Real-time estimation of the epidemic reproduction number: Scoping review of the applications and challenges

Simulation-based inference of the time-dependent reproduction number from temporally aggregated and under-reported disease incidence time series data

During infectious disease outbreaks, the time-dependent reproduction number ([Formula: see text]) can be estimated to monitor pathogen transmission. In previous work, we developed a simulation-based method for estimating [Formula: see text] from temporally aggregated disease incidence data (e.g. weekly case reports). While that approach is straightforward to use, it assumes implicitly that all cases are reported and the computation can be slow when applied to large datasets. In this article, we extend our previous approach and develop a computationally efficient simulation-based method for estimating [Formula: see text] in real-time accounting for both temporal aggregation of incidence data and under-reporting (with a fixed reporting probability per case). Using simulated data, we show that failing to consider stochastic under-reporting can lead to inappropriately precise estimates, including scenarios in which the true [Formula: see text] value lies outside inferred credible intervals more often than expected. We then apply our approach to data from the 2018 to 2020 Ebola outbreak in the Democratic Republic of the Congo (DRC), again exploring the effects of case under-reporting. Finally, we show how our method can be extended to account for temporal variations in reporting. Given information about the level of case reporting, our framework can be used to estimate [Formula: see text] during future outbreaks with under-reported and temporally aggregated case data.This article is part of the theme issue 'Uncertainty quantification for healthcare and biological systems (Part 2)'.

Estimating effective reproduction numbers using wastewater data from multiple sewersheds for SARS-CoV-2 in California counties

The effective reproduction number serves as a metric of population-wide, time-varying disease spread. During the early years of the COVID-19 pandemic, this metric was primarily derived from case data, which has varied in quality and representativeness due to changes in testing volume, test-seeking behavior, and resource constraints. Deriving nowcasting estimates from alternative data sources such as wastewater provides complementary information that could inform future public health responses. We estimated county-aggregated, sewershed-restricted wastewater-based SARS-CoV-2 effective reproduction numbers from May 1, 2022 to April 30, 2023 for five counties in California with heterogeneous population sizes, clinical testing rates, demographics, wastewater coverage, and sampling frequencies. We used two methods to produce sewershed-restricted effective reproduction numbers, both based on smoothed and deconvolved wastewater concentrations. We then population-weighted and aggregated these sewershed-level estimates to arrive at county-level effective reproduction numbers. Using mean absolute error (MAE), Spearman's rank correlation (ρ), confusion matrix classification, and cross-correlation analyses, we compared the timing and trajectory of our two wastewater-based models to: (1) a publicly available, county-level ensemble of case-based estimates, and (2) county-aggregated, sewershed-restricted case-based estimates. Both wastewater models demonstrated high concordance with the traditional case-based estimates, as indicated by low mean absolute errors (MAE ≤ 0.09), significant positive Spearman correlation (ρ ≥ 0.66), and high confusion matrix classification accuracy (≥ 0.81). The relative timings of wastewater- and case-based estimates were less clear, with cross-correlation analyses suggesting strong associations for a wide range of temporal lags that varied by county and wastewater model type. This methodology provides a generalizable, robust, and operationalizable framework for estimating county-level wastewater-based effective reproduction numbers. Our retrospective evaluation supports the potential usage of real-time wastewater-based nowcasting as a complementary epidemiological tool for surveillance by public health agencies at the state and local levels. Based on this research, we produced publicly available wastewater-based nowcasts for the California Communicable diseases Assessment Tool (calcat.cdph.ca.gov).

Evaluating a novel reproduction number estimation method: a comparative analysis

This paper presents practical methodologies for determining effective reproduction numbers, R(t), providing valuable insights for researchers and public health officials. It proposes multiple simplified approaches for estimating R(t) of infectious diseases and compares their effectiveness. These approaches include methods based on exponential, fixed (delta), normal, and gamma distributions for the generation time. The exponential and fixed generation time methods offer convenience as they rely solely on the mean generation time and the number of new infections. However, they are sensitive to the variance of the generation time distribution: the exponential method may underestimate R(t) when the variance is small, while the fixed generation time method may overestimate R(t) when the variance is large. The normal distribution method also risks underestimation, depending on the growth rate. In contrast, the gamma distribution method demonstrates greater robustness and accuracy across a variety of scenarios. A key contribution of this work is the consolidated presentation of these estimation methods, along with the novel derivation of an accurate R(t) formula based on the gamma distribution. This research offers practical guidance for selecting the most appropriate R(t) estimation method, emphasizing the importance of accounting for the specific characteristics of the infectious disease's generation time distribution.

