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A multi-method approach to modeling COVID-19 disease dynamics in the United States

In this paper, we proposed a multi-method modeling approach to community-level spreading of COVID-19 disease. Our methodology was composed of interconnected age-stratified system dynamics models in an agent-based modeling framework that allowed for a granular examination of the scale and severity of disease spread, including metrics such as infection cases, deaths, hospitalizations, and ICU usage. Model parameters were calibrated using an optimization technique with an objective function to minimize error associated with the cumulative cases of COVID-19 during a training period between March 15 and October 31, 2020. We outlined several case studies to demonstrate the model’s state- and local-level projection capabilities. We further demonstrated how model outcomes could be used to evaluate perceived levels of COVID-19 risk across different localities using a multi-criteria decision analysis framework. The model’s two, three, and four week out-of-sample projection errors varied on a state-by-state basis, and generally increased as the out-of-sample projection period was extended. Additionally, the prediction error in the state-level projections was generally due to an underestimation of cases and an overestimation of deaths. The proposed modeling approach can be used as a virtual laboratory to investigate a wide range of what-if scenarios and easily adapted to future high-consequence public health threats.

During the current COVID-19 pandemic, global efforts have taken place to contain the spread of the virus and develop effective non-therapeutic (e.g., social distancing, partial and full lockdowns) and therapeutic treatments (e.g., vaccination). As the COVID-19 pandemic has spread across the globe since early 2020, researchers have identified gaps in data and our understanding of ways in which the disease spreads within and between communities including its potential impacts on general and at-risk populations 1,2 .
Computational modeling has been long employed to further increase our understanding of complex infectious diseases as well as their development, spread dynamics, and potential treatments 3 . Using computational modeling, we have been able to identify common patterns in infectious diseases allowing us to leverage lessons learned through investigating past widespread disease events to predict who may get infected, where vaccination efforts should be prioritized, and how to limit the spread of infectious diseases in future events [4][5][6][7] .
Two methods, System Dynamics (SD) and Agent-Based Modeling (ABM), have been frequently used in recent years to investigate the complex nature of infectious diseases and their potential containment strategies. SD has a long history of being applied to the study of infectious disease epidemiology. This method operates at a high level of abstraction by compartmentalizing the population into different disease stages such as Susceptible (S), Infected (I), and Recovered (R), among others while assuming population homogeneity within each compartment 8,9 . Previous studies have identified limitations of SD in modeling infectious diseases such as inability to model multi-strain infections, deterministic nature, inability to model time-varying infectivity, and assumptions regarding population homogeneity, among others 10 . With the boom in computer processing capability in the twenty-first century, ABM has been recently used in modeling infectious disease dynamics 11,12 . ABM uses a bottom-up approach, where a complex dynamic system is described as interacting objects with their own behaviors such that systemic behavior can potentially emerge as a summary of the individual actions of agents 13,14 . ABM for infectious diseases focuses on incorporating individual information such as personal interactions, movements, and health information in an attempt to provide a more granular profile of disease spread as compared to the homogenous population of SD models. However, ABM is not without its limitations: (1) model parameters (e.g., reproduction number for infectious diseases) are often difficult to quantify; (2) model validation can be difficult to assess, particularly when modeling unobserved associations 15 ; (3) ABM can become Methods Model overview. We developed M 2 -CDRM as a highly customizable, evidence-based, and data-driven model by integrating an SD modeling approach within an ABM framework to study the COVID-19 transmission on multiple levels of aggregation in the United States (Fig. 1). The model is implemented in AnyLogic (Professional Edition, Version: 8.5.2, Link: https:// www. anylo gic. com), a modeling framework that integrates support for SD, ABM, and other dynamic computational methods. M 2 -CDRM included all 50 states as well as their individual counties with a simulation period between March 15 and December 31st, 2020.
