Phylodynamic Inference for Structured Epidemiological Models

Coalescent theory is routinely used to estimate past population dynamics and demographic parameters from genealogies. While early work in coalescent theory only considered simple demographic models, advances in theory have allowed for increasingly complex demographic scenarios to be considered. The success of this approach has lead to coalescent-based inference methods being applied to populations with rapidly changing population dynamics, including pathogens like RNA viruses. However, fitting epidemiological models to genealogies via coalescent models remains a challenging task, because pathogen populations often exhibit complex, nonlinear dynamics and are structured by multiple factors. Moreover, it often becomes necessary to consider stochastic variation in population dynamics when fitting such complex models to real data. Using recently developed structured coalescent models that accommodate complex population dynamics and population structure, we develop a statistical framework for fitting stochastic epidemiological models to genealogies. By combining particle filtering methods with Bayesian Markov chain Monte Carlo methods, we are able to fit a wide class of stochastic, nonlinear epidemiological models with different forms of population structure to genealogies. We demonstrate our framework using two structured epidemiological models: a model with disease progression between multiple stages of infection and a two-population model reflecting spatial structure. We apply the multi-stage model to HIV genealogies and show that the proposed method can be used to estimate the stage-specific transmission rates and prevalence of HIV. Finally, using the two-population model we explore how much information about population structure is contained in genealogies and what sample sizes are necessary to reliably infer parameters like migration rates.     

Talkin' 'bout my generation: environmental variability and cohort effects

In variable environments, it is probable that environmental conditions in the past can influence demographic performance now. Cohort effects occur when these delayed life-history effects are synchronized among groups of individuals in a population. Here we show how plasticity in density-dependent demographic traits throughout the life cycle can lead to cohort effects and that there can be substantial population dynamic consequences of these effects. We show experimentally that density and food conditions early in development can influence subsequent juvenile life-history traits. We also show that conditions early in development can interact with conditions at maturity to shape future adult performance. In fact, conditions such as food availability and density at maturity, like conditions early in development, can generate cohort effects in mature stages. Based on these data, and on current theory about the effects of plasticity generated by historical environments, we make predictions about the consequences of such changes on density-dependent demography and on mite population dynamics. We use a stochastic cohort effects model to generate a range of population dynamics. In accordance with the theory, we find the predicted changes in the strength of density dependence and associated changes in population dynamics and population variability.     

Spread of information through a population with socio-structural bias: III. Suggested experimental procedures

Experimental procedures are suggested to test the theory developed in previous papers which may have applications to predicting the spread of information by contact through a population. The experiment is designed to test the statistical properties of the “acquaintance net” of the population. Thus behavioral aspects are for the time being eliminated, and attention is focused exclusively in the properties of the potential communication net itself. Since the theory is not mathematically rigorous, it might be advisable to perform experiments with statistical materials only, such as playing cards, to test the validity of the assumptions and approximations of the theory in dealing with particular statistical processes. One such experiment has been performed. The results agree closely with those calculated from one of the derived equations.     

The Problem of Auto-Correlation in Parasitology

Explaining the contribution of host and pathogen factors in driving infection dynamics is a major ambition in parasitology. There is increasing recognition that analyses based on single summary measures of an infection (e.g., peak parasitaemia) do not adequately capture infection dynamics and so, the appropriate use of statistical techniques to analyse dynamics is necessary to understand infections and, ultimately, control parasites. However, the complexities of within-host environments mean that tracking and analysing pathogen dynamics within infections and among hosts poses considerable statistical challenges. Simple statistical models make assumptions that will rarely be satisfied in data collected on host and parasite parameters. In particular, model residuals (unexplained variance in the data) should not be correlated in time or space. Here we demonstrate how failure to account for such correlations can result in incorrect biological inference from statistical analysis. We then show how mixed effects models can be used as a powerful tool to analyse such repeated measures data in the hope that this will encourage better statistical practices in parasitology.     

