Estimating the time-varying reproduction number of COVID-19 with a state-space method

Stratified epidemic model using a latent marked Hawkes process

We extend the unstructured homogeneously mixing epidemic model introduced by Lamprinakou et al. (2023) to a finite population stratified by age bands. We model the actual unobserved infections using a latent marked Hawkes process and the reported aggregated infections as random quantities driven by the underlying Hawkes process. We apply a Kernel Density Particle Filter (KDPF) to infer the marked counting process, the instantaneous reproduction number for each age group and forecast the epidemic's trajectory in the near future. Taking into account the individual inhomogeneity in age does not increase significantly the computational cost of the proposed inference algorithm compared to the cost of the proposed algorithm for the homogeneously unstructured epidemic model. We demonstrate that considering the individual heterogeneity in age, we can derive the instantaneous reproduction numbers per age group that provide a real-time measurement of interventions and behavioural changes of the associated groups. We illustrate the performance of the proposed inference algorithm on synthetic data sets and COVID-19-reported cases in various local authorities in the UK, and benchmark our model to the unstructured homogeneously mixing epidemic model. Our paper is a "demonstration" of a methodology that might be applied to factors other than age for stratification.

Real-Time Epidemiology and Acute Care Need Monitoring and Forecasting for COVID-19 via Bayesian Sequential Monte Carlo-Leveraged Transmission Models

COVID-19 transmission models have conferred great value in informing public health understanding, planning, and response. However, the pandemic also demonstrated the infeasibility of basing public health decision-making on transmission models with pre-set assumptions. No matter how favourably evidenced when built, a model with fixed assumptions is challenged by numerous factors that are difficult to predict. Ongoing planning associated with rolling back and re-instituting measures, initiating surge planning, and issuing public health advisories can benefit from approaches that allow state estimates for transmission models to be continuously updated in light of unfolding time series. A model being continuously regrounded by empirical data in this way can provide a consistent, integrated depiction of the evolving underlying epidemiology and acute care demand, offer the ability to project forward such a depiction in a fashion suitable for triggering the deployment of acute care surge capacity or public health measures, and support quantitative evaluation of tradeoffs associated with prospective interventions in light of the latest estimates of the underlying epidemiology. We describe here the design, implementation, and multi-year daily use for public health and clinical support decision-making of a particle-filtered COVID-19 compartmental model, which served Canadian federal and provincial governments via regular reporting starting in June 2020. The use of the Bayesian sequential Monte Carlo algorithm of particle filtering allows the model to be regrounded daily and adapt to new trends within daily incoming data-including test volumes and positivity rates, endogenous and travel-related cases, hospital census and admissions flows, daily counts of dose-specific vaccinations administered, measured concentration of SARS-CoV-2 in wastewater, and mortality. Important model outputs include estimates (via sampling) of the count of undiagnosed infectives, the count of individuals at different stages of the natural history of frankly and pauci-symptomatic infection, the current force of infection, effective reproductive number, and current and cumulative infection prevalence. Following a brief description of the model design, we describe how the machine learning algorithm of particle filtering is used to continually reground estimates of the dynamic model state, support a probabilistic model projection of epidemiology and health system capacity utilization and service demand, and probabilistically evaluate tradeoffs between potential intervention scenarios. We further note aspects of model use in practice as an effective reporting tool in a manner that is parameterized by jurisdiction, including the support of a scripting pipeline that permits a fully automated reporting pipeline other than security-restricted new data retrieval, including automated model deployment, data validity checks, and automatic post-scenario scripting and reporting. As demonstrated by this multi-year deployment of the Bayesian machine learning algorithm of particle filtering to provide industrial-strength reporting to inform public health decision-making across Canada, such methods offer strong support for evidence-based public health decision-making informed by ever-current articulated transmission models whose probabilistic state and parameter estimates are continually regrounded by diverse data streams.

