Disease‐structured N‐mixture models: A practical guide to model disease dynamics using count data

Obtaining inferences on disease dynamics (e.g., host population size, pathogen prevalence, transmission rate, host survival probability) typically requires marking and tracking individuals over time. While multistate mark–recapture models can produce high‐quality inference, these techniques are difficult to employ at large spatial and long temporal scales or in small remnant host populations decimated by virulent pathogens, where low recapture rates may preclude the use of mark–recapture techniques. Recently developed N‐mixture models offer a statistical framework for estimating wildlife disease dynamics from count data. N‐mixture models are a type of state‐space model in which observation error is attributed to failing to detect some individuals when they are present (i.e., false negatives). The analysis approach uses repeated surveys of sites over a period of population closure to estimate detection probability. We review the challenges of modeling disease dynamics and describe how N‐mixture models can be used to estimate common metrics, including pathogen prevalence, transmission, and recovery rates while accounting for imperfect host and pathogen detection. We also offer a perspective on future research directions at the intersection of quantitative and disease ecology, including the estimation of false positives in pathogen presence, spatially explicit disease‐structured N‐mixture models, and the integration of other data types with count data to inform disease dynamics. Managers rely on accurate and precise estimates of disease dynamics to develop strategies to mitigate pathogen impacts on host populations. At a time when pathogens pose one of the greatest threats to biodiversity, statistical methods that lead to robust inferences on host populations are critically needed for rapid, rather than incremental, assessments of the impacts of emerging infectious diseases.     

Efficient Inference of Parsimonious Phenomenological Models of Cellular Dynamics Using S-Systems and Alternating Regression

The nonlinearity of dynamics in systems biology makes it hard to infer them from experimental data. Simple linear models are computationally efficient, but cannot incorporate these important nonlinearities. An adaptive method based on the S-system formalism, which is a sensible representation of nonlinear mass-action kinetics typically found in cellular dynamics, maintains the efficiency of linear regression. We combine this approach with adaptive model selection to obtain efficient and parsimonious representations of cellular dynamics. The approach is tested by inferring the dynamics of yeast glycolysis from simulated data. With little computing time, it produces dynamical models with high predictive power and with structural complexity adapted to the difficulty of the inference problem.     

Methods of modelling viral disease dynamics across the within- and between-host scales: the impact of virus dose on host population immunity

We study the epidemiology of a viral disease with dose-dependent replication and transmission by nesting a differential-equation model of the within-host viral dynamics inside a between-host epidemiological model. We use two complementary approaches for nesting the models: an agent-based (AB) simulation and a mean-field approximation called the growth-matrix (GM) model. We find that although infection rates and predicted case loads are somewhat different between the AB and GM models, several epidemiological parameters, e.g. mean immunity in the population and mean dose received, behave similarly across the methods. Further, through a comparison of our dose-dependent replication model against two control models that uncouple dose-dependent replication from transmission, we find that host immunity in a population after an epidemic is qualitatively different than when transmission depends on time-varying viral abundances within hosts. These results show that within-host dynamics and viral dose should not be neglected in epidemiological models, and that the simpler GM approach to model nesting provides a reasonable tradeoff between model complexity and accuracy of results.     

OutbreakFlow: Model-based Bayesian inference of disease outbreak dynamics with invertible neural networks and its application to the COVID-19 pandemics in Germany

