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

   https://doi.org/10.1186/s12874-024-02241-2  

   Chowell, Gerardo; Tariq, Amna; Dahal, Sushma; Bleichrodt, Amanda; Luo, Ruiyan; Hyman, James M (December 2024, BMC Medical Research Methodology)

   Abstract BackgroundDynamical 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. ResultsThis 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. ConclusionsWe 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. 

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2. Parameter estimation and forecasting with quantified uncertainty for ordinary differential equation models using QuantDiffForecast : A MATLAB toolbox and tutorial

   https://doi.org/10.1002/sim.10036  

   Chowell, Gerardo; Bleichrodt, Amanda; Luo, Ruiyan (April 2024, Statistics in Medicine)

   Mathematical models based on systems of ordinary differential equations (ODEs) are frequently applied in various scientific fields to assess hypotheses, estimate key model parameters, and generate predictions about the system's state. To support their application, we present a comprehensive, easy‐to‐use, and flexible MATLAB toolbox,QuantDiffForecast, and associated tutorial to estimate parameters and generate short‐term forecasts with quantified uncertainty from dynamical models based on systems of ODEs. We provide software (https://github.com/gchowell/paramEstimation_forecasting_ODEmodels/) and detailed guidance on estimating parameters and forecasting time‐series trajectories that are characterized using ODEs with quantified uncertainty through a parametric bootstrapping approach. It includes functions that allow the user to infer model parameters and assess forecasting performance for different ODE models specified by the user, using different estimation methods and error structures in the data. The tutorial is intended for a diverse audience, including students training in dynamic systems, and will be broadly applicable to estimate parameters and generate forecasts from models based on ODEs. The functions included in the toolbox are illustrated using epidemic models with varying levels of complexity applied to data from the 1918 influenza pandemic in San Francisco. A tutorial video that demonstrates the functionality of the toolbox is included. 

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3. Structural and Practical Identifiability of Phenomenological Growth Models for Epidemic Forecasting

   https://doi.org/10.3390/v17040496  

   Liyanage, Yuganthi R; Chowell, Gerardo; Pogudin, Gleb; Tuncer, Necibe (April 2025, Viruses)

   Phenomenological models are highly effective tools for forecasting disease dynamics using real-world data, particularly in scenarios where detailed knowledge of disease mechanisms is limited. However, their reliability depends on the model parameters’ structural and practical identifiability. In this study, we systematically analyze the identifiability of six commonly used growth models in epidemiology: the generalized growth model (GGM), the generalized logistic model (GLM), the Richards model, the generalized Richards model (GRM), the Gompertz model, and a modified SEIR model with inhomogeneous mixing. To address challenges posed by non-integer power exponents in these models, we reformulate them by introducing additional state variables. This enables rigorous structural identifiability analysis using the StructuralIdentifiability.jl package in JULIA. We validated the structural identifiability results by performing parameter estimation and forecasting using the GrowthPredict MATLAB Toolbox. This toolbox is designed to fit and forecast time series trajectories based on phenomenological growth models. We applied it to three epidemiological datasets: weekly incidence data for monkeypox, COVID-19, and Ebola. Additionally, we assessed practical identifiability through Monte Carlo simulations to evaluate parameter estimation robustness under varying levels of observational noise. Our results confirm that all six models are structurally identifiable under the proposed reformulation. Furthermore, practical identifiability analyses demonstrate that parameter estimates remain robust across different noise levels, though sensitivity varies by model and dataset. These findings provide critical insights into the strengths and limitations of phenomenological models to characterize epidemic trajectories, emphasizing their adaptability to real-world challenges and their role in informing public health interventions. 

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4. StatModPredict: A user-friendly R-Shiny interface for fitting and forecasting with statistical models

   https://doi.org/10.1371/journal.pone.0329791  

   Bleichrodt, Amanda; Phan, Amelia; Luo, Ruiyan; Kirpich, Alexander; Chowell, Gerardo (August 2025, PLOS One)

   Chen, Li-Pang (Ed.)

   BackgroundMany disciplines, such as public health, rely on statistical time series models for real-time and retrospective forecasting efforts; however, effectively implementing related methods often requires extensive programming knowledge. Therefore, such tools remain largely inaccessible to those with limited programming experience, including students training in modeling, as well as professionals and policymakers seeking to forecast an epidemic’s trajectory. To address the need for accessible and intuitive forecasting applications, we presentStatModPredict, an R-Shiny dashboard for conducting robust forecasting analysis utilizing auto-regressive integrated moving average (ARIMA), generalized linear models (GLM), generalized additive models (GAM), and Meta’s Prophet model. MethodsStatModPredictsupports robust real-time forecasting and retrospective model analysis, including fitting, forecasting, evaluation, visualization, and comparison of results from four popular models. After loading an incident time series data set into the interface, users can easily customize model parameters and forecasting options to obtain the desired output. Additionally,StatModPredictoffers multiple editable figures for, but not limited to, the time series data, the forecasts, and model fit and forecast metrics. Users can also upload external forecasts produced elsewhere and evaluate their performance alongside the dashboard’s built-in models, thereby enabling direct comparisons. We provide a detailed demonstration of the dashboard’s features using publicly available annual HIV case data in the US. A video tutorial is available athttps://www.youtube.com/watch?v=zgZOvqhvqw8. ConclusionsBy eliminating programming barriers,StatModPredictfacilitates exploration and use by students training in forecasting, as well as professionals and policymakers aiming to forecast epidemic trajectories. Additionally, the flexibility in the required input data structure and parameter specification process extends the application ofStatModPredictto any discipline that employs time series data. By offering this open-source interface, we aim to broaden access to forecasting tools, promote hands-on learning, and foster contributions from users across disciplines. 

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5. Ensemble bootstrap methodology for forecasting dynamic growth processes using differential equations: application to epidemic outbreaks

   https://doi.org/10.1186/s12874-021-01226-9  

   Chowell, Gerardo; Luo, Ruiyan (December 2021, BMC Medical Research Methodology)

   null (Ed.)

   Abstract Background Ensemble modeling aims to boost the forecasting performance by systematically integrating the predictive accuracy across individual models. Here we introduce a simple-yet-powerful ensemble methodology for forecasting the trajectory of dynamic growth processes that are defined by a system of non-linear differential equations with applications to infectious disease spread. Methods We propose and assess the performance of two ensemble modeling schemes with different parametric bootstrapping procedures for trajectory forecasting and uncertainty quantification. Specifically, we conduct sequential probabilistic forecasts to evaluate their forecasting performance using simple dynamical growth models with good track records including the Richards model, the generalized-logistic growth model, and the Gompertz model. We first test and verify the functionality of the method using simulated data from phenomenological models and a mechanistic transmission model. Next, the performance of the method is demonstrated using a diversity of epidemic datasets including scenario outbreak data of the Ebola Forecasting Challenge and real-world epidemic data outbreaks of including influenza, plague, Zika, and COVID-19. Results We found that the ensemble method that randomly selects a model from the set of individual models for each time point of the trajectory of the epidemic frequently outcompeted the individual models as well as an alternative ensemble method based on the weighted combination of the individual models and yields broader and more realistic uncertainty bounds for the trajectory envelope, achieving not only better coverage rate of the 95% prediction interval but also improved mean interval scores across a diversity of epidemic datasets. Conclusion Our new methodology for ensemble forecasting outcompete component models and an alternative ensemble model that differ in how the variance is evaluated for the generation of the prediction intervals of the forecasts. 

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