Estimating effective reproduction number revisited

Accurately estimating the effective reproduction number is crucial for characterizing the transmissibility of infectious diseases to optimize interventions and responses during epidemic outbreaks. In this study, we improve the estimation of the effective reproduction number through two main approaches. First, we derive a discrete model to represent a time series of case counts and propose an estimation method based on this framework. We also conduct numerical experiments to demonstrate the effectiveness of the proposed discretization scheme. By doing so, we enhance the accuracy of approximating the underlying epidemic process compared to previous methods, even when the counting period is similar to the mean generation time of an infectious disease. Second, we employ a negative binomial distribution to model the variability of count data to accommodate overdispersion. Specifically, given that observed incidence counts follow a negative binomial distribution, the posterior distribution of secondary infections is obtained as a Dirichlet multinomial distribution. With this formulation, we establish posterior uncertainty bounds for the effective reproduction number. Finally, we demonstrate the effectiveness of the proposed method using incidence data from the COVID-19 pandemic.

Improved time-varying reproduction numbers using the generation interval for COVID-19

Estimating key epidemiological parameters, such as incubation period, serial interval (SI), generation interval (GI) and latent period, is essential to quantify the transmissibility and effects of various interventions of COVID-19. These key parameters play a critical role in quantifying the basic reproduction number. With the hard work of epidemiological investigators in South Korea, estimating these key parameters has become possible based on infector-infectee surveillance data of COVID-19 between February 2020 and April 2021. Herein, the mean incubation period was estimated to be 4.9 days (95% CI: 4.2, 5.7) and the mean generation interval was estimated to be 4.3 days (95% CI: 4.2, 4.4). The mean serial interval was estimated to be 4.3, with a standard deviation of 4.2. It is also revealed that the proportion of presymptomatic transmission was ~57%, which indicates the potential risk of transmission before the disease onset. We compared the time-varying reproduction number based on GI and SI and found that the time-varying reproduction number based on GI may result in a larger estimation of Rt, which refers to the COVID-19 transmission potential around the rapid increase of cases. This highlights the importance of considering presymptomatic transmission and generation intervals when estimating the time-varying reproduction number.

Multivariate epidemic count time series model

An infectious disease spreads not only over a single population or community but also across multiple and heterogeneous communities. Moreover, its transmissibility varies over time because of various factors such as seasonality and epidemic control, which results in strongly nonstationary behavior. In conventional methods for assessing transmissibility trends or changes, univariate time-varying reproduction numbers are calculated without taking into account transmission across multiple communities. In this paper, we propose a multivariate-count time series model for epidemics. We also propose a statistical method for estimating the transmission of infections across multiple communities and the time-varying reproduction numbers of each community simultaneously from a multivariate time series of case counts. We apply our method to incidence data for the novel coronavirus disease 2019 (COVID-19) pandemic to reveal the spatiotemporal heterogeneity of the epidemic process.

Using a latent Hawkes process for epidemiological modelling

Understanding the spread of COVID-19 has been the subject of numerous studies, highlighting the significance of reliable epidemic models. Here, we introduce a novel epidemic model using a latent Hawkes process with temporal covariates for modelling the infections. Unlike other models, we model the reported cases via a probability distribution driven by the underlying Hawkes process. Modelling the infections via a Hawkes process allows us to estimate by whom an infected individual was infected. We propose a Kernel Density Particle Filter (KDPF) for inference of both latent cases and reproduction number and for predicting the new cases in the near future. The computational effort is proportional to the number of infections making it possible to use particle filter type algorithms, such as the KDPF. We demonstrate the performance of the proposed algorithm on synthetic data sets and COVID-19 reported cases in various local authorities in the UK, and benchmark our model to alternative approaches.

