High-resolution agent-based modeling of COVID-19 spreading in a small town
Agnieszka Truszkowska, Brandon Behring, Jalil Hasanyan, Lorenzo Zino, Sachit Butail, Emanuele Caroppo, Zhong-Ping Jiang, Alessandro Rizzo, Maurizio Porfiri
HHigh-resolution agent-based modeling of COVID-19spreading in a small town
Agnieszka Truszkowska , Brandon Behring , Jalil Hasanyan , Lorenzo Zino ,Sachit Butail , Emanuele Caroppo , , Zhong-Ping Jiang , Alessandro Rizzo , ,and Maurizio Porfiri , , Department of Mechanical and Aerospace Engineering and Department of BiomedicalEngineering, New York University Tandon School of Engineering, Brooklyn NY 11201, USA Faculty of Science and Engineering, University of Groningen, 9747 AG Groningen, Nether-lands Department of Mechanical Engineering, Northern Illinois University, DeKalb IL 60115, USA Mental Health Department, Local Health Unit ROMA 2, 00174 Rome, Italy University Research Center He.R.A., Universit`a Cattolica del Sacro Cuore, 00168 Rome,Italy Department of Electrical and Computer Engineering, New York University Tandon Schoolof Engineering, 370 Jay Street, Brooklyn NY 11201, USA Department of Electronics and Telecommunications, Politecnico di Torino, 10129 Turin,Italy Office of Innovation, New York University Tandon School of Engineering, Brooklyn NY11201, USA Department of Biomedical Engineering, New York University Tandon School of Engineering,Brooklyn NY 11201, USA Center for Urban Science and Progress, Tandon School of Engineering, New York Univer-sity, 370 Jay Street, Brooklyn, NY 11201, USACorrespondence should be addressed to: [email protected] a r X i v : . [ q - b i o . P E ] J a n bstract Amid the ongoing COVID-19 pandemic, public health authorities and the generalpopulation are striving to achieve a balance between safety and normalcy. Ever chang-ing conditions call for the development of theory and simulation tools to finely describemultiple strata of society while supporting the evaluation of “what-if” scenarios. Par-ticularly important is to assess the effectiveness of potential testing approaches andvaccination strategies. Here, an agent-based modeling platform is proposed to simulatethe spreading of COVID-19 in small towns and cities, with a single-individual reso-lution. The platform is validated on real data from New Rochelle, NY —one of thefirst outbreaks registered in the United States. Supported by expert knowledge andinformed by reported data, the model incorporates detailed elements of the spreadingwithin a statistically realistic population. Along with pertinent functionality such astesting, treatment, and vaccination options, the model accounts for the burden of otherillnesses with symptoms similar to COVID-19. Unique to the model is the possibilityto explore different testing approaches —in hospitals or drive-through facilities— andvaccination strategies that could prioritize vulnerable groups. Decision making by pub-lic authorities could benefit from the model, for its fine-grain resolution, open-sourcenature, and wide range of features.
In December 2019, COVID-19 was first observed in humans in Wuhan, the Hubei Province’scapital in China. The World Health Organization (WHO) declared this outbreak as a PublicHealth Emergency of International Concern on January 30, 2020, and later named it apandemic on March 11 th , 2020. As of December 23 rd , 2020, the WHO has reported 76,382,044cases globally, with 1,702,128 deaths. [1] In the US, the number of infected individuals keepsrising, with tens of thousands of newly infected cases discovered every day. The Centers forDisease Control and Prevention (CDC) has reported 17,974,303 cases as of December 23 rd ,2020. [2] Following an unprecedented containment campaign of lockdowns, most countries seeka delicate balance between safety and normalcy, aiming for a safe return to normal activitiesamidst less restrictive conditions.Timely case detection through efficient testing and contact tracing is among the key com-ponents required for lowering the COVID-19 spread before a vaccine becomes available. [3–6]
Important questions on testing pertain to the identification of infected individuals and theircontacts. Addressing these questions calls for an improved understanding of communitystructure, outbreak locations, and individual lifestyles. [7, 8]
Due to the scale of the COVID-19epidemic, an additional burden has been placed on traditional testing sites, such as hospitals,emergency rooms, and walk-in clinics, thereby challenging their safety. [9–11]
Computational models are powerful tools for understanding novel epidemics and evaluat-ing the effectiveness of potential countermeasures. [12–16]
Agent-based models (ABMs) are aclass of computational models that provide a high-resolution—both temporal and spatial—representation of the epidemic at the individual level. [4, 17–21]