Disease transmission models. With COVID-19, different subpopulations have been shown to be more or less susceptible, more or less likely to be infectious, and more or less likely to recover from the disease [21][22][23] . Therefore, treating the entire population with the same static assumptions about these rates can cause decision makers to miss key aspects of the disease's likely trajectory. M 2 -CDRM addresses this limitation by including five separate SD models to simulate COVID-19 disease dynamics in distinct age cohorts within each individual county: 0-17, 18-44, 45-64, 65-74, and 75 + years of age. While these cohorts were initially selected to stratify the population based on their ages, the model design is flexible and can accommodate any age stratification. Each SD model was defined using eight compartments, including Susceptible (S), Exposed (E), Asymptomatic Infection (AI), Mild Infection (MI), Severe Infection (SI), Critical Infection (CI), Recovered (R), and Death (D). In each model, severe infection and critical infection represented general admission to the hospital as well as ICU admission, respectively. During the early stage of the pandemic, the confirmed COVID-19 case counts in the U.S. did not capture the total burden of the pandemic. This was primarily because testing was restricted to   24 . Therefore, in order to correct for biased testing and imperfect diagnostic accuracy and provide a more realistic assessment of COVID-19 burden, we further adjusted simulated infection cases (I) by an under-reporting factor. For each state, the spread of COVID-19 in county j and for age cohort i was modeled based on the following set of differential equations ( Fig. 2): where S i,j represents susceptible population in age cohort i (i = 1,…,5) in county j, S j (0) represents initial susceptible population in county j across all age cohorts, E i,j represents exposed population in age cohort i in county j, I i,j represents symptomatic infectious population in age cohort i in county, AI i,j represents asymptomatic infectious population in age cohort i in county j, H i,j represents hospitalized population (severe infection) in age cohort i in county j, C i,j represents critically infected population (ICU admission) in age cohort i in county j, R i,j represents recovered (non-infectious) population in age cohort i in county j, D i,j represents deceased population in age cohort i in county j, IP represents incubation period (days), FR AI represents fraction of asymptomatic population, MIP H represents duration of mild infection prior to hospitalization (days), MIP represents duration of mild infection prior to recovery (days), AIP represents duration of asymptomatic infection (days), HR i represents hospitalization rate for age cohort i (i = 1,…,5), URF represents under-reporting factor of symptomatic infections, SIP ICU represents severe infection period prior to transfer to ICU (days), SIP represents severe infection period  ABM framework to connect SD models. Within each county, we defined population age cohorts (0-17, 18-44, 45-64, 65-74, 75 +) as individual agents. Each of these individual agents was then coupled with all other agents within the same county with explicit interactivity patterns. By focusing on micro-level interactions, this framework was able to explain emergent patterns such as transient dynamics on a system level and identify important mechanisms, taking into account heterogeneity of entities (e.g., individual age cohorts as agents) and spatial and temporal heterogeneity of processes (e.g., variability in disease dynamics across different counties). Additionally, the ABM structure allowed for the possibility of advanced data inputs such as age-specific reproduction numbers, interaction, and mobility patterns across age cohorts and counties, county-and age-specific adherence to social distancing policies, and what-if analysis such as customizable vaccine distribution networks. Outputs from our framework were timeseries of system-level variables further stratified by age cohorts, counties and states: (1) number of infected; (2) number of hospitalized; (3) number of ICU admissions; (4) number of deaths; and (5) hospital utilization considering available general and ICU beds in different counties.

Effective reproduction number.
Since a population will rarely be totally susceptible to an infection in the real world, the effective reproduction number, RE t , and not the basic reproduction number, R 0 , should be used as a measure of disease transmissibility at time t 25 . RE t represents the expected number of new infections caused by an infectious individual in a population where some individuals may no longer be susceptible. Estimates of RE t are typically used to assess how changes in policy, population immunity, and population behaviors, among other factors, have affected transmission at specific points in time [26][27][28][29] . Using observed number of daily cases of COVID-19 in county j, we calculated timeseries of RE t,j based on the methodology discussed in Cori et al. and implemented in the R-package EpiEstim 30 . This package implements a Bayesian approach for quantifying transmissibility over time during an epidemic and reports a 95% confidence interval for RE t . More specifically, it allows estimating the instantaneous and case reproduction numbers during an epidemic for which a timeseries of incidence is available and the distribution of the serial interval (time between symptoms onset in a primary case and symptoms onset in secondary case) is more or less precisely known. To calculate RE t,j , we assumed the median, mean, and standard deviation of the serial interval were 4.0, 4.7, and 2.9 days, respectively 31 . RE t,j was calculated as: where t 0,j represents time associated with the first observed case of illness in county j, t EC,j represents time associated with the end of model calibration period in county j (i.e., the last date with observed case of illness), and RE * t,j represents output from the EpiEstim package. β 0,j and β 1,j are coefficients from fitting an exponential regression model to the estimated RE * t,j values in the last two weeks, assuming that RE * t,j continue the same trend observed in the past two weeks. The minimum value of 0.3 represents the estimated reproduction number in the City of Wuhan after the lockdown of the region 26 .

Calibration of model parameters.
