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

Dynamic hierarchical state space forecasting

In this paper, we aim to both borrow information from existing units and incorporate the target unit's history data in time series forecasting. We consider a situation when we have time series data from multiple units that share similar patterns when aligned in terms of an internal time. The internal time is defined as an index according to evolving features of interest. When mapped back to the calendar time, these time series can span different time intervals that can include the future calendar time of the targeted unit, over which we can borrow the information from other units in forecasting the targeted unit. We first build a hierarchical state space model for the multiple time series data in terms of the internal time, where the shared components capture the similarities among different units while allowing for unit-specific deviations. A conditional state space model is then constructed to incorporate the information of existing units as the prior information in forecasting the targeted unit. By running the Kalman filtering based on the conditional state space model on the targeted unit, we incorporate both the information from the other units and the history of the targeted unit. The forecasts are then transformed from internal time back into calendar time for ease of interpretation. A simulation study is conducted to evaluate the finite sample performance. Forecasting state-level new COVID-19 cases in United States is used for illustration.

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.

Using neural ordinary differential equations to predict complex ecological dynamics from population density data

Simple models have been used to describe ecological processes for over a century. However, the complexity of ecological systems makes simple models subject to modelling bias due to simplifying assumptions or unaccounted factors, limiting their predictive power. Neural ordinary differential equations (NODEs) have surged as a machine-learning algorithm that preserves the dynamic nature of the data (Chen et al. 2018 Adv. Neural Inf. Process. Syst.). Although preserving the dynamics in the data is an advantage, the question of how NODEs perform as a forecasting tool of ecological communities is unanswered. Here, we explore this question using simulated time series of competing species in a time-varying environment. We find that NODEs provide more precise forecasts than autoregressive integrated moving average (ARIMA) models. We also find that untuned NODEs have a similar forecasting accuracy to untuned long-short term memory neural networks and both are outperformed in accuracy and precision by empirical dynamical modelling . However, we also find NODEs generally outperform all other methods when evaluating with the interval score, which evaluates precision and accuracy in terms of prediction intervals rather than pointwise accuracy. We also discuss ways to improve the forecasting performance of NODEs. The power of a forecasting tool such as NODEs is that it can provide insights into population dynamics and should thus broaden the approaches to studying time series of ecological communities.

COVID-19 pandemic waves: Identification and interpretation of global data

The mention of the COVID-19 waves is as prevalent as the pandemic itself. Identifying the beginning and end of the wave is critical to evaluating the impact of various COVID-19 variants and the different pharmaceutical and non-pharmaceutical (including economic, health and social, etc.) interventions. We demonstrate a scientifically robust method to identify COVID-19 waves and the breaking points at which they begin and end from January 2020 to June 2021. Employing the Break Least Square method, we determine the significance of COVID-19 waves for global-, regional-, and country-level data. The results show that the method works efficiently in detecting different breaking points. Identifying these breaking points is critical for evaluating the impact of the economic, health, social and other welfare interventions implemented during the pandemic crisis. Employing our method with high frequency data effectively determines the start and end points of the COVID-19 wave(s). Identifying waves at the country level is more relevant than at the global or regional levels. Our research results evidenced that the COVID-19 wave takes about 48 days on average to subside once it begins, irrespective of the circumstances.

Retrospective evaluation of short-term forecast performance of ensemble sub-epidemic frameworks and other time-series models: The 2022-2023 mpox outbreak across multiple geographical scales, July 14th, 2022, through February 26th, 2023

In May 2022, public health officials noted an unprecedented surge in mpox cases in non-endemic countries worldwide. As the epidemic accelerated, multi-model forecasts of the epidemic's trajectory were critical in guiding the implementation of public health interventions and determining policy. As the case levels have significantly decreased as of early September 2022, evaluating model performance is essential to advance the growing field of epidemic forecasting. Using laboratory-confirmed mpox case data from the Centers for Disease Control and Prevention (CDC) and Our World in Data (OWID) teams through the week of January 26th, 2023, we generated retrospective sequential weekly forecasts (e.g., 1-4-weeks) for Brazil, Canada, France, Germany, Spain, the United Kingdom, the USA, and at the global scale using models that require minimal input data including the auto-regressive integrated moving average (ARIMA), general additive model (GAM), simple linear regression (SLR), Facebook's Prophet model, as well as the sub-epidemic wave (spatial-wave) and n -sub-epidemic modeling frameworks. We assess forecast performance using average mean squared error (MSE), mean absolute error (MAE), weighted interval score (WIS), 95% prediction interval coverage (95% PI coverage), and skill scores. Average Winkler scores were used to calculate skill scores for 95% PI coverage. Overall, the n -sub-epidemic modeling framework outcompeted other models across most locations and forecasting horizons, with the unweighted ensemble model performing best across all forecasting horizons for most locations regarding average MSE, MAE, WIS, and 95% PI coverage. However, many locations had multiple models performing equally well for the average 95% PI coverage. The n -sub-epidemic and spatial-wave frameworks improved considerably in average MSE, MAE, and WIS, and Winkler scores (95% PI coverage) relative to the ARIMA model. Findings lend further support to sub-epidemic frameworks for short-term forecasting epidemics of emerging and re-emerging infectious diseases.

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.