A model of behavioural response to risk accurately predicts the statistical distribution of COVID-19 infection and reproduction numbers

One clear aspect of behaviour in the COVID-19 pandemic has been people's focus on, and response to, reported or observed infection numbers in their community. We describe a simple model of infectious disease spread in a pandemic situation where people's behaviour is influenced by the current risk of infection and where this behavioural response acts homeostatically to return infection risk to a certain preferred level. This homeostatic response is active until approximate herd immunity is reached: in this domain the model predicts that the reproduction rate R will be centred around a median of 1, that proportional change in infection numbers will follow the standard Cauchy distribution with location and scale parameters 0 and 1, and that high infection numbers will follow a power-law frequency distribution with exponent 2. To test these predictions we used worldwide COVID-19 data from 1st February 2020 to 30th June 2022 to calculate [Formula: see text] confidence interval estimates across countries for these R, location, scale and exponent parameters. The resulting median R estimate was [Formula: see text] (predicted value 1) the proportional change location estimate was [Formula: see text] (predicted value 0), the proportional change scale estimate was [Formula: see text] (predicted value 1), and the frequency distribution exponent estimate was [Formula: see text] (predicted value 2); in each case the observed estimate agreed with model predictions.

Role of multiresolution vulnerability indices in COVID-19 spread in India: a Bayesian model-based analysis

OBJECTIVES

COVID-19 has differentially affected countries, with health infrastructure and other related vulnerability indicators playing a role in determining the extent of its spread. Vulnerability of a geographical region to COVID-19 has been a topic of interest, particularly in low-income and middle-income countries like India to assess its multifactorial impact on incidence, prevalence or mortality. This study aims to construct a statistical analysis pipeline to compute such vulnerability indices and investigate their association with metrics of the pandemic growth.

DESIGN

Using publicly reported observational socioeconomic, demographic, health-based and epidemiological data from Indian national surveys, we compute contextual COVID-19 Vulnerability Indices (cVIs) across multiple thematic resolutions for different geographical and spatial administrative regions. These cVIs are then used in Bayesian regression models to assess their impact on indicators of the spread of COVID-19.

SETTING

This study uses district-level indicators and case counts data for the state of Odisha, India.

PRIMARY OUTCOME MEASURE

We use instantaneous R (temporal average of estimated time-varying reproduction number for COVID-19) as the primary outcome variable in our models.

RESULTS

Our observational study, focussing on 30 districts of Odisha, identified housing and hygiene conditions, COVID-19 preparedness and epidemiological factors as important indicators associated with COVID-19 vulnerability.

CONCLUSION

Having succeeded in containing COVID-19 to a reasonable level during the first wave, the second wave of COVID-19 made greater inroads into the hinterlands and peripheral districts of Odisha, burdening the already deficient public health system in these areas, as identified by the cVIs. Improved understanding of the factors driving COVID-19 vulnerability will help policy makers prioritise resources and regions, leading to more effective mitigation strategies for the present and future.

Optimal control design incorporating vaccination and treatment on six compartment pandemic dynamical system

In this paper, a mathematical model of the COVID-19 pandemic with lockdown that provides a more accurate representation of the infection rate has been analyzed. In this model, the total population is divided into six compartments: the susceptible class, lockdown class, exposed class, asymptomatic infected class, symptomatic infected class, and recovered class. The basic reproduction number (R0) is calculated using the next-generation matrix method and presented graphically based on different progression rates and effective contact rates of infective individuals. The COVID-19 epidemic model exhibits the disease-free equilibrium and endemic equilibrium. The local and global stability analysis has been done at the disease-free and endemic equilibrium based on R0. The stability analysis of the model shows that the disease-free equilibrium is both locally and globally stable when R0<1, and the endemic equilibrium is locally and globally stable when R0>1 under some conditions. A control strategy including vaccination and treatment has been studied on this pandemic model with an objective functional to minimize. Finally, numerical simulation of the COVID-19 outbreak in India is carried out using MATLAB, highlighting the usefulness of the COVID-19 pandemic model and its mathematical analysis.