Time-varying reproduction number estimation: fusing compartmental models with generalized additive models

The reproduction number, the mean number of secondary cases infected by each primary case, gives an indication of the effort required to control the disease. Beyond the well-known basic reproduction number, there are two natural extensions, namely the control and effective reproduction numbers. As behaviour, population immunity and viral characteristics can change with time, these reproduction numbers can vary over time. Real-world data can be complex, so in this work we consider a generalized additive model to smooth surveillance data through the explicit incorporation of day-of-the-week effects, to provide a simple measure of the time-varying growth rate associated with the data. Converting the resulting spline into an estimator for both the control and effective reproduction numbers requires assumptions on a model structure, which we here assume to be a compartmental model. The reproduction numbers calculated are based on both simulated and real-world data, and are compared with estimates from an already existing tool. The derived method for estimating the time-varying reproduction number is effective, efficient and comparable with other methods. It provides a useful alternative approach, which can be included as part of a toolbox of models, that is particularly apt at smoothing out day-of-the-week effects in surveillance.

Unlocking the power of time-since-infection models: data augmentation for improved instantaneous reproduction number estimation

The time-since-infection (TSI) models, which use disease surveillance data to model infectious diseases, have become increasingly popular due to their flexibility and capacity to address complex disease control questions. However, a notable limitation of TSI models is their primary reliance on incidence data. Even when hospitalization data are available, existing TSI models have not been crafted to improve the estimation of disease transmission or to estimate hospitalization-related parameters-metrics crucial for understanding a pandemic and planning hospital resources. Moreover, their dependence on reported infection data makes them vulnerable to variations in data quality. In this study, we advance TSI models by integrating hospitalization data, marking a significant step forward in modeling with TSI models. We introduce hospitalization propensity parameters to jointly model incidence and hospitalization data. We use a composite likelihood function to accommodate complex data structure and a Monte Carlo expectation-maximization algorithm to estimate model parameters. We analyze COVID-19 data to estimate disease transmission, assess risk factor impacts, and calculate hospitalization propensity. Our model improves the accuracy of estimating the instantaneous reproduction number in TSI models, particularly when hospitalization data is of higher quality than incidence data. It enables the estimation of key infectious disease parameters without relying on contact tracing data and provides a foundation for integrating TSI models with other infectious disease models.

The Impact of Public Health and Social Measures (PHSMs) on SARS-CoV-2 Transmission in the WHO European Region (2020-2022)

BACKGROUND

Between 2020 and 2022, countries used a range of different public health and social measures (PHSMs) to reduce the transmission of SARS-CoV-2. The impact of these PHSMs varied as the pandemic progressed, variants of concern (VOCs) emerged, vaccines rolled out and acceptance/uptake rates evolved. In this study, we assessed the impact of PHSMs in the World Health Organization (WHO) European Region during VOC phases.

METHODS

We relied on time series data on genome sequencing, PHSMs, health outcomes and physical contacts. Panel regression models were used to assess the association between PHSMs and SARS-CoV-2 transmission (approximated using time-varying reproduction numbers). The interpretation of these regression models was assisted by hierarchical clustering, which was used to detect the temporal co-occurrence of PHSMs. Generalised linear models were used to check if PHSMs are associated with physical contacts.

RESULTS

We identified four phases based on the dominating VOC in the WHO European Region: wild type (before early 2021), Alpha (early to mid-2021), Delta (mid-to-late 2021) and Omicron (after late 2021). 'School closure', 'stay-at-home requirement' and 'testing policy' were consistently associated with lower transmission across VOC phases. The impact of most PHSMs varied by VOC phases without clear increasing or decreasing trends as the pandemic progressed. Several PHSMs associated with lower transmission were not associated with fewer physical contacts.

CONCLUSIONS

The impact of PHSMs evolved as the pandemic progressed-although without clear trends. The specific mechanisms by which some PHSMs reduce SARS-CoV-2 transmission require further research.

