Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

AO_SCPLOWBSTRACTC_SCPLOWThe determination of the infection fatality rate (IFR) for the novel SARS-CoV-2 coronavirus is a key aim for many of the field studies that are currently being undertaken in response to the pandemic. The IFR together with the basic reproduction number R0, are the main epidemic parameters describing severity and transmissibility of the virus, respectively. The IFR can be also used as a basis for estimating and monitoring the number of infected individuals in a population, which may be subsequently used to inform policy decisions relating to public health interventions and lockdown strategies. The interpretation of IFR measurements requires the calculation of confidence intervals. We present a number of statistical methods that are relevant in this context and develop an inverse problem formulation to determine correction factors to mitigate time-dependent effects that can lead to biased IFR estimates. We also review a number of methods to combine IFR estimates from multiple independent studies, provide example calculations throughout this note and conclude with a summary and "best practice" recommendations. The developed code is available online.

We propose a novel method for the clustering of point-cloud data that originate from single-molecule localization microscopy (SMLM). Our scheme has the ability to infer a hierarchical structure from the data. It takes a particular relevance when quantitatively analyzing the biological particles of interest at different scales. It assumes a prior neither on the shape of particles nor on the background noise. Our multiscale clustering pipeline is built upon graph theory. At each scale, we first construct a weighted graph that represents the SMLM data. Next, we find clusters using spectral clustering. We then use the output of this clustering algorithm to build the graph in the next scale; in this way, we ensure consistency over different scales. We illustrate our method with examples that highlight some of its important properties.

Single-molecule localization microscopy methods such as dSTORM require specific buffer conditions to enable blinking and detection of individual emitters, making them incompatible with live cell imaging and expansion microscopy. An alternative approach to achieve super-resolution without blinking is to observe the fluctuations of the emitter intensity over time. Existing localization algorithms for high-emitter density make use of radial symmetry or use artificial neural networks trained on single high-density frames to predict emitter positions. Here, we aim to improve the resolution by using a larger temporal context. We combine the U-Net architecture used previously for image reconstruction with multi-head attention used in the Transformer architecture. We compare the results to DECODE and eSRRF as well as to traditional fitting algorithms on public benchmark data. A generic pre-trained model is provided together with a fast and robust simulator for training data and all scripts needed to train custom networks.

Diffusion coefficients often vary across regions, such as cellular membranes, and quantifying their variation can provide valuable insight into local membrane properties such as composition and stiffness. Toward quantifying diffusion coefficient spatial maps and uncertainties from particle tracks, we use a Bayesian method and place Gaussian Process (GP) Priors on the maps. For the sake of computational efficiency, we leverage inducing point methods on GPs arising from the mathematical structure of the data giving rise to non-conjugate likelihood-prior pairs. We analyze both synthetic data, where ground truth is known, as well as data drawn from live-cell singlemolecule imaging of membrane proteins. The resulting tool provides an unsupervised method to rigorously map diffusion coefficients continuously across membranes without data binning.

AO_SCPLOWBSTRACTC_SCPLOWInfectious disease models are used to predict the spread and impact of outbreaks of a disease. Like other complex models, they have parameters that need to be calibrated, and structural discrepancies from the reality that they simulate that should be accounted for in calibration and prediction. Whilst Uncertainty Quantification (UQ) techniques have been applied to infectious disease models before, they were not routinely used to inform policymakers in the UK during the COVID-19 pandemic. In this paper, we will argue that during a fast moving pandemic, models and policy are changing on timescales that make traditional UQ methods impractical, if not impossible to implement. We present an alternative formulation to the calibration problem that embeds model discrepancy within the structure of the model, and appropriately assimilates data within the simulation. We then show how UQ can be used to calibrate the model in real-time to produce disease trajectories accounting for parameter uncertainty and model discrepancy. We apply these ideas to an age-structured COVID-19 model for England and demonstrate the types of information it could have produced to feed into policy support prior to the lockdown of March 2020.

Fast testing can help mitigate the coronavirus disease 2019 (COVID-19) pandemic. Despite their accuracy for single sample analysis, infectious diseases diagnostic tools, like RT-PCR, require substantial resources to test large populations. We develop a scalable approach for determining the viral status of pooled patient samples. Our approach converts group testing to a linear inverse problem, where false positives and negatives are interpreted as generated by a noisy communication channel, and a message passing algorithm estimates the illness status of patients. Numerical results reveal that our approach estimates patient illness using fewer pooled measurements than existing noisy group testing algorithms. Our approach can easily be extended to various applications, including where false negatives must be minimized. Finally, in a Utopian world we would have collaborated with RT-PCR experts; it is difficult to form such connections during a pandemic. We welcome new collaborators to reach out and help improve this work!

