An integrated framework for building trustworthy data-driven epidemiological models: Application to the COVID-19 outbreak in New York City

Epidemiological models can provide the dynamic evolution of a pandemic but they are based on many assumptions and parameters that have to be adjusted over the time the pandemic lasts. However, often the available data are not sufficient to identify the model parameters and hence infer the unobserved dynamics. Here, we develop a general framework for building a trustworthy data-driven epidemiological model, consisting of a workflow that integrates data acquisition and event timeline, model development, identifiability analysis, sensitivity analysis, model calibration, model robustness analysis, and projection with uncertainties in different scenarios. In particular, we apply this framework to propose a modified susceptible–exposed–infectious–recovered (SEIR) model, including new compartments and model vaccination in order to project the transmission dynamics of COVID-19 in New York City (NYC). We find that we can uniquely estimate the model parameters and accurately project the daily new infection cases, hospitalizations, and deaths, in agreement with the available data from NYC’s government’s website. In addition, we employ the calibrated data-driven model to study the effects of vaccination and timing of reopening indoor dining in NYC.

This study aims to answer a fundamental question: given epidemiological data, how to 2 develop an appropriate model and identify which parameters we can accurately infer 3 that would, in turn, allow us to correctly predict the states of interest such as daily 4 cases, hospitalizations, and deaths. The objective of this work is to provide a systematic 5 way to model a pandemic accurately through carefully formulating a suitable model, 6 uniquely identifying the model parameters, and predicting outbreaks under 7 uncertainties based on the different epidemiological data available. To address the above 8 fundamental question, we propose a general integrated framework to approach the 9 problem systematically through identifiability analysis, sensitivity analysis, model 10 robustness analysis, and forecasting under uncertainties.

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Numerous modeling approaches have been used to gain insight into epidemic 12 disease's ever-evolving dynamics and the effects the interventions have had on 13 containing the spread. Mathematical modeling is an efficient way to test and evaluate 14 the effectiveness of hypothetical interventions that cannot be tested out due to practical 15 or ethical limitations. Describing the disease's development involves representing a interventions. Compartmental models are commonly applied in epidemiology for the 26 reason that they are simple and easily tractable. However, these models are local and 27 have limited capabilities to describe spatial dynamics. Partial differential equation 28 (PDE) models, such as diffusion-reaction models, describe dynamics in time and space 29 more naturally [12,13]. ODE models are the most commonly used models due to the 30 increased mathematical complexity and computational cost of PDE models. However, 31 their accuracy is constrained by parameter uncertainties and gaps in information about 32 the disease dynamics. Likewise, assumptions to maintain model simplicity may affect 33 the estimated values. For a long-lasting pandemic, the model parameters change with 34 time; hence the parameter identification problem becomes nontrivial given the fact that 35 typically a limited amount of relevant data is available. 36 In an ODE-based epidemiological model, the system parameters usually contain 37 critical information that often cannot be measured directly such as the transmission rate 38 which needs to be inferred from data. A necessary condition for the well-posedness of a 39 parameter estimation problem in ODE theory is the model's structural identifiability if 40 we assume noise-free data. The structural identifiability analysis can be performed 41 without any experimental data; it addresses whether the parameter estimation problem 42 is well-posed under ideal conditions. Should the postulated model not be structurally 43 identifiable, the parameters obtained will be unreliable. However, a model can be 44 structurally identifiable (a necessary condition) but may not be practically identifiable. 45 Thus, the structural identifiability analysis may conclude that a model's parameters are 46 uniquely determined, yet when real-life, noisy data are used, the estimated parameter 47 values could still be unreliable. To conduct the practical identifiability analysis, we 48 compute the correlation matrices of model parameters in different settings using Fisher 49 Information Matrices (FIMs) following lines of approach in [14,15]. 50 Non-identifiability is a problem frequently encountered in pandemics modeling since, 51 typically, not every state variable is available. In recent literature, model identifiability 52 issues have been studied due to the wide variation in model predictions in the context of 53 the COVID-19 pandemic [2,[16][17][18]. Tuncer et al. analyzed the structural and practical 54 identifiability of some of the most widely-used pandemic models, including SIR, SIR 55 with treatment, and SEIR, assuming only one observable is available using simulated 56 data [19]. Roda et al. extended these ideas by studying SIR and SEIR models' practical 57 identifiability using data from the COVID-19 outbreak in Wuhan, using only the counts 58 of infected individuals as the available data [2]. They found that complex mechanistic 59 models are more likely to have identifiability issues compared to simpler models. A general framework for building a trustworthy data-driven epidemiological model -an overview of the main contribution. In this work, we propose a general framework for building a trustworthy data-driven epidemiological model, which constructs a workflow to integrate data acquisition and event timeline, model development, identifiability analysis, sensitivity analysis, model calibration, model robustness analysis, and forecasting with uncertainties and scenarios. We first introduce a modified SEIR model that accommodates the pandemic data in New York City. Secondly, we study the structural identifiability, practical identifiability, and sensitivity to examine the relationship between the model's data and parameters. We then calibrate the identifiable model parameters using simulated annealing and MCMC simulation. Model robustness is then checked to study how the model behaves under random perturbations. In addition, we demonstrate the model's predictive capabilities with uncertainties. Finally, reopening scenarios are investigated as a reference for policymakers. parameters;

