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We derive and 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 the generation time distribution, w. Predominant approaches for tracking the dynamics of pathogen spread either infer R or the epidemic growth rate r. However, R is easily 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. Moreover, R and r may disagree on the relative transmissibility of two 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 in w while maintaining most of the interpretability of R. Additionally, we prove that {Omega} > 1 if and only if R > 1 and that {Omega} agrees with r on the relative transmissibility of pathogens. Estimating {Omega} is no harder than inferring R, uses existing software, and requires no generation time measurement. These advantages come at the expense of selecting one free parameter. We propose {Omega} as a useful statistic for tracking and comparing the spread of infectious diseases that may better reflect the impact of interventions when those interventions concurrently change both R and w or alter the relative risk of co-circulating pathogens.
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The effective reproduction number R is a prominent statistic for inferring the transmissibility of infectious diseases and effectiveness of interventions. R purportedly provides an easy-to-interpret threshold for deducing whether an epidemic will grow (R >1) or decline (R < 1). We posit that this interpretation can be misleading and statistically overconfident when applied to infections accumulated from groups featuring heterogeneous dynamics. In these settings, R implicitly weights the dynamics of groups by their number of circulating infections. We show that this weighting can cause delayed detection of outbreak resurgence and premature signalling of epidemic control because it underrepresents the risks from highly transmissible groups. Applying E-optimal experimental design theory, we develop a weighting algorithm to minimise these issues, yielding the risk averse reproduction number E. Using simulations, analytic approaches and a real-world case study, we find that E meaningfully summarises the dynamics across groups, balancing bias from the averaging underlying R with variance from using local group estimates. An E >1 generates timely resurgence signals (upweighting risky groups), while E < 1 ensures local outbreaks are under control. We propose E as an alternative to R for informing policy and assessing transmissibility at large scales (e.g., state-wide), where R is commonly computed but well-mixed assumptions break down.
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Reliably estimating the dynamics of transmissible diseases from noisy surveillance data is an enduring problem in modern epidemiology. Key parameters, such as the instantaneous reproduction number, Rt at time t, are often inferred from incident time series, with the aim of informing policymakers on the growth rate of outbreaks or testing hypotheses about the effectiveness of public health interventions. However, the reliability of these inferences depends critically on reporting errors and latencies innate to those time series. While studies have proposed corrections for these issues, methodology for formally assessing how these sources of noise degrade Rt estimate quality is lacking. By adapting Fisher information and experimental design theory, we develop an analytical framework to quantify the uncertainty induced by under-reporting and delays in reporting infections. This yields a novel metric, defined by the geometric means of reporting and cumulative delay probabilities, for ranking surveillance data informativeness. We apply this metric to two primary data sources for inferring Rt: epidemic case and death curves. We find that the assumption of death curves as more reliable, commonly made for acute infectious diseases such as COVID-19 and influenza, is not obvious and possibly untrue in many settings. Our framework clarifies and quantifies how actionable information about pathogen transmissibility is lost due to surveillance limitations.
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SARS-CoV-2 case data are primary sources for estimating epidemiological parameters and for modelling the dynamics of outbreaks. Understanding biases within case based data sources used in epidemiological analyses are important as they can detract from the value of these rich datasets. This raises questions of how variations in surveillance can affect the estimation of epidemiological parameters such as the case growth rates. We use standardised line list data of COVID-19 from Argentina, Brazil, Mexico and Colombia to estimate delay distributions of symptom-onset-to-confirmation, -hospitalisation and -death as well as hospitalisation-to-death at high spatial resolutions and throughout time. Using these estimates, we model the biases introduced by the delay from symptom-onset-to-confirmation on national and state level case growth rates (rt) using an adaptation of the Richardson-Lucy deconvolution algorithm. We find significant heterogeneities in the estimation of delay distributions through time and space with delay difference of up to 19 days between epochs at the state level. Further, we find that by changing the spatial scale, estimates of case growth rate can vary by up to 0.13 d-1. Lastly, we find that states with a high variance and/or mean delay in symptom-onset-to-diagnosis also have the largest difference between the rt estimated from raw and deconvolved case counts at the state level. We highlight the importance of high-resolution case based data in understanding biases in disease reporting and how these biases can be avoided by adjusting case numbers based on empirical delay distributions. Code and openly accessible data to reproduce analyses presented here are available.