Mathematical models in epidemiology are an indispensable tool to determine the dynamics and important characteristics of infectious diseases. Apart from their scientific merit, these models are often used to inform political decisions and interventional measures during an ongoing outbreak. However, reliably inferring the epidemical dynamics by connecting complex models to real data is still hard and requires either laborious manual parameter fitting or expensive optimization methods which have to be repeated from scratch for every application of a given model. In this work, we address this problem with a novel combination of epidemiological modeling with specialized neural networks. Our approach entails two computational phases: In an initial training phase, a mathematical model describing the epidemic is used as a coach for a neural network, which acquires global knowledge about the full range of possible disease dynamics. In the subsequent inference phase, the trained neural network processes the observed data of an actual outbreak and infers the parameters of the model in order to realistically reproduce the observed dynamics and reliably predict future progression. With its flexible framework, our simulation-based approach is applicable to a variety of epidemiological models. Moreover, since our method is fully Bayesian, it is designed to incorporate all available prior knowledge about plausible parameter values and returns complete joint posterior distributions over these parameters. Application of our method to the early Covid-19 outbreak phase in Germany demonstrates that we are able to obtain reliable probabilistic estimates for important disease characteristics, such as generation time, fraction of undetected infections, likelihood of transmission before symptom onset, and reporting delays using a very moderate amount of real-world observations.     

Inferential biases linked to unobservable states in complex occupancy models

Modeling of species distributions has undergone a shift from relying on equilibrium assumptions to recognizing transient system dynamics explicitly. This shift has necessitated more complex modeling techniques, but the performance of these dynamic models has not yet been assessed for systems where unobservable states exist. Our work is motivated by the impacts of the emerging infectious disease chytridiomycosis, a disease of amphibians that is associated with declines of many species worldwide. Using this host‐pathogen system as a general example, we first illustrate how misleading inferences can result from failing to incorporate pathogen dynamics into the modeling process, especially when the pathogen is difficult or impossible to survey in the absence of a host species. We found that traditional modeling techniques can underestimate the effect of a pathogen on host species occurrence and dynamics when the pathogen can only be detected in the host, and pathogen information is treated as a covariate. We propose a dynamic multistate modeling approach that is flexible enough to account for the detection structures that may be present in complex multistate systems, especially when the sampling design is limited by a species’ natural history or sampling technology. When multistate occupancy models are used and an unobservable state is present, parameter estimation can be influenced by model complexity, data sparseness, and the underlying dynamics of the system. We show that, even with large sample sizes, many models incorporating seasonal variation in vital rates may not generate reasonable estimates, indicating parameter redundancy. We found that certain types of missing data can greatly hinder inference, and we make study design recommendations to avoid these issues. Additionally, we advocate the use of time‐varying covariates to explain temporal trends in the data, and the development of sampling techniques that match the biology of the system to eliminate unobservable states when possible.     

Measuring complexity to infer changes in the dynamics of ecological systems under stress

Despite advances in our mechanistic understanding of ecological processes, the inherent complexity of real-world ecosystems still limits our ability in predicting ecological dynamics especially in the face of on-going environmental stress. Developing a model is frequently challenged by structure uncertainty, unknown parameters, and limited data for exploring out-of-sample predictions. One way to address this challenge is to look for patterns in the data themselves in order to infer the underlying processes of an ecological system rather than to build system-specific models. For example, it has been recently suggested that statistical changes in ecological dynamics can be used to infer changes in the stability of ecosystems as they approach tipping points. For computer scientists such inference is similar to the notion of a Turing machine: a computational device that could execute a program (the process) to produce the observed data (the pattern). Here, we make use of such basic computational ideas introduced by Alan Turing to recognize changing patterns in ecological dynamics in ecosystems under stress. To do this, we use the concept of Kolmogorov algorithmic complexity that is a measure of randomness. In particular, we estimate an approximation to Kolmogorov complexity based on the Block Decomposition Method (BDM). We apply BDM to identify changes in complexity in simulated time-series and spatial datasets from ecosystems that experience different types of ecological transitions. We find that in all cases, K_BDM complexity decreased before all ecological transitions both in time-series and spatial datasets. These trends indicate that loss of stability in the ecological models we explored is characterized by loss of complexity and the emergence of a regular and computable underlying structure. Our results suggest that Kolmogorov complexity may serve as tool for revealing changes in the dynamics of ecosystems close to ecological transitions.     

