A Quantitative Framework for Defining the End of an Infectious Disease Outbreak: Application to Ebola Virus Disease

Using real-time modelling to inform the 2017 Ebola outbreak response in DR Congo

Important policy questions during infections disease outbreaks include: i) How effective are particular interventions?; ii) When can resource-intensive interventions be removed? We used mathematical modelling to address these questions during the 2017 Ebola outbreak in Likati Health Zone, Democratic Republic of the Congo (DRC). Eight cases occurred before 15 May 2017, when the Ebola Response Team (ERT; co-ordinated by the World Health Organisation and DRC Ministry of Health) was deployed to reduce transmission. We used a branching process model to estimate that, pre-ERT arrival, the reproduction number was R = 1.49 (95% credible interval ( 0.67, 2.81 ) ). The risk of further cases occurring without the ERT was estimated to be 0.97 (97%). However, no cases materialised, suggesting that the ERT's measures were effective. We also estimated the risk of withdrawing the ERT in real-time. By the actual ERT withdrawal date (2 July 2017), the risk of future cases without the ERT was only 0.01, indicating that the ERT withdrawal decision was safe. We evaluated the sensitivity of our results to the estimated R value and considered different criteria for determining the ERT withdrawal date. This research provides an extensible modelling framework that can be used to guide decisions about when to relax interventions during future outbreaks.

Optimizing the timing of an end-of-outbreak declaration: Ebola virus disease in the Democratic Republic of the Congo

Following the apparent final case in an Ebola virus disease (EVD) outbreak, the decision to declare the outbreak over must balance societal benefits of relaxing interventions against the risk of resurgence. Estimates of the end-of-outbreak probability (the probability that no future cases will occur) provide quantitative evidence that can inform the timing of an end-of-outbreak declaration. An existing modeling approach for estimating the end-of-outbreak probability requires comprehensive contact tracing data describing who infected whom to be available, but such data are often unavailable or incomplete during outbreaks. Here, we develop a Markov chain Monte Carlo-based approach that extends the previous method and does not require contact tracing data. Considering data from two EVD outbreaks in the Democratic Republic of the Congo, we find that data describing who infected whom are not required to resolve uncertainty about when to declare an outbreak over.

Comparing the Performance of Three Computational Methods for Estimating the Effective Reproduction Number

The effective reproduction number ( R t ) is one of the most important epidemiological parameters, providing suggestions for monitoring the development trend of diseases and also for adjusting the prevention and control policies. However, a few studies have focused on the performance of some common computational methods for Rt. The purpose of this article is to compare the performance of three computational methods for Rt: the time-dependent (TD) method, the new time-varying (NT) method, and the sequential Bayesian (SB) method. Four evaluation methods-accuracy, correlation coefficient, similarity based on trend, and dynamic time warping distance-were used to compare the effectiveness of three computational methods for Rt under different time lags and time windows. The results showed that the NT method was a better choice for real-time monitoring and analysis of the epidemic in the middle and late stages of the infectious disease. The TD method could reflect the change of the number of cases stably and accurately, and was more suitable for monitoring the change of Rt during the whole process of the epidemic outbreak. When the data were relatively stable, the SB method could also provide a reliable estimate for Rt, while the error would increase when the fluctuation in the number of cases increased. The results would provide suggestions for selecting appropriate Rt estimation methods and making policy adjustments more timely and effectively according to the change of Rt.

Exact calculation of end-of-outbreak probabilities using contact tracing data

A key challenge for public health policymakers is determining when an infectious disease outbreak has finished. Following a period without cases, an estimate of the probability that no further cases will occur in future (the end-of-outbreak probability) can be used to inform whether or not to declare an outbreak over. An existing quantitative approach (the Nishiura method), based on a branching process transmission model, allows the end-of-outbreak probability to be approximated from disease incidence time series, the offspring distribution and the serial interval distribution. Here, we show how the end-of-outbreak probability under the same transmission model can be calculated exactly if data describing who-infected-whom (the transmission tree) are also available (e.g. from contact tracing studies). In that scenario, our novel approach (the traced transmission method) is straightforward to use. We demonstrate this by applying the method to data from previous outbreaks of Ebola virus disease and Nipah virus infection. For both outbreaks, the traced transmission method would have determined that the outbreak was over earlier than the Nishiura method. This highlights that collection of contact tracing data and application of the traced transmission method may allow stringent control interventions to be relaxed quickly at the end of an outbreak, with only a limited risk of outbreak resurgence.