Model calibration is the process of identifying the model parameter configurations that best explain the observed real-time values (e.g., observed cases of illness). While simple models with fewer parameters can be potentially calibrated by manually adjusting parameter values, calibration of complex models, such as M 2 -CDRM, requires extensive computational effort and resources. We used a simulation-based "optimization" method to calibrate selected model parameters by estimating their values and plausible ranges such that the model outcomes would closely match existing historic data of number of observed cases of illness.
The optimization engine in AnyLogic automatically finds the best values for different model parameters with respect to certain pre-defined constraints and requirements using the OptQuest Engine that incorporates metaheuristics to guide its search algorithm toward better solutions 32 . Inputs selected for model calibration including their ranges of plausible values are listed in Table 1. We performed the model calibration at both state

Multi-criteria framework for prioritizing counties based on the perceived risk of COVID-19.
We used an MCDA framework to generate risk maps for individual states that highlight counties where surveillance and disease control measures could be potentially targeted based on the perceived levels of COVID-19 risks. The methodological steps required in our MCDA approach encompassed: (i) selection of decision criteria; (ii) definition of criterion measures; (iii) definition of scores assigned to each decision criterion representing low (1), medium (3), and high (9) perceived levels of risk; (iv) attribution of weights to decision criteria and (v) aggregation of risk scores across all selected decision criteria to generate the spatial maps for perceived levels of risk in each state. Decision criteria, measures, and risk scores for ranking individual counties in each state are provided in Table 2 and briefly discussed in the following.
• New daily cases (NDC) this criterion, comparable to incidence in epidemiology represents the incident number of COVID-19 cases in a community. We considered a three-day average of the predicted new cases (across all age cohorts) and a cut-off value of less than five new cases per 100,000 residents to score this criterion. A risk score of low (1), medium (3), or high (9) was assigned to this criterion in each county if the cut-off value was met within 21 days since the training end date (October 31st, 2020), after 21 days since the training end date but before the end of the simulation period (December 31st, 2020), or was never met during the simulation period, respectively. • Decline in new daily deaths (NDD) we assumed that a county must experience a sustained decline in the threeday rolling average of predicted daily hospital deaths over the course of a 21-day period to be considered low risk. Alternatively, counties that have seen few COVID cases overall would satisfy this metric if the threeday rolling average of daily new hospital deaths has never exceeded one. We used three-day average of the projected number of deaths across all age cohorts in each county and scored the county as low (1), medium (3), or high (9) if the cut-off value was met within 21 days since the training end date, after 21 days since the training end date but before the end of the simulation period, or was never met during the simulation period, respectively. • New hospitalizations (NH) In addition to monitoring the decline in disease trajectory, it is important to monitor the absolute level of infection in each county. It is possible for a county that has seen a high level of infections to see a sustained decline in hospitalizations and deaths over a 21-day period while still having an underlying infection rate that is too high. Using the total number of projected new hospitalization cases across all age cohorts, each county needed to have fewer than two new hospitalizations per 100,000 residents to be considered low risk. We used three-day average of the projected number of new hospitalizations across all age cohorts in each county and scored the county as low (1), medium (3), or high (9) if the cut-off value was met within 21 days since the training end date, after 21 days since the training end date but before the end of the simulation period, or was never met during the simulation period, respectively. • ICU bed utilization (BU) It is critical that regional healthcare systems have sufficient capacity for ICU beds.
Taking into account the projected number of critically infected patients in each county across all ages and the ICU bed capacity in each county, we scored each county as low (1), medium (3), or high (9) if the cut-off value of 50% was met within 21 days since the training end date, after 21 days since the training end date but before the end of the simulation period, or was never met during the simulation period, respectively. Summary of the model inputs. Data used in M 2 -CDRM came from a variety of sources, grouped into three categories of disease impact, demographic data, and hospital resources. Summary data used in the model, including data sources is listed in Table 3.

