Estimating the instantaneous reproduction number (Rt) by using particle filter

Background

Monitoring the transmission of coronavirus disease 2019 (COVID-19) requires accurate estimation of the effective reproduction number (R t ). However, existing methods for calculating R t may yield biased estimates if important real-world factors, such as delays in confirmation, pre-symptomatic transmissions, or imperfect data observation, are not considered.

Method

To include real-world factors, we expanded the susceptible-exposed-infectious-recovered (SEIR) model by incorporating pre-symptomatic (P) and asymptomatic (A) states, creating the SEPIAR model. By utilizing both stochastic and deterministic versions of the model, and incorporating predetermined time series of R t , we generated simulated datasets that simulate real-world challenges in estimating R t . We then compared the performance of our proposed particle filtering method for estimating R t with the existing EpiEstim approach based on renewal equations.

Results

The particle filtering method accurately estimated R t even in the presence of data with delays, pre-symptomatic transmission, and imperfect observation. When evaluating via the root mean square error (RMSE) metric, the performance of the particle filtering method was better in general and was comparable to the EpiEstim approach if perfectly deconvolved infection time series were provided, and substantially better when R t exhibited short-term fluctuations and the data was right truncated.

Conclusions

The SEPIAR model, in conjunction with the particle filtering method, offers a reliable tool for predicting the transmission trend of COVID-19 and assessing the impact of intervention strategies. This approach enables enhanced monitoring of COVID-19 transmission and can inform public health policies aimed at controlling the spread of the disease.

Bagged filters for partially observed interacting systems

Bagging (i.e., bootstrap aggregating) involves combining an ensemble of bootstrap estimators. We consider bagging for inference from noisy or incomplete measurements on a collection of interacting stochastic dynamic systems. Each system is called a unit, and each unit is associated with a spatial location. A motivating example arises in epidemiology, where each unit is a city: the majority of transmission occurs within a city, with smaller yet epidemiologically important interactions arising from disease transmission between cities. Monte Carlo filtering methods used for inference on nonlinear non-Gaussian systems can suffer from a curse of dimensionality as the number of units increases. We introduce bagged filter (BF) methodology which combines an ensemble of Monte Carlo filters, using spatiotemporally localized weights to select successful filters at each unit and time. We obtain conditions under which likelihood evaluation using a BF algorithm can beat a curse of dimensionality, and we demonstrate applicability even when these conditions do not hold. BF can out-perform an ensemble Kalman filter on a coupled population dynamics model describing infectious disease transmission. A block particle filter also performs well on this task, though the bagged filter respects smoothness and conservation laws that a block particle filter can violate.

Modeling brain connectivity dynamics in functional magnetic resonance imaging via particle filtering

Interest in the studying of functional connections in the brain has grown considerably in the last decades, as many studies have pointed out that alterations in the interaction among brain areas can play a role as markers of neurological diseases. Most studies in this field treat the brain network as a system of connections stationary in time, but dynamic features of brain connectivity can provide useful information, both on physiology and pathological conditions of the brain. In this paper, we propose the application of a computational methodology, named Particle Filter (PF), to study non-stationarities in brain connectivity in functional Magnetic Resonance Imaging (fMRI). The PF algorithm estimates time-varying hidden parameters of a first-order linear time-varying Vector Autoregressive model (VAR) through a Sequential Monte Carlo strategy. On simulated time series, the PF approach effectively detected and enabled to follow time-varying hidden parameters and it captured causal relationships among signals. The method was also applied to real fMRI data, acquired in presence of periodic tactile or visual stimulations, in different sessions. On these data, the PF estimates were consistent with current knowledge on brain functioning. Most importantly, the approach enabled to detect statistically significant modulations in the cause-effect relationship between brain areas, which correlated with the underlying visual stimulation pattern presented during the acquisition.