These models afford consider-ation of multiple physical locations, such as businesses or schools, as well as unique featuresof communities, like human behavioral trends or local mobility patterns.2nce validated, ABMs can be used to test competing “what-if” scenarios that wouldotherwise require impractical and, potentially, unethical experiments. For example, Fergusonet al. developed an ABM to investigate the impact of non-pharmaceutical interventionsfor COVID-19, such as nation-wide confinement in the United Kingdom and the UnitedStates. [22]
Aleta et al. proposed an ABM for the entire Boston metropolitan area to elucidatethe role of different types of contact tracing measures in maintaining low levels of infection. [4]
Gressman and Peck created an ABM of a university campus to examine strategies for the safereopening of higher education institutions. [23]
Hinch et al. formulated an open-source agent-based modeling framework to support the analysis of select non-pharmaceutical interventionsand contact tracing schemes. [24]
The merits of ABMs have been recognized by a large numberof studies, which have shed light on technical aspects of their implementation as well as theirscalability across scenarios. [25–29]
The focus of existing ABMs is either small micro-environments or entire countries andlarge metropolitan areas, where the population is purposefully coarsened to enable numericalsimulation. Medium-size and highly resolved communities constitute an important, yet un-considered, modeling scale for COVID-19. America is a “nation of small towns,” as describedby the US Census, [30] , and after the initial wave of spreading in the New York metropolitanarea, we see that small and medium-size towns are increasingly hit by the pandemic. Ahigh-resolution ABM can closely capture real-world communities and interaction patternsat this intermediate scale, thereby allowing them to carefully reflect town-specific lifestyleswithout requiring coarsening to perform massive simulation campaigns. Thus, the syntheticgeneration of a one-to-one virtual population with its own individual buildings (residentialand public) opens a wide range of new possibilities in epidemiological analysis, which mayinform public health authorities to design accurate and targeted interventions. The analysiscould include lockdowns of different parts of the town and afford to quantify the effect oftesting practices, treatment prevalence, and vaccination strategies.Toward the study of these medium-size populations, we develop an agent-based modelingplatform of COVID-19 for the entire town of New Rochelle, located in Westchester Countyin New York, US. This location was chosen as it was one of the earliest COVID-19 outbreaksin the US and is representative of a typical small town. The ABM replicates, geographicallyand demographically, the town structure obtained from the US Census statistics. [31]
Themodel is based on the earlier ABM developed by Ferguson’s research group to study pan-demic influenza, [19–21] which has been recently adapted to study COVID-19. [22]
The proposedABM expands the original model by Ferguson et al. [19–21] along several directions. First, weincorporate two testing strategies: traditional, in-hospital testing with a non-negligible riskof infection, and a “safe” drive-through testing. [32–35]
Second, we account for temporal vari-ations in time-dependent testing capacity to reflect the change in resource allocation duringthe course of the pandemic. Third, our model explicitly includes multiple COVID-19 treat-ment types, such as home isolation, hospitalization, and hospitalization in intensive care units(ICUs). Fourth, we separately track individuals with COVID-19-like symptoms due to otherdiseases like seasonal influenza or the common cold. These individuals are expected to playan important role in the epidemic by imposing an additional burden on testing resources.Fifth, we individually model employees in schools, hospitals, and retirement homes, enablinga dedicated consideration of these professions. Finally, the model permits the selective studyof important interventions, such as business and school closures and their reopening and3accination strategies. After validation against real data, we explore alternative “what-if”vaccination scenarios, which may be relevant in the next several months. We consider aforeseeable situation in which a limited number of vaccines will be available. In additionto random vaccination, we explore the possibility of prioritizing vaccination for groups ofindividuals whose status or profession makes them particularly vulnerable to COVID-19 in-fection: healthcare, school, and retirement home workers, as well as residents. Along withselective vaccination of hospital employees, we assess the potential benefit of schools andbusinesses’ concurrent closures.