Variational Phylodynamic Inference Using Pandemic-scale Data

The ongoing global pandemic has sharply increased the amount of data available to researchers in epidemiology and public health. Unfortunately, few existing analysis tools are capable of exploiting all of the information contained in a pandemic-scale data set, resulting in missed opportunities for improved surveillance and contact tracing. In this paper, we develop the variational Bayesian skyline (VBSKY), a method for fitting Bayesian phylodynamic models to very large pathogen genetic data sets. By combining recent advances in phylodynamic modeling, scalable Bayesian inference and differentiable programming, along with a few tailored heuristics, VBSKY is capable of analyzing thousands of genomes in a few minutes, providing accurate estimates of epidemiologically relevant quantities such as the effective reproduction number and overall sampling effort through time. We illustrate the utility of our method by performing a rapid analysis of a large number of SARS-CoV-2 genomes, and demonstrate that the resulting estimates closely track those derived from alternative sources of public health data.     

A model-based Bayesian estimation of the rate of evolution of VNTR loci in Mycobacterium tuberculosis

Variable numbers of tandem repeats (VNTR) typing is widely used for studying the bacterial cause of tuberculosis. Knowledge of the rate of mutation of VNTR loci facilitates the study of the evolution and epidemiology of Mycobacterium tuberculosis. Previous studies have applied population genetic models to estimate the mutation rate, leading to estimates varying widely from around to per locus per year. Resolving this issue using more detailed models and statistical methods would lead to improved inference in the molecular epidemiology of tuberculosis. Here, we use a model-based approach that incorporates two alternative forms of a stepwise mutation process for VNTR evolution within an epidemiological model of disease transmission. Using this model in a Bayesian framework we estimate the mutation rate of VNTR in M. tuberculosis from four published data sets of VNTR profiles from Albania, Iran, Morocco and Venezuela. In the first variant, the mutation rate increases linearly with respect to repeat numbers (linear model); in the second, the mutation rate is constant across repeat numbers (constant model). We find that under the constant model, the mean mutation rate per locus is (95% CI: ,)and under the linear model, the mean mutation rate per locus per repeat unit is (95% CI: ,). These new estimates represent a high rate of mutation at VNTR loci compared to previous estimates. To compare the two models we use posterior predictive checks to ascertain which of the two models is better able to reproduce the observed data. From this procedure we find that the linear model performs better than the constant model. The general framework we use allows the possibility of extending the analysis to more complex models in the future.     

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.     

Assessing the variability of stochastic epidemics.

In predicting the course of individual realizations of an epidemic it is important to know the magnitude of the variability of such realizations about their mean. In this paper and in the context of the general stochastic epidemic, some methods of obtaining approximate estimates of this variability are investigated; one is a multivariate normal approximation based on an asymptotic Gaussian diffusion process, and another uses an approximating linear stochastic process. The extension of these methods to the more detailed models used to describe the transmission dynamics of HIV infection and AIDS is discussed.     

Estimating epidemiological parameters for bovine tuberculosis in British cattle using a Bayesian partial-likelihood approach

Fitting models with Bayesian likelihood-based parameter inference is becoming increasingly important in infectious disease epidemiology. Detailed datasets present the opportunity to identify subsets of these data that capture important characteristics of the underlying epidemiology. One such dataset describes the epidemic of bovine tuberculosis (bTB) in British cattle, which is also an important exemplar of a disease with a wildlife reservoir (the Eurasian badger). Here, we evaluate a set of nested dynamic models of bTB transmission, including individual- and herd-level transmission heterogeneity and assuming minimal prior knowledge of the transmission and diagnostic test parameters. We performed a likelihood-based bootstrapping operation on the model to infer parameters based only on the recorded numbers of cattle testing positive for bTB at the start of each herd outbreak considering high- and low-risk areas separately. Models without herd heterogeneity are preferred in both areas though there is some evidence for super-spreading cattle. Similar to previous studies, we found low test sensitivities and high within-herd basic reproduction numbers (R₀), suggesting that there may be many unobserved infections in cattle, even though the current testing regime is sufficient to control within-herd epidemics in most cases. Compared with other, more data-heavy approaches, the summary data used in our approach are easily collected, making our approach attractive for other systems.     