Results
State-level predictions. Tables 4 and 5 summarize the model predictions for number of COVID-19 cases aggregated across all age cohorts in the 20 most populous states in the United States. We reported a range of values for two-week (November 14, 2020), three-week (November 21, 2020), and four-week (November 28, 2020) out-of-sample model predictions based on the 95% confidence intervals reported for RE t . We also reported the cumulative observed values for COVID-19 cases by selected dates and % error calculated by comparing the observed values with mean predictions. For each of these states, selected model parameters (listed in Table 1) were calibrated to replicate the observed cumulative number of cases between March 15 and October 31, 2020 across the whole state. We further used the state-wide calibrated model parameters for all individual counties in the selected state assuming no change in disease epidemiology in different localities (e.g., no change in critical infection rate for a particular age cohort across different counties in California). Summary results typically showed underestimated number of COVID-19 cases with variability in % error across different states. Furthermore, we observed relative decrease in model accuracy when period of out-of-sample predictions was increased from two to four weeks. For example, average % error for two-week out-of-sample prediction of cases was − 6.7% across all 20 states with a range of values between − 1.1% (California) and -16.9% (Michigan). We observed lower accuracy for the four-week out-of-sample case predictions with an average % error value of − 16.2% across all 20 states and a range of values between − 7.1 and − 32.4% for California and Michigan, respectively. Model results showed similar patterns for predicted number of COVID-19 deaths across these selected states (Table 5); however, the prediction accuracies were typically higher for cumulative number of deaths by selected dates. For example, average % error for two-week out-of-sample prediction of deaths across selected states was 3.2% (compared to − 6.7% error for prediction of cases) with a range of values between 0.1 and 15.2% for Missouri and Washington, respectively. Tables 4 and 5, our model generated results for each individual county within a state, allowing for analysis of the heterogenous disease growth patterns across localities. Although each county used an independent predicted timeseries for RE t based on the county-specific observed cases of illness, a simplifying assumption was made that calibrated disease parameters (listed in Table 1) were homogenous across all counties in a particular state when model was trained to replicate the state-level observed cumulative number of cases and deaths between March 15, 2020-October 31, 2020. We further investigated the impact of this assumption on the model prediction accuracy by conducting a county-level calibration experiment across three localities in Virginia, including Richmond City, Montgomery County, and Norfolk City. The experiment included two scenarios to evaluate the out-of-sample model prediction accuracy between November 1 and 28 based on: (1) calibrated model parameters using state-level observed data (223,568 and 3973 for observed cumulative cases of illness and deaths in Virginia, respectively); and (2)  Figure 3 shows the resulting timeseries for the out-of-sample model predictions between November 1 and 28, 2020 for selected localities in Virginia including cumulative number of observed COVID-19 cases during the same time period. Each predicted timeseries represents model results for the cumulative COVID-19 cases based on the mean RE t value as well as range of cases based on the 95% confidence interval associated with RE t (shaded areas). Results indicated that conducting county-level model calibration led to increase in model accuracy. For example, % errors for four-week out-of-sample predictions were − 11.3%, − 15.4%, and − 8.4% for Richmond City, Montgomery County, and Norfolk City, respectively, when model parameters were calibrated using statelevel cumulative number of observed cases. When model parameters were calibrated for each individual county, % errors reduced to − 6.9%, − 7.8%, and − 4.0% for the selected counties.

County-level predictions. For each of the state-level predictions listed in
State-level risk maps using MCDA. In addition to out-of-sample case and death predictions across different localities in individual states, we utilized various county-level model outputs, including three-day rolling average of new daily cases per 100,000 residents, three-day rolling average of daily new hospital deaths, three-day rolling average of new hospitalizations per 100,000 residents, and ICU bed utilization percentages, and time to meet their cut-off values (listed in Table 2   We also calculated the aggregated risk scores across selected decision criteria for all counties in Virginia. The risk map based on the aggregated scores is shown in Fig. 8. Aggregated risk scores showed spatial variability with an average value of 14.3 across all counties and minimum and maximum values of 4 and 30, respectively. The model typically predicted higher aggregated risk scores (15 or higher) in the southwestern localities while lower scores (15 or lower) in the northern and eastern localities of the state, primarily due to additional hospital resources (e.g., number of general and ICU beds) in those counties.

Discussion
The COVID-19 pandemic has resulted in a global health crisis with unprecedented growing economic, social, and health impacts not seen since the 1918 Spanish flu pandemic. Computational models have played an important role in the ongoing crisis by providing insights regarding the disease spread dynamics as well as the potential impacts of public policies at the local, national, and global levels. Different models with a wide range of underlying methodologies have been used by policy makers and public health officials to assess the evolution of the COVID-19 pandemic, design and analyze control measures, and study various what-if scenarios. For example, the Centers for Disease Control and Prevention (CDC) has been working with different partners to bring together weekly COVID-19 forecasts based on statistical and mathematical models aiming to predict national and state numbers of new and total COVID-19 deaths as well as cases of infection and hospitalization 34 . Table 6 provides a summary of selected COVID-19 computational models available from the CDC website including their key features, geographic scope, methodology, frequency of updates, and ability to conduct what-if scenario analysis. The majority of these models have adapted different forms of the SD-based models (e.g., SEIR) with geographical scopes typically limited to the national or state level predictions. All models faced challenges due to availability of data, rapidly evolving pandemic and unprecedented control measures put in place. Despite these challenges, we believe that mathematical models can provide useful and timely information to the policy makers.