Sub-ms 3D tracking of individual molecules in living cells is an important goal for microscopy since it will enable measurements at the scale of diffusion limited macromolecular interactions. Here, we present a 3D tracking principle based on the true excitation point spread function and cross-entropy minimization for position localization of moving fluorescent reporters that approaches the relevant regime. When tested on beads moved on a stage, we reached 67nm lateral and 109nm axial precision with a time resolution of 0.84 ms at a photon count rate of 60kHz, coming close to the theoretical and simulated predictions. A critical step in the implementation was a new method for microsecond 3D PSF positioning that combines 3D holographic beam shaping and electro-optical deflection. For the analysis of tracking data, a new point estimator for diffusion was derived and evaluated by a detailed simulation of the 3D tracking principle applied to a fictive reaction-diffusion process in an E. coli-like geometry. Finally, we successfully applied these methods to track the Trigger Factor protein in living bacterial cells. Overall our results show that it is possible to reach sub-millisecond live-cell single-molecule tracking, but that it is still hard to resolve state transitions based on diffusivity at this time scale.

Disease surveillance is an integral component of government policy, allowing public health professionals to monitor transmission of infectious diseases and appropriately apply interventions. To aid with surveillance efforts, there has been extensive development of mathematical models to help inform policy decisions, However, these mathematical models rely upon data streams that are expensive and often only practical for high income countries. With a growing focus on equitable public health tools there is a dire need for development of mathematical models that are equipped to handle the data stream challenges prevalent in low and middle income countries, where data is often incomplete and subject to aggregation. To address this need, we develop a mathematical model for the joint estimation of the effective reproduction number and daily incidence of an infectious disease using incomplete and aggregated data. Our investigation demonstrates that this novel mathematical model is robust across a variety of reduced data streams, making it suitable for application in diverse regions. Author summaryMonitoring the transmission of infectious diseases is an important part of government policy that is often hindered by limitations in data streams. This is especially true in low and middle income countries where health sectors have less funding. In this work we develop a mathematical model to enhance disease surveillance by overcoming these data limitations, providing accurate inferences of relevant epidemiological parameters.

We address the problem of inferring the number of independently blinking fluorescent light emitters, when only their combined intensity contributions can be observed at each timepoint. This problem occurs regularly in light microscopy of objects that are smaller than the diffraction limit, where one wishes to count the number of fluorescently labelled subunits. Our proposed solution directly models the photo-physics of the system, as well as the blinking kinetics of the fluorescent emitters as a fully differentiable hidden Markov model. Given a trace of intensity over time, our model jointly estimates the parameters of the intensity distribution per emitter, their blinking rates, as well as a posterior distribution of the total number of fluorescent emitters. We show that our model is consistently more accurate and increases the range of countable subunits by a factor of two compared to current state-of-the-art methods, which count based on autocorrelation and blinking frequency. Furthermore, we demonstrate that our model can be used to investigate the effect of blinking kinetics on counting ability, and therefore can inform experimental conditions that will maximize counting accuracy.

Fluorescence imaging with spectrally variant fluorophores allows the spatial mapping of biological structures with exquisite cellular and molecular specificity. However, the ability to robustly discriminate multiple fluorophores in any single imaging experiment is greatly hindered by the broad emission spectra of bio-compatible fluorophores and the large contribution of noise in low-energy regime fluorescence microscopy. In this study, we propose a novel machine learning framework, Bleaching-Excitation-Emission Photodynamics (BEEP) learning, that exploits multiple discriminatory features of fluorescent dyes to greatly expand the number of distinguishable objects in an image by integrating emission spectra, excitation variability, and bleaching dynamics into a unified multi-view, fluorescence unmixing approach. Our method is built upon a rank-one-tensor-based generalized linear model and leverages two biophysically grounded assumptions: consistent spectral and bleaching behaviors under fixed excitation, and invariant fluorophore abundances across excitations. We first extract excitation-specific spectral and bleaching signatures from reference images, and then use them to estimate abundances in complex mixtures. Experimental results on both simulated and real images of microbial populations demonstrate that our approach significantly outperforms conventional and partially multi-view methods, offering improved robustness and accuracy in highly multiplexed fluorescence imaging.

