Population mixing and the adaptive divergence of quantitative traits in discrete populations: a theoretical framework for empirical tests

Empirical tests for the importance of population mixing in constraining adaptive divergence have not been well grounded in theory for quantitative traits in spatially discrete populations. We develop quantitative-genetic models to examine the equilibrium difference between two populations that are experiencing different selective regimes and exchanging individuals. These models demonstrate that adaptive divergence is negatively correlated with the rate of population mixing (m, most strongly so when m is low), positively correlated with the difference in phenotypic optima between populations, and positively correlated with the amount of additive genetic variance (G, most strongly so when G is low). The approach to equilibrium is quite rapid (fewer than 50 generations for two populations to evolve 90% of the distance to equilibrium) when either heritability or mixing are not too low (h2 > 0.2 or m > 0.05). The theory can be used to aid empirical tests that: (1) compare observed divergence to that predicted using estimates of population mixing, additive genetic variance/covariance, and selection; (2) test for a negative correlation between population mixing and adaptive divergence across multiple independent population pairs; and (3) experimentally manipulate the rate of mixing. Application of the first two of these approaches to data from two well-studied natural systems suggests that population mixing has constrained adaptive divergence for color patterns in Lake Erie water snakes (Nerodia sipedon), but not for trophic traits in sympatric pairs of benthic and limnetic stickleback (Gasterosteus aculeatus). The theoretical framework we outline should provide an improved basis for future empirical tests of the role of population mixing in adaptive divergence.     

Inferring the Structure of Social Contacts from Demographic Data in the Analysis of Infectious Diseases Spread

Social contact patterns among individuals encode the transmission route of infectious diseases and are a key ingredient in the realistic characterization and modeling of epidemics. Unfortunately, the gathering of high quality experimental data on contact patterns in human populations is a very difficult task even at the coarse level of mixing patterns among age groups. Here we propose an alternative route to the estimation of mixing patterns that relies on the construction of virtual populations parametrized with highly detailed census and demographic data. We present the modeling of the population of 26 European countries and the generation of the corresponding synthetic contact matrices among the population age groups. The method is validated by a detailed comparison with the matrices obtained in six European countries by the most extensive survey study on mixing patterns. The methodology presented here allows a large scale comparison of mixing patterns in Europe, highlighting general common features as well as country-specific differences. We find clear relations between epidemiologically relevant quantities (reproduction number and attack rate) and socio-demographic characteristics of the populations, such as the average age of the population and the duration of primary school cycle. This study provides a numerical approach for the generation of human mixing patterns that can be used to improve the accuracy of mathematical models in the absence of specific experimental data.     

On the existence of stable pairing distributions

We formulate and analyze pair-formation models for multiple groups with general pairing rates and arbitrary mixing probabilities. Under the assumption of constant recruitment rates and equal average duration of all types of partnerships, we have shown that the dynamics are relatively simple because of the monotonicity properties of the dynamical system associated with the pairing/mixing of heterogeneous populations of male and female individuals. In fact, we have shown that the corresponding asymptotic stable paired distribution is given precisely by the asymptotic values of the matrices that prescribe the mixing/contact structure. In other words, if the sizes of the mixing subpopulations of males and females are asymptotically constant and if the average durations of partnerships are about the same regardless of type, then the matrices that describe the mixing between subpopulations also characterize the distribution of paired types. Alternatively, if the distribution of the average duration of relationships between individuals has a large variance then it may be impossible to detect any relationship between the mixing/contact structure and the observed distribution of paired types. The study of models with constant per-capita recruitment rates give rise to homogeneous systems of degree one. The analysis of the dynamics of pairs for models with exponentially growing populations of singles is complicated. So far, we are only able to classify the stability of all non-strictly positive boundary exponential solutions. From our incomplete analysis, it is not possible to detect necessary and sufficient conditions for the existence and stability of strictly interior exponential solutions. We cannot rule out the possibility of oscillations. The mathematical problems associated with the stability of exponential solutions of dynamical systems of degree one are of relevance in demography, epidemiology, and population dynamics.On leave from University of Alabama in Huntsville     

Modeling optimal treatment strategies in a heterogeneous mixing model

Background

Many mathematical models assume random or homogeneous mixing for various infectious diseases. Homogeneous mixing can be generalized to mathematical models with multi-patches or age structure by incorporating contact matrices to capture the dynamics of the heterogeneously mixing populations. Contact or mixing patterns are difficult to measure in many infectious diseases including influenza. Mixing patterns are considered to be one of the critical factors for infectious disease modeling.