Individual risk-aversion responses tune epidemics to critical transmissibility (R = 1)

Changes in human behaviour are a major determinant of epidemic dynamics. Collective activity can be modified through imposed control measures, but spontaneous changes can also arise as a result of uncoordinated individual responses to the perceived risk of contagion. Here, we introduce a stochastic epidemic model implementing population responses driven by individual time-varying risk aversion. The model reveals an emergent mechanism for the generation of multiple infection waves of decreasing amplitude that progressively tune the effective reproduction number to its critical value R = 1. In successive waves, individuals with gradually lower risk propensity are infected. The overall mechanism shapes well-defined risk-aversion profiles over the whole population as the epidemic progresses. We conclude that uncoordinated changes in human behaviour can by themselves explain major qualitative and quantitative features of the epidemic process, like the emergence of multiple waves and the tendency to remain around R = 1 observed worldwide after the first few waves of COVID-19.

Fundamental limits on inferring epidemic resurgence in real time using effective reproduction numbers

We find that epidemic resurgence, defined as an upswing in the effective reproduction number (R) of the contagion from subcritical to supercritical values, is fundamentally difficult to detect in real time. Inherent latencies in pathogen transmission, coupled with smaller and intrinsically noisier case incidence across periods of subcritical spread, mean that resurgence cannot be reliably detected without significant delays of the order of the generation time of the disease, even when case reporting is perfect. In contrast, epidemic suppression (where R falls from supercritical to subcritical values) may be ascertained 5-10 times faster due to the naturally larger incidence at which control actions are generally applied. We prove that these innate limits on detecting resurgence only worsen when spatial or demographic heterogeneities are incorporated. Consequently, we argue that resurgence is more effectively handled proactively, potentially at the expense of false alarms. Timely responses to recrudescent infections or emerging variants of concern are more likely to be possible when policy is informed by a greater quality and diversity of surveillance data than by further optimisation of the statistical models used to process routine outbreak data.

Modeling COVID-19 Incidence by the Renewal Equation after Removal of Administrative Bias and Noise

Simple Summary

In the past two years, the COVID-19 incidence curves and reproduction number Rt have been the main metrics used by policy makers and journalists to monitor the spread of this global pandemic. However, these metrics are not always reliable in the short term, because of a combination of delay in detection, administrative delays and random noise. In this article, we present a complete model of COVID-19 incidence, faithfully reconstructing the incidence curve and reproduction number from the renewal equation of the disease and precisely estimating the biases associated with periodic weekly bias, festive day bias and residual noise.

Abstract

The sanitary crisis of the past two years has focused the public’s attention on quantitative indicators of the spread of the COVID-19 pandemic. The daily reproduction number Rt, defined by the average number of new infections caused by a single infected individual at time t, is one of the best metrics for estimating the epidemic trend. In this paper, we provide a complete observation model for sampled epidemiological incidence signals obtained through periodic administrative measurements. The model is governed by the classic renewal equation using an empirical reproduction kernel, and subject to two perturbations: a time-varying gain with a weekly period and a white observation noise. We estimate this noise model and its parameters by extending a variational inversion of the model recovering its main driving variable Rt. Using Rt, a restored incidence curve, corrected of the weekly and festive day bias, can be deduced through the renewal equation. We verify experimentally on many countries that, once the weekly and festive days bias have been corrected, the difference between the incidence curve and its expected value is well approximated by an exponential distributed white noise multiplied by a power of the magnitude of the restored incidence curve.

Reconstruction of Epidemiological Data in Hungary Using Stochastic Model Predictive Control

In this paper, we propose a model-based method for the reconstruction of not directly measured epidemiological data. To solve this task, we developed a generic optimization-based approach to compute unknown time-dependent quantities (such as states, inputs, and parameters) of discrete-time stochastic nonlinear models using a sequence of output measurements. The problem was reformulated as a stochastic nonlinear model predictive control computation, where the unknown inputs and parameters were searched as functions of the uncertain states, such that the model output followed the observations. The unknown data were approximated by Gaussian distributions. The predictive control problem was solved over a relatively long time window in three steps. First, we approximated the expected trajectories of the unknown quantities through a nonlinear deterministic problem. In the next step, we fixed the expected trajectories and computed the corresponding variances using closed-form expressions. Finally, the obtained mean and variance values were used as an initial guess to solve the stochastic problem. To reduce the estimated uncertainty of the computed states, a closed-loop input policy was considered during the optimization, where the state-dependent gain values were determined heuristically. The applicability of the approach is illustrated through the estimation of the epidemiological data of the COVID-19 pandemic in Hungary. To describe the epidemic spread, we used a slightly modified version of a previously published and validated compartmental model, in which the vaccination process was taken into account. The mean and the variance of the unknown data (e.g., the number of susceptible, infected, or recovered people) were estimated using only the daily number of hospitalized patients. The problem was reformulated as a finite-horizon predictive control problem, where the unknown time-dependent parameter, the daily transmission rate of the disease, was computed such that the expected value of the computed number of hospitalized patients fit the truly observed data as much as possible.

