Forecasting the 2001 Foot-and-Mouth Disease Epidemic in the UK

SpatialWavePredict: a tutorial-based primer and toolbox for forecasting growth trajectories using the ensemble spatial wave sub-epidemic modeling framework

BACKGROUND

Dynamical mathematical models defined by a system of differential equations are typically not easily accessible to non-experts. However, forecasts based on these types of models can help gain insights into the mechanisms driving the process and may outcompete simpler phenomenological growth models. Here we introduce a friendly toolbox, SpatialWavePredict, to characterize and forecast the spatial wave sub-epidemic model, which captures diverse wave dynamics by aggregating multiple asynchronous growth processes and has outperformed simpler phenomenological growth models in short-term forecasts of various infectious diseases outbreaks including SARS, Ebola, and the early waves of the COVID-19 pandemic in the US.

RESULTS

This tutorial-based primer introduces and illustrates a user-friendly MATLAB toolbox for fitting and forecasting time-series trajectories using an ensemble spatial wave sub-epidemic model based on ordinary differential equations. Scientists, policymakers, and students can use the toolbox to conduct real-time short-term forecasts. The five-parameter epidemic wave model in the toolbox aggregates linked overlapping sub-epidemics and captures a rich spectrum of epidemic wave dynamics, including oscillatory wave behavior and plateaus. An ensemble strategy aims to improve forecasting performance by combining the resulting top-ranked models. The toolbox provides a tutorial for forecasting time-series trajectories, including the full uncertainty distribution derived through parametric bootstrapping, which is needed to construct prediction intervals and evaluate their accuracy. Functions are available to assess forecasting performance, estimation methods, error structures in the data, and forecasting horizons. The toolbox also includes functions to quantify forecasting performance using metrics that evaluate point and distributional forecasts, including the weighted interval score.

CONCLUSIONS

We have developed the first comprehensive toolbox to characterize and forecast time-series data using an ensemble spatial wave sub-epidemic wave model. As an epidemic situation or contagion occurs, the tools presented in this tutorial can facilitate policymakers to guide the implementation of containment strategies and assess the impact of control interventions. We demonstrate the functionality of the toolbox with examples, including a tutorial video, and is illustrated using daily data on the COVID-19 pandemic in the USA.

SubEpiPredict: A tutorial-based primer and toolbox for fitting and forecasting growth trajectories using the ensemble n-sub-epidemic modeling framework

An ensemble n-sub-epidemic modeling framework that integrates sub-epidemics to capture complex temporal dynamics has demonstrated powerful forecasting capability in previous works. This modeling framework can characterize complex epidemic patterns, including plateaus, epidemic resurgences, and epidemic waves characterized by multiple peaks of different sizes. In this tutorial paper, we introduce and illustrate SubEpiPredict, a user-friendly MATLAB toolbox for fitting and forecasting time series data using an ensemble n-sub-epidemic modeling framework. The toolbox can be used for model fitting, forecasting, and evaluation of model performance of the calibration and forecasting periods using metrics such as the weighted interval score (WIS). We also provide a detailed description of these methods including the concept of the n-sub-epidemic model, constructing ensemble forecasts from the top-ranking models, etc. For the illustration of the toolbox, we utilize publicly available daily COVID-19 death data at the national level for the United States. The MATLAB toolbox introduced in this paper can be very useful for a wider group of audiences, including policymakers, and can be easily utilized by those without extensive coding and modeling backgrounds.

GrowthPredict: A toolbox and tutorial-based primer for fitting and forecasting growth trajectories using phenomenological growth models

Simple dynamic modeling tools can help generate real-time short-term forecasts with quantified uncertainty of the trajectory of diverse growth processes unfolding in nature and society, including disease outbreaks. An easy-to-use and flexible toolbox for this purpose is lacking. This tutorial-based primer introduces and illustrates GrowthPredict, a user-friendly MATLAB toolbox for fitting and forecasting time-series trajectories using phenomenological dynamic growth models based on ordinary differential equations. This toolbox is accessible to a broad audience, including students training in mathematical biology, applied statistics, and infectious disease modeling, as well as researchers and policymakers who need to conduct short-term forecasts in real-time. The models included in the toolbox capture exponential and sub-exponential growth patterns that typically follow a rising pattern followed by a decline phase, a common feature of contagion processes. Models include the 1-parameter exponential growth model and the 2-parameter generalized-growth model, which have proven useful in characterizing and forecasting the ascending phase of epidemic outbreaks. It also includes the 2-parameter Gompertz model, the 3-parameter generalized logistic-growth model, and the 3-parameter Richards model, which have demonstrated competitive performance in forecasting single peak outbreaks. We provide detailed guidance on forecasting time-series trajectories and available software ( https://github.com/gchowell/forecasting_growthmodels ), including the full uncertainty distribution derived through parametric bootstrapping, which is needed to construct prediction intervals and evaluate their accuracy. Functions are available to assess forecasting performance across different models, estimation methods, error structures in the data, and forecasting horizons. The toolbox also includes functions to quantify forecasting performance using metrics that evaluate point and distributional forecasts, including the weighted interval score. This tutorial and toolbox can be broadly applied to characterizing and forecasting time-series data using simple phenomenological growth models. As a contagion process takes off, the tools presented in this tutorial can help create forecasts to guide policy regarding implementing control strategies and assess the impact of interventions. The toolbox functionality is demonstrated through various examples, including a tutorial video, and the examples use publicly available data on the monkeypox (mpox) epidemic in the USA.

Weekly Forecasting of Yellow Fever Occurrence and Incidence via Eco-Meteorological Dynamics

Yellow Fever (YF), a mosquito-borne disease, requires ongoing surveillance and prevention due to its persistence and ability to cause major epidemics, including one that began in Brazil in 2016. Forecasting based on factors influencing YF risk can improve efficiency in prevention. This study aimed to produce weekly forecasts of YF occurrence and incidence in Brazil using weekly meteorological and ecohydrological conditions. Occurrence was forecast as the probability of observing any cases, and incidence was forecast to represent morbidity if YF occurs. We fit gamma hurdle models, selecting predictors from several meteorological and ecohydrological factors, based on forecast accuracy defined by receiver operator characteristic curves and mean absolute error. We fit separate models for data before and after the start of the 2016 outbreak, forecasting occurrence and incidence for all municipalities of Brazil weekly. Different predictor sets were found to produce most accurate forecasts in each time period, and forecast accuracy was high for both time periods. Temperature, precipitation, and previous YF burden were most influential predictors among models. Minimum, maximum, mean, and range of weekly temperature, precipitation, and humidity contributed to forecasts, with optimal lag times of 2, 6, and 7 weeks depending on time period. Results from this study show the use of environmental predictors in providing regular forecasts of YF burden and producing nationwide forecasts. Weekly forecasts, which can be produced using the forecast model developed in this study, are beneficial for informing immediate preparedness measures.