Predicting the process of extinction in experimental microcosms and accounting for interspecific interactions in single‐species time series

Predicting population extinction risk is a fundamental application of ecological theory to the practice of conservation biology. Here, we compared the prediction performance of a wide array of stochastic, population dynamics models against direct observations of the extinction process from an extensive experimental data set. By varying a series of biological and statistical assumptions in the proposed models, we were able to identify the assumptions that affected predictions about population extinction. We also show how certain autocorrelation structures can emerge due to interspecific interactions, and that accounting for the stochastic effect of these interactions can improve predictions of the extinction process. We conclude that it is possible to account for the stochastic effects of community interactions on extinction when using single‐species time series.     

The effect of correlations on the population dynamics of lymphocytes

Recent studies of the population dynamics of a system of lymphocytes in an in vitro immune response have reported strong correlations in cell division times, both between parents and their progeny, and between those of sibling cells. The data also show a high level of correlation in the ultimate number of divisions achieved by cells within the same clone. Such correlations are often ignored in mathematical models of cell dynamics as they violate a standard assumption in the theory of branching processes, that of the statistical independence of cells. In this article we present a model in which these correlations can be incorporated, and have used this model to study the effect of these correlations on the population dynamics of a system of cells. We found that correlation in the division times between parents and their progeny can alter the mean population size of clones within the system, while all of the correlations can affect the variance in the sizes of different clones. The model was then applied to experimental data obtained from time-lapse video microscopy of a system of CpG stimulated B lymphocytes and it was found that inclusion of the correct correlation structure is necessary to accurately reproduce the observed population dynamics. We conclude that correlations in the dynamics of cells within an ensemble will affect the population dynamics of the system, and the effects will become more pronounced as the number of divisions increases.     

The population growth consequences of variation in individual heterozygosity

Heterozygosity has been associated with components of fitness in numerous studies across a wide range of taxa. Because heterozygosity is associated with individual performance it is also expected to be associated with population dynamics. However, investigations into the association between heterozygosity and population dynamics have been rare because of difficulties in linking evolutionary and ecological processes. The choice of heterozygosity measure is a further issue confounding such studies as it can be biased by individual differences in the frequencies of the alleles studied, the number of alleles at each locus as well as the total number of loci typed. In this study, we first examine the differences between the principal metrics used to calculate heterozygosity using long-term data from a marked population of Soay sheep (Ovis aries). Next, by means of statistical transformation of the homozygosity weighted by loci index, we determine how heterozygosity contributes to population growth in Soay sheep by modelling individual contributions to population growth (p(t(i))) as a function of several covariates, including sex, weight and faecal egg count--a surrogate of parasitic nematode burden in the gut. We demonstrate that although heterozygosity is associated with some components of fitness, most notably adult male reproductive success, in general it is only weakly associated with population growth.     

The Distribution of Fitness Effects of Beneficial Mutations in Pseudomonas aeruginosa

Understanding how beneficial mutations affect fitness is crucial to our understanding of adaptation by natural selection. Here, using adaptation to the antibiotic rifampicin in the opportunistic pathogen Pseudomonas aeruginosa as a model system, we investigate the underlying distribution of fitness effects of beneficial mutations on which natural selection acts. Consistent with theory, the effects of beneficial mutations are exponentially distributed where the fitness of the wild type is moderate to high. However, when the fitness of the wild type is low, the data no longer follow an exponential distribution, because many beneficial mutations have large effects on fitness. There is no existing population genetic theory to explain this bias towards mutations of large effects, but it can be readily explained by the underlying biochemistry of rifampicin–RNA polymerase interactions. These results demonstrate the limitations of current population genetic theory for predicting adaptation to severe sources of stress, such as antibiotics, and they highlight the utility of integrating statistical and biophysical approaches to adaptation.     