A modelling analysis of a new multi-stage pathway for classifying achievement of public health milestones for leprosy

Several countries have come close to eliminating leprosy, but leprosy cases continue to be detected at low levels. Due to the long, highly variable delay from infection to detection, the relationship between observed cases and transmission is uncertain. The World Health Organization's new technical guidance provides a path for countries to reach elimination. We use a simple probabilistic model to simulate the stochastic dynamics of detected cases as transmission declines, and evaluate progress through the new public health milestones. In simulations where transmission is halted, 5 years of zero incidence in autochthonous children, combined with 3 years of zero incidence in all ages is a flawed indicator that transmission has halted (54% correctly classified). A further 10 years of only occasional sporadic cases is associated with a high probability of having interrupted transmission (99%). If, however, transmission continues at extremely low levels, it is possible that cases could be misidentified as historic cases from the tail of the incubation period distribution, although misleadingly achieving all three milestones is unlikely (less than 1% probability across a 15-year period of ongoing low-level transmission). These results demonstrate the feasibility and challenges of a phased progression of milestones towards interruption of transmission, allowing assessment of programme status. This article is part of the theme issue 'Challenges and opportunities in the fight against neglected tropical diseases: a decade from the London Declaration on NTDs'.

Accounting for the Potential of Overdispersion in Estimation of the Time-varying Reproduction Number

BACKGROUND

The time-varying reproduction number, Rt, is commonly used to monitor the transmissibility of an infectious disease during an epidemic, but standard methods for estimating Rt seldom account for the impact of overdispersion on transmission.

METHODS

We developed a negative binomial framework to estimate Rt and a time-varying dispersion parameter (kt). We applied the framework to COVID-19 incidence data in Hong Kong in 2020 and 2021. We conducted a simulation study to compare the performance of our model with the conventional Poisson-based approach.

RESULTS

Our framework estimated an Rt peaking around 4 (95% credible interval = 3.13, 4.30), similar to that from the Poisson approach but with a better model fit. Our approach further estimated kt <0.5 at the start of both waves, indicating appreciable heterogeneity in transmission. We also found that kt decreased sharply to around 0.4 when a large cluster of infections occurred.

CONCLUSIONS

Our proposed approach can contribute to the estimation of Rt and monitoring of the time-varying dispersion parameters to quantify the role of superspreading.

A quantitative method to project the probability of the end of an epidemic: Application to the COVID-19 outbreak in Wuhan, 2020

The end-of-outbreak declaration is an important part of epidemic control, marking the relaxation or cancellation of prevention and control measures. We propose a probability model to retrospectively quantify the confidence of giving the end-of-outbreak declaration during the COVID-19 epidemic in early 2020 in Wuhan. By using the linear spline, we firstly estimates the time-varying proportion of cases who miss the nonpharmaceutical interventions (NPIs) among all reported cases. Assuming the reproduction numbers being 1.5, 2.0, 3.0, 4.0, 5.0 and 6.0, the respective probability of the end of the COVID-19 outbreak with time after the last reported case can be iteratively computed. Consequently, the varying reproduction numbers produce slightly different increasing patterns of NPI effectiveness, and the end-of-outbreak declarations with 95% confidence are projected consistently earlier than the day when the lockdown was actually lifted. The reason for the timing discrepancy is discussed as well.

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

The time-varying reproduction number (Rt) is an important measure of transmissibility during outbreaks. Estimating whether and how rapidly an outbreak is growing (Rt > 1) or declining (Rt < 1) can inform the design, monitoring and adjustment of control measures in real-time. We use a popular R package for Rt estimation, EpiEstim, as a case study to evaluate the contexts in which Rt estimation methods have been used and identify unmet needs which would enable broader applicability of these methods in real-time. A scoping review, complemented by a small EpiEstim user survey, highlight issues with the current approaches, including the quality of input incidence data, the inability to account for geographical factors, and other methodological issues. We summarise the methods and software developed to tackle the problems identified, but conclude that significant gaps remain which should be addressed to enable easier, more robust and applicable estimation of Rt during epidemics.

Inferring the reproduction number using the renewal equation in heterogeneous epidemics

Real-time estimation of the reproduction number has become the focus of modelling groups around the world as the SARS-CoV-2 pandemic unfolds. One of the most widely adopted means of inference of the reproduction number is via the renewal equation, which uses the incidence of infection and the generation time distribution. In this paper, we derive a multi-type equivalent to the renewal equation to estimate a reproduction number which accounts for heterogeneity in transmissibility including through asymptomatic transmission, symptomatic isolation and vaccination. We demonstrate how use of the renewal equation that misses these heterogeneities can result in biased estimates of the reproduction number. While the bias is small with symptomatic isolation, it can be much larger with asymptomatic transmission or transmission from vaccinated individuals if these groups exhibit substantially different generation time distributions to unvaccinated symptomatic transmitters, whose generation time distribution is often well defined. The bias in estimate becomes larger with greater population size or transmissibility of the poorly characterized group. We apply our methodology to Ebola in West Africa in 2014 and the SARS-CoV-2 in the UK in 2020-2021.