An algebraic formula, deep learning and a novel SEIR-type model for the COVID-19 pandemic

The most extensively used mathematical models in epidemiology are the susceptible-exposed-infectious-recovered (SEIR) type models with constant coefficients. For the first wave of the COVID-19 epidemic, such models predict that at large times equilibrium is reached exponentially. However, epidemiological data from Europe suggest that this approach is algebraic. Indeed, accurate long-term predictions have been obtained via a forecasting model only if it uses an algebraic as opposed to the standard exponential formula. In this work, by allowing those parameters of the SEIR model that reflect behavioural aspects (e.g. spatial distancing) to vary nonlinearly with the extent of the epidemic, we construct a model which exhibits asymptoticly algebraic behaviour. Interestingly, the emerging power law is consistent with the typical dynamics observed in various social settings. In addition, using reliable epidemiological data, we solve in a numerically robust way the inverse problem of determining all model parameters characterizing our novel model. Finally, using deep learning, we demonstrate that the algebraic forecasting model used earlier is optimal.

A MATLAB toolbox to fit and forecast growth trajectories using phenomenological growth models: Application to epidemic outbreaks

Background

Simple dynamic modeling tools can be useful for generating real-time short-term forecasts with quantified uncertainty of the trajectory of diverse growth processes unfolding in nature and society, including disease outbreaks. An easy-to-use and flexible toolbox for this purpose is lacking.

Results

In this tutorial-based primer, we introduce and illustrate a user-friendly MATLAB toolbox for fitting and forecasting time-series trajectories using phenomenological dynamic growth models based on ordinary differential equations. This toolbox is accessible to various audiences, including students training in time-series forecasting, dynamic growth modeling, parameter estimation, parameter uncertainty and identifiability, model comparison, performance metrics, and forecast evaluation, as well as researchers and policymakers who need to conduct short-term forecasts in real-time. The models included in the toolbox capture exponential and sub-exponential growth patterns that typically follow a rising pattern followed by a decline phase, a common feature of contagion processes. Models include the 2-parameter generalized-growth model, which has proved useful to characterize and forecast the ascending phase of epidemic outbreaks, and the Gompertz model as well as the 3-parameter generalized logistic-growth model and the Richards model, which have demonstrated competitive performance in forecasting single peak outbreaks.The toolbox provides a tutorial for forecasting time-series trajectories that include the full uncertainty distribution, derived through parametric bootstrapping, which is needed to construct prediction intervals and evaluate their accuracy. Functions are available to assess forecasting performance across different models, estimation methods, error structures in the data, and forecasting horizons. The toolbox also includes functions to quantify forecasting performance using metrics that evaluate point and distributional forecasts, including the weighted interval score.

Conclusions

We have developed the first comprehensive toolbox to characterize and forecast time-series data using simple phenomenological growth models. As a contagion process takes off, the tools presented in this tutorial can facilitate policymaking to guide the implementation of control strategies and assess the impact of interventions. The toolbox functionality is demonstrated through various examples, including a tutorial video, and is illustrated using weekly data on the monkeypox epidemic in the USA.

A MATLAB toolbox to fit and forecast growth trajectories using phenomenological growth models: Application to epidemic outbreaks

BACKGROUND

Simple dynamic modeling tools can be useful for generating real-time short-term forecasts with quantified uncertainty of the trajectory of diverse growth processes unfolding in nature and society, including disease outbreaks. An easy-to-use and flexible toolbox for this purpose is lacking.

RESULTS

In this tutorial-based primer, we introduce and illustrate a user-friendly MATLAB toolbox for fitting and forecasting time-series trajectories using phenomenological dynamic growth models based on ordinary differential equations. This toolbox is accessible to various audiences, including students training in time-series forecasting, dynamic growth modeling, parameter estimation, parameter uncertainty and identifiability, model comparison, performance metrics, and forecast evaluation, as well as researchers and policymakers who need to conduct short-term forecasts in real-time. The models included in the toolbox capture exponential and sub-exponential growth patterns that typically follow a rising pattern followed by a decline phase, a common feature of contagion processes. Models include the 2-parameter generalized-growth model, which has proved useful to characterize and forecast the ascending phase of epidemic outbreaks, and the Gompertz model as well as the 3-parameter generalized logistic-growth model and the Richards model, which have demonstrated competitive performance in forecasting single peak outbreaks.The toolbox provides a tutorial for forecasting time-series trajectories that include the full uncertainty distribution, derived through parametric bootstrapping, which is needed to construct prediction intervals and evaluate their accuracy. Functions are available to assess forecasting performance across different models, estimation methods, error structures in the data, and forecasting horizons. The toolbox also includes functions to quantify forecasting performance using metrics that evaluate point and distributional forecasts, including the weighted interval score.

CONCLUSIONS

We have developed the first comprehensive toolbox to characterize and forecast time-series data using simple phenomenological growth models. As a contagion process takes off, the tools presented in this tutorial can facilitate policymaking to guide the implementation of control strategies and assess the impact of interventions. The toolbox functionality is demonstrated through various examples, including a tutorial video, and is illustrated using weekly data on the monkeypox epidemic in the USA.

Real-time forecasting the trajectory of monkeypox outbreaks at the national and global levels, July-October 2022

BACKGROUND

Beginning May 7, 2022, multiple nations reported an unprecedented surge in monkeypox cases. Unlike past outbreaks, differences in affected populations, transmission mode, and clinical characteristics have been noted. With the existing uncertainties of the outbreak, real-time short-term forecasting can guide and evaluate the effectiveness of public health measures.

METHODS

We obtained publicly available data on confirmed weekly cases of monkeypox at the global level and for seven countries (with the highest burden of disease at the time this study was initiated) from the Our World in Data (OWID) GitHub repository and CDC website. We generated short-term forecasts of new cases of monkeypox across the study areas using an ensemble n-sub-epidemic modeling framework based on weekly cases using 10-week calibration periods. We report and assess the weekly forecasts with quantified uncertainty from the top-ranked, second-ranked, and ensemble sub-epidemic models. Overall, we conducted 324 weekly sequential 4-week ahead forecasts across the models from the week of July 28th, 2022, to the week of October 13th, 2022.