A framework for counterfactual analysis, strategy evaluation, and control of epidemics using reproduction number estimates

During pandemics, countries, regions, and communities develop various epidemic models to evaluate spread and guide mitigation policies. However, model uncertainties caused by complex transmission behaviors, contact-tracing networks, time-varying parameters, human factors, and limited data present significant challenges to model-based approaches. To address these issues, we propose a novel framework that centers around reproduction number estimates to perform counterfactual analysis, strategy evaluation, and feedback control of epidemics. The framework 1) introduces a mechanism to quantify the impact of the testing-for-isolation intervention strategy on the basic reproduction number. Building on this mechanism, the framework 2) proposes a method to reverse engineer the effective reproduction number under different strengths of the intervention strategy. In addition, based on the method that quantifies the impact of the testing-for-isolation strategy on the basic reproduction number, the framework 3) proposes a closed-loop control algorithm that uses the effective reproduction number both as feedback to indicate the severity of the spread and as the control goal to guide adjustments in the intensity of the intervention. We illustrate the framework, along with its three core methods, by addressing three key questions and validating its effectiveness using data collected during the COVID-19 pandemic at the University of Illinois Urbana-Champaign (UIUC) and Purdue University: 1) How severe would an outbreak have been without the implemented intervention strategies? 2) What impact would varying the intervention strength have had on an outbreak? 3) How can we adjust the intervention intensity based on the current state of an outbreak?

Bayesian estimation of the time-varying reproduction number for pulmonary tuberculosis in Iran: A registry-based study from 2018 to 2022 using new smear-positive cases

Introduction

Tuberculosis (TB) is one of the most prevalent infectious diseases in the world, causing major public health problems in developing countries. The rate of TB incidence in Iran was estimated to be 13 per 100,000 in 2021. This study aimed to estimate the reproduction number and serial interval for pulmonary tuberculosis in Iran.

Material and methods

The present national historical cohort study was conducted from March 2018 to March 2022 based on data from the National Tuberculosis and Leprosy Registration Center of Iran's Ministry of Health and Medical Education (MOHME). The study included 30,762 tuberculosis cases and 16,165 new smear-positive pulmonary tuberculosis patients in Iran. We estimated the reproduction number of pulmonary tuberculosis in a Bayesian framework, which can incorporate uncertainty in estimating it. Statistical analyses were accomplished in R software.

Results

The mean age at diagnosis of patients was 52.3 ± 21.2 years, and most patients were in the 35-63 age group (37.1%). Among the data, 9121 (56.4%) cases were males, and 7044 (43.6%) were females. Among patients, 7459 (46.1%) had a delayed diagnosis between 1 and 3 months. Additionally, 3039 (18.8%) cases were non-Iranians, and 2978 (98%) were Afghans. The time-varying reproduction number for pulmonary tuberculosis disease was calculated at an average of 1.06 ± 0.05 (95% Crl 0.96-1.15).

Conclusions

In this study, the incidence and the time-varying reproduction number of pulmonary tuberculosis showed the same pattern. The mean of the time-varying reproduction number indicated that each infected person is causing at least one new infection over time, and the chain of transmission is not being disrupted.

Increasing situational awareness through nowcasting of the reproduction number

Background

The time-varying reproduction number R is a critical variable for situational awareness during infectious disease outbreaks; however, delays between infection and reporting of cases hinder its accurate estimation in real-time. A number of nowcasting methods, leveraging available information on data consolidation delays, have been proposed to mitigate this problem.

Methods

In this work, we retrospectively validate the use of a nowcasting algorithm during 18 months of the COVID-19 pandemic in Italy by quantitatively assessing its performance against standard methods for the estimation of R.

Results

Nowcasting significantly reduced the median lag in the estimation of R from 13 to 8 days, while concurrently enhancing accuracy. Furthermore, it allowed the detection of periods of epidemic growth with a lead of between 6 and 23 days.

Conclusions

Nowcasting augments epidemic awareness, empowering better informed public health responses.