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• We treat the transmission rate, hospitalization ratio, and death from hospital • We specifically investigate the effects of indoor dining reopening and vaccination 128 scenarios as a reference for policymakers. 129 General Framework and Workflow 130 Holmdahl and Buckee, in [26], discussed different types of models for the COVID-19 131 epidemic as well as the distinct challenges in these approaches. The authors highlighted 132 that "models are a way to formalize what we know about the viral transmission and 133 explore possible futures of a system that involves nonlinear interactions, something that 134 is almost impossible to do using intuition alone." They further elaborate that "models 135 will be useful for exploring possibilities rather than making strong predictions about 136 longer-term disease dynamics." Thus, a systematic way of designing an effective 137 data-driven model is essential for assessing ongoing control strategies endowed with 138 uncertainty quantification.

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To systematically design an effective data-driven model, we propose a general 140 framework for building a trustworthy data-driven epidemiological model, which  168 We evaluate the effectiveness of the proposed framework by applying all the outlined 169 steps to the outbreak dataset in NYC. One of the advantages of the framework's 170 generality is that it is not limited to a single dataset or model. In general, it provides a 171 guideline on how to build an effective and trustworthy epidemiological model with the 172 available data. For illustration purposes, we show how our framework can handle 173 different data types by assuming scenarios when some observables in the NYC dataset 174 are missing. However, for practical use, one should determine the data that are fed to 175 the model at the very beginning.

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(I) Data acquisition and event timeline 177 We use the data consisting of daily cases, hospitalizations, and deaths between February 178 29, 2020, and February 4, 2021, to fit the model's parameters. All the data used in this 179 paper were extracted from the NYC's government's website and collected by the NYC 180 Health Department [25]. Hospitalization data were collected from several sources, such 181 as NYC public hospitals, non-public hospitals, and the Health Department's syndromic 182 surveillance database, which track hospital admissions across NYC. The data on the  The stay-at-home order was effective in bringing down the disease's incidence. As 214 a result, a four-phase reopening plan was developed, taking into account seven 215 health metrics the city needed to meet before reopening [30]. NYC entered Phase 216 1 of reopening on June 8, Phase 2 on June 22, Phase 3 on July 6, and Phase 4 on 217 July 20. Each phase had specific policies that determined what businesses could 218 reopen and in what capacity. Industries that posed the lowest risk of infection for 219 employees and customers were allowed to reopen in Phase 1. These included but 220 were not limited to construction, manufacturing, and wholesale supply-chain 221 businesses and retailers for curbside pickup, in-store pickup, or drop-off [31]. were eased, and meetings of up to 50 people were allowed. Indoor religious 229 meetings were allowed to resume at 33% capacity. Malls, zoos, and botanical 230 gardens were also permitted to reopen in this phase. NYC stayed at Phase 4 until 231 September 30, 2020, when indoor dining at 25% capacity was allowed. The careful reopening process, which followed the strict stay-at-home order,  In December 2020, the increasing rate of virus transmission in NYC threatened to 243 overwhelm hospital capacity. Although contact tracing data from NYC placed 244 indoor dining as the fifth source of new infections in the state, the CDC 245 designated indoor dining as a "high risk" activity [34]. The governor's decision to 246 ban indoor dining was an attempt to halt the steep increase in cases and avoid a 247 broader shutdown. In the same week when the governor's office closed indoor 248 dining again, the first coronavirus vaccine was administered in Queens on 249 December 14, 2020 [35].   These modifications allow a more accurate description of the biology of the disease. 284 Since this study focuses on a single outbreak, births and other deaths are not 285 considered. When the epidemic begins, all individuals are susceptible and transit to the 286 exposed class via contact with presymptomatic, symptomatic, or asymptomatic 287 individuals. Moreover, we assume that there is no reinfection. In other words, once an 288 individual recovers, they do not become susceptible again. We divide the total 289 population into nine different compartments: susceptible (S), exposed (E), individual. Similar to [39], we assume that presymptomatic individuals (P ) are just as 296 infectious as asymptomatic individuals (A). The transmission rate of the disease is 297 denoted by β. After a latent period (1/d E ), an exposed individual becomes 298 presymptomatic infectious. A presymptomatic infectious individual develops symptoms 299 with probability δ or is asymptomatic with probability (1 − δ) after (1/d P ) days.