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SARS-CoV-2 virus genomes are currently being sequenced at an unprecedented pace. The choice of viral sequences used in genetic and epidemiological analysis is important as it can induce biases that detract from the value of these rich datasets. This raises questions about how a set of sequences should be chosen for analysis, and which epidemiological parameters derived from genomic data are sensitive or robust to changes in sampling. We provide initial insights on these largely understudied problems using SARS-CoV-2 genomic sequences from Hong Kong, China, and the Amazonas State, Brazil. We consider sampling schemes that select sequences uniformly, in proportion or reciprocally with case incidence and which simply use all available sequences (unsampled). We apply Birth-Death Skyline and Skygrowth methods to estimate the time-varying reproduction number (Rt) and growth rate (rt) under these strategies as well as related R0 and date of origin parameters. We compare these to estimates from case data derived from EpiFilter, which we use as a reference for assessing bias. We find that both Rt and rt are sensitive to changes in sampling whilst R0 and the date of origin are relatively robust. Moreover, we find that analysis using unsampled datasets, which reflect an opportunistic sampling scheme, result in the most biased Rt and rt estimates for both our Hong Kong and Amazonas case studies. We highlight that sampling strategy choices may be an influential yet neglected component of sequencing analysis pipelines. More targeted attempts at genomic surveillance and epidemic analyses, particularly in settings with limited sequencing capabilities, are necessary to maximise the informativeness of virus genomic datasets.
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Mathematical models can provide insights into the control of pandemic COVID-19, which remains a global priority. The dynamics of directly-transmitted infectious diseases, such as COVID-19, are usually described by compartmental models where individuals are classified as susceptible, infected and removed. These SIR models typically assume homogenous transmission of infection, even in large populations, a simplification that is convenient but inconsistent with observations. Here we use original data on the dynamics of COVID-19 spread in a Brazilian city to investigate the structure of the transmission network. We find that transmission can be described by a network in which each infectious individual has a small number of susceptible contacts, of the order of 2-5, which is independent of total population size. Compared with standard models of homogenous mixing, this scale-free, fractal infection process gives a better description of COVID-19 dynamics through time. In addition, the contact process explains the geographically localized clusters of disease seen in this Brazilian city. Our scale-free model can help refine criteria for physical and social distancing in order to more effectively mitigate the spread of COVID-19. We propose that scale-free COVID-19 dynamics could be a widespread phenomenon, a topic for further investigation.
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Genomic surveillance of SARS-CoV-2 has played a decisive role in understanding the transmission and evolution of the virus during its emergence and continued circulation. However, limited genomic sampling in many high-incidence countries has impeded detailed studies of SARS-CoV-2 genomic epidemiology. Consequently, critical questions remain about the generation and global distribution of virus genetic diversity. To address this gap, we investigated SARS-CoV-2 transmission dynamics in Gujarat, India, during its first epidemic wave and shed light on virus spread in one of the pandemics hardest-hit regions. By integrating regional case data and 434 whole virus genome sequences sampled across 20 districts from March to July 2020, we reconstructed the epidemic dynamics and spatial spread of SARS-CoV-2 in Gujarat, India. Our findings revealed that global and regional connectivity, along with population density, were significant drivers of the Gujarat SARS-CoV-2 outbreak. The three most populous districts in Gujarat accounted [~]84% of total cases during the first wave. Moreover, we detected over 100 virus lineage introductions, which were primarily associated with international travel. Within Gujarat, virus dissemination occurred predominantly from densely populated regions to geographically proximate locations with low-population density. Our findings suggest SARS-CoV-2 transmission follows a gravity model in India, with urban centres contributing disproportionately to onward virus spread.