Estimating time-varying cholera transmission and oral cholera vaccine effectiveness in Haiti and Cameroon, 2021-2023

In 2023, cholera affected approximately 1 million people and caused more than 5000 deaths globally, predominantly in low-income and conflict settings. In recent years, the number of new cholera outbreaks has grown rapidly. Further, ongoing cholera outbreaks have been exacerbated by conflict, climate change, and poor infrastructure, resulting in prolonged crises. As a result, the demand for treatment and intervention is quickly outpacing existing resource availability. Prior to improved water and sanitation systems, cholera, a disease primarily transmitted via contaminated water sources, also routinely ravaged high-income countries. Crumbling infrastructure and climate change are now putting new locations at risk - even in high-income countries. Thus, understanding the transmission and prevention of cholera is critical. Combating cholera requires multiple interventions, the two most common being behavioral education and water treatment. Two-dose oral cholera vaccination (OCV) is often used as a complement to these interventions. Due to limited supply, countries have recently switched to single-dose vaccines (OCV1). One challenge lies in understanding where to allocate OCV1 in a timely manner, especially in settings lacking well-resourced public health surveillance systems. As cholera occurs and propagates in such locations, timely, accurate, and openly accessible outbreak data are typically inaccessible for disease modeling and subsequent decision-making. In this study, we demonstrated the value of open-access data to rapidly estimate cholera transmission and vaccine effectiveness. Specifically, we obtained non-machine readable (NMR) epidemic curves for recent cholera outbreaks in two countries, Haiti and Cameroon, from figures published in situation and disease outbreak news reports. We used computational digitization techniques to derive weekly counts of cholera cases, resulting in nominal differences when compared against the reported cumulative case counts (i.e., a relative error rate of 5.67% in Haiti and 0.54% in Cameroon). Given these digitized time series, we leveraged EpiEstim-an open-source modeling platform-to derive rapid estimates of time-varying disease transmission via the effective reproduction number (R t ). To compare OCV1 effectiveness in the two considered countries, we additionally used VaxEstim, a recent extension of EpiEstim that facilitates the estimation of vaccine effectiveness via the relation among three inputs: the basic reproduction number (R 0 ),R t , and vaccine coverage. Here, with Haiti and Cameroon as case studies, we demonstrated the first implementation of VaxEstim in low-resource settings. Importantly, we are the first to use VaxEstim with digitized data rather than traditional epidemic surveillance data. In the initial phase of the outbreak, weekly rolling average estimates ofR t were elevated in both countries: 2.60 in Haiti [95% credible interval: 2.42-2.79] and 1.90 in Cameroon [1.14-2.95]. These values are largely consistent with previous estimates ofR 0 in Haiti, where average values have ranged from 1.06 to 3.72, and in Cameroon, where average values have ranged from 1.10 to 3.50. In both Haiti and Cameroon, this initial period of high transmission preceded a longer period during whichR t oscillated around the critical threshold of 1. Our results derived from VaxEstim suggest that Haiti had higher OCV1 effectiveness than Cameroon (75.32% effective [54.00-86.39%] vs. 54.88% [18.94-84.90%]). These estimates of OCV1 effectiveness are generally aligned with those derived from field studies conducted in other countries. Thus, our case study reinforces the validity of VaxEstim as an alternative to costly, time-consuming field studies of OCV1 effectiveness. Indeed, prior work in South Sudan, Bangladesh, and the Democratic Republic of the Congo reported OCV1 effectiveness ranging from approximately 40% to 80%. This work underscores the value of combining NMR sources of outbreak case data with computational techniques and the utility of VaxEstim for rapid, inexpensive estimation of vaccine effectiveness in data-poor outbreak settings.

Challenges in the case-based surveillance of infectious diseases

To effectively inform infectious disease control strategies, accurate knowledge of the pathogen's transmission dynamics is required. Since the timings of infections are rarely known, estimates of the infection incidence, which is crucial for understanding the transmission dynamics, often rely on measurements of other quantities amenable to surveillance. Case-based surveillance, in which infected individuals are identified by a positive test, is the predominant form of surveillance for many pathogens, and was used extensively during the COVID-19 pandemic. However, there can be many biases present in case-based surveillance indicators due to, for example test sensitivity, changing testing behaviours and the co-circulation of pathogens with similar symptom profiles. Here, we develop a mathematical description of case-based surveillance of infectious diseases. By considering realistic epidemiological parameters and situations, we demonstrate many of the potential biases in common surveillance indicators based on case-based surveillance data. Crucially, we find that many of these common surveillance indicators (e.g. case numbers, test-positive proportion) are heavily biased by circulating pathogens with similar symptom profiles. Future surveillance strategies could be designed to minimize these sources of bias and uncertainty, providing more accurate estimates of a pathogen's transmission dynamics and, ultimately, more targeted application of public health measures.