Methods

A two-group influenza model is considered to evaluate the impact of heterogeneous mixing on the influenza transmission dynamics. Heterogeneous mixing between two groups with two different activity levels includes proportionate mixing, preferred mixing and like-with-like mixing. Furthermore, the optimal control problem is formulated in this two-group influenza model to identify the group-specific optimal treatment strategies at a minimal cost. We investigate group-specific optimal treatment strategies under various mixing scenarios.

Results

The characteristics of the two-group influenza dynamics have been investigated in terms of the basic reproduction number and the final epidemic size under various mixing scenarios. As the mixing patterns become proportionate mixing, the basic reproduction number becomes smaller; however, the final epidemic size becomes larger. This is due to the fact that the number of infected people increases only slightly in the higher activity level group, while the number of infected people increases more significantly in the lower activity level group. Our results indicate that more intensive treatment of both groups at the early stage is the most effective treatment regardless of the mixing scenario. However, proportionate mixing requires more treated cases for all combinations of different group activity levels and group population sizes.

Conclusions

Mixing patterns can play a critical role in the effectiveness of optimal treatments. As the mixing becomes more like-with-like mixing, treating the higher activity group in the population is almost as effective as treating the entire populations since it reduces the number of disease cases effectively but only requires similar treatments. The gain becomes more pronounced as the basic reproduction number increases. This can be a critical issue which must be considered for future pandemic influenza interventions, especially when there are limited resources available.

     

Analysis of Population Structure: A Unifying Framework and Novel Methods Based on Sparse Factor Analysis

We consider the statistical analysis of population structure using genetic data. We show how the two most widely used approaches to modeling population structure, admixture-based models and principal components analysis (PCA), can be viewed within a single unifying framework of matrix factorization. Specifically, they can both be interpreted as approximating an observed genotype matrix by a product of two lower-rank matrices, but with different constraints or prior distributions on these lower-rank matrices. This opens the door to a large range of possible approaches to analyzing population structure, by considering other constraints or priors. In this paper, we introduce one such novel approach, based on sparse factor analysis (SFA). We investigate the effects of the different types of constraint in several real and simulated data sets. We find that SFA produces similar results to admixture-based models when the samples are descended from a few well-differentiated ancestral populations and can recapitulate the results of PCA when the population structure is more “continuous,” as in isolation-by-distance models.     

On the effects of the spatial distribution in an epidemic model based on cellular automaton

Several theoretical studies on disease propagation assume that individuals belonging to different groups regarding their health conditions are homogeneously distributed over the space. This is the well-known homogenous mixing assumption, which supports epidemiological models written in terms of ordinary differential or difference equations. Here, we consider that the host population infected by a contagious pathogen is composed by two groups with distinct traits and habits, which can be homogeneously mixed or not. The pathogen propagation is modeled by using an asynchronous probabilistic cellular automaton. Our main goal is to examine how a heterogeneous spatial distribution of these groups affects the endemic state. We noted that homogeneous distribution favors the occurrence of oscillations in the population composition. Surprisingly, we found out that the propagation dynamics of the heterogeneous distribution can also be described by a set of ordinary difference equations.     

Importance of correlations among matrix entries in stochastic models in relation to number of transition matrices

Stochastic matrix models are used to predict population viability and the risk of extinction. Different stochastic methods require different amounts of estimation effort and may lead to divergent estimates. We used 16 transition matrices collected from ten populations of the perennial herb Primula veris to compare population estimates produced by different stochastic methods, such as selection of matrices, selection of vital rates, selection of matrix elements, and Tuljapurkar's approximation. Specifically, we tested the reliability of the methods using different numbers of transition matrices, and examined the importance of correlations among matrix entries. When correlations among matrix entries were included in the models, selection of vital rates produced the lowest and Tuljapurkar's approximation produced the highest estimates of mean population growth rates. Selection of matrices and matrix elements often produced nearly similar population estimates. Simulations based on incompletely estimated correlations among matrix entries considerably differed from those based on all correlations estimated, particularly when correlations were strong. The magnitude of correlations among matrix entries depended on the number of matrices, which made it difficult to generalize correlations within a species. Given that selection of vital rates or matrix elements is used, correlations among matrix entries should usually be included in the model, and they should preferably be estimated from the present data rather than according to other information of the species.     