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Asymptomatic individuals recover after an infective period of (1/d A ). A proportion p of 301 symptomatic individuals is hospitalized after an infective period of (1/d I ), while the 302 rest of them are isolated (for example, isolated at home) but do not go to the hospital. 303 The isolation period is (1/d Q ). The hospitalization period is (1/d H ), at the end of 304 which, a proportion q of the hospitalized individuals die while the rest of them recover. 305 Asymptomatic cases pose a challenge to identify since there is no widespread systematic 306 February 10, 2021 8/39 . CC-BY-NC 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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The copyright holder for this preprint this version posted February 24, 2021. ;  Table 1 for the notations and the initial values. See Table 2 for the parameters. See Eq. (1) for the corresponding ODE system.
where P * is the number of cases from day 1 to day 2/d P (Mar 1-5, 2020) I Symptomatic 1 February 10, 2021 9/39 . CC-BY-NC 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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The copyright holder for this preprint this version posted February 24, 2021. ; https://doi.org/10.1101/2021.02.22.21252255 doi: medRxiv preprint testing of the population [41]. Thus, there is a wide range of estimates for the 307 proportion of symptomatic individuals δ. In this study, we use the best estimate of δ The ODE system for our model (see Fig 3) is the following: Note that the compartments (I, H, D) represent the current symptomatic 319 individuals, hospitalizations, and deaths. In order to represent the daily cases I new , 320 hospitalizations H new , and deaths D new , we add the following ODEs that take into 321 account the inflow of the these compartments to record the cumulative cases I sum ,

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hospitalizations H sum , and deaths D sum : Now, the daily numbers are just the increments of the cumulative numbers: for t = 1, 2, 3, · · · .

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The copyright holder for this preprint this version posted February 24, 2021. The transmission rate of a disease, β, is the per capita rate of infection when a contact 328 occurs. Directly measuring the transmission rate is not possible for most infections [48]. 329 Nevertheless, if we want to quantify the effects of public health policies that directly function is a simplification that allows us to estimate the impact each policy has in each 339 stage. Similarly, we define the piecewise constant hospitalization ratio p(t) and death 340 from hospital ratio q(t), which also exhibit varying values over time: We fit the parameters in each stage defined by policy changes, such as the stay-at-home 342 order and the subsequent reopening processes; see  reproduction number of the model is: In this particular study, given that the model parameters are defined in a piecewise 349 fashion and there was no control in Stage 1, the basic reproduction number, R 0 , is 350 computed by using β = β 1 .

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The transmission of the disease slows down when there are more immune individuals. 352 Since R c is the number in an entirely susceptible population, we can calculate the 353 effective reproduction number: By setting R e = 1, we obtain the immunity threshold of the ODE system, which is the 355 critical portion of the population needed to be immune to stop the transmission of the 356 disease: The herd immunity threshold (HIT) is calculated by substituting R c with R 0 . A higher 358 R 0 results in a higher HIT.