A simulation-based approach for estimating the time-dependent reproduction number from temporally aggregated disease incidence time series data

Tracking pathogen transmissibility during infectious disease outbreaks is essential for assessing the effectiveness of public health measures and planning future control strategies. A key measure of transmissibility is the time-dependent reproduction number, which has been estimated in real-time during outbreaks of a range of pathogens from disease incidence time series data. While commonly used approaches for estimating the time-dependent reproduction number can be reliable when disease incidence is recorded frequently, such incidence data are often aggregated temporally (for example, numbers of cases may be reported weekly rather than daily). As we show, commonly used methods for estimating transmissibility can be unreliable when the timescale of transmission is shorter than the timescale of data recording. To address this, here we develop a simulation-based approach involving Approximate Bayesian Computation for estimating the time-dependent reproduction number from temporally aggregated disease incidence time series data. We first use a simulated dataset representative of a situation in which daily disease incidence data are unavailable and only weekly summary values are reported, demonstrating that our method provides accurate estimates of the time-dependent reproduction number under such circumstances. We then apply our method to two outbreak datasets consisting of weekly influenza case numbers in 2019-20 and 2022-23 in Wales (in the United Kingdom). Our simple-to-use approach will allow accurate estimates of time-dependent reproduction numbers to be obtained from temporally aggregated data during future infectious disease outbreaks.

Nowcasting and forecasting the 2022 U.S. mpox outbreak: Support for public health decision making and lessons learned

In June of 2022, the U.S. Centers for Disease Control and Prevention (CDC) Mpox Response wanted timely answers to important epidemiological questions which can now be answered more effectively through infectious disease modeling. Infectious disease models have shown to be valuable tools for decision making during outbreaks; however, model complexity often makes communicating the results and limitations of models to decision makers difficult. We performed nowcasting and forecasting for the 2022 mpox outbreak in the United States using the R package EpiNow2. We generated nowcasts/forecasts at the national level, by Census region, and for jurisdictions reporting the greatest number of mpox cases. Modeling results were shared for situational awareness within the CDC Mpox Response and publicly on the CDC website. We retrospectively evaluated forecast predictions at four key phases (early, exponential growth, peak, and decline) during the outbreak using three metrics, the weighted interval score, mean absolute error, and prediction interval coverage. We compared the performance of EpiNow2 with a naïve Bayesian generalized linear model (GLM). The EpiNow2 model had less probabilistic error than the GLM during every outbreak phase except for the early phase. We share our experiences with an existing tool for nowcasting/forecasting and highlight areas of improvement for the development of future tools. We also reflect on lessons learned regarding data quality issues and adapting modeling results for different audiences.

Incorporating testing volume into estimation of effective reproduction number dynamics

Branching process inspired models are widely used to estimate the effective reproduction number-a useful summary statistic describing an infectious disease outbreak-using counts of new cases. Case data is a real-time indicator of changes in the reproduction number, but is challenging to work with because cases fluctuate due to factors unrelated to the number of new infections. We develop a new model that incorporates the number of diagnostic tests as a surveillance model covariate. Using simulated data and data from the SARS-CoV-2 pandemic in California, we demonstrate that incorporating tests leads to improved performance over the state of the art.