Calibration of complex stochastic infectious disease models is challenging. These often have high-dimensional input and output spaces, with the models exhibiting complex, non-linear dynamics. Coupled with a paucity of necessary data, this results in a large number of non-ignorable hidden states that must be handled by the inference routine. Likelihood-based approaches to this missing data problem are very flexible, but challenging to scale, due to having to monitor and update these hidden states. Methods based on simulating the hidden states directly from the model-of-interest have an advantage that they are often more straightforward to code, and thus are easier to implement and adapt in real-time. However, these often require evaluating very large numbers of simulations, rendering them infeasible for many large-scale problems. We present a framework for using emulation-based methods to calibrate a large-scale, stochastic, age-structured, spatial meta-population model of COVID-19 transmission in England and Wales. By embedding a model discrepancy process into the simulation model, and combining this with particle filtering, we show that it is possible to calibrate complex models to high-dimensional data by emulating the log-likelihood surface instead of individual data points. The use of embedded model discrepancy also helps to alleviate other key challenges, such as the introduction of infection across space and time. We conclude with a discussion of major challenges remaining and key areas for future work.

In oblique plane microscopy, a remote refocussing system is used to create an aberration-free 3D image of the object, from which an inclined plane is sampled using a tilted imaging system. As a result, unlike in conventional microscopes, the point spread function (PSF) and optical transfer function (OTF) are tilted with respect to the optical axis of the imaging system. This suggests that a small sample-scanning step size is required to satisfy the Nyquist-Shannon sampling criterion. Recently, we have shown that a careful choice of apparent undersampling leads to aliased OTF copies that can be separated and stitched together in Fourier space to form a properly sampled OTF. In our demonstrations, this led to acquisition speed gains of between twoand four-fold. Here, we introduce a deconvolution-based method to reconstruct properly sampled volumes from apparently undersampled datasets that outperforms our previous Fourier-stitching approach.

In epidemic models, the effective reproduction number is of central importance to assess the transmission dynamics of an infectious disease and to orient health intervention strategies. Publicly shared data during an outbreak often suffers from two sources of misreporting (underreporting and delay in reporting) that should not be overlooked when estimating epidemiological parameters. The main statistical challenge in models that intrinsically account for a misreporting process lies in the joint estimation of the time-varying reproduction number and the delay/underreporting parameters. Existing Bayesian approaches typically rely on Markov chain Monte Carlo (MCMC) algorithms that are extremely costly from a computational perspective. We propose a much faster alternative based on Laplacian-P-splines (LPS) that combines Bayesian penalized B-splines for flexible and smooth estimation of the time-varying reproduction number and Laplace approximations to selected posterior distributions for fast computation. Assuming a known generation interval distribution, the incidence at a given calendar time is governed by the epidemic renewal equation and the delay structure is specified through a composite link framework. Laplace approximations to the conditional posterior of the spline vector are obtained from analytical versions of the gradient and Hessian of the log-likelihood, implying a drastic speed-up in the computation of posterior estimates. Furthermore, the proposed LPS approach can be used to obtain point estimates and approximate credible intervals for the delay and reporting probabilities. Simulation of epidemics with different combinations for the underreporting rate and delay structure (one-day, two-day and weekend delays) show that the proposed LPS methodology delivers fast and accurate estimates outperforming existing methods that do not take into account underreporting and delay patterns. Finally, LPS is illustrated on two real case studies of epidemic outbreaks.

We introduce the angular reproduction number {Omega}, which measures time-varying changes in epidemic transmissibility resulting from variations in both the effective reproduction number R, and generation time distribution w. Predominant approaches for tracking pathogen spread either infer R or the epidemic growth rate r. However, R is biased by mismatches between the assumed and true w, while r is difficult to interpret in terms of the individual-level branching process underpinning transmission. R and r may also disagree on the relative transmissibility of epidemics or variants (i.e., rA>rB does not imply RA>RB for variants A and B). We find that {Omega} responds meaningfully to mismatches and time-variations in w while mostly maintaining the interpretability of R. We prove that {Omega}>1 implies R>1 and that {Omega} agrees with r on the relative transmissibility of pathogens. Estimating {Omega} is no more difficult than inferring R, uses existing software, and requires no generation time measurements. These advantages come at the expense of selecting one free parameter. We propose {Omega} as complementary statistic to R and r that improves transmissibility estimates when w is misspecified or time-varying and better reflects the impact of interventions, when those interventions concurrently change R and w or alter the relative risk of co-circulating pathogens.

Coronavirus disease (COVID-19) has highlighted both the shortcomings and value of modelling infectious diseases. Infectious disease models can serve as critical tools to predict the development of cases and associated healthcare demand and to determine the set of non-pharmaceutical interventions (NPI) that is most effective in slowing the spread of the infectious agent. Current approaches to estimate NPI effects typically focus on relatively short time periods and either on the number of reported cases, deaths, intensive care occupancy or hospital occupancy as a single indicator of disease transmission. In this work, we propose a Bayesian hierarchical model that integrates multiple outcomes and complementary sources of information in the estimation of the true and unknown number of infections while accounting for time-varying under-reporting and weekday-specific delays in reported cases and deaths, allowing us to estimate the number of infections on a daily basis rather than having to smooth the data. Using information from the entire course of the pandemic, we account for the spread of variants of concern, seasonality and vaccination coverage in the model. We implement a Markov Chain Monte Carlo algorithm to conduct Bayesian inference and estimate the effect of NPIs for 20 European countries. The approach shows good performance on simulated data and produces posterior predictions that show a good fit to reported cases, deaths, hospital and intensive care occupancy.