RESULTS

The last 10 of 12 forecasting periods (starting the week of August 11th, 2022) show either a plateauing or declining trend of monkeypox cases for all models and areas of study. According to our latest 4-week ahead forecast from the top-ranked model, a total of 6232 (95% PI 487.8, 12,468.0) cases could be added globally from the week of 10/20/2022 to the week of 11/10/2022. At the country level, the top-ranked model predicts that the USA will report the highest cumulative number of new cases for the 4-week forecasts (median based on OWID data: 1806 (95% PI 0.0, 5544.5)). The top-ranked and weighted ensemble models outperformed all other models in short-term forecasts.

CONCLUSIONS

Our top-ranked model consistently predicted a decreasing trend in monkeypox cases on the global and country-specific scale during the last ten sequential forecasting periods. Our findings reflect the potential impact of increased immunity, and behavioral modification among high-risk populations.

An ensemble n-sub-epidemic modeling framework for short-term forecasting epidemic trajectories: Application to the COVID-19 pandemic in the USA

We analyze an ensemble of n-sub-epidemic modeling for forecasting the trajectory of epidemics and pandemics. These ensemble modeling approaches, and models that integrate sub-epidemics to capture complex temporal dynamics, have demonstrated powerful forecasting capability. This modeling framework can characterize complex epidemic patterns, including plateaus, epidemic resurgences, and epidemic waves characterized by multiple peaks of different sizes. We systematically assess their calibration and short-term forecasting performance in short-term forecasts for the COVID-19 pandemic in the USA from late April 2020 to late February 2022. We compare their performance with two commonly used statistical ARIMA models. The best fit sub-epidemic model and three ensemble models constructed using the top-ranking sub-epidemic models consistently outperformed the ARIMA models in terms of the weighted interval score (WIS) and the coverage of the 95% prediction interval across the 10-, 20-, and 30-day short-term forecasts. In our 30-day forecasts, the average WIS ranged from 377.6 to 421.3 for the sub-epidemic models, whereas it ranged from 439.29 to 767.05 for the ARIMA models. Across 98 short-term forecasts, the ensemble model incorporating the top four ranking sub-epidemic models (Ensemble(4)) outperformed the (log) ARIMA model 66.3% of the time, and the ARIMA model, 69.4% of the time in 30-day ahead forecasts in terms of the WIS. Ensemble(4) consistently yielded the best performance in terms of the metrics that account for the uncertainty of the predictions. This framework can be readily applied to investigate the spread of epidemics and pandemics beyond COVID-19, as well as other dynamic growth processes found in nature and society that would benefit from short-term predictions.

An ensemble n -sub-epidemic modeling framework for short-term forecasting epidemic trajectories: Application to the COVID-19 pandemic in the USA

We analyze an ensemble of n -sub-epidemic modeling for forecasting the trajectory of epidemics and pandemics. These ensemble modeling approaches, and models that integrate sub-epidemics to capture complex temporal dynamics, have demonstrated powerful forecasting capability. This modeling framework can characterize complex epidemic patterns, including plateaus, epidemic resurgences, and epidemic waves characterized by multiple peaks of different sizes. We systematically assess their calibration and short-term forecasting performance in short-term forecasts for the COVID-19 pandemic in the USA from late April 2020 to late February 2022. We compare their performance with two commonly used statistical ARIMA models. The best fit sub-epidemic model and three ensemble models constructed using the top-ranking sub-epidemic models consistently outperformed the ARIMA models in terms of the weighted interval score (WIS) and the coverage of the 95% prediction interval across the 10-, 20-, and 30-day short-term forecasts. In the 30-day forecasts, the average WIS ranged from 377.6 to 421.3 for the sub-epidemic models, whereas it ranged from 439.29 to 767.05 for the ARIMA models. Across 98 short-term forecasts, the ensemble model incorporating the top four ranking sub-epidemic models (Ensemble(4)) outperformed the (log) ARIMA model 66.3% of the time, and the ARIMA model 69.4% of the time in 30-day ahead forecasts in terms of the WIS. Ensemble(4) consistently yielded the best performance in terms of the metrics that account for the uncertainty of the predictions. This framework could be readily applied to investigate the spread of epidemics and pandemics beyond COVID-19, as well as other dynamic growth processes found in nature and society that would benefit from short-term predictions.

Summary

The COVID-19 pandemic has highlighted the urgent need to develop reliable tools to forecast the trajectory of epidemics and pandemics in near real-time. We describe and apply an ensemble n -sub-epidemic modeling framework for forecasting the trajectory of epidemics and pandemics. We systematically assess its calibration and short-term forecasting performance in weekly 10-30 days ahead forecasts for the COVID-19 pandemic in the USA from late April 2020 to late February 2022 and compare its performance with two different statistical ARIMA models. This framework demonstrated reliable forecasting performance and substantially outcompeted the ARIMA models. The forecasting performance was consistently best for the ensemble sub-epidemic models incorporating a higher number of top-ranking sub-epidemic models. The ensemble model incorporating the top four ranking sub-epidemic models consistently yielded the best performance, particularly in terms of the coverage rate of the 95% prediction interval and the weighted interval score. This framework can be applied to forecast other growth processes found in nature and society including the spread of information through social media.

An investigation of spatial-temporal patterns and predictions of the coronavirus 2019 pandemic in Colombia, 2020-2021

Colombia announced the first case of severe acute respiratory syndrome coronavirus 2 on March 6, 2020. Since then, the country has reported a total of 5,002,387 cases and 127,258 deaths as of October 31, 2021. The aggressive transmission dynamics of SARS-CoV-2 motivate an investigation of COVID-19 at the national and regional levels in Colombia. We utilize the case incidence and mortality data to estimate the transmission potential and generate short-term forecasts of the COVID-19 pandemic to inform the public health policies using previously validated mathematical models. The analysis is augmented by the examination of geographic heterogeneity of COVID-19 at the departmental level along with the investigation of mobility and social media trends. Overall, the national and regional reproduction numbers show sustained disease transmission during the early phase of the pandemic, exhibiting sub-exponential growth dynamics. Whereas the most recent estimates of reproduction number indicate disease containment, with Rt<1.0 as of October 31, 2021. On the forecasting front, the sub-epidemic model performs best at capturing the 30-day ahead COVID-19 trajectory compared to the Richards and generalized logistic growth model. Nevertheless, the spatial variability in the incidence rate patterns across different departments can be grouped into four distinct clusters. As the case incidence surged in July 2020, an increase in mobility patterns was also observed. On the contrary, a spike in the number of tweets indicating the stay-at-home orders was observed in November 2020 when the case incidence had already plateaued, indicating the pandemic fatigue in the country.

