Aaron A. King

Phylogenetic comparative analysis: A modeling approach for adaptive evolution

Biologists employ phylogenetic comparative methods to study adaptive evolution. However, none of the popular methods model selection directly. We explain and develop a method based on the Ornstein‐Uhlenbeck (OU) process, first proposed by Hansen. Ornstein‐Uhlenbeck models incorporate both selection and drift and are thus qualitatively different from, and more general than, pure drift models based on Brownian motion. Most importantly, OU models possess selective optima that formalize the notion of adaptive zone. In this article, we develop the method for one quantitative character, discuss interpretations of its parameters, and provide code implementing the method. Our approach allows us to translate hypotheses regarding adaptation in different selective regimes into explicit models, to test the models against data using maximum‐likelihood‐based model selection techniques, and to infer details of the evolutionary process. We illustrate the method using two worked examples. Relative to existing approaches, the direct modeling approach we demonstrate allows one to explore more detailed hypotheses and to utilize more of the information content of comparative data sets than existing methods. Moreover, the use of a model selection framework to simultaneously compare a variety of hypotheses advances our ability to assess alternative evolutionary explanations.

Inapparent infections and cholera dynamics.

In many infectious diseases, an unknown fraction of infections produce symptoms mild enough to go unrecorded, a fact that can seriously compromise the interpretation of epidemiological records. This is true for cholera, a pandemic bacterial disease, where estimates of the ratio of asymptomatic to symptomatic infections have ranged from 3 to 100 (refs 1–5). In the absence of direct evidence, understanding of fundamental aspects of cholera transmission, immunology and control has been based on assumptions about this ratio and about the immunological consequences of inapparent infections. Here we show that a model incorporating high asymptomatic ratio and rapidly waning immunity, with infection both from human and environmental sources, explains 50 yr of mortality data from 26 districts of Bengal, the pathogen’s endemic home. We find that the asymptomatic ratio in cholera is far higher than had been previously supposed and that the immunity derived from mild infections wanes much more rapidly than earlier analyses have indicated. We find, too, that the environmental reservoir (free-living pathogen) is directly responsible for relatively few infections but that it may be critical to the disease’s endemicity. Our results demonstrate that inapparent infections can hold the key to interpreting the patterns of disease outbreaks. New statistical methods, which allow rigorous maximum likelihood inference based on dynamical models incorporating multiple sources and outcomes of infection, seasonality, process noise, hidden variables and measurement error, make it possible to test more precise hypotheses and obtain unexpected results. Our experience suggests that the confrontation of time-series data with mechanistic models is likely to revise our understanding of the ecology of many infectious diseases.

Inference for nonlinear dynamical systems

Nonlinear stochastic dynamical systems are widely used to model systems across the sciences and engineering. Such models are natural to formulate and can be analyzed mathematically and numerically. However, difficulties associated with inference from time-series data about unknown parameters in these models have been a constraint on their application. We present a new method that makes maximum likelihood estimation feasible for partially-observed nonlinear stochastic dynamical systems (also known as state-space models) where this was not previously the case. The method is based on a sequence of filtering operations which are shown to converge to a maximum likelihood parameter estimate. We make use of recent advances in nonlinear filtering in the implementation of the algorithm. We apply the method to the study of cholera in Bangladesh. We construct confidence intervals, perform residual analysis, and apply other diagnostics. Our analysis, based upon a model capturing the intrinsic nonlinear dynamics of the system, reveals some effects overlooked by previous studies.

Contact Network Structure Explains the Changing Epidemiology of Pertussis

Coughing Back Resurgence of whooping cough (or pertussis) is problematic because of the risk of infant fatality, but it has been occurring despite the availability of a vaccine. Data from Sweden, where national vaccination for whooping cough was discontinued for 17 years, before resumption in 1996, was used by Rohani et al. (p. 982) to build a model that showed that age-specific social contacts were an important influence on the spread of infection. The model helped to explain why resumption of vaccination was so successful in curbing infant whooping cough but had no effect on pertussis in adolescents. The model thus demonstrated how potentially important it is to take account of age-stratification in a population when considering public health policy. Age-specific social networks can influence transmission of infection and should be considered in health policy planning. The epidemiology of whooping cough (pertussis) remains enigmatic. A leading cause of infant mortality globally, its resurgence in several developed nations—despite the availability and use of vaccines for many decades—has caused alarm. We combined data from a singular natural experiment and a detailed contact network study to show that age-specific contact patterns alone can explain shifts in prevalence and age-stratified incidence in the vaccine era. The practical implications of our results are notable: Ignoring age-structured contacts is likely to result in misinterpretation of epidemiological data and potentially costly policy missteps.

Plug-and-play inference for disease dynamics: measles in large and small populations as a case study

Statistical inference for mechanistic models of partially observed dynamic systems is an active area of research. Most existing inference methods place substantial restrictions upon the form of models that can be fitted and hence upon the nature of the scientific hypotheses that can be entertained and the data that can be used to evaluate them. In contrast, the so-called plug-and-play methods require only simulations from a model and are thus free of such restrictions. We show the utility of the plug-and-play approach in the context of an investigation of measles transmission dynamics. Our novel methodology enables us to ask and answer questions that previous analyses have been unable to address. Specifically, we demonstrate that plug-and-play methods permit the development of a modelling and inference framework applicable to data from both large and small populations. We thereby obtain novel insights into the nature of heterogeneity in mixing and comment on the importance of including extra-demographic stochasticity as a means of dealing with environmental stochasticity and model misspecification. Our approach is readily applicable to many other epidemiological and ecological systems.