Receptive field inference with localized priors

The linear receptive field describes a mapping from sensory stimuli to a one-dimensional variable governing a neuron's spike response. However, traditional receptive field estimators such as the spike-triggered average converge slowly and often require large amounts of data. Bayesian methods seek to overcome this problem by biasing estimates towards solutions that are more likely a priori, typically those with small, smooth, or sparse coefficients. Here we introduce a novel Bayesian receptive field estimator designed to incorporate locality, a powerful form of prior information about receptive field structure. The key to our approach is a hierarchical receptive field model that flexibly adapts to localized structure in both spacetime and spatiotemporal frequency, using an inference method known as empirical Bayes. We refer to our method as automatic locality determination (ALD), and show that it can accurately recover various types of smooth, sparse, and localized receptive fields. We apply ALD to neural data from retinal ganglion cells and V1 simple cells, and find it achieves error rates several times lower than standard estimators. Thus, estimates of comparable accuracy can be achieved with substantially less data. Finally, we introduce a computationally efficient Markov Chain Monte Carlo (MCMC) algorithm for fully Bayesian inference under the ALD prior, yielding accurate Bayesian confidence intervals for small or noisy datasets.     

A Note on Probabilistic Models over Strings: The Linear Algebra Approach

Probabilistic models over strings have played a key role in developing methods that take into consideration indels as phylogenetically informative events. There is an extensive literature on using automata and transducers on phylogenies to do inference on these probabilistic models, in which an important theoretical question is the complexity of computing the normalization of a class of string-valued graphical models. This question has been investigated using tools from combinatorics, dynamic programming, and graph theory, and has practical applications in Bayesian phylogenetics. In this work, we revisit this theoretical question from a different point of view, based on linear algebra. The main contribution is a set of results based on this linear algebra view that facilitate the analysis and design of inference algorithms on string-valued graphical models. As an illustration, we use this method to give a new elementary proof of a known result on the complexity of inference on the “TKF91” model, a well-known probabilistic model over strings. Compared to previous work, our proving method is easier to extend to other models, since it relies on a novel weak condition, triangular transducers, which is easy to establish in practice. The linear algebra view provides a concise way of describing transducer algorithms and their compositions, opens the possibility of transferring fast linear algebra libraries (for example, based on GPUs), as well as low rank matrix approximation methods, to string-valued inference problems.     

Estimating and interpreting migration of Amazonian forests using spatially implicit and semi‐explicit neutral models

With many sophisticated methods available for estimating migration, ecologists face the difficult decision of choosing for their specific line of work. Here we test and compare several methods, performing sanity and robustness tests, applying to large‐scale data and discussing the results and interpretation. Five methods were selected to compare for their ability to estimate migration from spatially implicit and semi‐explicit simulations based on three large‐scale field datasets from South America (Guyana, Suriname, French Guiana and Ecuador). Space was incorporated semi‐explicitly by a discrete probability mass function for local recruitment, migration from adjacent plots or from a metacommunity. Most methods were able to accurately estimate migration from spatially implicit simulations. For spatially semi‐explicit simulations, estimation was shown to be the additive effect of migration from adjacent plots and the metacommunity. It was only accurate when migration from the metacommunity outweighed that of adjacent plots, discrimination, however, proved to be impossible. We show that migration should be considered more an approximation of the resemblance between communities and the summed regional species pool. Application of migration estimates to simulate field datasets did show reasonably good fits and indicated consistent differences between sets in comparison with earlier studies. We conclude that estimates of migration using these methods are more an approximation of the homogenization among local communities over time rather than a direct measurement of migration and hence have a direct relationship with beta diversity. As betadiversity is the result of many (non)‐neutral processes, we have to admit that migration as estimated in a spatial explicit world encompasses not only direct migration but is an ecological aggregate of these processes. The parameter m of neutral models then appears more as an emerging property revealed by neutral theory instead of being an effective mechanistic parameter and spatially implicit models should be rejected as an approximation of forest dynamics.     