"Exact" and Approximate Methods for Bayesian Inference: Stochastic Volatility Case Study

We conduct a case study in which we empirically illustrate the performance of different classes of Bayesian inference methods to estimate stochastic volatility models. In particular, we consider how different particle filtering methods affect the variance of the estimated likelihood. We review and compare particle Markov Chain Monte Carlo (MCMC), RMHMC, fixed-form variational Bayes, and integrated nested Laplace approximation to estimate the posterior distribution of the parameters. Additionally, we conduct the review from the point of view of whether these methods are (1) easily adaptable to different model specifications; (2) adaptable to higher dimensions of the model in a straightforward way; (3) feasible in the multivariate case. We show that when using the stochastic volatility model for methods comparison, various data-generating processes have to be considered to make a fair assessment of the methods. Finally, we present a challenging specification of the multivariate stochastic volatility model, which is rarely used to illustrate the methods but constitutes an important practical application.

Hindsight is 2020 vision: a characterisation of the global response to the COVID-19 pandemic

BACKGROUND

The global impact of COVID-19 and the country-specific responses to the pandemic provide an unparalleled opportunity to learn about different patterns of the outbreak and interventions. We model the global pattern of reported COVID-19 cases during the primary response period, with the aim of learning from the past to prepare for the future.

METHODS

Using Bayesian methods, we analyse the response to the COVID-19 outbreak for 158 countries for the period 22 January to 9 June 2020. This encompasses the period in which many countries imposed a variety of response measures and initial relaxation strategies. Instead of modelling specific intervention types and timings for each country explicitly, we adopt a stochastic epidemiological model including a feedback mechanism on virus transmission to capture complex nonlinear dynamics arising from continuous changes in community behaviour in response to rising case numbers. We analyse the overall effect of interventions and community responses across diverse regions. This approach mitigates explicit consideration of issues such as period of infectivity and public adherence to government restrictions.

RESULTS

Countries with the largest cumulative case tallies are characterised by a delayed response, whereas countries that avoid substantial community transmission during the period of study responded quickly. Countries that recovered rapidly also have a higher case identification rate and small numbers of undocumented community transmission at the early stages of the outbreak. We also demonstrate that uncertainty in numbers of undocumented infections dramatically impacts the risk of multiple waves. Our approach is also effective at pre-empting potential flare-ups.

CONCLUSIONS

We demonstrate the utility of modelling to interpret community behaviour in the early epidemic stages. Two lessons learnt that are important for the future are: i) countries that imposed strict containment measures early in the epidemic fared better with respect to numbers of reported cases; and ii) broader testing is required early in the epidemic to understand the magnitude of undocumented infections and recover rapidly. We conclude that clear patterns of containment are essential prior to relaxation of restrictions and show that modelling can provide insights to this end.

Efficient Real-Time Monitoring of an Emerging Influenza Pandemic: How Feasible?

A prompt public health response to a new epidemic relies on the ability to monitor and predict its evolution in real time as data accumulate. The 2009 A/H1N1 outbreak in the UK revealed pandemic data as noisy, contaminated, potentially biased and originating from multiple sources. This seriously challenges the capacity for real-time monitoring. Here, we assess the feasibility of real-time inference based on such data by constructing an analytic tool combining an age-stratified SEIR transmission model with various observation models describing the data generation mechanisms. As batches of data become available, a sequential Monte Carlo (SMC) algorithm is developed to synthesise multiple imperfect data streams, iterate epidemic inferences and assess model adequacy amidst a rapidly evolving epidemic environment, substantially reducing computation time in comparison to standard MCMC, to ensure timely delivery of real-time epidemic assessments. In application to simulated data designed to mimic the 2009 A/H1N1 epidemic, SMC is shown to have additional benefits in terms of assessing predictive performance and coping with parameter nonidentifiability.

Efficient inference in state-space models through adaptive learning in online Monte Carlo expectation maximization

Expectation maximization (EM) is a technique for estimating maximum-likelihood parameters of a latent variable model given observed data by alternating between taking expectations of sufficient statistics, and maximizing the expected log likelihood. For situations where sufficient statistics are intractable, stochastic approximation EM (SAEM) is often used, which uses Monte Carlo techniques to approximate the expected log likelihood. Two common implementations of SAEM, Batch EM (BEM) and online EM (OEM), are parameterized by a "learning rate", and their efficiency depend strongly on this parameter. We propose an extension to the OEM algorithm, termed Introspective Online Expectation Maximization (IOEM), which removes the need for specifying this parameter by adapting the learning rate to trends in the parameter updates. We show that our algorithm matches the efficiency of the optimal BEM and OEM algorithms in multiple models, and that the efficiency of IOEM can exceed that of BEM/OEM methods with optimal learning rates when the model has many parameters. Finally we use IOEM to fit two models to a financial time series. A Python implementation is available at https://github.com/luntergroup/IOEM.git.