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BackgroundAs of July 2021, more than 180,000,000 cases of COVID-19 have been reported across the world, with more than 4 million deaths. Mathematical modelling and forecasting efforts have been widely used to inform policy-making and to create situational awareness. Methods and FindingsFrom 8th March to 29th November 2020, we produced weekly estimates of SARS-CoV-2 transmissibility and forecasts of deaths due to COVID-19 for countries with evidence of sustained transmission. The estimates and forecasts were based on an ensemble model comprising of three models that were calibrated using only the reported number of COVID-19 cases and deaths in each country. We also developed a novel heuristic to combine weekly estimates of transmissibility and potential changes in population immunity due to infection to produce forecasts over a 4-week horizon. We evaluated the robustness of the forecasts using relative error, coverage probability, and comparisons with null models. ConclusionsDuring the 39-week period covered by this study, we produced short- and medium-term forecasts for 81 countries. Both the short- and medium-term forecasts captured well the epidemic trajectory across different waves of COVID-19 infections with small relative errors over the forecast horizon. The model was well calibrated with 56.3% and 45.6% of the observations lying in the 50% Credible Interval in 1-week and 4-week ahead forecasts respectively. We could accurately characterise the overall phase of the epidemic up to 4-weeks ahead in 84.9% of country-days. The medium-term forecasts can be used in conjunction with the short-term forecasts of COVID-19 mortality as a useful planning tool as countries continue to relax stringent public health measures that were implemented to contain the pandemic.
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BackgroundBrazil is one of the countries worst affected by the COVID-19 pandemic with over 20 million cases and 557,000 deaths reported. Comparison of real-time local COVID-19 data between areas is essential for understanding transmission, measuring the effects of interventions and predicting the course of the epidemic, but are often challenging due to different population sizes and structures. MethodsWe describe the development of a new app for the real-time visualisation of COVID-19 data in Brazil at the municipality level. In the CLIC-Brazil app, daily updates of case and death data are downloaded, age standardised and used to estimate reproduction number (Rt). We show how such platforms can perform real-time regression analyses to identify factors associated with the rate of initial spread and early reproduction number. We also use survival methods to predict the likelihood of occurrence of a new peak of COVID-19 incidence. FindingsAfter an initial introduction in Sao Paulo and Rio de Janeiro states in early March 2020, the epidemic spread to Northern states and then to highly populated coastal regions and the Central-West. Municipalities with higher metrics of social development experienced earlier arrival of COVID-19 (decrease of 11{middle dot}1 days [95% CI:13{middle dot}2,8{middle dot}9] in the time to arrival for each 10% increase in the social development index). Differences in the initial epidemic intensity (mean Rt) were largely driven by geographic location and the date of local onset. InterpretationThis study demonstrates that platforms that monitor, standardise and analyse the epidemiological data at a local level can give useful real-time insights into outbreak dynamics that can be used to better adapt responses to the current and future pandemics. FundingThis project was supported by a Medical Research Council UK (MRC-UK) -Sao Paulo Research Foundation (FAPESP) CADDE partnership award (MR/S0195/1 and FAPESP 18/14389-0)
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Inferring the transmission potential of an infectious disease during low-incidence periods following epidemic waves is crucial for preparedness. In such periods, scarce data may hinder existing inference methods, blurring early-warning signals essential for discriminating between the likelihoods of resurgence versus elimination. Advanced insight into whether elevating caseloads (requiring swift community-wide interventions) or local elimination (allowing controls to be relaxed or refocussed on case-importation) might occur, can separate decisive from ineffective policy. By generalising and fusing recent approaches, we propose a novel early-warning framework that maximises the information extracted from low-incidence data to robustly infer the chances of sustained local-transmission or elimination in real time, at any scale of investigation (assuming sufficiently good surveillance). Applying this framework, we decipher hidden disease-transmission signals in prolonged low-incidence COVID-19 data from New Zealand, Hong Kong and Victoria, Australia. We uncover how timely interventions associate with averting resurgent waves, support official elimination declarations and evidence the effectiveness of the rapid, adaptive COVID-19 responses employed in these regions.
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The UKs COVID-19 epidemic during early 2020 was one of worlds largest and unusually well represented by virus genomic sampling. Here we reveal the fine-scale genetic lineage structure of this epidemic through analysis of 50,887 SARS-CoV-2 genomes, including 26,181 from the UK sampled throughout the countrys first wave of infection. Using large-scale phylogenetic analyses, combined with epidemiological and travel data, we quantify the size, spatio-temporal origins and persistence of genetically-distinct UK transmission lineages. Rapid fluctuations in virus importation rates resulted in >1000 lineages; those introduced prior to national lockdown were larger and more dispersed. Lineage importation and regional lineage diversity declined after lockdown, whilst lineage elimination was size-dependent. We discuss the implications of our genetic perspective on transmission dynamics for COVID-19 epidemiology and control.
