A mechanistic and data-driven reconstruction of the time-varying reproduction number: Application to the COVID-19 epidemic

Bayesian workflow for time-varying transmission in stratified compartmental infectious disease transmission models

Compartmental models that describe infectious disease transmission across subpopulations are central for assessing the impact of non-pharmaceutical interventions, behavioral changes and seasonal effects on the spread of respiratory infections. We present a Bayesian workflow for such models, including four features: (1) an adjustment for incomplete case ascertainment, (2) an adequate sampling distribution of laboratory-confirmed cases, (3) a flexible, time-varying transmission rate, and (4) a stratification by age group. Within the workflow, we benchmarked the performance of various implementations of two of these features (2 and 3). For the second feature, we used SARS-CoV-2 data from the canton of Geneva (Switzerland) and found that a quasi-Poisson distribution is the most suitable sampling distribution for describing the overdispersion in the observed laboratory-confirmed cases. For the third feature, we implemented three methods: Brownian motion, B-splines, and approximate Gaussian processes (aGP). We compared their performance in terms of the number of effective samples per second, and the error and sharpness in estimating the time-varying transmission rate over a selection of ordinary differential equation solvers and tuning parameters, using simulated seroprevalence and laboratory-confirmed case data. Even though all methods could recover the time-varying dynamics in the transmission rate accurately, we found that B-splines perform up to four and ten times faster than Brownian motion and aGPs, respectively. We validated the B-spline model with simulated age-stratified data. We applied this model to 2020 laboratory-confirmed SARS-CoV-2 cases and two seroprevalence studies from the canton of Geneva. This resulted in detailed estimates of the transmission rate over time and the case ascertainment. Our results illustrate the potential of the presented workflow including stratified transmission to estimate age-specific epidemiological parameters. The workflow is freely available in the R package HETTMO, and can be easily adapted and applied to other infectious diseases.

Dynamic parameterization of a modified SEIRD model to analyze and forecast the dynamics of COVID-19 outbreaks in the United States

The rapid spread of the numerous outbreaks of the coronavirus disease 2019 (COVID-19) pandemic has fueled interest in mathematical models designed to understand and predict infectious disease spread, with the ultimate goal of contributing to the decision making of public health authorities. Here, we propose a computational pipeline that dynamically parameterizes a modified SEIRD (susceptible-exposed-infected-recovered-deceased) model using standard daily series of COVID-19 cases and deaths, along with isolated estimates of population-level seroprevalence. We test our pipeline in five heavily impacted states of the US (New York, California, Florida, Illinois, and Texas) between March and August 2020, considering two scenarios with different calibration time horizons to assess the update in model performance as new epidemiologic data become available. Our results show a median normalized root mean squared error (NRMSE) of 2.38% and 4.28% in calibrating cumulative cases and deaths in the first scenario, and 2.41% and 2.30% when new data are assimilated in the second scenario, respectively. Then, 2-week (4-week) forecasts of the calibrated model resulted in median NRMSE of cumulative cases and deaths of 5.85% and 4.68% (8.60% and 17.94%) in the first scenario, and 1.86% and 1.93% (2.21% and 1.45%) in the second. Additionally, we show that our method provides significantly more accurate predictions of cases and deaths than a constant parameterization in the second scenario (p < 0.05). Thus, we posit that our methodology is a promising approach to analyze the dynamics of infectious disease outbreaks, and that our forecasts could contribute to designing effective pandemic-arresting public health policies.

Supplementary Information

The online version contains supplementary material available at 10.1007/s00366-023-01816-9.

A comparative study on the three calculation methods for reproduction numbers of COVID-19

Objective

This study uses four COVID-19 outbreaks as examples to calculate and compare merits and demerits, as well as applicational scenarios, of three methods for calculating reproduction numbers.

Method

The epidemiological characteristics of the COVID-19 outbreaks are described. Through the definition method, the next-generation matrix-based method, and the epidemic curve and serial interval (SI)-based method, corresponding reproduction numbers were obtained and compared.