Like other computational modeling methods, commonly used SD-based models can be especially useful when invoked for the right task, however they are not appropriate for all forecasting, prediction, and scenario simulations. These models operate at an elevated level of abstraction, assume population homogeneity, and typically lack the ability to update underlying model parameters once new, real-time data become available. In this study, we developed a multi-method modeling approach by using an ABM framework to combine thousands of age-stratified and location-specific SEIR models that could potentially capture essential virus transmission Table 5. Model performance for two-, three-, and four-week out-of-sample predictions of the cumulative COVID-19 deaths in the top 20 populous states.   forecasted outcomes with user-defined health metrics in a multi-criteria decision framework. While the current case study is focused on COVID-19, the modular framework of our solution easily allows future adaptation to any high-consequence public health threats. We have also addressed some of the key limitations of SD-based epidemiological models. First, current SDbased epidemiological models typically approximate the spread of COVID-19 at the state and national level. These models do not account for the effect of mitigation policies, population demographics, or cohort behaviors on disease spread dynamics at local levels. Our multi-method approach provided enhanced precision and fidelity at the local level. Second, existing SD-based models typically focus on the constant value of the basic reproduction number (R 0 ) as a measure of disease transmissibility. We used potential changes in R 0 over time, represented by R E , which reflected how the disease transmission within the population changed over time. We used this dynamic adjustment to assess how changes in mitigation policies, population immunity, and population behaviors, among other factors, could potentially affect COVID-19 transmission at specific time and location points. Lastly, most SD-based models fail to account for the effect of population demographics (e.g., age), particularly at the county and local levels. We believe that characterizing model parameters such as disease transmission, hospitalization, critical infection, and fatality rates based on the population demographics potentially mitigates the bias for under-represented segments of the population.
We are also aware that computational models are approximations of the real-life scenarios. There are currently no predictive models that generate a highly accurate picture of the COVID-19 disease spread or its clinical impacts, including ours, as too many factors can potentially affect the spread of the disease. For example, our model showed to underestimate cases and overestimate deaths. Modelling exercises tend to carry forward certain distortions that are inherent to the complex and dynamic characteristics of real-world reporting systems We also acknowledge that there were multiple sources of uncertainty in our model resulting in prediction inaccuracies and errors as reported in Tables 4 and 5. Key sources of uncertainty in our model potentially included model structure (e.g., set of differential equations identified for disease dynamics), model detail (e.g., simplifying assumptions related to reinfection as well as between-county population movements), model calibrations (e.g., state versus county-level parameter calibration), and scenario reasonableness (e.g., assumption of homogenous age-stratified reproduction numbers. There are areas for improvement in our modeling approach that can potentially reduce the above uncertainties and enhance the prediction accuracies. For example, alternative sets of scientific or technical assumptions might be available for developing the complex dynamics of COVID-19 disease spread. The implications of these alternative foundations may be evaluated by constructing alternative models and comparing results across different solutions. It may be possible to potentially parameterize alternative model structures into a higher order model, and to evaluate the impact of modeling assumptions using sensitivity analysis. Also, while we used the observed daily cases of COVID-19 to characterize location-specific timeseries for R E , future values were approximated using exponential regression models fitted to the latest two weeks of data. This approximation may potentially pose bias and limitations in forecasting the disease dynamics in populous areas where changes in behaviors (e.g., lack of social distancing, limited stay-at-home restrictions) can significantly impact the disease spread   [35][36][37] . Such methodologies can be potentially coupled with our modeling approach. Furthermore, our model relies on the current body of evidence with regards to the chances of reinfection. In this sense, recovered patients are considered to be immune to future COVID-19 infections. These assumptions are being revised as new viral variants are identified, which might imply the need to redefine the basic assumptions of the model. Also, the current approach for calibrating the model parameters is largely an ad-hoc simulationbased procedure based on the state-level observed cases of infection as well as death. Although computationally intensive, we demonstrated that the model accuracy could be substantially improved when calibrations were conducted at the local levels (e.g., individual counties). Finally, we did not estimate age-stratified timeseries for R E because reported daily cases of COVID-19 currently do not contain demographic data including age. Accounting for heterogeneity in transmission due to demographic factors and also estimating age-stratified reproduction numbers could provide insight into differences in transmission potential by age and other factors. In addition, although the use of age serves as proxy of several risk factors and health conditions, subsequent improvements of this modeling approach could account for other epidemiological and demographic population characteristics that are highly correlated with COVID-19 transmission and outcomes. This is the case for co-morbidities, mobility patterns, population density, and climate, among others.