A New Method for Estimating Growth Transition Matrices

Summary The vast majority of population models work using age or stage not length but there are many cases where animals cannot be aged sensibly or accurately. For these cases length‐based models form the logical alternative but there has been little work done to develop and compare different methods of estimating growth transition matrices to be used in such models. This article demonstrates how a consistent Bayesian framework for estimating growth parameters and a novel method for constructing length transition matrices accounts for variation in growth in a clear and consistent manner and avoids potential subjective choices required using more established methods. The inclusion of the resultant growth uncertainty in population assessment models and the potential impact on management decisions is also addressed.     

Dynamics of a feline retrovirus (FeLV) in host populations with variable spatial structure.

The predictions of epidemic models are remarkably affected by the underlying assumptions concerning host population dynamics and the relation between host density and disease transmission. Furthermore, hypotheses underlying distinct models are rarely tested. Domestic cats (Felis catus) can be used to compare models and test their predictions, because cat populations show variable spatial structure that probably results in variability in the relation between density and disease transmission. Cat populations also exhibit various dynamics. We compare four epidemiological models of Feline Leukaemia Virus (FeLV). We use two different incidence terms, i.e. proportionate mixing and pseudo-mass action. Population dynamics are modelled as logistic or exponential growth. Compared with proportionate mixing, mass action incidence with logistic growth results in a threshold population size under which the virus cannot persist in the population. Exponential growth of host populations results in systems where FeLV persistence at a steady prevalence and depression of host population growth are biologically unlikely to occur. Predictions of our models account for presently available data on FeLV dynamics in various populations of cats. Thus, host population dynamics and spatial structure can be determinant parameters in parasite transmission, host population depression, and disease control.     

A cautionary note on elasticity analyses in a ternary plot using randomly generated population matrices

Matrix population models are one of the most common mathematical models in ecology, which describe the dynamics of stage-structured populations and provide us many population statistics. One of the statistics, elasticity onto population growth rate, is frequently used and represents the degree of the relative impact of life history parameters to the population growth rate. Due to the utility of elasticities for cross-taxonomic comparisons, Silvertown and his coauthors have published multiple papers and reported the relationship between elasticities and life forms (or life history) in multiple plant species, using a triangle map (called “ternary plot”). To understand why their elasticities are located in specific regions of the ternary plot, we constructed four archetypes of population matrices, from which we simulated 24,000 randomly generated population matrices and obtained the consequent elasticities. We found a large discrepancy when comparing our results to those in Silvertown et al.'s study (Conserv Biol 10:591–597, 1996): for our simulated matrices where rapid transitions were not allowed (e.g., trees), the elasticity distribution resulted in a line across the ternary plot. We provided the mathematical proof for this result, and found that its slope depends on matrix dimension. We also used 1230 matrices from the COMPADRE Plant Matrix Database and calculated the elasticities. Our simulated results were validated with field data from COMPADRE: two straight lines appeared in the ternary plot. Furthermore, we answered several addressed questions, such as, “Is there any special elasticity distribution in matrices with high population growth rates?” and “Why are the elasticities of natural populations concentrated in the upper half of the ternary plot?”.     

Towards the virtual brain: network modeling of the intact and the damaged brain

Neurocomputational models of large-scale brain dynamics utilizing realistic connectivity matrices have advanced our understanding of the operational network principles in the brain. In particular, spontaneous or resting state activity has been studied on various scales of spatial and temporal organization including those that relate to physiological, encephalographic and hemodynamic data. In this article we focus on the brain from the perspective of a dynamic network and discuss the role of its network constituents in shaping brain dynamics. These constituents include the brain's structural connectivity, the population dynamics of its network nodes and the time delays involved in signal transmission. In addition, no discussion of brain dynamics would be complete without considering noise and stochastic effects. In fact, there is mounting evidence that the interaction between noise and dynamics plays an important functional role in shaping key brain processes. In particular, we discuss a unifying theoretical framework that explains how structured spatio-temporal resting state patterns emerge from noise driven explorations of unstable or stable oscillatory states. Embracing this perspective, we explore the consequences of network manipulations to understand some of the brain's dysfunctions, as well as network effects that offer new insights into routes towards therapy, recovery and brain repair. These collective insights will be at the core of a new computational environment, the Virtual Brain, which will allow flexible incorporation of empirical data constraining the brain models to integrate, unify and predict network responses to incipient pathological processes.     