Phenomenological and mechanistic models for predicting early transmission data of COVID-19

Forecasting future epidemics helps inform policy decisions regarding interventions. During the early coronavirus disease 2019 epidemic period in January-February 2020, limited information was available, and it was too challenging to build detailed mechanistic models reflecting population behavior. This study compared the performance of phenomenological and mechanistic models for forecasting epidemics. For the former, we employed the Richards model and the approximate solution of the susceptible-infected-recovered (SIR) model. For the latter, we examined the exponential growth (with lockdown) model and SIR model with lockdown. The phenomenological models yielded higher root mean square error (RMSE) values than the mechanistic models. When using the numbers from reported data for February 1 and 5, the Richards model had the highest RMSE, whereas when using the February 9 data, the SIR approximation model was the highest. The exponential model with a lockdown effect had the lowest RMSE, except when using the February 9 data. Once interventions or other factors that influence transmission patterns are identified, they should be additionally taken into account to improve forecasting.

Transmission Dynamics and Short-Term Forecasts of COVID-19: Nepal 2020/2021

Nepal was hard hit by a second wave of COVID-19 from April-May 2021. We investigated the transmission dynamics of COVID-19 at the national and provincial levels by using data on laboratory-confirmed RT-PCR positive cases from the official national situation reports. We performed 8 week-to-week sequential forecasts of 10-days and 20-days at national level using three dynamic phenomenological growth models from 5 March 2021-22 May 2021. We also estimated effective and instantaneous reproduction numbers at national and provincial levels using established methods and evaluated the mobility trends using Google's mobility data. Our forecast estimates indicated a declining trend of COVID-19 cases in Nepal as of June 2021. Sub-epidemic and Richards models provided reasonable short-term projections of COVID-19 cases based on standard performance metrics. There was a linear pattern in the trajectory of COVID-19 incidence during the first wave (deceleration of growth parameter (p) = 0.41-0.43, reproduction number (Rt) at 1.1 (95% CI: 1.1, 1.2)), and a sub-exponential growth pattern in the second wave (p = 0.61 (95% CI: 0.58, 0.64)) and Rt at 1.3 (95% CI: 1.3, 1.3)). Across provinces, Rt ranged from 1.2 to 1.5 during the early growth phase of the second wave. The instantaneous Rt fluctuated around 1.0 since January 2021 indicating well sustained transmission. The peak in mobility across different areas coincided with an increasing incidence trend of COVID-19. In conclusion, we found that the sub-epidemic and Richards models yielded reasonable short-terms projections of the COVID-19 trajectory in Nepal, which are useful for healthcare utilization planning.

Characterization of bat coronaviruses: a latent global threat

It has been speculated that bats serve as reservoirs of a huge variety of emerging coronaviruses (CoVs) that have been responsible for severe havoc in human health systems as well as negatively affecting human economic and social systems. A prime example is the currently active severe acute respiratory syndrome (SARS)-CoV2, which presumably originated from bats, demonstrating that the risk of a new outbreak of bat coronavirus is always latent. Therefore, an in-depth investigation to better comprehend bat CoVs has become an important issue within the international community, a group that aims to attenuate the consequences of future outbreaks. In this review, we present a concise introduction to CoVs found in bats and discuss their distribution in Southeast Asia. We also discuss the unique adaptation features in bats that confer the ability to be a potential coronavirus reservoir. In addition, we review the bat coronavirus-linked diseases that have emerged in the last two decades. Finally, we propose key factors helpful in the prediction of a novel coronavirus outbreak and present the most recent methods used to forecast an evolving outbreak.

Transmission dynamics and forecasts of the COVID-19 pandemic in Mexico, March-December 2020

Mexico has experienced one of the highest COVID-19 mortality rates in the world. A delayed implementation of social distancing interventions in late March 2020 and a phased reopening of the country in June 2020 has facilitated sustained disease transmission in the region. In this study we systematically generate and compare 30-day ahead forecasts using previously validated growth models based on mortality trends from the Institute for Health Metrics and Evaluation for Mexico and Mexico City in near real-time. Moreover, we estimate reproduction numbers for SARS-CoV-2 based on the methods that rely on genomic data as well as case incidence data. Subsequently, functional data analysis techniques are utilized to analyze the shapes of COVID-19 growth rate curves at the state level to characterize the spatiotemporal transmission patterns of SARS-CoV-2. The early estimates of the reproduction number for Mexico were estimated between Rt ~1.1-1.3 from the genomic and case incidence data. Moreover, the mean estimate of Rt has fluctuated around ~1.0 from late July till end of September 2020. The spatial analysis characterizes the state-level dynamics of COVID-19 into four groups with distinct epidemic trajectories based on epidemic growth rates. Our results show that the sequential mortality forecasts from the GLM and Richards model predict a downward trend in the number of deaths for all thirteen forecast periods for Mexico and Mexico City. However, the sub-epidemic and IHME models perform better predicting a more realistic stable trajectory of COVID-19 mortality trends for the last three forecast periods (09/21-10/21, 09/28-10/27, 09/28-10/27) for Mexico and Mexico City. Our findings indicate that phenomenological models are useful tools for short-term epidemic forecasting albeit forecasts need to be interpreted with caution given the dynamic implementation and lifting of social distancing measures.