Introduction to particle Markov-chain Monte Carlo for disease dynamics modellers

The particle Markov-chain Monte Carlo (PMCMC) method is a powerful tool to efficiently explore high-dimensional parameter space using time-series data. We illustrate an overall picture of PMCMC with minimal but sufficient theoretical background to support the readers in the field of biomedical/health science to apply PMCMC to their studies. Some working examples of PMCMC applied to infectious disease dynamic models are presented with R code.

A Computationally Efficient Labeled Multi-Bernoulli Smoother for Multi-Target Tracking

A forward–backward labeled multi-Bernoulli (LMB) smoother is proposed for multi-target tracking. The proposed smoother consists of two components corresponding to forward LMB filtering and backward LMB smoothing, respectively. The former is the standard LMB filter and the latter is proved to be closed under LMB prior. It is also shown that the proposed LMB smoother can improve both the cardinality estimation and the state estimation, and the major computational complexity is linear with the number of targets. Implementation based on the Sequential Monte Carlo method in a representative scenario has demonstrated the effectiveness and computational efficiency of the proposed smoother in comparison to existing approaches.

Panel Data Analysis via Mechanistic Models

Panel data, also known as longitudinal data, consist of a collection of time series. Each time series, which could itself be multivariate, comprises a sequence of measurements taken on a distinct unit. Mechanistic modeling involves writing down scientifically motivated equations describing the collection of dynamic systems giving rise to the observations on each unit. A defining characteristic of panel systems is that the dynamic interaction between units should be negligible. Panel models therefore consist of a collection of independent stochastic processes, generally linked through shared parameters while also having unit-specific parameters. To give the scientist flexibility in model specification, we are motivated to develop a framework for inference on panel data permitting the consideration of arbitrary nonlinear, partially observed panel models. We build on iterated filtering techniques that provide likelihood-based inference on nonlinear partially observed Markov process models for time series data. Our methodology depends on the latent Markov process only through simulation; this plug-and-play property ensures applicability to a large class of models. We demonstrate our methodology on a toy example and two epidemiological case studies. We address inferential and computational issues arising due to the combination of model complexity and dataset size. Supplementary materials for this article are available online.

Bayesian multi-dipole modelling in the frequency domain

BACKGROUND

Magneto- and Electro-encephalography record the electromagnetic field generated by neural currents with high temporal frequency and good spatial resolution, and are therefore well suited for source localization in the time and in the frequency domain. In particular, localization of the generators of neural oscillations is very important in the study of cognitive processes in the healthy and in the pathological brain.

NEW METHOD

We introduce the use of a Bayesian multi-dipole localization method in the frequency domain. Given the Fourier Transform of the data at one or multiple frequencies and/or trials, the algorithm approximates numerically the posterior distribution with Monte Carlo techniques.

RESULTS

We use synthetic data to show that the proposed method behaves well under a wide range of experimental conditions, including low signal-to-noise ratios and correlated sources. We use dipole clusters to mimic the effect of extended sources. In addition, we test the algorithm on real MEG data to confirm its feasibility.

COMPARISON WITH EXISTING METHOD(S)

Throughout the whole study, DICS (Dynamic Imaging of Coherent Sources) is used systematically as a benchmark. The two methods provide similar general pictures; the posterior distributions of the Bayesian approach contain much richer information at the price of a higher computational cost.

CONCLUSIONS

The Bayesian method described in this paper represents a reliable approach for localization of multiple dipoles in the frequency domain.

SeqClone: sequential Monte Carlo based inference of tumor subclones

Background

Tumor samples are heterogeneous. They consist of varying cell populations or subclones and each subclone is characterized with a distinct single nucleotide variant (SNV) profile. This explains the source of genetic heterogeneity observed in tumor sequencing data. To make precise prognosis and design effective therapy for cancer, ascertaining the subclonal composition of a tumor is of great importance.