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The copyright holder for this preprint this version posted February 24, 2021. In this section we address whether a set of unknown parameters in the proposed model 361 is globally identifiable from the available data. Fitting a model to the data is not 362 sufficient to show how reliable the estimated parameters are. Insufficient or noisy data 363 can produce drastically different sets of parameters without affecting the fit to data if a 364 model is non-identifiable [2]. Furthermore, depending on the available data 365 (observables), different models may be appropriate.

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Formally speaking, a parameter in a dynamical system is considered to be  In our framework, we analyze both structural and practical identifiability, and use There are 11 undetermined parameters in the proposed model, and it is impossible to fit 391 every parameter without fixing some of the values. For example, it is unnecessary to fit 392 biologically determined parameters such as the time an individual spends in the exposed 393 or infected classes. Since d E , d P , d I , d A , d H , and d Q are determined by the biology of 394 the disease, we fix these values according to [39]. 395 We analyze the structural identifiability of the rest of the parameters when different 396 types of data are given. Note that for general use of our modeling framework, one 397 should fix the dataset at step (I). Here we consider all the scenarios just for illustration 398 purpose. Specifically, we assume that the data are given as the cumulative cases I sum , 399 cumulative hospitalizations H sum , and cumulative deaths D sum , or a subset of the three 400 aforementioned observables, because these quantities can be calculated directly from 401 daily quantities I new , H new , and D new . In other words, it is equivalent to assume X new 402 or X sum to be given as one of the observables, where X can be I, H, and D. The 403 effectively vaccinated population v is treated as the input variable to the system.

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According to Table 3, when only two out of three observables are available, the model is 405 not identifiable and the fitting result will not be unique. The results are to be 406 interpreted in the following way. In the case of lacking deceased individual counts, the 407 death rate q cannot be inferred accurately. If the hospitalization data are not available, 408 both the hospitalization ratio p and death from hospital ratio q cannot be inferred   proportion of disease-related deaths when one of the observables considered in this 417 paper is missing. One of the main differences between the proposed model and most 418 other existing SEIR-based models is that our model integrates information of infectious, 419 hospitalized, and deceased populations simultaneously, therefore producing more 420 reliable results on these estimated parameter values. Since we have data for all three 421 observables in NYC, we should utilize all of them.

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In practice, hospitalization data could be reported in different ways; some databases 423 provide daily reports of the number of hospitalized individuals, whereas others register 424 the number of currently hospitalized individuals. Regardless of the data type available, 425 structurally identifiability of the model remains the same according to S2 Table. 426 Practical identifiability 427 We then proceed with fitting 5 undetermined parameters using all the available data,  Table 4. We see that the values of and δ vary a lot among different 431 stages, which is inconsistent with reality. This poses a question on the practical 432 identifiability of the model.