Mathematical and Statistical Analysis of Doubling Times to Investigate the Early Spread of Epidemics: Application to the COVID-19 Pandemic

Simple mathematical tools are needed to quantify the threat posed by emerging and re-emerging infectious disease outbreaks using minimal data capturing the outbreak trajectory. Here we use mathematical analysis, simulation and COVID-19 epidemic data to demonstrate a novel approach to numerically and mathematically characterize the rate at which the doubling time of an epidemic is changing over time. For this purpose, we analyze the dynamics of epidemic doubling times during the initial epidemic stage, defined as the sequence of times at which the cumulative incidence doubles. We introduce new methodology to characterize epidemic threats by analyzing the evolution of epidemics as a function of (1) the number of times the epidemic doubles until the epidemic peak is reached and (2) the rate at which the doubling times increase. In our doubling-time approach, the most dangerous epidemic threats double in size many times and the doubling times change at a relatively low rate (e.g., doubling times remain nearly invariant) whereas the least transmissible threats double in size only a few times and the doubling times rapidly increases in the period of emergence. We derive analytical formulas and test and illustrate our methodology using synthetic and COVID-19 epidemic data. Our mathematical analysis demonstrates that the series of epidemic doubling times increase approximately according to an exponential function with a rate that quantifies the rate of change of the doubling times. Our analytic results are in excellent agreement with numerical results. Our methodology offers a simple and intuitive approach that relies on minimal outbreak trajectory data to characterize the threat posed by emerging and re-emerging infectious diseases.

Transmission dynamics and control of COVID-19 in Chile, March-October, 2020

Since the detection of the first case of COVID-19 in Chile on March 3rd, 2020, a total of 513,188 cases, including ~14,302 deaths have been reported in Chile as of November 2nd, 2020. Here, we estimate the reproduction number throughout the epidemic in Chile and study the effectiveness of control interventions especially the effectiveness of lockdowns by conducting short-term forecasts based on the early transmission dynamics of COVID-19. Chile's incidence curve displays early sub-exponential growth dynamics with the deceleration of growth parameter, p, estimated at 0.8 (95% CI: 0.7, 0.8) and the reproduction number, R, estimated at 1.8 (95% CI: 1.6, 1.9). Our findings indicate that the control measures at the start of the epidemic significantly slowed down the spread of the virus. However, the relaxation of restrictions and spread of the virus in low-income neighborhoods in May led to a new surge of infections, followed by the reimposition of lockdowns in Greater Santiago and other municipalities. These measures have decelerated the virus spread with R estimated at ~0.96 (95% CI: 0.95, 0.98) as of November 2nd, 2020. The early sub-exponential growth trend (p ~0.8) of the COVID-19 epidemic transformed into a linear growth trend (p ~0.5) as of July 7th, 2020, after the reimposition of lockdowns. While the broad scale social distancing interventions have slowed the virus spread, the number of new COVID-19 cases continue to accrue, underscoring the need for persistent social distancing and active case detection and isolation efforts to maintain the epidemic under control.

Transmission dynamics and control of COVID-19 in Chile, March-October, 2020

Since the detection of the first case of COVID-19 in Chile on March 3rd, 2020, a total of 513188 cases, including ~14302 deaths have been reported in Chile as of November 2nd, 2020. Here, we estimate the reproduction number throughout the epidemic in Chile and study the effectiveness of control interventions especially the effectiveness of lockdowns by conducting short-term forecasts based on the early transmission dynamics of COVID-19. Chile's incidence curve displays early sub-exponential growth dynamics with the deceleration of growth parameter, p, estimated at 0.8 (95% CI: 0.7, 0.8) and the reproduction number, R, estimated at 1.8 (95% CI: 1.6, 1.9). Our findings indicate that the control measures at the start of the epidemic significantly slowed down the spread of the virus. However, the relaxation of restrictions and spread of the virus in low-income neighborhoods in May led to a new surge of infections, followed by the reimposition of lockdowns in Greater Santiago and other municipalities. These measures have decelerated the virus spread with R estimated at ~0.96( 95% CI: 0.95, 0.98) as of November 2nd, 2020. The early sub-exponential growth trend (p ~0.8) of the COVID-19 epidemic transformed into a linear growth trend (p ~0.5) as of July 7th, 2020, after the reimposition of lockdowns. While the broad scale social distancing interventions have slowed the virus spread, the number of new COVID-19 cases continue to accrue, underscoring the need for persistent social distancing and active case detection and isolation efforts to maintain the epidemic under control.

Systematic comparison of epidemic growth patterns using two different estimation approaches

Background

Different estimation approaches are frequently used to calibrate mathematical models to epidemiological data, particularly for analyzing infectious disease outbreaks. Here, we use two common methods to estimate parameters that characterize growth patterns using the generalized growth model (GGM) calibrated to real outbreak datasets.

Materials and methods

Data from 31 outbreaks are used to fit the GGM to the ascending phase of each outbreak and estimate the parameters using both least squares (LSQ) and maximum likelihood estimation (MLE) methods. We utilize parametric bootstrapping to construct confidence intervals for parameter estimates. We compare the results including RMSE, Anscombe residual, and 95% prediction interval coverage. We also evaluate the correlation between the estimates from both methods.

Results

Comparing LSQ and MLE estimates, most outbreaks have similar parameter estimates, RMSE, Anscombe, and 95% prediction interval coverage. Parameter estimates do not differ across methods when the model yields a good fit to the early growth phase. However, for two outbreaks, there are systematic deviations in model fit to the data that explain differences in parameter estimates (e.g., residuals represent random error rather than systematic deviation).

Conclusion

Our findings indicate that utilizing LSQ and MLE methods produce similar results in the context of characterizing epidemic growth patterns with the GGM, provided that the model yields a good fit to the data.

COVID-19, flattening the curve, and Benford's law

For many countries attempting to control the fast-rising number of coronavirus cases and deaths, the race is on to "flatten the curve," since the spread of coronavirus disease 2019 (COVID-19) has taken on pandemic proportions. In the absence of significant control interventions, the curve could be steep, with the number of COVID-19 cases growing exponentially. In fact, this level of proliferation may already be happening, since the number of patients infected in Italy closely follows an exponential trend. Thus, we propose a test. When the numbers are taken from an exponential distribution, it has been demonstrated that they automatically follow Benford's Law (BL). As a result, if the current control interventions are successful and we flatten the curve (i.e., we slow the rate below an exponential growth rate), then the number of infections or deaths will not obey BL. For this reason, BL may be useful for assessing the effects of the current control interventions and may be able to answer the question, "How flat is flat enough?" In this study, we used an epidemic growth model in the presence of interventions to describe the potential for a flattened curve, and then investigated whether the epidemic growth model followed BL for ten selected countries with a relatively high mortality rate. Among these countries, South Korea showed a particularly high degree of control intervention. Although all of the countries have aggressively fought the epidemic, our analysis shows that all countries except for Japan satisfied BL, indicating the growth rates of COVID-19 were close to an exponential trend. Based on the simulation table in this study, BL test shows that the data from Japan is incorrect.