Results

In this paper, we propose a state-space formulation of the feature allocation model. This model is interpreted as the blind deconvolution of the expected variant allele fractions (VAFs). VAFs are deconvolved into a binary matrix of genotypes and a matrix of genotype proportions in the samples. Specifically, we consider a sequential construction of the genotype matrix which we model by Indian buffet process (IBP). We describe an efficient sequential Monte Carlo (SMC) algorithm, SeqClone, that jointly estimates the genotypes of subclones and their proportions in the samples. When compared to other methods for resolving tumor heterogeneity, SeqClone provides comparable and sometimes, better estimates of model parameters. By design, SeqClone conveniently handles any number of probed SNVs in the samples. In particular, we can analyze VAFs from newly probed SNVs to improve existing estimates, an attribute not present in existing solutions.

Conclusions

We show that the SMC algorithm for deconvolving VAFs from tumor sequencing data is a robust and promising alternative for explaining the observed genetic heterogeneity in tumor samples.

Electronic supplementary material

The online version of this article (10.1186/s12859-018-2562-y) contains supplementary material, which is available to authorized users.

Characterization of tumor heterogeneity by latent haplotypes: a sequential Monte Carlo approach

Tumor samples obtained from a single cancer patient spatially or temporally often consist of varying cell populations, each harboring distinct mutations that uniquely characterize its genome. Thus, in any given samples of a tumor having more than two haplotypes, defined as a scaffold of single nucleotide variants (SNVs) on the same homologous genome, is evidence of heterogeneity because humans are diploid and we would therefore only observe up to two haplotypes if all cells in a tumor sample were genetically homogeneous. We characterize tumor heterogeneity by latent haplotypes and present state-space formulation of the feature allocation model for estimating the haplotypes and their proportions in the tumor samples. We develop an efficient sequential Monte Carlo (SMC) algorithm that estimates the states and the parameters of our proposed state-space model, which are equivalently the haplotypes and their proportions in the tumor samples. The sequential algorithm produces more accurate estimates of the model parameters when compared with existing methods. Also, because our algorithm processes the variant allele frequency (VAF) of a locus as the observation at a single time-step, VAF from newly sequenced candidate SNVs from next-generation sequencing (NGS) can be analyzed to improve existing estimates without re-analyzing the previous datasets, a feature that existing solutions do not possess.

An extended Kalman filter for mouse tracking

Animal tracking is an important tool for observing behavior, which is useful in various research areas. Animal specimens can be tracked using dynamic models and observation models that require several types of data. Tracking mouse has several barriers due to the physical characteristics of the mouse, their unpredictable movement, and cluttered environments. Therefore, we propose a reliable method that uses a detection stage and a tracking stage to successfully track mouse. The detection stage detects the surface area of the mouse skin, and the tracking stage implements an extended Kalman filter to estimate the state variables of a nonlinear model. The changes in the overall shape of the mouse are tracked using an oval-shaped tracking model to estimate the parameters for the ellipse. An experiment is conducted to demonstrate the performance of the proposed tracking algorithm using six video images showing various types of movement, and the ground truth values for synthetic images are compared to the values generated by the tracking algorithm. A conventional manual tracking method is also applied to compare across eight experimenters. Furthermore, the effectiveness of the proposed tracking method is also demonstrated by applying the tracking algorithm with actual images of mouse. Graphical abstract.

Bayesian estimation of scaled mutation rate under the coalescent: a sequential Monte Carlo approach

BACKGROUND

Samples of molecular sequence data of a locus obtained from random individuals in a population are often related by an unknown genealogy. More importantly, population genetics parameters, for instance, the scaled population mutation rate Θ=4N _e μ for diploids or Θ=2N _e μ for haploids (where N _e is the effective population size and μ is the mutation rate per site per generation), which explains some of the evolutionary history and past qualities of the population that the samples are obtained from, is of significant interest.