Differences between the true reproduction number and the apparent reproduction number of an epidemic time series

The time-varying reproduction number R(t) measures the number of new infections per infectious individual and is closely correlated with the time series of infection incidence by definition. The timings of actual infections are rarely known, and analysis of epidemics usually relies on time series data for other outcomes such as symptom onset. A common implicit assumption, when estimating R(t) from an epidemic time series, is that R(t) has the same relationship with these downstream outcomes as it does with the time series of incidence. However, this assumption is unlikely to be valid given that most epidemic time series are not perfect proxies of incidence. Rather they represent convolutions of incidence with uncertain delay distributions. Here we define the apparent time-varying reproduction number, RA(t), the reproduction number calculated from a downstream epidemic time series and demonstrate how differences between RA(t) and R(t) depend on the convolution function. The mean of the convolution function sets a time offset between the two signals, whilst the variance of the convolution function introduces a relative distortion between them. We present the convolution functions of epidemic time series that were available during the SARS-CoV-2 pandemic. Infection prevalence, measured by random sampling studies, presents fewer biases than other epidemic time series. Here we show that additionally the mean and variance of its convolution function were similar to that obtained from traditional surveillance based on mass-testing and could be reduced using more frequent testing, or by using stricter thresholds for positivity. Infection prevalence studies continue to be a versatile tool for tracking the temporal trends of R(t), and with additional refinements to their study protocol, will be of even greater utility during any future epidemics or pandemics.

Monitoring the reproductive number of COVID-19 in France: Comparative estimates from three datasets

BACKGROUND

The effective reproduction number (Rt) quantifies the average number of secondary cases caused by one person with an infectious disease. Near-real-time monitoring of Rt during an outbreak is a major indicator used to monitor changes in disease transmission and assess the effectiveness of interventions. The estimation of Rt usually requires the identification of infected cases in the population, which can prove challenging with the available data, especially when asymptomatic people or with mild symptoms are not usually screened. The purpose of this study was to perform sensitivity analysis of Rt estimates for COVID-19 surveillance in France based on three data sources with different sensitivities and specificities for identifying infected cases.

METHODS

We applied a statistical method developed by Cori et al. to estimate Rt using (1) confirmed cases identified from positive virological tests in the population, (2) suspected cases recorded by a national network of emergency departments, and (3) COVID-19 hospital admissions recorded by a national administrative system to manage hospital organization.

RESULTS

Rt estimates in France from May 27, 2020, to August 12, 2022, showed similar temporal trends regardless of the dataset. Estimates based on the daily number of confirmed cases provided an earlier signal than the two other sources, with an average lag of 3 and 6 days for estimates based on emergency department visits and hospital admissions, respectively.

CONCLUSION

The COVID-19 experience confirmed that monitoring temporal changes in Rt was a key indicator to help the public health authorities control the outbreak in real time. However, gaining access to data on all infected people in the population in order to estimate Rt is not straightforward in practice. As this analysis has shown, the opportunity to use more readily available data to estimate Rt trends, provided that it is highly correlated with the spread of infection, provides a practical solution for monitoring the COVID-19 pandemic and indeed any other epidemic.

estimateR: an R package to estimate and monitor the effective reproductive number

BACKGROUND

Accurate estimation of the effective reproductive number ([Formula: see text]) of epidemic outbreaks is of central relevance to public health policy and decision making. We present estimateR, an R package for the estimation of the reproductive number through time from delayed observations of infection events. Such delayed observations include confirmed cases, hospitalizations or deaths. The package implements the methodology of Huisman et al. but modularizes the [Formula: see text] estimation procedure to allow easy implementation of new alternatives to the currently available methods. Users can tailor their analyses according to their particular use case by choosing among implemented options.

RESULTS

The estimateR R package allows users to estimate the effective reproductive number of an epidemic outbreak based on observed cases, hospitalization, death or any other type of event documenting past infections, in a fast and timely fashion. We validated the implementation with a simulation study: estimateR yielded estimates comparable to alternative publicly available methods while being around two orders of magnitude faster. We then applied estimateR to empirical case-confirmation incidence data for COVID-19 in nine countries and for dengue fever in Brazil; in parallel, estimateR is already being applied (i) to SARS-CoV-2 measurements in wastewater data and (ii) to study influenza transmission based on wastewater and clinical data in other studies. In summary, this R package provides a fast and flexible implementation to estimate the effective reproductive number for various diseases and datasets.

CONCLUSIONS

The estimateR R package is a modular and extendable tool designed for outbreak surveillance and retrospective outbreak investigation. It extends the method developed for COVID-19 by Huisman et al. and makes it available for a variety of pathogens, outbreak scenarios, and observation types. Estimates obtained with estimateR can be interpreted directly or used to inform more complex epidemic models (e.g. for forecasting) on the value of [Formula: see text].