An Accessible Method for Implementing Hierarchical Models with Spatio-Temporal Abundance Data

A common goal in ecology and wildlife management is to determine the causes of variation in population dynamics over long periods of time and across large spatial scales. Many assumptions must nevertheless be overcome to make appropriate inference about spatio-temporal variation in population dynamics, such as autocorrelation among data points, excess zeros, and observation error in count data. To address these issues, many scientists and statisticians have recommended the use of Bayesian hierarchical models. Unfortunately, hierarchical statistical models remain somewhat difficult to use because of the necessary quantitative background needed to implement them, or because of the computational demands of using Markov Chain Monte Carlo algorithms to estimate parameters. Fortunately, new tools have recently been developed that make it more feasible for wildlife biologists to fit sophisticated hierarchical Bayesian models (i.e., Integrated Nested Laplace Approximation, ‘INLA’). We present a case study using two important game species in North America, the lesser and greater scaup, to demonstrate how INLA can be used to estimate the parameters in a hierarchical model that decouples observation error from process variation, and accounts for unknown sources of excess zeros as well as spatial and temporal dependence in the data. Ultimately, our goal was to make unbiased inference about spatial variation in population trends over time.     

State-space models of individual animal movement

Detailed observation of the movement of individual animals offers the potential to understand spatial population processes as the ultimate consequence of individual behaviour, physiological constraints and fine-scale environmental influences. However, movement data from individuals are intrinsically stochastic and often subject to severe observation error. Linking such complex data to dynamical models of movement is a major challenge for animal ecology. Here, we review a statistical approach, state-space modelling, which involves changing how we analyse movement data and draw inferences about the behaviours that shape it. The statistical robustness and predictive ability of state-space models make them the most promising avenue towards a new type of movement ecology that fuses insights from the study of animal behaviour, biogeography and spatial population dynamics.     

The evolution of stability in a competitive system

The characteristics governing the dynamics of populations can evolve and this evolution can either be towards stability or chaos. Yet it is not obvious how or why such population characteristics can evolve through selection on individuals. In this paper we construct a mathematical model, inspired by experimental results, illustrating the dynamics of a population of competing Drosophila. We demonstrate how selection of life history characteristics and stability influence one another as a population interacts with its environment. We generalize this result and show that population stability can evolve as a consequence of selection on individuals.     

Non-Linear Analysis Indicates Chaotic Dynamics and Reduced Resilience in Model-Based Daphnia Populations Exposed to Environmental Stress

In this study we present evidence that anthropogenic stressors can reduce the resilience of age-structured populations. Enhancement of disturbance in a model-based Daphnia population lead to a repression of chaotic population dynamics at the same time increasing the degree of synchrony between the population''s age classes. Based on the theory of chaos-mediated survival an increased risk of extinction was revealed for this population exposed to high concentrations of a chemical stressor. The Lyapunov coefficient was supposed to be a useful indicator to detect disturbance thresholds leading to alterations in population dynamics. One possible explanation could be a discrete change in attractor orientation due to external disturbance. The statistical analysis of Lyapunov coefficient distribution is proposed as a methodology to test for significant non-linear effects of general disturbance on populations. Although many new questions arose, this study forms a theoretical basis for a dynamical definition of population recovery.     

MAXIMUM-LIKELIHOOD ESTIMATION OF POPULATION DIVERGENCE TIMES AND POPULATION PHYLOGENY IN MODELS WITHOUT MUTATION

In this paper we present a method for estimating population divergence times by maximum likelihood in models without mutation. The maximum-likelihood estimator is compared to a commonly applied estimator based on Wright's F_ST statistic. Simulations suggest that the maximum-likelihood estimator is less biased and has a lower variance than the F_ST-based estimator. The maximum-likelihood estimator provides a statistical framework for the analysis of population history given genetic data. We demonstrate how maximum-likelihood estimates of the branching pattern of divergence of multiple populations may be obtained. We also describe how the method may be applied to test hypotheses such as whether populations have maintained equal population sizes. We illustrate the method by applying it to two previously published sets of human restriction fragment length polymorphism (RFLP) data.     