We collected and organized a database of geographical coordinates, type, capacity, residentialbuildings, and public spaces in the town of New Rochelle, NY, USA. The database was createdby manually collecting geographical coordinates and characteristics of each residential andpublic building in the town using OpenStreetMap [36] and Google Maps. [37]
The databaseand the code for creating the town and its population are available through our repository( https://github.com/Dynamical-Systems-Laboratory/NR-ABM-population ). Thepopulation data were collected in March and April 2020 from US Census using 2018 5-yearaverage tables for New Rochelle. [31]
Since then, some of the data changed slightly, and thereader is referred to our repository for the exact datasets. Data regarding the number ofstudents and employees in local schools was obtained from the National Center of EducationStatistics, [38] while the hospital staff and patient number estimates were based on the recordsfrom the New York State Department of Health [39] and the American Hospital Directory. [40]
Figure 1 shows the locations considered in the model.
Households were assigned to agents using Census data on household and family structure,vacancy rate, and the number of sub-units and floors in multi-unit buildings. Figure 2a)shows the population distribution of the model set at the start of the simulation (see SectionS3 of the Supporting Information), which mirrors the increased density registered toward thesouthern part of the town. [41]
As demonstrated in Figure 2b), the distribution of householdsizes and the mean size is in good agreement with Census data. As evidenced in Figures 2c)and 2d), we closely match the distribution of employed family members and the overall agedistribution, accurately resolving several age groups. As reported in Table 1, we are successfulin maintaining similar percentages of households with agents who are above 60 years old. Thisaspect is particularly important for realistic predictions of COVID-19, which is significantlymore severe and fatal among the older population. Aside from age, the modeled communitypreserves the ratio of families with children and of single-parent households, as shown inTable 1.All children aged 5-17 were assigned to schools of appropriate levels. The number ofstudents in a school was proportional to school size based on the data from the National4able 1: Modeled properties of the population and US Census data for New Rochelle, NY.Category Model US Census valueMean household size 2.77 2.71Households with one ormore individuals 60 years and older 47.0% 42.3%Families with children 39.1% 34.2%Single parent families 18.7% 25.0%Center for Education Statistics. [38]
Accordingly, the initial distribution of students in themodel, representing time before the pandemic, was set so that a portion of children youngerthan five years old was placed in daycares, and similarly, 35% of individuals between 18–21years old attended the town’s higher education institutions.A portion of agents older than 16 were allowed to work and study according to a set ofrules based on the estimated workplace and school sizes (see Section S3 of the SupportingInformation). Agents’ workplace distribution was generated using the US Census data onthe percentage of the population working in a given industry while the number of employeesat hospitals, schools, and retirement homes was estimated. The initial numbers of retirementhomes residents were estimated based on institution size. The initial number of hospitalpatients with a condition other than COVID-19 was set to occupy one-sixth of the totalavailable beds. The exact steps for creating the population from building information andcensus data are outlined in Section S3 of the Supporting Information.
The number of total and active New Rochelle cases were collected manually from officialreports and videos available for Westchester County. [42]
The mortality and testing statisticswere obtained from the Official Twitter account of the Westchester County, [42] and the NewYork State Department of Health, [43] respectively, for Westchester County as a whole, andwere then scaled to the population of New Rochelle. All the data is available through ourrepository ( https://github.com/Dynamical-Systems-Laboratory/NR-ABM-population ). During a day, agents transition between different locations (identified in the New Rochelledatabase: households, workplaces, schools, retirement homes, and hospitals). In each ofthese locations, they can interact with other agents, thereby supporting the transmission ofCOVID-19. Agents can be healthy, undergoing testing, or be under treatment. We assumethat the town is isolated, such that the agents cannot leave the town, and new agents cannotenter during the simulation.Motivated by [19, 20, 22], the COVID-19 progression model consists of five states: sus-ceptible ( S ), exposed —which includes infectious individuals who have not yet developed5ymptoms— ( E ), infectious-symptomatic ( Sy ), removed-healed ( R ), and removed-dead ( D );with a detailed outline of states, their variants, and transitions as shown in Figure 3. Uponinfection, susceptible agents become exposed ( E ) and remain so for a latency period. Follow-ing the COVID-19 infectiousness profile, we assume that exposed agents are not infectiousduring the initial part of the latency period. When the latency period is over, exposed agentsdevelop symptoms and become infectious-symptomatic ( Sy ). [4] Some exposed agents mayrecover without ever developing symptoms (that is, they are asymptomatic); in this case,their latency period is extended to match the expected COVID-19 recovery period.Exposed agents and agents showing symptoms —whether from COVID-19 or anothercondition— have the possibility of being tested which can be performed either in a hospital( T Hs ) or a car ( T C ). Testing in a hospital carries the possibility of infecting hospital staffand patients with a condition other than COVID-19. Additionally, if the agent is COVID-19-negative, then the agent is at risk of becoming infected during testing. We assume thatdrive-through testing does not carry the risk of infection based on the work of Upham. [34] The outcomes of a test can be true positive or negative or false positive or negative.After testing, agents are assigned treatment from one of the following three treatmenttypes: home isolation ( I Hm ), routine hospitalization ( H N ), and hospitalization in an ICU( H ICU ). An exposed agent who has tested positive for COVID-19 will always undergo homeisolation until developing symptoms, at which point their treatment can potentially changeto H N or H ICU . In contrast, a symptomatic agent can be assigned to one of the threetreatments. A symptomatic agent can transition between different treatment types duringthe course of the disease. All infected agents are removed through either recovery ( R ) ordeath ( D ). A removed agent no longer contributes to the spread of the infection.A symptomatic agent who is untested will not undergo treatment, and their removal isdetermined following similar rules to the tested agents. However, an untested agent requiringICU treatment has an increased probability of dying, to account for the higher mortality in theabsence of a diagnosis. While the agents undergoing testing are routinely quarantined, thosewho are not deemed to be tested retain their normal activities. However, when developingsymptoms, these agents will refrain from going to work or school, thereby reducing thecontribution to contagion in public areas.Susceptible agents can have symptoms similar to COVID-19 due to non-COVID-19 dis-eases, such as a common cold or seasonal influenza. [44] We assume that this population isconstant throughout the duration of the simulation. Since these susceptible agents are sus-pected of having COVID-19, they can undergo testing and thus introduce two additionalelements in our model: i) they increase the count of people being tested, and therefore theburden on testing sites, and ii) they can contract COVID-19 upon interacting with infectedpeople at the testing site. Finally, such agents can be erroneously tested positive duringtheir recovery from the non-COVID-19 disease. However, after recovery, these agents arestill susceptible to COVID-19.To simulate realistic COVID-19 epidemic conditions, we introduce school closures, a state-wide lockdown, and the three reopening phases I, II, and III. School closure is modeled byomitting the contribution of schools to the transmission of COVID-19. Similarly, lockdownand reopening phases are characterized by tuning the contributions of the workplaces to thetransmission. The number of tests performed per day can be time-dependent, following realpractices. [43] .2 COVID-19 transmission dynamics The proposed epidemiological model consists of COVID-19 transmission through agents in-teracting at their residences and public places. The agents reside in households, retirementhomes, or hospitals when being treated for conditions other than COVID-19. They can alsoattend schools and go to work; employees of schools, retirement homes, and hospitals aremodeled explicitly for their high-risk and critical role. Following the original work of Fergu-son, there is no distinction between times of the day, for example day versus night. At eachsimulation step, ∆ t , an agent may contract the disease or infect other agents at home, school,workplace, or hospital. [19, 20] For example, if a susceptible agent is a high school student whoalso works part-time, their probability of being infected with COVID-19 is computed basedon their contacts at their school, workplace, and household. When agents are being testedor hospitalized, they do not infect those in their households, schools, or workplaces.The model comprises a set of agents N = { , ..., n } and a set of locations L = { , ..., L } .According to the town database, an agent is associated with a subset of locations deter-mined