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As of 1st June 2020, the US Centers for Disease Control and Prevention reported 104,232 confirmed or probable COVID-19-related deaths in the US. This was more than twice the number of deaths reported in the next most severely impacted country. We jointly modelled the US epidemic at the state-level, using publicly available death data within a Bayesian hierarchical semi-mechanistic framework. For each state, we estimate the number of individuals that have been infected, the number of individuals that are currently infectious and the time-varying reproduction number (the average number of secondary infections caused by an infected person). We used changes in mobility to capture the impact that non-pharmaceutical interventions and other behaviour changes have on the rate of transmission of SARS-CoV-2. Nationally, we estimated 3.7% [3.4%-4.0%] of the population had been infected by 1st June 2020, with wide variation between states, and approximately 0.01% of the population was infectious. We also demonstrated that good model forecasts of deaths for the next 3 weeks with low error and good coverage of our credible intervals.
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1Brazil is currently reporting the second highest number of COVID-19 deaths in the world. Here we characterise the initial dynamics of COVID-19 across the country and assess the impact of non-pharmaceutical interventions (NPIs) that were implemented using a semi-mechanistic Bayesian hierarchical modelling approach. Our results highlight the significant impact these NPIs had across states, reducing an average Rt > 3 to an average of 1.5 by 9-May-2020, but that these interventions failed to reduce Rt < 1, congruent with the worsening epidemic Brazil has experienced since. We identify extensive heterogeneity in the epidemic trajectory across Brazil, with the estimated number of days to reach 0.1% of the state population infected since the first nationally recorded case ranging from 20 days in Sao Paulo compared to 60 days in Goias, underscoring the importance of sub-national analyses in understanding asynchronous state-level epidemics underlying the national spread and burden of COVID-19.
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Italy was the first European country to experience sustained local transmission of COVID-19. As of 1st May 2020, the Italian health authorities reported 28,238 deaths nationally. To control the epidemic, the Italian government implemented a suite of non-pharmaceutical interventions (NPIs), including school and university closures, social distancing and full lockdown involving banning of public gatherings and non essential movement. In this report, we model the effect of NPIs on transmission using data on average mobility. We estimate that the average reproduction number (a measure of transmission intensity) is currently below one for all Italian regions, and significantly so for the majority of the regions. Despite the large number of deaths, the proportion of population that has been infected by SARS-CoV-2 (the attack rate) is far from the herd immunity threshold in all Italian regions, with the highest attack rate observed in Lombardy (13.18% [10.66%-16.70%]). Italy is set to relax the currently implemented NPIs from 4th May 2020. Given the control achieved by NPIs, we consider three scenarios for the next 8 weeks: a scenario in which mobility remains the same as during the lockdown, a scenario in which mobility returns to pre-lockdown levels by 20%, and a scenario in which mobility returns to pre-lockdown levels by 40%. The scenarios explored assume that mobility is scaled evenly across all dimensions, that behaviour stays the same as before NPIs were implemented, that no pharmaceutical interventions are introduced, and it does not include transmission reduction from contact tracing, testing and the isolation of confirmed or suspected cases. New interventions, such as enhanced testing and contact tracing are going to be introduced and will likely contribute to reductions in transmission; therefore our estimates should be viewed as pessimistic projections. We find that, in the absence of additional interventions, even a 20% return to pre-lockdown mobility could lead to a resurgence in the number of deaths far greater than experienced in the current wave in several regions. Future increases in the number of deaths will lag behind the increase in transmission intensity and so a second wave will not be immediately apparent from just monitoring of the daily number of deaths. Our results suggest that SARS-CoV-2 transmission as well as mobility should be closely monitored in the next weeks and months. To compensate for the increase in mobility that will occur due to the relaxation of the currently implemented NPIs, adherence to the recommended social distancing measures alongside enhanced community surveillance including swab testing, contact tracing and the early isolation of infections are of paramount importance to reduce the risk of resurgence in transmission. SUGGESTED CITATIONMichaela A. C. Vollmer, Swapnil Mishra, H Juliette T Unwin, Axel Gandy et al. Using mobility to estimate the transmission intensity of COVID-19 in Italy: a subnational analysis with future scenarios. Imperial College London (2020) doi:https://doi.org/10.25561/78677 This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