Target reproduction numbers for reaction-diffusion population models

A very important population threshold quantity is the target reproduction number, which is a measure of control effort required for a target prevention, intervention or control. This concept, as a generalization of type reproduction number, was first introduced in Shuai et al. (J Math Biol 67:1067–1082, 2013) for nonnegative matrices with immediate applications to compartmental population models of ordinary differential equations. The current paper is devoted to the study of all target reproduction numbers for reaction-diffusion population models with compartmental structure. It turns out that the target reproduction number can be regarded as the basic reproduction number of a modified system, where the state of newborn individuals is limited to the target control set and the offspring from the non-target set is regarded as a part of the transition. In other words, the target reproduction number can be interpreted as the expected number of offspring in a specific target set that a primary newborn individual of the same set would produce during its lifetime. We also characterize the target reproduction number so that it can be easily computed numerically for reaction-diffusion models. At the end, we demonstrate our theoretical observations using two examples.

     

Temporal quota corrections based on timing of harvest in a small game species

Theoretical models have shown that the effect of removing a given proportion of the population can be profoundly different if the harvest takes place late in the season compared to early. We explore the effect of these differences using theoretical models based on the concept of demographic value and empirical data on seasonal patterns of natural mortality risk in two contrasting populations of willow ptarmigan in Norway. Based on the theoretical models, we found that changes in the timing of harvest have a much stronger effect in populations with relatively low annual survival compared to populations characterized by longevity typical for species with slow life histories. Also, the timing of harvest is more influential in cases with constant mortality hazards compared to a situation with density-dependent natural mortality. Empirical data from two study populations of willow ptarmigan showed large deviations from the theoretical predictions of models with both constant and density-dependent mortality hazards. There were also large differences in both the temporal pattern and magnitude of annual survival between the two ptarmigan populations (54 vs 26% annual survival). Site differences illustrate the importance of knowledge of both the magnitude and temporal pattern of natural mortality hazard to be able to correctly predict the effect of changing the timing of harvest in a population. In the two ptarmigan populations, we show how harvest quotas can be adjusted in accordance to the empirical estimates of natural mortality risk and how this determines the effects of shifting from harvesting early to late in the annual cycle.     

The exact values of the probability of fixation of underdominant chromosomal rearrangements.

A numerical analysis of the probability of fixation of a chromosomal mutation with partial sterility of the heterozygote in a single population is performed. Three different genetic models are considered: the first model entails constant selection against the heterozygote and is the model almost universally used in previous works; in the other two models selection against the heterozygote depends on its frequency. The exact values of the fixation probability are found by iterating transition matrices with genotype specification. Differences in results among models are small. The exact values found in the first model are compared to estimates obtained from approximations. Solutions based on diffusion models give good approximations when selection against the heterozygote is low, especially if the population is very small. For the higher values of the selection coefficient against the heterozygote, the estimates are rather imprecise, especially when the populations are not very small.     

On the methods to assess significance in nestedness analyses

Use of Z values to evaluate nestedness significance is a common procedure. An appealing alternative to the use of Z values is that of using a value of relative nestedness (RN). However, there is no agreement on the preferable procedures to generate the null matrices needed to compute both Z and RN. In general, it is recommended to use restrictive null models that take into account row and column totals. The two most widely used null models of this kind, namely, FF and CE [that generate matrices with row and column sums equal (FF) or proportional (CE) to the row and column totals of the original matrix, respectively], are very different in terms of restrictiveness. We performed a set of comparative analyses on both theoretical and real matrices to investigate the differences between the use of Z and RN values, and between the use of FF and CE null models, when NODF (Nestedness metric based on overlap and decreasing fill) or ρ(A) (i.e., the largest eigenvalue of the adjacency matrix) are used to measure nestedness. We found no difference in the use of Z or RN values. On the other hand, we found that different combinations of nestedness measures and null models may lead to inconsistent outcomes. Our results offer some clarity on a few issues that, despite playing a central role in the practical application of nestedness analysis, have been little explored, and highlight the need for the definition of some commonly accepted standards.     