Mycoplasma bovis outbreak in New Zealand cattle: An assessment of transmission trends using surveillance data

Mycoplasma bovis most likely infected New Zealand cattle in the latter half of 2015. Infection was detected in mid-2017 after which control activities were implemented. An official eradication programme commenced in mid-2018, which is ongoing. We examined farm-level tracing and surveillance data to describe the outbreak, analyse transmission trends and make inference on progress towards eradication. Results indicate that cattle movements were the primary means of spread. Although case farms were distributed throughout both islands of New Zealand, most animal movements off infected farms did not result in newly infected farms, indicating Mycoplasma bovis is not highly transmissible between farms. To describe and analyse outbreak trends, we undertook a standard descriptive outbreak investigation, including construction of an epidemic curve and calculation of estimated dissemination ratios. We then employed three empirical models-a non-linear growth model, time series model and branching process model based on time-varying effective reproduction numbers-to further analyse transmission trends and provide short-term forecasts of farm-level incidence. Our analyses suggest that Mycoplasma bovis transmission in New Zealand has declined and progress towards eradication has been made. Few incident cases were forecast for the period between 8 September and 17 December 2019. To date, no case farms with an estimated infection date assigned to this period have been detected; however, case detection is ongoing, and these results need to be interpreted cautiously considering model validation and other important contextual information on performance of the eradication programme, such as the time between infection, detection and implementation of movement controls on case farms.

Phenomenological Modelling of COVID-19 Epidemics in Sri Lanka, Italy, the United States, and Hebei Province of China

The COVID-19 pandemic has resulted in increasing number of infections and deaths every day. Lack of specialized treatments for the disease demands preventive measures based on statistical/mathematical models. The analysis of epidemiological curve fitting, on number of daily infections across affected countries, provides useful insights on the characteristics of the epidemic. A variety of phenomenological models are available to capture the dynamics of disease spread and growth. The number of daily new infections and cumulative number of infections in COVID-19 over four selected countries, namely, Sri Lanka, Italy, the United States, and Hebei province of China, from the first day of appearance of cases to 2^nd July 2020 were used in the study. Gompertz, logistic, Weibull, and exponential growth curves were fitted on the cumulative number of infections across countries. AIC, BIC, RMSE, and R² were used to determine the best fitting curve for each country. Results revealed that the most appropriate growth curves for Sri Lanka, Italy, the United States, and China (Hebei) are the logistic, Gompertz, Weibull, and Gompertz curves, respectively. Country-wise, overall growth rate, final epidemic size, and short-term forecasts were evaluated using the selected model. Daily log incidences in each country were regressed before and after the identified peak time of the respective outbreak of epidemic. Hence, doubling time/halving time together with daily growth rates and predictions was estimated. Findings and relevant interpretations demonstrate that the outbreak seems to be extinct in Hebei, China, whereas further transmissions are possible in the United States. In Italy and Sri Lanka, current outbreaks transmit in a decreasing rate.

Multi-model forecasts of the ongoing Ebola epidemic in the Democratic Republic of Congo, March-October 2019

The 2018–2020 Ebola outbreak in the Democratic Republic of the Congo is the first to occur in an armed conflict zone. The resulting impact on population movement, treatment centres and surveillance has created an unprecedented challenge for real-time epidemic forecasting. Most standard mathematical models cannot capture the observed incidence trajectory when it deviates from a traditional epidemic logistic curve. We fit seven dynamic models of increasing complexity to the incidence data published in the World Health Organization Situation Reports, after adjusting for reporting delays. These models include a simple logistic model, a Richards model, an endemic Richards model, a double logistic growth model, a multi-model approach and two sub-epidemic models. We analyse model fit to the data and compare real-time forecasts throughout the ongoing epidemic across 29 weeks from 11 March to 23 September 2019. We observe that the modest extensions presented allow for capturing a wide range of epidemic behaviour. The multi-model approach yields the most reliable forecasts on average for this application, and the presented extensions improve model flexibility and forecasting accuracy, even in the context of limited epidemiological data.

Predicting the Epidemiological Outbreak of the Coronavirus Disease 2019 (COVID-19) in Saudi Arabia

The coronavirus diseases 2019 (COVID-19) outbreak continues to spread rapidly across the world and has been declared as pandemic by World Health Organization (WHO). Saudi Arabia was among the countries that was affected by the deadly and contagious virus. Using a real-time data from 2 March 2020 to 15 May 2020 collected from Saudi Ministry of Health, we aimed to give a local prediction of the epidemic in Saudi Arabia. We used two models: the Logistic Growth and the Susceptible-Infected-Recovered for real-time forecasting the confirmed cases of COVID-19 across Saudi Arabia. Our models predicted that the epidemics of COVID-19 will have total cases of 69,000 to 79,000 cases. The simulations also predicted that the outbreak will entering the final-phase by end of June 2020.

Real-time monitoring the transmission potential of COVID-19 in Singapore, March 2020

BACKGROUND

As of March 31, 2020, the ongoing COVID-19 epidemic that started in China in December 2019 is now generating local transmission around the world. The geographic heterogeneity and associated intervention strategies highlight the need to monitor in real time the transmission potential of COVID-19. Singapore provides a unique case example for monitoring transmission, as there have been multiple disease clusters, yet transmission remains relatively continued.

METHODS

Here we estimate the effective reproduction number, Rt, of COVID-19 in Singapore from the publicly available daily case series of imported and autochthonous cases by date of symptoms onset, after adjusting the local cases for reporting delays as of March 17, 2020. We also derive the reproduction number from the distribution of cluster sizes using a branching process analysis that accounts for truncation of case counts.

RESULTS

The local incidence curve displays sub-exponential growth dynamics, with the reproduction number following a declining trend and reaching an estimate at 0.7 (95% CI 0.3, 1.0) during the first transmission wave by February 14, 2020, while the overall R based on the cluster size distribution as of March 17, 2020, was estimated at 0.6 (95% CI 0.4, 1.02). The overall mean reporting delay was estimated at 6.4 days (95% CI 5.8, 6.9), but it was shorter among imported cases compared to local cases (mean 4.3 vs. 7.6 days, Wilcoxon test, p < 0.001).

CONCLUSION

The trajectory of the reproduction number in Singapore underscores the significant effects of successful containment efforts in Singapore, but it also suggests the need to sustain social distancing and active case finding efforts to stomp out all active chains of transmission.