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The correlation matrices are then computed based on parameters obtained in Stage 1 434 to help determine the practical identifiability. In practice, one can obtain the correlation 435 matrix from the FIM of model parameters, with details given in S4 Text. As shown in 436 Fig 4(b), there is a strong correlation between , δ, and β, while either p or q is need to be fixed, while the rest and p, q can be fitted. In this paper, we fix , δ and fit 441 β, p, q for the following reason: β, which represents the transmission rate, is highly 442 affected by local government policy and does not have a stable and universal value 443 compared to and δ. This means that the value of β could be very different across 444 different datasets and is hard to determine a priori. Therefore, we fix and δ according 445 to [38]. After that, the model becomes practically identifiable as shown in S3 Fig. A 446 summary of the reasoning is shown in Fig 4(a). Variance-based sensitivity analysis, also called Sobol sensitivity analysis, is a global method that measures sensitivity across the whole input space. It decomposes the model's output variance into fractions that can be attributed to individual inputs or groups of inputs [20]. Suppose we are given a black box model: where y ∈ R is the output and Θ = [θ 1 , θ 2 , · · · , θ k ] ∈ [0, 1] k are independent and 449 uniformly distributed uncertain inputs. If some components of Θ are not within [0, 1], 450 we may transform Θ into the unit hypercube. The parameter β is the most important parameter for all three quantities of interest in every stage of the pandemic. The parameter p has no influence on I sum . The parameter q has no influence on I sum or H sum .
First-order sensitivity index measures the contribution to the output variance by a single input θ i alone: where Θ ∼i = [θ 1 , · · · , θ i−1 , θ i+1 , · · · , θ k ]. Total-order sensitivity index measures the 452 contribution to the output variance by an input, including its first-order effect and all 453 higher-order interactions with other inputs: Note that by definition, and k i=1 S T i (y) ≥ 1 since the interaction between θ i and θ j is counted in both S T i (y) and S T j (y). The vaccination data of COVID-19 in NYC are given in the form of the number of first 477 and second doses administered from December 14, 2020 to February 4, 2021 (see Fig 2). 478 The vaccine efficacy is 52% for only one dose and 95% for both doses. As the vaccines 479 are not 100% effective, in order to calculate the number of effectively vaccinated where D 1 is the number of individuals who receive the first dose and D 2 is the number 483 of individuals who receive the second dose on that day. As the vaccines provide parameter v in our model (see Fig 3). Before any vaccine is effective, we have v ≡ 0.  Table 1, and parameters in Table 2, 496 we fit the transmission rate β, hospitalization ratio p, and death from hospital ratio q 497 within the range [0, 1]. The fitting is split into seven stages defined by the public 498 policies described in Fig 2. In each stage, the parameters (β, p, q) are assumed to be 499 constant as in Eq. (4). We use simulated annealing, a global optimization algorithm, to 500 search for the optimal parameter values in each stage. The objective is to minimize the 501 following loss function: is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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Fig 8. Estimation of parameters and reproduction numbers. (a) Estimated time-dependent transmission rate β(t).
(b) Estimated time-dependent hospitalization ratio p(t), compared with daily hospitalizations over daily cases calculated from the raw data. (c) Estimated time-dependent death from hospital ratio q(t), compared with daily deaths over daily hospitalizations calculated from the raw data. (d) Estimated control reproduction number R c and effective reproduction number R e calculated by the estimated parameters, compared with 1/2 of the logarithm of daily cases. . CC-BY-NC 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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in the region β, p, q ∈ [0, 1], where "MSE" stands for mean squared error. The estimated 503 final value in the previous stage is the initial value for the next stage. The results of all 504 compartments are plotted in Fig 6 and Fig 7. The time-dependent parameters (β, p, q) 505 and the control reproduction number R c are shown in Fig 8, Table 5, and Table 6.

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The control reproduction number R c , whose expression is in Eq. (5), depends on six 507 parameters: β, , δ, d P , d I , and d A . Since , δ, d P , d I , and d A are fixed as in Table 2, 508 R c is proportional to the transmission rate β; see Using the fitted parameter values in Table 5, we perform the Monte Carlo simulation to 535 check the robustness of our model to perturbations, adapting ideas from [19,57]. The 536 major difference between their approach and ours is that we incorporate sensitivity 537 analysis results in our method. 538 We first multiply the daily increase in the calibrated data (a subset of 539 {I sum , H sum , D sum }) by independent and identically distributed Gaussian random noise 540 of mean 1 and standard deviation σ to generate a new dataset, which looks like our 541 original dataset with measurement error. Then, we estimate the parameters by fitting 542 the model to the artificially generated dataset and compare the result with the 543 parameter values obtained in Table 5. The same procedure is repeated for M = 1000 544 times, and we compute the average error between the parameter values estimated from 545 the original and the generated datasets. We then rescale this error by a sensitivity Sensitivity-based Average Relative Error (SARE) of (β, p, q) in different observable settings. Each row corresponds to a standard deviation level of random noise multiplied to the observables. Each column represents an observable setting. Note that when SARE of a parameter is 0, it means the provided data are not sensitive to that parameter. When (I sum , H sum , D sum ) are given, SARE is lower than the threshold 1. Therefore, our model is robust to noise in the NYC dataset. In some of the missing observable cases, our model is also robust to perturbations at different noise levels. (SARE): where θ i is the fitted value of the ith parameter (i.e., β, p, q) on the original dataset, where k is the total number of parameters to be estimated. If MSARE < 1, we say the 561 model is robust to perturbation. The algorithm is detailed in S5 Text. This method is a 562 complement to the identifiability analysis in the sense that the identifiability analysis 563 only takes care of the uniqueness of model parameters but not their significance to the 564 data. An insensitive non-identifiable parameter would not influence the predictability of 565 the model too much. On the other hand, robustness does not guarantee the correctness 566 February 10, 2021 20/39 . CC-BY-NC 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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The copyright holder for this preprint this version posted February 24, 2021. ; https://doi.org/10.1101/2021.02.22.21252255 doi: medRxiv preprint of the fitted parameters or the inferred compartmental values apart from the particular 567 data. One has to combine identifiability with robustness to draw conclusions.