by the model input. Formally, we define a set of functions f q : N → L , with q ∈ { H , W , S , Rh , Hsp } , so that function f q associates each agent i ∈ V to the correspondinglocation (cid:96) ∈ L of type q . The types of locations are households (H), workplaces (W), schools(S), retirement homes (Rh), and hospitals (Hsp). Note that each agent may not be associatedto all the types of locations. To denote that agent i is not associated to a location of type q ,we write f q ( i ) = ∅ . We denote by n (cid:96) the number of agents associated with location (cid:96) .At every simulation step (of duration ∆ t ), infected agents assigned to location (cid:96) con-tribute to the probability of infection for susceptible agents at that location. Specifically, theprobability that an agent i that is susceptible at time t becomes infected in the followingtime-step is equal to p i ( t ) := 1 − e − ∆ t Λ i ( t ) , (1)where the non-negative time-varying parameter Λ i ( t ) quantifies the contagion risk at all thelocations associated with the agent and it is equal toΛ i ( t ) := λ H ,f H ( i ) ( t ) + λ W ,f W ( i ) ( t ) + λ S ,f S i ( t ) + λ Rh ,f Rh ( i ) ( t ) + λ Hsp ,f Hsp ( i ) ( t ) , (2)where each contribution represents the so-called infectiousness function of each type of lo-cation associated with agent i , with the understanding that λ q, ∅ = 0, that is, if agent i isnot associated with any location of type q , than locations of type q do not contribute to theagent’s contagion risk.The infectiousness function of a location (cid:96) of type q at time t , λ q,(cid:96) ( t ), due to all the agents(indexed by k ) at that location, is defined as λ q,(cid:96) ( t ) := 1 n α q (cid:96) n (cid:96) (cid:88) k =1 ( E k ρ k β q,k + Sy k ψ (cid:96) c k ρ k β q,k ) . (3)The sum is performed over the n (cid:96) agents who are associated with location (cid:96) and it representsa weighted ratio between the number of exposed and infected agents and all the agents at thelocation; E k is an indicator function that is equal to 1 if agent k is in the exposed ( E ) andhas become infectious and 0 otherwise; Sy k is equal to 1 if agent k is symptomatic ( Sy ) and0 otherwise. The parameter ρ k ≥ k > α q ≤ ψ (cid:96) ∈ [0 ,
1] is an absenteeism correction for workplaces and schools, which is usedto model reduction of agent presence upon developing symptoms; β q,k ≥ q and on the activity of agent k at thatparticular location (for example, the transmission rate for an agent who is being tested ata hospital is different from an agent who works there). Further details of the model areprovided in Section S1 of the Supporting Information.When an agent becomes exposed, they undergo an incubation period. The latency of theincubation is drawn from a log-normal distribution, which allows for the possibility of someagents to not spread the virus until they develop symptoms. Once the incubation ends, anagent transitions from exposed to symptomatic. The model allows for a portion of exposedagents to recover without symptoms (commonly referred to as asymptomatic individuals) byincluding their recovery time within their incubation period. Both exposed and symptomatic agents can undergo testing for COVID-19 according to twodifferent probabilities. When an agent is scheduled to be tested, they are placed under homeisolation and randomly assigned to a testing location — a drive-through or a hospital. Weassume that the test is performed for a fixed amount of time after the decision to be tested,and similarly that the result of a test appears after a fixed delay following the test. Thetest result can be either positive (true or false) or negative (true or false), with negativeresults causing the agent to return from home isolation to the community. An exposed agentconfirmed positive for COVID-19 remains in home isolation, while a symptomatic one is givenan initial treatment.Testing is performed differently for exposed hospital employees and patients originallyadmitted for non-COVID-19-related diagnosis (such as a car accident or cancer treatment).These agents do not undergo home isolation, and their testing is always performed in thehospitals they work or reside in, without the option of a test car. The symptomatic hospitalstaff is home isolated prior to receiving the test results, while the non-COVID-19 patientsstay in the hospital and, upon developing COVID-19 symptoms, they are counted amonghospitalized COVID-19 cases. This fine level of detail is needed to capture evidence ofextensive COVID-19 spreading in the early stage of the pandemic in hospitals. [45–47]
Afterconfirming COVID-19, agents are assigned treatment.With the exception of hospital employees and patients who develop disease symptoms,the model does not apply any explicit contact tracing. Instead, case detection is implementedin an average sense. Whether an agent will be tested is determined by stochastic samplingof a uniform distribution, followed by a comparison with testing prevalence at that time.
