend in vegetation condition evident in nearby forested areas where possum densities were not reduced. Other measures of possum impact (browsed foliage, stem damage) improved significantly within a year of the poison operation, and may prove more sensitive short-term indicators of reduced possum-related damage. The apparent ability of residual possum populations to retard vegetation recovery has important implications for forest health strategies based on intermittent poison operations. While a large initial population reduction is required to halt vegetation decline, an ability to maintain residual possum populations at very low levels appears necessary for significant vegetation recovery. Targetting the increased resources now available for reducing possum populations to fewer areas of higher conservation value may well return greater ecological dividends than attempts to sustain long-term reduction of possum populations over large areas of more modified indigenous forest