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The first column in Fig 9 shows that when (I sum , H sum , D sum ) are given, our model 569 is robust to noise, which justifies the correctness of the fitting result (since β, p, q are 570 also identifiable) and provides a theoretical backup for the forecasting in the next 571 section. The other columns in Fig 9 show that even when some observables are missing, 572 the model could still be robust to perturbation at different noise levels, so that one can 573 use our model to make predictions in some missing observable cases. For example, only 574 infectious and hospitalized data (I sum , H sum ), only hospitalized and deceased data The situation in NYC evolves day by day. The city reinstated indoor dining restrictions 582 in mid-December due to the steady increase in the virus incidence. The ever-changing 583 policies add a high level of uncertainty to any long term forecast we can make. Here, we 584 explore our model's ability to predict the number of daily cases, hospitalizations, and 585 deaths in the city with uncertainty.

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The MCMC simulation provides us with a way to quantify uncertainty. We may 587 sample from the posterior distribution of the parameters in the last stage (see S11 Fig) 588 and run the model after that to obtain a distribution of the predicted daily cases, 589 hospitalizations, and deaths. However, this approach assumes that the situation remains 590 the same after the last stage, which may not be the case. There might be policy changes 591 or other events. As a result, we perturb the transmission rate β by a percentage to 592 reflect future policy changes or other events. We sample from the posterior distribution 593 of the parameters (β, p, q) in the last stage. Then, we multiply β by a random number 594 drawn from a uniform distribution (U(0.95, 1.05) or U(0.85, 1.15)). In other words, we 595 perturb β by 5% or 15%. As a reference, see Table 6 for the historical changes of β 596 between contiguous stages. We run the model with the initial value as the ending value 597 of the last stage to predict daily cases, hospitalizations, and deaths. After repeating 598 4000 times, we obtain a distribution of the daily cases, hospitalizations, and deaths at 599 every timestamp after the last stage. Then, the 95% confidence interval at every 600 timestamp is plotted.

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The COVID-19 epidemic is an unprecedented worldwide public health challenge, 614 especially in densely populated areas such as New York City (NYC). Epidemiological 615 models can provide the dynamic evolution of a pandemic but they are based on many 616 assumptions and parameters that have to be adjusted over the time when the pandemic 617 lasts. However, the available data might not be sufficient to identify the model 618 parameters and hence infer the unobserved dynamics. This is typical of any past 619 epidemics or pandemics, and hence a systematic integrated framework is required to 620 make existing or modified models useful for designing health policies.

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To this end and after studying the current pandemic for almost a year, we have   2. Identify the flow between the components and link them with directed arrows.

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Assign the rate of the flow with one parameter per arrow except for S → E, which 886 is defined by the transmission of the disease. See S1 Fig.   887 3. Change parameters such that every parameter has physical meaning. We use nine 888 parameters (p, q, δ, d E , d P , d I , d A , d H , d Q ) to replace (w 1 , · · · , w 9 ). See Eq. (13) 889 for the formulas. Note that there is a one-to-one correspondence between the nine 890 parameters we use and (w 1 , · · · , w 9 ).