Estimating the epidemic reproduction number from temporally aggregated incidence data: A statistical modelling approach and software tool

The time-varying reproduction number (Rt) is an important measure of epidemic transmissibility that directly informs policy decisions and the optimisation of control measures. EpiEstim is a widely used opensource software tool that uses case incidence and the serial interval (SI, time between symptoms in a case and their infector) to estimate Rt in real-time. The incidence and the SI distribution must be provided at the same temporal resolution, which can limit the applicability of EpiEstim and other similar methods, e.g. for contexts where the time window of incidence reporting is longer than the mean SI. In the EpiEstim R package, we implement an expectation-maximisation algorithm to reconstruct daily incidence from temporally aggregated data, from which Rt can then be estimated. We assess the validity of our method using an extensive simulation study and apply it to COVID-19 and influenza data. For all datasets, the influence of intra-weekly variability in reported data was mitigated by using aggregated weekly data. Rt estimated on weekly sliding windows using incidence reconstructed from weekly data was strongly correlated with estimates from the original daily data. The simulation study revealed that Rt was well estimated in all scenarios and regardless of the temporal aggregation of the data. In the presence of weekend effects, Rt estimates from reconstructed data were more successful at recovering the true value of Rt than those obtained from reported daily data. These results show that this novel method allows Rt to be successfully recovered from aggregated data using a simple approach with very few data requirements. Additionally, by removing administrative noise when daily incidence data are reconstructed, the accuracy of Rt estimates can be improved.

New Insights into the Estimation of Reproduction Numbers during an Epidemic

In this paper, we deal with the problem of estimating the reproduction number Rt during an epidemic, as it represents one of the most used indicators to study and control this phenomenon. In particular, we focus on two issues. First, to estimate Rt, we consider the use of positive test case data as an alternative to the first symptoms data, which are typically used. We both theoretically and empirically study the relationship between the two approaches. Second, we modify a method for estimating Rt during an epidemic that is widely used by public institutions in several countries worldwide. Our procedure is not affected by the problems deriving from the hypothesis of Rt local constancy, which is assumed in the standard approach. We illustrate the results obtained by applying the proposed methodologies to real and simulated SARS-CoV-2 datasets. In both cases, we also apply some specific methods to reduce systematic and random errors affecting the data. Our results show that the Rt during an epidemic can be estimated by using the positive test data, and that our estimator outperforms the standard estimator that makes use of the first symptoms data. It is hoped that the techniques proposed here could help in the study and control of epidemics, particularly the current SARS-CoV-2 pandemic.

EpiLPS: A fast and flexible Bayesian tool for estimation of the time-varying reproduction number

In infectious disease epidemiology, the instantaneous reproduction number [Formula: see text] is a time-varying parameter defined as the average number of secondary infections generated by an infected individual at time t. It is therefore a crucial epidemiological statistic that assists public health decision makers in the management of an epidemic. We present a new Bayesian tool (EpiLPS) for robust estimation of the time-varying reproduction number. The proposed methodology smooths the epidemic curve and allows to obtain (approximate) point estimates and credible intervals of [Formula: see text] by employing the renewal equation, using Bayesian P-splines coupled with Laplace approximations of the conditional posterior of the spline vector. Two alternative approaches for inference are presented: (1) an approach based on a maximum a posteriori argument for the model hyperparameters, delivering estimates of [Formula: see text] in only a few seconds; and (2) an approach based on a Markov chain Monte Carlo (MCMC) scheme with underlying Langevin dynamics for efficient sampling of the posterior target distribution. Case counts per unit of time are assumed to follow a negative binomial distribution to account for potential overdispersion in the data that would not be captured by a classic Poisson model. Furthermore, after smoothing the epidemic curve, a "plug-in'' estimate of the reproduction number can be obtained from the renewal equation yielding a closed form expression of [Formula: see text] as a function of the spline parameters. The approach is extremely fast and free of arbitrary smoothing assumptions. EpiLPS is applied on data of SARS-CoV-1 in Hong-Kong (2003), influenza A H1N1 (2009) in the USA and on the SARS-CoV-2 pandemic (2020-2021) for Belgium, Portugal, Denmark and France.