Edward L. Ionides

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.

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.

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.

The reversal of the relation between economic growth and health progress: Sweden in the 19th and 20th centuries

Health progress, as measured by the decline in mortality rates and the increase in life expectancy, is usually conceived as related to economic growth, especially in the long run. In this investigation it is shown that economic growth is positively associated with health progress in Sweden throughout the 19th century. However, the relation becomes weaker as time passes and is completely reversed in the second half of the 20th century, when economic growth negatively affects health progress. The effect of the economy on health occurs mostly at lag 0 in the 19th century and is lagged up to 2 years in the 20th century. No evidence is found for economic effects on mortality at greater lags. These findings are shown to be robustly consistent across a variety of statistical procedures, including linear regression, spectral analysis, cross-correlation, and lag regression models. Models using inflation and unemployment as economic indicators reveal similar results. Evidence for reverse effects of health progress on economic growth is weak, and unobservable in the second half of the 20th century.

Forcing Versus Feedback: Epidemic Malaria and Monsoon Rains in Northwest India

Malaria epidemics in regions with seasonal windows of transmission can vary greatly in size from year to year. A central question has been whether these interannual cycles are driven by climate, are instead generated by the intrinsic dynamics of the disease, or result from the resonance of these two mechanisms. This corresponds to the more general inverse problem of identifying the respective roles of external forcings vs. internal feedbacks from time series for nonlinear and noisy systems. We propose here a quantitative approach to formally compare rival hypotheses on climate vs. disease dynamics, or external forcings vs. internal feedbacks, that combines dynamical models with recently developed, computational inference methods. The interannual patterns of epidemic malaria are investigated here for desert regions of northwest India, with extensive epidemiological records for Plasmodium falciparum malaria for the past two decades. We formulate a dynamical model of malaria transmission that explicitly incorporates rainfall, and we rely on recent advances on parameter estimation for nonlinear and stochastic dynamical systems based on sequential Monte Carlo methods. Results show a significant effect of rainfall in the inter-annual variability of epidemic malaria that involves a threshold in the disease response. The model exhibits high prediction skill for yearly cases in the malaria transmission season following the monsoonal rains. Consideration of a more complex model with clinical immunity demonstrates the robustness of the findings and suggests a role of infected individuals that lack clinical symptoms as a reservoir for transmission. Our results indicate that the nonlinear dynamics of the disease itself play a role at the seasonal, but not the interannual, time scales. They illustrate the feasibility of forecasting malaria epidemics in desert and semi-arid regions of India based on climate variability. This approach should be applicable to malaria in other locations, to other infectious diseases, and to other nonlinear systems under forcing.

Forecasting Models of Emergency Department Crowding

OBJECTIVES The authors investigated whether models using time series methods can generate accurate short-term forecasts of emergency department (ED) bed occupancy, using traditional historical averages models as comparison. METHODS From July 2005 through June 2006, retrospective hourly ED bed occupancy values were collected from three tertiary care hospitals. Three models of ED bed occupancy were developed for each site: 1) hourly historical average, 2) seasonal autoregressive integrated moving average (ARIMA), and 3) sinusoidal with an autoregression (AR)-structured error term. Goodness of fits were compared using log likelihood and Akaikes Information Criterion (AIC). The accuracies of 4- and 12-hour forecasts were evaluated by comparing model forecasts to actual observed bed occupancy with root mean square (RMS) error. Sensitivity of prediction errors to model training time was evaluated, as well. RESULTS The seasonal ARIMA outperformed the historical average in complexity adjusted goodness of fit (AIC). Both AR-based models had significantly better forecast accuracy for the 4- and the 12-hour forecasts of ED bed occupancy (analysis of variance [ANOVA] p < 0.01), compared to the historical average. The AR-based models did not differ significantly from each other in their performance. Model prediction errors did not show appreciable sensitivity to model training times greater than 7 days. CONCLUSIONS Both a sinusoidal model with AR-structured error term and a seasonal ARIMA model were found to robustly forecast ED bed occupancy 4 and 12 hours in advance at three different EDs, without needing data input beyond bed occupancy in the preceding hours.

Individual Joblessness, Contextual Unemployment, and Mortality Risk

Longitudinal studies at the level of individuals find that employees who lose their jobs are at increased risk of death. However, analyses of aggregate data find that as unemployment rates increase during recessions, population mortality actually declines. We addressed this paradox by using data from the US Department of Labor and annual survey data (1979-1997) from a nationally representative longitudinal study of individuals-the Panel Study of Income Dynamics. Using proportional hazards (Cox) regression, we analyzed how the hazard of death depended on 1) individual joblessness and 2) state unemployment rates, as indicators of contextual economic conditions. We found that 1) compared with the employed, for the unemployed the hazard of death was increased by an amount equivalent to 10 extra years of age, and 2) each percentage-point increase in the state unemployment rate reduced the mortality hazard in all individuals by an amount equivalent to a reduction of 1 year of age. Our results provide evidence that 1) joblessness strongly and significantly raises the risk of death among those suffering it, and 2) periods of higher unemployment rates, that is, recessions, are associated with a moderate but significant reduction in the risk of death among the entire population.

Truncated Importance Sampling

Importance sampling is a fundamental Monte Carlo technique. It involves generating a sample from a proposal distribution in order to estimate some property of a target distribution. Importance sampling can be highly sensitive to the choice of proposal distribution, and fails if the proposal distribution does not sufficiently well approximate the target. Procedures that involve truncation of large importance sampling weights are shown theoretically to improve on standard importance sampling by being less sensitive to the proposal distribution and having lower mean squared estimation error.Consistency is shown under weak conditions, and optimal truncation rates found under more specific conditions. Truncation at rate n1/2 is shown to be a good general choice. An adaptive truncation threshold, based on minimizing an unbiased risk estimate, is also presented. As an example, truncation is found to be effective for calculating the likelihood of partially observed multivariate diffusions. It is demonstrated as a component of a sequential importance sampling scheme for a continuous time population disease model. Truncation is most valuable for computationally intensive, multidimensional situations in which finding a proposal distribution that is everywhere a good approximation to the target distribution is challenging.

Inference for dynamic and latent variable models via iterated, perturbed Bayes maps

Significance Many scientific challenges involve the study of stochastic dynamic systems for which only noisy or incomplete measurements are available. Inference for partially observed Markov process models provides a framework for formulating and answering questions about these systems. Except when the system is small, or approximately linear and Gaussian, state-of-the-art statistical methods are required to make efficient use of available data. Evaluation of the likelihood for a partially observed Markov process model can be formulated as a filtering problem. Iterated filtering algorithms carry out repeated Monte Carlo filtering operations to maximize the likelihood. We develop a new theoretical framework for iterated filtering and construct a new algorithm that dramatically outperforms previous approaches on a challenging inference problem in disease ecology. Iterated filtering algorithms are stochastic optimization procedures for latent variable models that recursively combine parameter perturbations with latent variable reconstruction. Previously, theoretical support for these algorithms has been based on the use of conditional moments of perturbed parameters to approximate derivatives of the log likelihood function. Here, a theoretical approach is introduced based on the convergence of an iterated Bayes map. An algorithm supported by this theory displays substantial numerical improvement on the computational challenge of inferring parameters of a partially observed Markov process.