RESULTS

In this paper, we present the evolution of sequence data in a Bayesian framework and the approximation of the posterior distributions of the unknown parameters of the model, which include Θ via the sequential Monte Carlo (SMC) samplers for static models. Specifically, we approximate the posterior distributions of the unknown parameters with a set of weighted samples i.e., the set of highly probable genealogies out of the infinite set of possible genealogies that describe the sampled sequences. The proposed SMC algorithm is evaluated on simulated DNA sequence datasets under different mutational models and real biological sequences. In terms of the accuracy of the estimates, the proposed SMC method shows a comparable and sometimes, better performance than the state-of-the-art MCMC algorithms.

CONCLUSIONS

We showed that the SMC algorithm for static model is a promising alternative to the state-of-the-art approach for simulating from the posterior distributions of population genetics parameters.

A second-order iterated smoothing algorithm

Simulation-based inference for partially observed stochastic dynamic models is currently receiving much attention due to the fact that direct computation of the likelihood is not possible in many practical situations. Iterated filtering methodologies enable maximization of the likelihood function using simulation-based sequential Monte Carlo filters. Doucet et al. (2013) developed an approximation for the first and second derivatives of the log likelihood via simulation-based sequential Monte Carlo smoothing and proved that the approximation has some attractive theoretical properties. We investigated an iterated smoothing algorithm carrying out likelihood maximization using these derivative approximations. Further, we developed a new iterated smoothing algorithm, using a modification of these derivative estimates, for which we establish both theoretical results and effective practical performance. On benchmark computational challenges, this method beat the first-order iterated filtering algorithm. The method's performance was comparable to a recently developed iterated filtering algorithm based on an iterated Bayes map. Our iterated smoothing algorithm and its theoretical justification provide new directions for future developments in simulation-based inference for latent variable models such as partially observed Markov process models.

A computational method for the investigation of multistable systems and its application to genetic switches

Background

Genetic switches exhibit multistability, form the basis of epigenetic memory, and are found in natural decision making systems, such as cell fate determination in developmental pathways. Synthetic genetic switches can be used for recording the presence of different environmental signals, for changing phenotype using synthetic inputs and as building blocks for higher-level sequential logic circuits. Understanding how multistable switches can be constructed and how they function within larger biological systems is therefore key to synthetic biology.

Results

Here we present a new computational tool, called StabilityFinder, that takes advantage of sequential Monte Carlo methods to identify regions of parameter space capable of producing multistable behaviour, while handling uncertainty in biochemical rate constants and initial conditions. The algorithm works by clustering trajectories in phase space, and iteratively minimizing a distance metric. Here we examine a collection of models of genetic switches, ranging from the deterministic Gardner toggle switch to stochastic models containing different positive feedback connections. We uncover the design principles behind making bistable, tristable and quadristable switches, and find that rate of gene expression is a key parameter. We demonstrate the ability of the framework to examine more complex systems and examine the design principles of a three gene switch. Our framework allows us to relax the assumptions that are often used in genetic switch models and we show that more complex abstractions are still capable of multistable behaviour.

Conclusions

Our results suggest many ways in which genetic switches can be enhanced and offer designs for the construction of novel switches. Our analysis also highlights subtle changes in correlation of experimentally tunable parameters that can lead to bifurcations in deterministic and stochastic systems. Overall we demonstrate that StabilityFinder will be a valuable tool in the future design and construction of novel gene networks.

Electronic supplementary material

The online version of this article (doi:10.1186/s12918-016-0375-z) contains supplementary material, which is available to authorized users.

BCM: toolkit for Bayesian analysis of Computational Models using samplers

BACKGROUND

Computational models in biology are characterized by a large degree of uncertainty. This uncertainty can be analyzed with Bayesian statistics, however, the sampling algorithms that are frequently used for calculating Bayesian statistical estimates are computationally demanding, and each algorithm has unique advantages and disadvantages. It is typically unclear, before starting an analysis, which algorithm will perform well on a given computational model.

RESULTS

We present BCM, a toolkit for the Bayesian analysis of Computational Models using samplers. It provides efficient, multithreaded implementations of eleven algorithms for sampling from posterior probability distributions and for calculating marginal likelihoods. BCM includes tools to simplify the process of model specification and scripts for visualizing the results. The flexible architecture allows it to be used on diverse types of biological computational models. In an example inference task using a model of the cell cycle based on ordinary differential equations, BCM is significantly more efficient than existing software packages, allowing more challenging inference problems to be solved.