植物种群生存力分析研究进展

对十多年来国外植物PVA的研究进行了综合评述;具体分析了影响植物种群生存力的各种随机性因子及确定性因子;总结了植物PVA研究的方法步骤及采用的模拟模型;探讨了植物PVA的难点,PVA对管理措施的评价效果;并提出对今后植物PVA的研究展望,认为PVA是研究濒危植物种群灭绝及评价管理或保护措施的有力工具;发展描述复杂种间关系的多种种的PVA模型以及包含多个影响因素的PVA应用模型是未来植物PVA的研究方向。     

Modeling protein evolution with several amino Acid replacement matrices depending on site rates

Most protein substitution models use a single amino acid replacement matrix summarizing the biochemical properties of amino acids. However, site evolution is highly heterogeneous and depends on many factors that influence the substitution patterns. In this paper, we investigate the use of different substitution matrices for different site evolutionary rates. Indeed, the variability of evolutionary rates corresponds to one of the most apparent heterogeneity factors among sites, and there is no reason to assume that the substitution patterns remain identical regardless of the evolutionary rate. We first introduce LG4M, which is composed of four matrices, each corresponding to one discrete gamma rate category (of four). These matrices differ in their amino acid equilibrium distributions and in their exchangeabilities, contrary to the standard gamma model where only the global rate differs from one category to another. Next, we present LG4X, which also uses four different matrices, but leaves aside the gamma distribution and follows a distribution-free scheme for the site rates. All these matrices are estimated from a very large alignment database, and our two models are tested using a large sample of independent alignments. Detailed analysis of resulting matrices and models shows the complexity of amino acid substitutions and the advantage of flexible models such as LG4M and LG4X. Both significantly outperform single-matrix models, providing gains of dozens to hundreds of log-likelihood units for most data sets. LG4X obtains substantial gains compared with LG4M, thanks to its distribution-free scheme for site rates. Since LG4M and LG4X display such advantages but require the same memory space and have comparable running times to standard models, we believe that LG4M and LG4X are relevant alternatives to single replacement matrices. Our models, data, and software are available from http://www.atgc-montpellier.fr/models/lg4x.     

Equivalence of aggregated Markov models of ion-channel gating

One cannot always distinguish different Markov models of ion-channel kinetics solely on the basis of steady-state kinetic data. If two generator (or transition) matrices are related by a similarity transformation that does not combine states with different conductances, then the models described by these generator matrices have the same observable steady-state statistics. This result suggests a procedure for expressing the model in a unique form, and sometimes reducing the number of parameters in a model. I apply the similarity transformation procedure to a number of simple models. When a model specifies the dependence of the rates of transition on an experimentally variable parameter such as the concentration of a ligand or the membrane potential, the class of equivalent models may be further restricted, but a model is not always uniquely determined even under these conditions. Voltage-step experiments produce non-stationary data that can also be used to distinguish models.     

Equivalence of truncated count mixture distributions and mixtures of truncated count distributions

This article is about modeling count data with zero truncation. A parametric count density family is considered. The truncated mixture of densities from this family is different from the mixture of truncated densities from the same family. Whereas the former model is more natural to formulate and to interpret, the latter model is theoretically easier to treat. It is shown that for any mixing distribution leading to a truncated mixture, a (usually different) mixing distribution can be found so that the associated mixture of truncated densities equals the truncated mixture, and vice versa. This implies that the likelihood surfaces for both situations agree, and in this sense both models are equivalent. Zero-truncated count data models are used frequently in the capture-recapture setting to estimate population size, and it can be shown that the two Horvitz-Thompson estimators, associated with the two models, agree. In particular, it is possible to achieve strong results for mixtures of truncated Poisson densities, including reliable, global construction of the unique NPMLE (nonparametric maximum likelihood estimator) of the mixing distribution, implying a unique estimator for the population size. The benefit of these results lies in the fact that it is valid to work with the mixture of truncated count densities, which is less appealing for the practitioner but theoretically easier. Mixtures of truncated count densities form a convex linear model, for which a developed theory exists, including global maximum likelihood theory as well as algorithmic approaches. Once the problem has been solved in this class, it might readily be transformed back to the original problem by means of an explicitly given mapping. Applications of these ideas are given, particularly in the case of the truncated Poisson family.