Pathogen invasion and host extinction in lattice structured populations

We examined the propagation of an infectious disease and the eventual extinction of the host population in a lattice-structured population. Both the host colonization and pathogen transmission processes are assumed to be restricted to act between the nearest neighbor sites. The model is analyzed by an improved version of pair approximation (IPA). Pair approximation is a technique to trace the dynamics of the number of nearest neighbor pairs having particular states, and IPA takes account of the clustering property of lattice models more precisely. The results are checked by computer simulations. The analysis shows: (i) in a one-dimensional lattice population, a pathogen cannot invade a host population no matter how large is the transmission rate; (ii) in a two-dimensional lattice population, pathogens will drive the host to extinction if the transmission rate is larger than a threshold. These results indicate that spatially structured population models may give qualitatively different results from conventional population models, such as Lotka-Volterra ones, without spatial structure.     

Inferring the temperature dependence of population parameters: the effects of experimental design and inference algorithm

Understanding and quantifying the temperature dependence of population parameters, such as intrinsic growth rate and carrying capacity, is critical for predicting the ecological responses to environmental change. Many studies provide empirical estimates of such temperature dependencies, but a thorough investigation of the methods used to infer them has not been performed yet. We created artificial population time series using a stochastic logistic model parameterized with the Arrhenius equation, so that activation energy drives the temperature dependence of population parameters. We simulated different experimental designs and used different inference methods, varying the likelihood functions and other aspects of the parameter estimation methods. Finally, we applied the best performing inference methods to real data for the species Paramecium caudatum. The relative error of the estimates of activation energy varied between 5% and 30%. The fraction of habitat sampled played the most important role in determining the relative error; sampling at least 1% of the habitat kept it below 50%. We found that methods that simultaneously use all time series data (direct methods) and methods that estimate population parameters separately for each temperature (indirect methods) are complementary. Indirect methods provide a clearer insight into the shape of the functional form describing the temperature dependence of population parameters; direct methods enable a more accurate estimation of the parameters of such functional forms. Using both methods, we found that growth rate and carrying capacity of Paramecium caudatum scale with temperature according to different activation energies. Our study shows how careful choice of experimental design and inference methods can increase the accuracy of the inferred relationships between temperature and population parameters. The comparison of estimation methods provided here can increase the accuracy of model predictions, with important implications in understanding and predicting the effects of temperature on the dynamics of populations.     

Successful Reconstruction of a Physiological Circuit with Known Connectivity from Spiking Activity Alone

Identifying the structure and dynamics of synaptic interactions between neurons is the first step to understanding neural network dynamics. The presence of synaptic connections is traditionally inferred through the use of targeted stimulation and paired recordings or by post-hoc histology. More recently, causal network inference algorithms have been proposed to deduce connectivity directly from electrophysiological signals, such as extracellularly recorded spiking activity. Usually, these algorithms have not been validated on a neurophysiological data set for which the actual circuitry is known. Recent work has shown that traditional network inference algorithms based on linear models typically fail to identify the correct coupling of a small central pattern generating circuit in the stomatogastric ganglion of the crab Cancer borealis. In this work, we show that point process models of observed spike trains can guide inference of relative connectivity estimates that match the known physiological connectivity of the central pattern generator up to a choice of threshold. We elucidate the necessary steps to derive faithful connectivity estimates from a model that incorporates the spike train nature of the data. We then apply the model to measure changes in the effective connectivity pattern in response to two pharmacological interventions, which affect both intrinsic neural dynamics and synaptic transmission. Our results provide the first successful application of a network inference algorithm to a circuit for which the actual physiological synapses between neurons are known. The point process methodology presented here generalizes well to larger networks and can describe the statistics of neural populations. In general we show that advanced statistical models allow for the characterization of effective network structure, deciphering underlying network dynamics and estimating information-processing capabilities.     