The serial interval of an infectious disease is a key instrument to understand transmission dynamics. Estimation of the serial interval distribution from illness onset data extracted from transmission pairs is challenging due to the presence of censoring and state-of-the-art frequentist or Bayesian methods mostly rely on parametric models. We present a fully data-driven methodology to estimate the serial interval distribution based on (coarse) serial interval data. The proposal combines a nonparametric estimator of the cumulative distribution function with the bootstrap and yields point and interval estimates of any desired feature of the serial interval distribution. Algorithms underlying our approach are simple, fast and stable, and are thus easily implementable in any programming language most desired by modelers from the infectious disease community. The nonparametric routines are included in the EpiLPS package for ease of implementation. Our method complements existing parametric approaches for serial interval estimation and permits to straightforwardly analyze past, current, or future illness onset data streams.

Severity rates like the case-fatality rate and infection-fatality rate are key metrics in public health. To guide decision-making in response to changes like new variants or vaccines, it is imperative to understand how these rates shift in real time. In practice, time-varying severity rates are typically estimated using a ratio of aggregate counts. We demonstrate that these estimators are capable of exhibiting large statistical biases, with concerning implications for public health practice, as they may fail to detect heightened risks or falsely signal nonexistent surges. We supplement our mathematical analyses with experimental results on real and simulated COVID-19 data. Finally, we briefly discuss strategies to mitigate this bias, drawing connections with effective reproduction number (Rt) estimation.

We investigate the influence of different accuracy-detection rate trade-offs on image reconstruction in single molecule localization microscopy. Our main focus is the investigation of image artifacts experienced when using low localization accuracy, especially in the presence of sample drift and inhomogeneous background. In this context we present a newly developed SMLM software termed FIRESTORM which is optimized for high accuracy reconstruction. For our analysis we used in silico SMLM data and compared the reconstructed images to the ground truth data. We observe two discriminable reconstruction populations of which only one shows the desired localization behavior.

Single molecule fluorescence localization with minimum photon flux imaging (MINFLUX) can achieve localization precisions in the small nanometer range or better under suitable conditions. Potentially adverse conditions, such as a fixed fluorescence dipole or optical aberrations, that could cause systematic localization errors, have received little attention up to now. Here, we study these effects in simulation. We find that biases occur for fluorophores with a fixed absorption dipole tilted out of the imaging plane. These become larger (up to about 25% of the diameter of the circle spanned by the doughnut center positions) the larger the tilt angle gets. As a rule of thumb the spread in bias is smaller than 5 nm in case the dipole orientation is less than 30{degrees} out of plane for the typical case of a doughnut probing circle of diameter 100 nm. For freely rotating dipoles only the primary aberrations astigmatism and coma contribute to bias. This bias depends on the position of the fluorophore inside the circular probing area of MINFLUX and can be significantly larger than the localization precision. We show that increasing the number of measurements over the circle from a triangular to a hexagonal pattern is beneficial for reducing bias in all cases. Iterative shrinking of the probing area can eliminate the position dependent bias completely, but a strong dependence on dipole orientation of the bias at the center of the probing area remains.

Methods for the quantification of rhythmic biological signals have been essential for the discovery of function and design of biological oscillators. Advances in live measurements have allowed recordings of unprecedented resolution revealing a new world of complex heterogeneous oscillations with multiple noisy non-stationary features. However, our understanding of the underlying mechanisms regulating these oscillations has been lagging behind, partially due to the lack of simple tools to reliably quantify these complex non-stationary features. With this challenge in mind, we have developed pyBOAT, a Python-based fully automatic stand-alone software that integrates multiple steps of non-stationary oscillatory time series analysis into an easy-to-use graphical user interface. pyBOAT implements continuous wavelet analysis which is specifically designed to reveal time-dependent features. In this work we illustrate the advantages of our tool by analyzing complex non-stationary time-series profiles. Our approach integrates data-visualization, optimized sinc-filter detrending, amplitude envelope removal and a subsequent continuous-wavelet based time-frequency analysis. Finally, using analytical considerations and numerical simulations we discuss unexpected pitfalls in commonly used smoothing and detrending operations.