Add vaccination dynamics
S2 Text. Parameter settings. Parameter values for the fixed parameters are summarized in Table 2. The percentage of infected people who never show the disease's symptoms is extracted from the CDC's current best estimate for this value [38]. The parameter δ is obtained by averaging the lower and upper bounds of the estimates for this value from different sources: We use data from the hospitalization surveillance network used by the CDC to estimate the median number of days an individual spends hospitalized due to the disease [58]. The numbers provided are specific to individuals admitted to the ICU and not admitted to the ICU divided into age groups; therefore, we perform a weighted average of those values considering the demographic composition of NYC. According to S1 Table,    is the disease-free equilibrium state of the system. We have Then, Let ρ(F V −1 ) denote the dominant eigenvalue of F V −1 . The control reproduction 893 number is given by R c = ρ(F V −1 ).

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S4 Text. Definition and algorithm for identifiability analysis.

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Structural identifiability 896 Suppose we are give a dynamical system of the following abstract form where X = (X 1 , · · · , X n ) represents the state variables, y = (y 1 , · · · , y m ) represents the 898 observables, Θ = (θ 1 , · · · , θ k ) contains the parameters to identify, and u(t) represents 899 the input variable to the system. A parameter set Θ is called structurally identifiable if 900 g(X(t), Θ, u(t)) = g(X(t), Φ, u(t)) =⇒ Θ = Φ February 10, 2021 30/39 . CC-BY-NC 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.
The structural identifiability, defined above, is referred to as global identifiability 903 in [51][52][53][54]. Global identifiability is stronger than the so-called local identifiability, which 904 only requires Eq. (15) or Eq. (16) to hold in a neighbourhood N (Θ) of Θ. Structural 905 identifiability analysis is usually conducted before the fitting of the model and it does 906 not rely on any data. 907 We perform the structural identifiability analysis using the software SIAN [56], 908 which is based on differential algebra and Taylor series expansion. Detailed 909 documentations on the theory behind the software can be found in [60], and information 910 regarding the algorithm can be obtained from [56].

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SIAN is a randomized algorithm that requires users to choose a hyperparameter that 912 defines the probability of correctness. In this paper we choose 0.999. The algorithm first 913 replaces the observables with their truncated Taylor series and reduces the identifiability 914 problem to the problem of studying the generic fiber of the map between the parameter 915 space and the space of truncated observables, which is a map between finite-dimensional 916 varieties. Due to the hardness of analyzing the generic fiber, a step to simplify 917 computation is applied in the algorithm by studying fiber at a specific point instead of 918 generic ones. The correctness of this step is controlled by the hyperparameter 919 aforementioned. The problem is then turned into checking the consistency of a system of 920 algebraic equations. The Buchberger algorithm allows the computation of the Grobner 921 basis of the system. This basis shows whether the algebraic relations' parameters have a 922 single solution/finitely many solutions/infinitely many solutions, which implies the 923 model would be structurally globally identifiable/locally identifiable/not identifiable.  [19,57].

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Statistical correlation can be used to describe some of the practical non-identifiability phenomena. Specifically, one can use the Fisher Information Matrix (FIM) to calculate the correlation matrix of Θ in Eq. (14). FIM is defined as: where V is covariance matrix of the measurements error. S i = ∂ỹ(ti) ∂Θ is the sensitivity matrix at time t i , which can be calculated by solving the adjoint equation: using automatic differentiation libraries. It is proved that the correlation matrix is equal 933 to F −1 by the Cramér-Rao theorem [61]. Once the correlation matrix is obtained, one 934 can determine whether a parameter is identifiable by checking whether it is correlated 935 with the other parameters. If θ i is strongly correlated with θ j , then neither of these 936 parameters is practically identifiable because they cannot be uniquely determined from 937 the data.

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February 10, 2021 31/39 . CC-BY-NC 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. . The sensitivity-based average relative error (SARE) for θ i is defined as: where S T i (y l ) is the total-order sensitivity, which is the contribution to the variance of y l by θ i . In other words, it quantifies the influence of θ i on the output variable. For more important parameters which have a higher S T i , we require more robustness. This is taken into account by the multiplier If MSARE < 1, we say that the model is robust to perturbation. 947 S1  . CC-BY-NC 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted February 24, 2021. ; https://doi.org/10.1101/2021.02.22.21252255 doi: medRxiv preprint