Early intervention is the key to success in COVID-19 control

New Zealand responded to the COVID-19 pandemic with a combination of border restrictions and an Alert Level (AL) system that included strict stay-at-home orders. These interventions were successful in containing an outbreak and ultimately eliminating community transmission of COVID-19 in June 2020. The timing of interventions is crucial to their success. Delaying interventions may reduce their effectiveness and mean that they need to be maintained for a longer period. We use a stochastic branching process model of COVID-19 transmission and control to simulate the epidemic trajectory in New Zealand's March-April 2020 outbreak and the effect of its interventions. We calculate key measures, including the number of reported cases and deaths, and the probability of elimination within a specified time frame. By comparing these measures under alternative timings of interventions, we show that changing the timing of AL4 (the strictest level of restrictions) has a far greater impact than the timing of border measures. Delaying AL4 restrictions results in considerably worse outcomes. Implementing border measures alone, without AL4 restrictions, is insufficient to control the outbreak. We conclude that the early introduction of stay-at-home orders was crucial in reducing the number of cases and deaths, enabling elimination.

Sub-spreading events limit the reliable elimination of heterogeneous epidemics

We show that sub-spreading events, i.e. transmission events in which an infection propagates to few or no individuals, can be surprisingly important for defining the lifetime of an infectious disease epidemic and hence its waiting time to elimination or fade-out, measured from the time-point of its last observed case. While limiting super-spreading promotes more effective control when cases are growing, we find that when incidence is waning, curbing sub-spreading is more important for achieving reliable elimination of the epidemic. Controlling super-spreading in this low-transmissibility phase offers diminishing returns over non-selective, population-wide measures. By restricting sub-spreading, we efficiently dampen remaining variations among the reproduction numbers of infectious events, which minimizes the risk of premature and late end-of-epidemic declarations. Because case-ascertainment or reporting rates can be modelled in exactly the same way as control policies, we concurrently show that the under-reporting of sub-spreading events during waning phases will engender overconfident assessments of epidemic elimination. While controlling sub-spreading may not be easily realized, the likely neglecting of these events by surveillance systems could result in unexpectedly risky end-of-epidemic declarations. Super-spreading controls the size of the epidemic peak but sub-spreading mediates the variability of its tail.

A hospital-related outbreak of SARS-CoV-2 associated with variant Epsilon (B.1.429) in Taiwan: transmission potential and outbreak containment under intensified contact tracing, January-February 2021

OBJECTIVES

A hospital-related cluster of 22 cases of coronavirus disease 2019 (COVID-19) occurred in Taiwan in January-February 2021. Rigorous control measures were introduced and could only be relaxed once the outbreak was declared over. Each day after the apparent outbreak end, we estimated the risk of future cases occurring in order to inform decision-making.

METHODS

Probabilistic transmission networks were reconstructed, and transmission parameters (the reproduction number R and overdispersion parameter k) were estimated. The reporting delay during the outbreak was estimated (Scenario 1). In addition, a counterfactual scenario with less effective interventions characterized by a longer reporting delay was considered (Scenario 2). Each day, the risk of future cases was estimated under both scenarios.

RESULTS

The values of R and k were estimated to be 1.30 ((95% credible interval (CI) 0.57-3.80) and 0.38 (95% CI 0.12-1.20), respectively. The mean reporting delays considered were 2.5 days (Scenario 1) and 7.8 days (Scenario 2). Following the final case, ttthe inferred probability of future cases occurring declined more quickly in Scenario 1 than Scenario 2.

CONCLUSIONS

Rigorous control measures allowed the outbreak to be declared over quickly following outbreak containment. This highlights the need for effective interventions, not only to reduce cases during outbreaks but also to allow outbreaks to be declared over with confidence.

How epidemics end

As COVID-19 drags on and new vaccines promise widespread immunity, the world's attention has turned to predicting how the present pandemic will end. How do societies know when an epidemic is over and normal life can resume? What criteria and markers indicate such an end? Who has the insight, authority, and credibility to decipher these signs? Detailed research on past epidemics has demonstrated that they do not end suddenly; indeed, only rarely do the diseases in question actually end. This article examines the ways in which scholars have identified and described the end stages of previous epidemics, pointing out that significantly less attention has been paid to these periods than to origins and climaxes. Analysis of the ends of epidemics illustrates that epidemics are as much social, political, and economic events as they are biological; the "end," therefore, is as much a process of social and political negotiation as it is biomedical. Equally important, epidemics end at different times for different groups, both within one society and across regions. Multidisciplinary research into how epidemics end reveals how the end of an epidemic shifts according to perspective, whether temporal, geographic, or methodological. A multidisciplinary analysis of how epidemics end suggests that epidemics should therefore be framed not as linear narratives-from outbreak to intervention to termination-but within cycles of disease and with a multiplicity of endings.