Coalescent inference for infectious disease: meta-analysis of hepatitis C

Genetic analysis of pathogen genomes is a powerful approach to investigating the population dynamics and epidemic history of infectious diseases. However, the theoretical underpinnings of the most widely used, coalescent methods have been questioned, casting doubt on their interpretation. The aim of this study is to develop robust population genetic inference for compartmental models in epidemiology. Using a general approach based on the theory of metapopulations, we derive coalescent models under susceptible–infectious (SI), susceptible–infectious–susceptible (SIS) and susceptible–infectious–recovered (SIR) dynamics. We show that exponential and logistic growth models are equivalent to SI and SIS models, respectively, when co-infection is negligible. Implementing SI, SIS and SIR models in BEAST, we conduct a meta-analysis of hepatitis C epidemics, and show that we can directly estimate the basic reproductive number (R₀) and prevalence under SIR dynamics. We find that differences in genetic diversity between epidemics can be explained by differences in underlying epidemiology (age of the epidemic and local population density) and viral subtype. Model comparison reveals SIR dynamics in three globally restricted epidemics, but most are better fit by the simpler SI dynamics. In summary, metapopulation models provide a general and practical framework for integrating epidemiology and population genetics for the purposes of joint inference.     

Travel Time Estimation Using Freeway Point Detector Data Based on Evolving Fuzzy Neural Inference System

Travel time is an important measurement used to evaluate the extent of congestion within road networks. This paper presents a new method to estimate the travel time based on an evolving fuzzy neural inference system. The input variables in the system are traffic flow data (volume, occupancy, and speed) collected from loop detectors located at points both upstream and downstream of a given link, and the output variable is the link travel time. A first order Takagi-Sugeno fuzzy rule set is used to complete the inference. For training the evolving fuzzy neural network (EFNN), two learning processes are proposed: (1) a K-means method is employed to partition input samples into different clusters, and a Gaussian fuzzy membership function is designed for each cluster to measure the membership degree of samples to the cluster centers. As the number of input samples increases, the cluster centers are modified and membership functions are also updated; (2) a weighted recursive least squares estimator is used to optimize the parameters of the linear functions in the Takagi-Sugeno type fuzzy rules. Testing datasets consisting of actual and simulated data are used to test the proposed method. Three common criteria including mean absolute error (MAE), root mean square error (RMSE), and mean absolute relative error (MARE) are utilized to evaluate the estimation performance. Estimation results demonstrate the accuracy and effectiveness of the EFNN method through comparison with existing methods including: multiple linear regression (MLR), instantaneous model (IM), linear model (LM), neural network (NN), and cumulative plots (CP).     

The impact of non-lethal synergists on the population and evolutionary dynamics of host-pathogen interactions

Multiple pathogenic infections can influence disease transmission and virulence, and have important consequences for understanding the community ecology and epidemiology of host-pathogen interactions. Here the population and evolutionary dynamics of a host-pathogen interaction with free-living stages are explored in the presence of a non-lethal synergist that hosts must tolerate. Through the coupled effects on pathogen transmission, host mass gain and allometry it is shown how investing in tolerance to a non-lethal synergist can lead to a broad range of different population dynamics. The effects of the synergist on pathogen fitness are explored through a series of life-history trait trade-offs. Coupling trade-offs between pathogen yield and pathogen speed of kill and the presence of a synergist favour parasites that have faster speeds of kill. This evolutionary change in pathogen characteristics is predicted to lead to stable population dynamics. Evolutionary analysis of tolerance of the synergist (strength of synergy) and lethal pathogen yield show that decreasing tolerance allows alternative pathogen strategies to invade and replace extant strategies. This evolutionary change is likely to destabilise the host-pathogen interaction leading to population cycles. Correlated trait effects between speed of kill and tolerance (strength of synergy) show how these traits can interact to affect the potential for the coexistence of multiple pathogen strategies. Understanding the consequences of these evolutionary relationships is important for the both the evolutionary and population dynamics of host-pathogen interactions.