Model with transmission delays for COVID-19 control: Theory and empirical assessment

The paper focuses on modeling of public health measures to control the COVID-19 pandemic. The authors suggest a flexible integral model with distributed lags, which realistically describes COVID-19 infectiousness period from clinical data. It contains susceptible-infectious-recovered (SIR), susceptible-exposed-infectious-recovered (SEIR), and other epidemic models as special cases. The model is used for assessing how government decisions to lockdown and reopen the economy affect epidemic spread. The authors demonstrate essential differences in transition and asymptotic dynamics of the integral model and the SIR model after lockdown. The provided simulation on real data accurately describes several waves of the COVID-19 epidemic in the United States and is in good correspondence with government actions to curb the epidemic.

What Can We Learn from the Time Evolution of COVID-19 Epidemic in Slovenia?

A recent work indicates that temporarily splitting larger populations into smaller groups can efficiently mitigate the spread of SARS‐CoV‐2 virus. The fact that, soon afterward, the two million people Slovenia was the first European country proclaiming the end of COVID‐19 epidemic within national borders may be relevant from this perspective. Motivated by this evolution, in this paper the time dynamics of coronavirus cases in Slovenia is investigated with emphasis on how efficient various containment measures act to diminish the number of COVID‐19 infections. Noteworthily, the present analysis does not rely on any speculative theoretical assumption; it is solely based on raw epidemiological data. Out of the results presented here, the most important one is perhaps the finding that, while imposing drastic curfews and travel restrictions reduce the infection rate κ by a factor of four with respect to the unrestricted state, they only improve the κ‐value by ≈15% as compared to the much bearable state of social and economic life wherein wearing face masks and social distancing rules are enforced/followed. Significantly for behavioral and social science, our analysis may reveal an interesting self‐protection instinct of the population, which became manifest even before the official lockdown enforcement.

Simple discrete-time self-exciting models can describe complex dynamic processes: A case study of COVID-19

Hawkes processes are a form of self-exciting process that has been used in numerous applications, including neuroscience, seismology, and terrorism. While these self-exciting processes have a simple formulation, they can model incredibly complex phenomena. Traditionally Hawkes processes are a continuous-time process, however we enable these models to be applied to a wider range of problems by considering a discrete-time variant of Hawkes processes. We illustrate this through the novel coronavirus disease (COVID-19) as a substantive case study. While alternative models, such as compartmental and growth curve models, have been widely applied to the COVID-19 epidemic, the use of discrete-time Hawkes processes allows us to gain alternative insights. This paper evaluates the capability of discrete-time Hawkes processes by modelling daily mortality counts as distinct phases in the COVID-19 outbreak. We first consider the initial stage of exponential growth and the subsequent decline as preventative measures become effective. We then explore subsequent phases with more recent data. Various countries that have been adversely affected by the epidemic are considered, namely, Brazil, China, France, Germany, India, Italy, Spain, Sweden, the United Kingdom and the United States. These countries are all unique concerning the spread of the virus and their corresponding response measures. However, we find that this simple model is useful in accurately capturing the dynamics of the process, despite hidden interactions that are not directly modelled due to their complexity, and differences both within and between countries. The utility of this model is not confined to the current COVID-19 epidemic, rather this model could explain many other complex phenomena. It is of interest to have simple models that adequately describe these complex processes with unknown dynamics. As models become more complex, a simpler representation of the process can be desirable for the sake of parsimony.