CONCLUSIONS

BCM represents an efficient one-stop-shop for computational modelers wishing to use sampler-based Bayesian statistics.

A probabilistic framework for the exploration of enzymatic capabilities based on feasible kinetics and control analysis

BACKGROUND

Analysis of limiting steps within enzyme-catalyzed reactions is fundamental to understand their behavior and regulation. Methods capable of unravelling control properties and exploring kinetic capabilities of enzymatic reactions would be particularly useful for protein and metabolic engineering. While single-enzyme control analysis formalism has previously been applied to well-studied enzymatic mechanisms, broader application of this formalism is limited in practice by the limited amount of kinetic data and the difficulty of describing complex allosteric mechanisms.

METHODS

To overcome these limitations, we present here a probabilistic framework enabling control analysis of previously unexplored mechanisms under uncertainty. By combining a thermodynamically consistent parameterization with an efficient Sequential Monte Carlo sampler embedded in a Bayesian setting, this framework yields insights into the capabilities of enzyme-catalyzed reactions with modest kinetic information, provided that the catalytic mechanism and a thermodynamic reference point are defined.

RESULTS

The framework was used to unravel the impact of thermodynamic affinity, substrate saturation levels and effector concentrations on the flux control and response coefficients of a diverse set of enzymatic reactions.

CONCLUSIONS

Our results highlight the importance of the metabolic context in the control analysis of isolated enzymes as well as the use of statistically sound methods for their interpretation.

GENERAL SIGNIFICANCE

This framework significantly expands our current capabilities for unravelling the control properties of general reaction kinetics with limited amount of information. This framework will be useful for both theoreticians and experimentalists in the field.

Labeled RFS-Based Track-Before-Detect for Multiple Maneuvering Targets in the Infrared Focal Plane Array

The problem of jointly detecting and tracking multiple targets from the raw observations of an infrared focal plane array is a challenging task, especially for the case with uncertain target dynamics. In this paper a multi-model labeled multi-Bernoulli (MM-LMB) track-before-detect method is proposed within the labeled random finite sets (RFS) framework. The proposed track-before-detect method consists of two parts-MM-LMB filter and MM-LMB smoother. For the MM-LMB filter, original LMB filter is applied to track-before-detect based on target and measurement models, and is integrated with the interacting multiple models (IMM) approach to accommodate the uncertainty of target dynamics. For the MM-LMB smoother, taking advantage of the track labels and posterior model transition probability, the single-model single-target smoother is extended to a multi-model multi-target smoother. A Sequential Monte Carlo approach is also presented to implement the proposed method. Simulation results show the proposed method can effectively achieve tracking continuity for multiple maneuvering targets. In addition, compared with the forward filtering alone, our method is more robust due to its combination of forward filtering and backward smoothing.

Sequential Monte Carlo for Maximum Weight Subgraphs with Application to Solving Image Jigsaw Puzzles

We consider a problem of finding maximum weight subgraphs (MWS) that satisfy hard constraints in a weighted graph. The constraints specify the graph nodes that must belong to the solution as well as mutual exclusions of graph nodes, i.e., pairs of nodes that cannot belong to the same solution. Our main contribution is a novel inference approach for solving this problem in a sequential monte carlo (SMC) sampling framework. Usually in an SMC framework there is a natural ordering of the states of the samples. The order typically depends on observations about the states or on the annealing setup used. In many applications (e.g., image jigsaw puzzle problems), all observations (e.g., puzzle pieces) are given at once and it is hard to define a natural ordering. Therefore, we relax the assumption of having ordered observations about states and propose a novel SMC algorithm for obtaining maximum a posteriori estimate of a high-dimensional posterior distribution. This is achieved by exploring different orders of states and selecting the most informative permutations in each step of the sampling. Our experimental results demonstrate that the proposed inference framework significantly outperforms loopy belief propagation in solving the image jigsaw puzzle problem. In particular, our inference quadruples the accuracy of the puzzle assembly compared to that of loopy belief propagation.