Interactions between serotypes of dengue highlight epidemiological impact of cross-immunity

Dengue, a mosquito-borne virus of humans, infects over 50 million people annually. Infection with any of the four dengue serotypes induces protective immunity to that serotype, but does not confer long-term protection against infection by other serotypes. The immunological interactions between serotypes are of central importance in understanding epidemiological dynamics and anticipating the impact of dengue vaccines. We analysed a 38-year time series with 12 197 serotyped dengue infections from a hospital in Bangkok, Thailand. Using novel mechanistic models to represent different hypothesized immune interactions between serotypes, we found strong evidence that infection with dengue provides substantial short-term cross-protection against other serotypes (approx. 1–3 years). This is the first quantitative evidence that short-term cross-protection exists since human experimental infection studies performed in the 1950s. These findings will impact strategies for designing dengue vaccine studies, future multi-strain modelling efforts, and our understanding of evolutionary pressures in multi-strain disease systems.

Time series analysis via mechanistic models

The purpose of time series analysis via mechanistic models is to reconcile the known or hypothesized structure of a dynamical system with observations collected over time. We develop a framework for constructing nonlinear mechanistic models and carrying out inference. Our framework permits the consideration of implicit dynamic models, meaning statistical models for stochastic dynamical systems which are specified by a simulation algorithm to generate sample paths. Inference procedures that operate on implicit models are said to have the plug-and-play property. Our work builds on recently developed plug-and-play inference methodology for partially observed Markov models. We introduce a class of implicitly specified Markov chains with stochastic transition rates, and we demonstrate its applicability to open problems in statistical inference for biological systems. As one example, these models are shown to give a fresh perspective on measles transmission dynamics. As a second example, we present a mechanistic analysis of cholera incidence data, involving interaction between two competing strains of the pathogen Vibrio cholerae.

Natural immune boosting in pertussis dynamics and the potential for long-term vaccine failure.

Incidence of whooping cough, unlike many other childhood diseases for which there is an efficacious vaccine, has been increasing over the past twenty years despite high levels of vaccine coverage. Its reemergence has been particularly noticeable among teenagers and adults. Many hypotheses have been put forward to explain these two patterns, but parsimonious reconciliation of clinical data on the limited duration of immunity with both pre- and postvaccine era age-specific incidence remains a challenge. We consider the immunologically relevant, yet epidemiologically largely neglected, possibility that a primed immune system can respond to a lower dose of antigen than a naive one. We hypothesize that during the prevaccine era teenagers’ and adults’ primed immunity was frequently boosted by reexposure, so maintaining herd immunity in the face of potentially eroding individual immunity. In contrast, low pathogen circulation in the current era, except during epidemic outbreaks, allows immunity to be lost before reexposure occurs. We develop and analyze an age-structured model that encapsulates this hypothesis. We find that immune boosting must be more easily triggered than primary infection to account for age-incidence data. We make age-specific and dynamical predictions through bifurcation analysis and simulation. The boosting model proposed here parsimoniously captures four key features of pertussis data from highly vaccinated countries: (i) the shift in age-specific incidence, (ii) reemergence with high vaccine coverage, (iii) the possibility for cyclic dynamics in the pre- and postvaccine eras, and (iv) the apparent shift from susceptible-infectious-recovered (SIR)-like to susceptible-infectious-recovered-susceptible (SIRS)-like phenomenology of infection and immunity to Bordetella pertussis.

THE GEOMETRY OF A POPULATION CYCLE: A MECHANISTIC MODEL OF SNOWSHOE HARE DEMOGRAPHY

The phenomenology and causes of snowshoe hare cycles are addressed via construction of a three-trophic-level population dynamics model in which hare populations are limited by the availability of winter browse from below and by predation from above. In the absence of predators, the model predicts annual oscillations, the magnitude of which depends on habitat quality. With predators in the system, a wide range of additional dynamics are possible: multi-annual cycles of various periods, quasiperiodicity, and chaos. Parameterizing the model from the literature leads to the conclusion that the model is compatible with the principal features of the cycle in nature: its regularity, mean period, and the observed range of peak-to-trough amplitudes. The model also points to circumstances that can result in the cycles abolition as observed, for example, at the southern edge of the hares range. The model predicts that the increase phase of the cycle is brought to a halt by food limitation, while the decline from p...

Highly localized sensitivity to climate forcing drives endemic cholera in a megacity

The population dynamics of endemic cholera in urban environments—in particular interannual variation in the size and distribution of seasonal outbreaks—remain poorly understood and highly unpredictable. In part, this situation is due to the considerable demographic, socioeconomic, and environmental heterogeneity of large and growing urban centers. Despite this heterogeneity, the influence of climate variability on the population dynamics of infectious diseases is considered a large-scale, regional, phenomenon, and as such has been previously addressed for cholera only with temporal models that do not incorporate spatial structure. Here we show that a probabilistic spatial model can explain cholera dynamics in the megacity of Dhaka, Bangladesh, and afford a basis for cholera forecasts at lead times of 11 mo. Critically, we find that the action of climate variability (El Niño southern oscillation and flooding) is quite localized: There is a climate-sensitive urban core that acts to propagate risk to the rest of the city. The modeling framework presented here should be applicable to cholera in other cities, as well as to other infectious diseases in urban settings and other biological systems with spatiotemporal interactions.