Early transmission dynamics of COVID-19 in a southern hemisphere setting: Lima-Peru: February 29th-March 30th, 2020

The COVID-19 pandemic that emerged in Wuhan China has generated substantial morbidity and mortality impact around the world during the last four months. The daily trend in reported cases has been rapidly rising in Latin America since March 2020 with the great majority of the cases reported in Brazil followed by Peru as of April 15th, 2020. Although Peru implemented a range of social distancing measures soon after the confirmation of its first case on March 6th, 2020, the daily number of new COVID-19 cases continues to accumulate in this country. We assessed the early COVID-19 transmission dynamics and the effect of social distancing interventions in Lima, Peru. We estimated the reproduction number, R, during the early transmission phase in Lima from the daily series of imported and autochthonous cases by the date of symptoms onset as of March 30th, 2020. We also assessed the effect of social distancing interventions in Lima by generating short-term forecasts grounded on the early transmission dynamics before interventions were put in place. Prior to the implementation of the social distancing measures in Lima, the local incidence curve by the date of symptoms onset displays near exponential growth dynamics with the mean scaling of growth parameter, p, estimated at 0.96 (95% CI: 0.87, 1.0) and the reproduction number at 2.3 (95% CI: 2.0, 2.5). Our analysis indicates that school closures and other social distancing interventions have helped slow down the spread of the novel coronavirus, with the nearly exponential growth trend shifting to an approximately linear growth trend soon after the broad scale social distancing interventions were put in place by the government. While the interventions appear to have slowed the transmission rate in Lima, the number of new COVID-19 cases continue to accumulate, highlighting the need to strengthen social distancing and active case finding efforts to mitigate disease transmission in the region.

Early transmission dynamics of COVID-19 in a southern hemisphere setting: Lima-Peru: February 29th-March 30th, 2020

The COVID-19 pandemic that emerged in Wuhan China has generated substantial morbidity and mortality impact around the world during the last four months. The daily trend in reported cases has been rapidly rising in Latin America since March 2020 with the great majority of the cases reported in Brazil followed by Peru as of April 15th, 2020. Although Peru implemented a range of social distancing measures soon after the confirmation of its first case on March 6th, 2020, the daily number of new COVID-19 cases continues to accumulate in this country. We assessed the early COVID-19 transmission dynamics and the effect of social distancing interventions in Lima, Peru. We estimated the reproduction number, R, during the early transmission phase in Lima from the daily series of imported and autochthonous cases by the date of symptoms onset as of March 30th, 2020. We also assessed the effect of social distancing interventions in Lima by generating short-term forecasts grounded on the early transmission dynamics before interventions were put in place. Prior to the implementation of the social distancing measures in Lima, the local incidence curve by the date of symptoms onset displays near exponential growth dynamics with the mean scaling of growth parameter, p, estimated at 0.9 (95%CI: 0.9,1.0) and the reproduction number at 2.3 (95% CI: 2.0, 2.5). Our analysis indicates that school closures and other social distancing interventions have helped slow down the spread of the novel coronavirus, with the nearly exponential growth trend shifting to an approximately linear growth trend soon after the broad scale social distancing interventions were put in place by the government. While the interventions appear to have slowed the transmission rate in Lima, the number of new COVID-19 cases continue to accumulate, highlighting the need to strengthen social distancing and active case finding efforts to mitigate disease transmission in the region.

Real-time monitoring the transmission potential of COVID-19 in Singapore, March 2020

Background

As of March 31, 2020 the ongoing COVID-19 epidemic that started in China in December 2019 is now generating local transmission around the world. The geographic heterogeneity and associated intervention strategies highlight the need to monitor in real time the transmission potential of COVID-19. Singapore provides a unique case example for monitoring transmission, as there have been multiple disease clusters, yet transmission remains relatively continued.

Methods

Here we estimate the effective reproduction number, Rt, of COVID-19 in Singapore from the publicly available daily case series of imported and autochthonous cases by date of symptoms onset, after adjusting the local cases for reporting delays as of March 17, 2020. We also derive the reproduction number from the distribution of cluster sizes using a branching process analysis that accounts for truncation of case counts.

Results

The local incidence curve displays sub-exponential growth dynamics, with the reproduction number following a declining trend and reaching an estimate at 0.7 (95% CI: 0.3, 1.0) during the first transmission wave by February 14, 2020 while the overall R based on the cluster size distribution as of March 17, 2020 was estimated at 0.6 (95% CI: 0.4, 1.02). The overall mean reporting delay was estimated at 6.4 days (95% CI: 5.8, 6.9), but it was shorter among imported cases compared to local cases (mean 4.3 vs. 7.6 days, Wilcoxon test, p<0.001).

Conclusion

The trajectory of the reproduction number in Singapore underscores the significant effects of successful containment efforts in Singapore, but it also suggests the need to sustain social distancing and active case finding efforts to stomp out all active chains of transmission.

Real-time forecasts of the COVID-19 epidemic in China from February 5th to February 24th, 2020

The initial cluster of severe pneumonia cases that triggered the COVID-19 epidemic was identified in Wuhan, China in December 2019. While early cases of the disease were linked to a wet market, human-to-human transmission has driven the rapid spread of the virus throughout China. The Chinese government has implemented containment strategies of city-wide lockdowns, screening at airports and train stations, and isolation of suspected patients; however, the cumulative case count keeps growing every day. The ongoing outbreak presents a challenge for modelers, as limited data are available on the early growth trajectory, and the epidemiological characteristics of the novel coronavirus are yet to be fully elucidated. We use phenomenological models that have been validated during previous outbreaks to generate and assess short-term forecasts of the cumulative number of confirmed reported cases in Hubei province, the epicenter of the epidemic, and for the overall trajectory in China, excluding the province of Hubei. We collect daily reported cumulative confirmed cases for the 2019-nCoV outbreak for each Chinese province from the National Health Commission of China. Here, we provide 5, 10, and 15 day forecasts for five consecutive days, February 5th through February 9th, with quantified uncertainty based on a generalized logistic growth model, the Richards growth model, and a sub-epidemic wave model. Our most recent forecasts reported here, based on data up until February 9, 2020, largely agree across the three models presented and suggest an average range of 7409-7496 additional confirmed cases in Hubei and 1128-1929 additional cases in other provinces within the next five days. Models also predict an average total cumulative case count between 37,415 and 38,028 in Hubei and 11,588-13,499 in other provinces by February 24, 2020. Mean estimates and uncertainty bounds for both Hubei and other provinces have remained relatively stable in the last three reporting dates (February 7th - 9th). We also observe that each of the models predicts that the epidemic has reached saturation in both Hubei and other provinces. Our findings suggest that the containment strategies implemented in China are successfully reducing transmission and that the epidemic growth has slowed in recent days.