Effects of weather and climate on the dynamics of animal population time series

Weather is one of the most basic factors impacting animal populations, but the typical strength of such impacts on population dynamics is unknown. We incorporate weather and climate index data into analysis of 492 time series of mammals, birds and insects from the global population dynamics database. A conundrum is that a multitude of weather data may a priori be considered potentially important and hence present a risk of statistical over-fitting. We find that model selection or averaging alone could spuriously indicate that weather provides strong improvements to short-term population prediction accuracy. However, a block randomization test reveals that most improvements result from over-fitting. Weather and climate variables do, in general, improve predictions, but improvements were barely detectable despite the large number of datasets considered. Climate indices such as North Atlantic Oscillation are not better predictors of population change than local weather variables. Insect time series are typically less predictable than bird or mammal time series, although all taxonomic classes display low predictability. Our results are in line with the view that population dynamics is often too complex to allow resolving mechanisms from time series, but we argue that time series analysis can still be useful for estimating net environmental effects.     

Controlling false discoveries in genome scans for selection

Population differentiation (PD) and ecological association (EA) tests have recently emerged as prominent statistical methods to investigate signatures of local adaptation using population genomic data. Based on statistical models, these genomewide testing procedures have attracted considerable attention as tools to identify loci potentially targeted by natural selection. An important issue with PD and EA tests is that incorrect model specification can generate large numbers of false‐positive associations. Spurious association may indeed arise when shared demographic history, patterns of isolation by distance, cryptic relatedness or genetic background are ignored. Recent works on PD and EA tests have widely focused on improvements of test corrections for those confounding effects. Despite significant algorithmic improvements, there is still a number of open questions on how to check that false discoveries are under control and implement test corrections, or how to combine statistical tests from multiple genome scan methods. This tutorial study provides a detailed answer to these questions. It clarifies the relationships between traditional methods based on allele frequency differentiation and EA methods and provides a unified framework for their underlying statistical tests. We demonstrate how techniques developed in the area of genomewide association studies, such as inflation factors and linear mixed models, benefit genome scan methods and provide guidelines for good practice while conducting statistical tests in landscape and population genomic applications. Finally, we highlight how the combination of several well‐calibrated statistical tests can increase the power to reject neutrality, improving our ability to infer patterns of local adaptation in large population genomic data sets.     

Nonrandom association of genetic markers in a sample of South African blacks

According to classical genetic theory, allelic genes at one locus are expected to segregate and be manifested independently of allelic genes at another locus. At the population level any significant deviation from this general hypothesis resulting from specific biologic and genetic effects can be recognized in the form of nonrandom associations between genetic markers. The present data, consisting of 24 genetic polymorphisms determined from a sample of 998 unselected and unrelated South African blacks, offers an opportunity to test whether or not any such nonrandom associations exist between the genetic markers. After appropriate statistical calculations on the population data, we found that 13 pairs of genetic polymorphisms demonstrate a nonrandom association (statistically significant). Because the results cannot be explained in terms of known biologic mechanisms, we conclude that the associations observed could be due to random statistical effects (repeated application of the chi-square test) and/or to real (as yet unknown) biologic phenomena in the population studied. This tentative conclusion can serve as a guideline for more specific investigations.     

A Method for Detecting Positive Growth Autocorrelation without Marking Individuals

In most ecological studies, within-group variation is a nuisance that obscures patterns of interest and reduces statistical power. However, patterns of within-group variability often contain information about ecological processes. In particular, such patterns can be used to detect positive growth autocorrelation (consistent variation in growth rates among individuals in a cohort across time), even in samples of unmarked individuals. Previous methods for detecting autocorrelated growth required data from marked individuals. We propose a method that requires only estimates of within-cohort variance through time, using maximum likelihood methods to obtain point estimates and confidence intervals of the correlation parameter. We test our method on simulated data sets and determine the loss in statistical power due to the inability to identify individuals. We show how to accommodate nonlinear growth trajectories and test the effects of size-dependent mortality on our method''s accuracy. The method can detect significant growth autocorrelation at moderate levels of autocorrelation with moderate-sized cohorts (for example, statistical power of 80% to detect growth autocorrelation ρ ² = 0.5 in a cohort of 100 individuals measured on 16 occasions). We present a case study of growth in the red-eyed tree frog. Better quantification of the processes driving size variation will help ecologists improve predictions of population dynamics. This work will help researchers to detect growth autocorrelation in cases where marking is logistically infeasible or causes unacceptable decreases in the fitness of marked individuals.     

