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.

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.

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.

A general modeling framework for exploring the impact of individual concern and personal protection on vector-borne disease dynamics.

BACKGROUND: As climate variability and extreme weather events associated with climate change become more prevalent, public health authorities can expect to face an expanding spectrum of vector-borne diseases with increasing incidence and geographical spread. Common interventions include the use of larvicides and adulticides, as well as targeted communications to increase public awareness regarding the need for personal protective measures, such as mosquito repellant, protective clothing, and mosquito nets. Here, we propose a simplified compartmental model of mosquito-borne disease dynamics that incorporates the use of personal protection against mosquito bites influenced by two key individual-level behavioral drivers-concern for being bitten by mosquitos as a nuisance and concern for mosquito-borne disease transmission. METHODS: We propose a modified compartmental model that describes the dynamics of vector-borne disease spread in a naïve population while considering the public demand for community-level control and, importantly, the effects of personal-level protection on population-level outbreak dynamics. We consider scenarios at low, medium, and high levels of community-level vector control, and at each level, we consider combinations of low, medium, and high levels of motivation to use personal protection, namely concern for disease transmission and concern for being bitten in general. RESULTS: When there is very little community-level vector control, nearly the entire population is quickly infected, regardless of personal protection use. When vector control is at an intermediate level, both concerns that motivate the use of personal protection play an important role in reducing disease burden. When authorities have the capacity for high-level community vector control through pesticide use, the motivation to use personal protection to reduce disease transmission has little additional effect on the outbreak. CONCLUSIONS: While results show that personal-level protection alone is not enough to significantly impact an outbreak, personal protective measures can significantly reduce the severity of an outbreak in conjunction with community-level control. Furthermore, the model provides insight for targeting public health messaging to increase the use of personal protection based on concerns related to being bitten by mosquitos or vector-borne disease transmission.

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.

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.

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.

Doubling Time of the COVID-19 Epidemic by Chinese Province.

COVID-19 epidemic doubling time by Chinese province was increasing from January 20 through February 9, 2020. The harmonic mean of the arithmetic mean doubling time estimates ranged from 1.4 (Hunan, 95% CI, 1.2-2.0) to 3.1 (Xinjiang, 95% CI, 2.1-4.8), with an estimate of 2.5 days (95% CI, 2.4-2.6) for Hubei.

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 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.

Forecasting the impact of coronavirus disease during delivery hospitalization: an aid for resource utilization.

Background: The ongoing coronavirus disease 2019 pandemic has severely affected the United States. During infectious disease outbreaks, forecasting models are often developed to inform resource utilization. Pregnancy and delivery pose unique challenges, given the altered maternal immune system and the fact that most American women choose to deliver in the hospital setting. Objective: This study aimed to forecast the first pandemic wave of coronavirus disease 2019 in the general population and the incidence of severe, critical, and fatal coronavirus disease 2019 cases during delivery hospitalization in the United States. Study Design: We used a phenomenological model to forecast the incidence of the first wave of coronavirus disease 2019 in the United States. Incidence data from March 1, 2020, to April 14, 2020, were used to calibrate the generalized logistic growth model. Subsequently, Monte Carlo simulation was performed for each week from March 1, 2020, to estimate the incidence of coronavirus disease 2019 for delivery hospitalizations during the first pandemic wave using the available data estimate. Results: From March 1, 2020, our model forecasted a total of 860,475 cases of coronavirus disease 2019 in the general population across the United States for the first pandemic wave. The cumulative incidence of coronavirus disease 2019 during delivery hospitalization is anticipated to be 16,601 (95% confidence interval, 9711-23,491) cases, 3308 (95% confidence interval, 1755-4861) cases of which are expected to be severe, 681 (95% confidence interval, 1324-1038) critical, and 52 (95% confidence interval, 23-81) fatal. Assuming similar baseline maternal mortality rate as the year 2018, we projected an increase in maternal mortality rate in the United States to at least 18.7 (95% confidence interval, 18.0-19.5) deaths per 100,000 live births as a direct result of coronavirus disease 2019. Conclusion: Coronavirus disease 2019 in pregnant women is expected to severely affect obstetrical care. From March 1, 2020, we forecast 3308 severe and 681 critical cases with about 52 coronavirus disease 2019-related maternal mortalities during delivery hospitalization for the first pandemic wave in the United States. These results are significant for informing counseling and resource allocation.

Doubling Time of the COVID-19 Epidemic by Province, China.

In China, the doubling time of the coronavirus disease epidemic by province increased during January 20-February 9, 2020. Doubling time estimates ranged from 1.4 (95% CI 1.2-2.0) days for Hunan Province to 3.1 (95% CI 2.1-4.8) days for Xinjiang Province. The estimate for Hubei Province was 2.5 (95% CI 2.4-2.6) days.

Short-term Forecasts of the COVID-19 Epidemic in Guangdong and Zhejiang, China: February 13-23, 2020.