When a symptomatic agent is confirmed COVID-19 positive, they are assigned to one of thethree formal treatments according to a probabilistic mechanism: home isolation, regular hos-pitalization, and hospitalization in an ICU. [22, 24]
Afterward, the agent can change treatment8ypes depending on their recovery status and clinically observed COVID-19 progression. Theagent’s initial treatment is chosen based on the probability of normal hospitalization andhospitalization in an ICU obtained from clinical data depending on their age. [22, 48]
In bothcases, agents are assigned to a random hospital. If hospitalized in an ICU, their recoverystatus is recomputed based on an agent’s probability of dying in an ICU. [22]
All other agentsare placed in home isolation.Treatment changes include all possible transfers except direct transfer from an ICU tohome isolation, as outlined in Figure 3. In our model, whether an agent dies or recovers isdetermined upfront when the agent develops symptoms of the disease. Treatment transitionsare closely related to an agent’s future recovery outlooks. An agent originally determined todie and treated in an ICU will die in the ICU. [49]
Upon recovering, agents will be transferredto normal hospitalization after an amount of time decided a priori. [22]
Any dying agent whowas previously confirmed COVID-19 positive would be placed in an ICU for a predeterminednumber of days before death. [50]
A recovering agent initially hospitalized outside an ICU canbecome home isolated if their recovery time exceeds their hospital stay. [49]
Finally, an agentrecovering while isolated at home can become hospitalized for a certain amount of time, acommonly observed course for the disease. [49, 51]
At the beginning of the simulation, we initialize the entire population as susceptible, assumingno prior immunity in a virtually COVID-19-free population. Then, a predefined number ofagents are assigned the exposed health state. These agents can only be tested after developingsymptoms.Part of the susceptible population can also be vaccinated and thus becomes immune. Thevaccines are distributed by two modes: i) randomly throughout the entire population, or ii)to a specific type of agent, such as healthcare workers or retirement home employees.
Disease progression can have two possible outcomes: recovery or death. The outcome isdetermined using age-based mortality data, the agent’s treatment requirements, and currenttesting prevalence. An agent’s mortality also depends on whether they are tested and receiveproper medical attention. [50, 52–55]
Specifically, while asymptomatic agents always recover, wedistinguish between two events that can occur to symptomatic agents and influence theirprobability of dying.The model decides if an agent needs ICU care upon exhibiting symptoms. An agent whoneeds an ICU will be admitted upon being tested. Not all the agents who need an ICU willbe admitted to one; those who are not tested, and therefore not diagnosed, will die. Amongthe agents who do not need an ICU, a fraction may still die, for example, due to heart failure,stroke, or a rapid decline in condition; some of these individuals will die in their homes, butsome others will formally be admitted to ICU, despite not needing it, based on the agent’sexpected lifetime.All these probabilities are available from empirical observations except the probabilityof dying without the need of an ICU. In order to estimate this quantity, we perform the9ollowing calculations. We expand the overall probability that a symptomatic agent dies, P ( D | Sy ), using the law of total probability with respect to the conditioning on whether thesymptomatic agent needs ICU treatment, N , and the event that the symptomatic agent doesnot need an ICU, N , P ( D | Sy ) = P ( D | N ) P ( N ) + P ( D | N ) P ( N ) . (4)By rearranging Equation (4), we obtain the following closed-form expression for the requiredprobability P ( D | N ) = P ( D | Sy ) − P ( D | N ) P ( N )1 − P ( N ) . (5)In the following, we derive the three expressions for P ( D | Sy ), P ( D | N ), and P ( N ), whichare needed to compute the formula.First, the probability of dying for symptomatic agents, P ( D | Sy ), is inferred from the in-fection fatality ratio ( IF R ) available in the literature. [22]
Since the
IF R is based on serology-informed estimations, it reflects the probability of dying for an infected agents (regardlessof whether the agent is symptomatic or not). Assuming that asymptomatic agents do notdie, we can compute the overall time-averaged probability that a symptomatic agent dies byre-scaling the