A novel sub-epidemic modeling framework for short-term forecasting epidemic waves

BACKGROUND

Simple phenomenological growth models can be useful for estimating transmission parameters and forecasting epidemic trajectories. However, most existing phenomenological growth models only support single-peak outbreak dynamics whereas real epidemics often display more complex transmission trajectories.

METHODS

We develop and apply a novel sub-epidemic modeling framework that supports a diversity of epidemic trajectories including stable incidence patterns with sustained or damped oscillations to better understand and forecast epidemic outbreaks. We describe how to forecast an epidemic based on the premise that the observed coarse-scale incidence can be decomposed into overlapping sub-epidemics at finer scales. We evaluate our modeling framework using three outbreak datasets: Severe Acute Respiratory Syndrome (SARS) in Singapore, plague in Madagascar, and the ongoing Ebola outbreak in the Democratic Republic of Congo (DRC) and four performance metrics.

RESULTS

The sub-epidemic wave model outperforms simpler growth models in short-term forecasts based on performance metrics that account for the uncertainty of the predictions namely the mean interval score (MIS) and the coverage of the 95% prediction interval. For example, we demonstrate how the sub-epidemic wave model successfully captures the 2-peak pattern of the SARS outbreak in Singapore. Moreover, in short-term sequential forecasts, the sub-epidemic model was able to forecast the second surge in case incidence for this outbreak, which was not possible using the simple growth models. Furthermore, our findings support the view that the national incidence curve of the Ebola epidemic in DRC follows a stable incidence pattern with periodic behavior that can be decomposed into overlapping sub-epidemics.

CONCLUSIONS

Our findings highlight how overlapping sub-epidemics can capture complex epidemic dynamics, including oscillatory behavior in the trajectory of the epidemic wave. This observation has significant implications for interpreting apparent noise in incidence data where the oscillations could be dismissed as a result of overdispersion, rather than an intrinsic part of the epidemic dynamics. Unless the oscillations are appropriately modeled, they could also give a false positive, or negative, impression of the impact from public health interventions. These preliminary results using sub-epidemic models can help guide future efforts to better understand the heterogenous spatial and social factors shaping sub-epidemic patterns for other infectious diseases.

A novel sub-epidemic modeling framework for short-term forecasting epidemic waves

BACKGROUND

Simple phenomenological growth models can be useful for estimating transmission parameters and forecasting epidemic trajectories. However, most existing phenomenological growth models only support single-peak outbreak dynamics whereas real epidemics often display more complex transmission trajectories.

METHODS

We develop and apply a novel sub-epidemic modeling framework that supports a diversity of epidemic trajectories including stable incidence patterns with sustained or damped oscillations to better understand and forecast epidemic outbreaks. We describe how to forecast an epidemic based on the premise that the observed coarse-scale incidence can be decomposed into overlapping sub-epidemics at finer scales. We evaluate our modeling framework using three outbreak datasets: Severe Acute Respiratory Syndrome (SARS) in Singapore, plague in Madagascar, and the ongoing Ebola outbreak in the Democratic Republic of Congo (DRC) and four performance metrics.

RESULTS

The sub-epidemic wave model outperforms simpler growth models in short-term forecasts based on performance metrics that account for the uncertainty of the predictions namely the mean interval score (MIS) and the coverage of the 95% prediction interval. For example, we demonstrate how the sub-epidemic wave model successfully captures the 2-peak pattern of the SARS outbreak in Singapore. Moreover, in short-term sequential forecasts, the sub-epidemic model was able to forecast the second surge in case incidence for this outbreak, which was not possible using the simple growth models. Furthermore, our findings support the view that the national incidence curve of the Ebola epidemic in DRC follows a stable incidence pattern with periodic behavior that can be decomposed into overlapping sub-epidemics.

CONCLUSIONS

Our findings highlight how overlapping sub-epidemics can capture complex epidemic dynamics, including oscillatory behavior in the trajectory of the epidemic wave. This observation has significant implications for interpreting apparent noise in incidence data where the oscillations could be dismissed as a result of overdispersion, rather than an intrinsic part of the epidemic dynamics. Unless the oscillations are appropriately modeled, they could also give a false positive, or negative, impression of the impact from public health interventions. These preliminary results using sub-epidemic models can help guide future efforts to better understand the heterogenous spatial and social factors shaping sub-epidemic patterns for other infectious diseases.

The Final Size of a Serious Epidemic

In an epidemic of a serious disease, there is likely to be behavioral response that decreases the epidemic size considerably, and taking this into account may lead to estimates of the final epidemic size that are much smaller and more realistic than estimates that do not take this into account.

Assessing the relationship between epidemic growth scaling and epidemic size: The 2014-16 Ebola epidemic in West Africa

We assess the relationship between epidemic size and the scaling of epidemic growth of Ebola epidemics at the level of administrative areas during the 2014-16 Ebola epidemic in West Africa. For this purpose, we quantify growth scaling parameters from the ascending phase of Ebola outbreaks comprising at least 7 weeks of epidemic growth. We then study how these parameters are associated with observed epidemic sizes. For validation purposes, we also analyse two historic Ebola outbreaks. We find a high monotonic association between the scaling of epidemic growth parameter and the observed epidemic size. For example, scaling of growth parameters around 0.3-0.4, 0.4-0.6 and 0.6 are associated with epidemic sizes on the order of 350-460, 460-840 and 840-2500 cases, respectively. These results are not explained by differences in epidemic onset across affected areas. We also find the relationship between the scaling of epidemic growth parameter and the observed epidemic size to be consistent for two past Ebola outbreaks in Congo (1976) and Uganda (2000). Signature features of epidemic growth could become useful to assess the risk of observing a major epidemic outbreak, generate improved diseases forecasts and enhance the predictive power of epidemic models. Our results indicate that the epidemic growth scaling parameter is a useful indicator of epidemic size, which may have significant implications to guide control of Ebola outbreaks and possibly other infectious diseases.