The ongoing COVID-19 epidemic continues to spread within and outside of China, despite several social distancing measures implemented by the Chinese government. Limited epidemiological data are available, and recent changes in case definition and reporting further complicate our understanding of the impact of the epidemic, particularly in the epidemic's epicenter. Here we use previously validated phenomenological models to generate short-term forecasts of cumulative reported cases in Guangdong and Zhejiang, China. Using daily reported cumulative case data up until 13 February 2020 from the National Health Commission of China, we report 5- and 10-day ahead forecasts of cumulative case reports. Specifically, we generate forecasts using a generalized logistic growth model, the Richards growth model, and a sub-epidemic wave model, which have each been previously used to forecast outbreaks due to different infectious diseases. Forecasts from each of the models suggest the outbreaks may be nearing extinction in both Guangdong and Zhejiang; however, the sub-epidemic model predictions also include the potential for further sustained transmission, particularly in Zhejiang. Our 10-day forecasts across the three models predict an additional 65-81 cases (upper bounds: 169-507) in Guangdong and an additional 44-354 (upper bounds: 141-875) cases in Zhejiang by February 23, 2020. In the best-case scenario, current data suggest that transmission in both provinces is slowing down.

Spatial variability in the reproduction number of Ebola virus disease, Democratic Republic of the Congo, January-September 2019.

The ongoing Ebola virus disease epidemic (August 2018─October 2019) in the Democratic Republic of the Congo, has been exacerbated by deliberate attacks on healthcare workers despite vaccination efforts. Using a mathematical/statistical modelling framework, we present the quantified effective reproduction number (Rt) at national and regional levels as at 29 September. The weekly trend in Rt displays fluctuations while our recent national-level Rt falls slightly above 1.0 with substantial uncertainty, which suggests improvements in epidemic control.

Comparative assessment of parameter estimation methods in the presence of overdispersion: a simulation study.

The Poisson distribution is commonly assumed as the error structure for count data; however, empirical data may exhibit greater variability than expected based on a given statistical model. Greater variability could point to model misspecification, such as missing crucial information about the epidemiology of the disease or changes in population behavior. When the mechanism producing the apparent overdispersion is unknown, it is typically assumed that the variance in the data exceeds the mean (by some scaling factor). Thus, a probability distribution that allows for overdispersion (negative binomial, for example) may better represent the data. Here, we utilize simulation studies to assess how misspecifying the error structure affects parameter estimation results, specifically bias and uncertainty, as a function of the level of random noise in the data. We compare results for two parameter estimation methods: nonlinear least squares and maximum likelihood estimation with Poisson error structure. We analyze two phenomenological models the generalized growth model and generalized logistic growth model to assess how results of parameter estimation are affected by the level of overdispersion underlying in the data. We use simulation to obtain confidence intervals and mean squared error of parameter estimates. We also analyze the impact of the amount of data, or ascending phase length, on the results of the generalized growth model for increasing levels of overdispersion. The results show a clear pattern of increasing uncertainty, or confidence interval width, as the overdispersion in the data increases. While maximum likelihood estimation consistently yields narrower confidence intervals and smaller mean squared error, differences between the two methods were minimal and not practically significant. At moderate levels of overdispersion, both estimation methods yielded similar performance. Importantly, it is shown that issues of parameter uncertainty and bias in the presence of overdispersion can be mitigated with the inclusion of more data.

Assessing parameter identifiability in compartmental dynamic models using a computational approach: application to infectious disease transmission models.

BACKGROUND: Mathematical modeling is now frequently used in outbreak investigations to understand underlying mechanisms of infectious disease dynamics, assess patterns in epidemiological data, and forecast the trajectory of epidemics. However, the successful application of mathematical models to guide public health interventions lies in the ability to reliably estimate model parameters and their corresponding uncertainty. Here, we present and illustrate a simple computational method for assessing parameter identifiability in compartmental epidemic models. METHODS: We describe a parametric bootstrap approach to generate simulated data from dynamical systems to quantify parameter uncertainty and identifiability. We calculate confidence intervals and mean squared error of estimated parameter distributions to assess parameter identifiability. To demonstrate this approach, we begin with a low-complexity SEIR model and work through examples of increasingly more complex compartmental models that correspond with applications to pandemic influenza, Ebola, and Zika. RESULTS: Overall, parameter identifiability issues are more likely to arise with more complex models (based on number of equations/states and parameters). As the number of parameters being jointly estimated increases, the uncertainty surrounding estimated parameters tends to increase, on average, as well. We found that, in most cases, R0 is often robust to parameter identifiability issues affecting individual parameters in the model. Despite large confidence intervals and higher mean squared error of other individual model parameters, R0 can still be estimated with precision and accuracy. CONCLUSIONS: Because public health policies can be influenced by results of mathematical modeling studies, it is important to conduct parameter identifiability analyses prior to fitting the models to available data and to report parameter estimates with quantified uncertainty. The method described is helpful in these regards and enhances the essential toolkit for conducting model-based inferences using compartmental dynamic models.

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.