Comparison of Parameters from Rhythmometric Models with Multiple Components on Hybrid Data

Population multiple components is a statistical tool useful for the analysis of time-dependent hybrid data. With a small number of parameters, it is possible to model and to predict the periodic behavior of a population. In this article, we propose two methods to compare among populations rhythmometric parameters obtained by multiple component analysis. The first is a parametric method based in the usual statistical techniques for comparison of mean vectors in multivariate normal populations. The method, through MANOVA analysis, allows comparison of the MESOR and amplitude-acrophase pair of each component among two or more populations. The second is a nonparametric method, based in bootstrap techniques, to compare parameters from two populations. This test allows one to compare the MESOR, the amplitude, and the acrophase of each fitted component, as well as the global amplitude, orthophase, and bathyphase estimated when all fitted components are harmonics of a fundamental period. The idea is to calculate a confidence interval for the difference of the parameters of interest. If this interval does not contain zero, it can be concluded that the parameters from the two models are different with high probability. An estimation of p-value for the corresponding test can also be calculated. Both methods are illustrated with an example, based on clinical data. The nonparametric test can also be applied to paired data, a special situation of great interest in practice. By the use of similar bootstrap techniques, we illustrate how to construct confidence intervals for any rhythmometric parameter estimated from population multiple components models, including the orthophase, bathyphase, and global amplitude. These tests for comparison of parameters among populations are a needed tool when modeling the nonsinusoidal rhythmic behavior of hybrid data by population multiple component analysis.     

Spatial heterogeneity stabilizes predator–prey interactions at the microscale while patch connectivity controls their outcome

Natural landscapes are both fragmented and heterogeneous, affecting the distribution of organisms, and their interactions. While predation in homogeneous environments increases the probability of population extinction, fragmentation/heterogeneity promotes coexistence and enhances community stability as shown by experimentation with animals and microorganisms, and supported by theory. Patch connectivity can modulate such effects but how microbial predatory interactions are affected by water-driven connectivity is unknown. In soil, patch habitability by microorganisms, and their connectivity depend upon the water saturation degree (SD). Here, using the obligate bacterial predator Bdellovibrio bacteriovorus, and a Burkholderia prey, we show that soil spatial heterogeneity profoundly affects predatory dynamics, enhancing long-term co-existence of predator and prey in a SD-threshold dependent-manner. However, as patches and connectors cannot be distinguished in these soil matrices, metapopulations cannot be invoked to explain the dynamics of increased persistence. Using a set of experiments combined with statistical and physical models we demonstrate and quantify how under full connectivity, predation is independent of water content but depends on soil microstructure characteristics. In contrast, the SD below which predation is largely impaired corresponds to a threshold below which the water network collapses and water connectivity breaks down, preventing the bacteria to move within the soil matrix.     

Inference for nonlinear epidemiological models using genealogies and time series

Phylodynamics - the field aiming to quantitatively integrate the ecological and evolutionary dynamics of rapidly evolving populations like those of RNA viruses - increasingly relies upon coalescent approaches to infer past population dynamics from reconstructed genealogies. As sequence data have become more abundant, these approaches are beginning to be used on populations undergoing rapid and rather complex dynamics. In such cases, the simple demographic models that current phylodynamic methods employ can be limiting. First, these models are not ideal for yielding biological insight into the processes that drive the dynamics of the populations of interest. Second, these models differ in form from mechanistic and often stochastic population dynamic models that are currently widely used when fitting models to time series data. As such, their use does not allow for both genealogical data and time series data to be considered in tandem when conducting inference. Here, we present a flexible statistical framework for phylodynamic inference that goes beyond these current limitations. The framework we present employs a recently developed method known as particle MCMC to fit stochastic, nonlinear mechanistic models for complex population dynamics to gene genealogies and time series data in a Bayesian framework. We demonstrate our approach using a nonlinear Susceptible-Infected-Recovered (SIR) model for the transmission dynamics of an infectious disease and show through simulations that it provides accurate estimates of past disease dynamics and key epidemiological parameters from genealogies with or without accompanying time series data.