Results

Reproduction numbers (R eff ), obtained by the definition method of the four regions, are 1.20, 1.14, 1.66, and 1.12. Through the next generation matrix method, in region H R eff = 4.30, 0.44; region P R eff = 6.5, 1.39, 0; region X R eff = 6.82, 1.39, 0; and region Z R eff = 2.99, 0.65. Time-varying reproduction numbers (R t ), which are attained by SI of onset dates, are decreasing with time. Region H reached its highest R t = 2.8 on July 29 and decreased to R t < 1 after August 4; region P reached its highest R t = 5.8 on September 9 and dropped to R t < 1 by September 14; region X had a fluctuation in the R t and R t < 1 after September 22; R t in region Z reached a maximum of 1.8 on September 15 and decreased continuously to R t < 1 on September 19.

Conclusion

The reproduction number obtained by the definition method is optimal in the early stage of epidemics with a small number of cases that have clear transmission chains to predict the trend of epidemics accurately. The effective reproduction number R eff , calculated by the next generation matrix, could assess the scale of the epidemic and be used to evaluate the effectiveness of prevention and control measures used in epidemics with a large number of cases. Time-varying reproduction number R t , obtained via epidemic curve and SI, can give a clear picture of the change in transmissibility over time, but the conditions of use are more rigorous, requiring a greater sample size and clear transmission chains to perform the calculation. The rational use of the three methods for reproduction numbers plays a role in the further study of the transmissibility of COVID-19.

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.

Modelling COVID-19 dynamics and potential for herd immunity by vaccination in Austria, Luxembourg and Sweden

Against the COVID-19 pandemic, non-pharmaceutical interventions have been widely applied and vaccinations have taken off. The upcoming question is how the interplay between vaccinations and social measures will shape infections and hospitalizations. Hence, we extend the Susceptible-Exposed-Infectious-Removed (SEIR) model including these elements. We calibrate it to data of Luxembourg, Austria and Sweden until 15 December 2020. Sweden results having the highest fraction of undetected, Luxembourg of infected and all three being far from herd immunity in December. We quantify the level of social interaction, showing that a level around 1/3 of before the pandemic was still required in December to keep the effective reproduction number Refft below 1, for all three countries. Aiming to vaccinate the whole population within 1 year at constant rate would require on average 1,700 fully vaccinated people/day in Luxembourg, 24,000 in Austria and 28,000 in Sweden, and could lead to herd immunity only by mid summer. Herd immunity might not be reached in 2021 if too slow vaccines rollout speeds are employed. The model thus estimates which vaccination rates are too low to allow reaching herd immunity in 2021, depending on social interactions. Vaccination will considerably, but not immediately, help to curb the infection; thus limiting social interactions remains crucial for the months to come.

Dynamics of the COVID-19 epidemic in Ireland under mitigation

BACKGROUND

In Ireland and across the European Union the COVID-19 epidemic waves, driven mainly by the emergence of new variants of the SARS-CoV-2 have continued their course, despite various interventions from governments. Public health interventions continue in their attempts to control the spread as they wait for the planned significant effect of vaccination.

METHODS

To tackle this challenge and the observed non-stationary aspect of the epidemic we used a modified SEIR stochastic model with time-varying parameters, following Brownian process. This enabled us to reconstruct the temporal evolution of the transmission rate of COVID-19 with the non-specific hypothesis that it follows a basic stochastic process constrained by the available data. This model is coupled with Bayesian inference (particle Markov Chain Monte Carlo method) for parameter estimation and utilized mainly well-documented Irish hospital data.

RESULTS

In Ireland, mitigation measures provided a 78-86% reduction in transmission during the first wave between March and May 2020. For the second wave in October 2020, our reduction estimation was around 20% while it was 70% for the third wave in January 2021. This third wave was partly due to the UK variant appearing in Ireland. In June 2020 we estimated that sero-prevalence was 2.0% (95% CI: 1.2-3.5%) in complete accordance with a sero-prevalence survey. By the end of April 2021, the sero-prevalence was greater than 17% due in part to the vaccination campaign. Finally we demonstrate that the available observed confirmed cases are not reliable for analysis owing to the fact that their reporting rate has as expected greatly evolved.

CONCLUSION

We provide the first estimations of the dynamics of the COVID-19 epidemic in Ireland and its key parameters. We also quantify the effects of mitigation measures on the virus transmission during and after mitigation for the three waves. Our results demonstrate that Ireland has significantly reduced transmission by employing mitigation measures, physical distancing and lockdown. This has to date avoided the saturation of healthcare infrastructures, flattened the epidemic curve and likely reduced mortality. However, as we await for a full roll out of a vaccination programme and as new variants potentially more transmissible and/or more infectious could continue to emerge and mitigation measures change silent transmission, challenges remain.