IF R by the probability of developing symptoms once contracting COVID-19, P ( Sy | CoV), obtaining P ( D | Sy ) = IF RP ( Sy | CoV) . (6)Second, the probability of dying if an agent needs an ICU, P ( D | N ), in Equation (5) iscomputed depending on whether they are tested once they become symptomatic. Specifically,by means of the law of total probability with respect to the conditioning on the event T , wehave P ( D | N ) = 1 − (1 − P ( D | N, T )) · P ( T | Sy ) , (7)where P ( T | Sy ) is the probability that a symptomatic agent is tested. Since the IF R usedin Equation 6 is a temporal average over the entire duration of the pandemic, the probability P ( T | Sy ) is also estimated as an average over the entire duration of the pandemic, from datareported in Supplementary Table S9.Third, the probability that a symptomatic agent needs ICU care, P ( N ), is computedfrom empirical data on the probability that an agent needs to be hospitalized, P ( H ), andthe probability that symptomatic hospitalized agents need ICU care, P ( N | H ), yielding P ( N ) = P ( N | H ) P ( H ) , (8)where these empirical data are reported in Supplementary Table S7.Finally, the required probability is computed by substituting Equations (6), (7), and (8)into Equation (5).When an agent dies, they are removed from all public places, hospitals, or current res-idences. Recovered agents become active in public locations, and if hospitalized, return totheir households or retirement homes. Recovered agents who were previously hospitalizedfor a condition other than COVID-19 are readmitted to the hospital.10 .7 Susceptible agents with COVID-19-like symptoms Susceptible agents with COVID-19-like symptoms do not exist in the model until the onsetof testing. Once testing efforts begin, these agents are assigned as a fraction of those whoare still susceptible. A portion of these agents will undergo testing and receive either a falsepositive or a true negative result. Therefore, the probability of a susceptible agent withCOVID-19-like symptoms to be tested, P ( T, Sy,
CoV), is given as P ( T, Sy,
CoV) = (cid:0) P (true negative | Sy ) + P (false positive | Sy ) (cid:1) P ( T ) . (9)We assume that the probability of getting a false negative among COVID-19-infected symp-tomatic agents when the epidemic prevalence is low is negligible. [56, 57] We can then approx-imate the probability of a true negative by the probability of receiving a negative result.If scheduled to be tested, the agent is assigned a test time and a testing site (either arandomly chosen hospital or a drive-through test). When the test occurs, the time is selectedfrom a Gamma distribution to avoid these agents undergoing home isolation and testingsimultaneously. Similar to the procedure for an infected agent undergoing testing, the agentdisplaying COVID-19-like symptoms is placed under home isolation for a certain amountof time before testing occurs. Home isolation lasts until the agent is confirmed negative orreaching “recovery” after a false-positive result. The duration of the home isolation beforethe test and the subsequent wait time for results is the same for these agents as for theinfected ones.If an agent with COVID-19-like symptoms contracts COVID-19, they become an exposedagent. To maintain a fixed fraction of such agents in the population, a new susceptible agentis then randomly chosen to take their place, provided such agents are still present in thepopulation.
Our model provides options to simulate school closure, lockdown, and three reopening phases,1, 2, and 3. School closures are simulated by zeroing the transmission rates of studentsand employees. The business closure and reopening are implemented through user-definedreduction or increase of the initial workplace transmission rates, respectively. The model alsoallows for adjusting the absenteeism correction of a workplace, that is, ψ (cid:96) in Equation (3),to a lockdown value, valid through the reopening phases. The transmission parameters forhouseholds, hospitals, and retirement homes remain unchanged throughout the simulation. Parameters originate from several sources: established literature data used in otherABMs [19–22] , clinical data on COVID-19 [22, 49, 51, 58] , and information from a clinical consultantwho is part of the team. In addition, due to the lack of concrete data, some of our parametersare informed estimates, in line with the current understanding of COVID-19 from scientificliterature and the media. Furthermore, some parameter types are identified from reporteddata through model calibration. These latter parameters are the number of initially infectedagents, time-varying testing prevalence, COVID-19 transmission changes following closures11nd reopening phases, and asymptomatic agents’ age distribution. Parameters, data sources,and assumptions are listed and indicated in Section S2 of the Supporting Information.There are four groups of model parameters: COVID-19 transmission dynamics param-eters, testing parameters, parameters related to closure and reopening events, and otherparameters, all listed as Tables in the Supporting Information. Transmission dynamics origi-nates from the COVID-19 agent-based model in Ferguson et al. [22]
While not explicitly stated,the transmission dynamics parameters used therein mirror those previously developed for in-fluenza by the same research group. Such a choice is justified since COVID-19 is a respiratorydisease that spreads [19, 20] similarly to influenza. However, to make the transmission ratesmore representative, we further scale them by the ratio of reproductive numbers, R , for thesetwo diseases. R represents the average number of secondary infections directly caused by asingle infected individual. [59] Following analogous models and procedures, R for COVID-19was estimated to be 2 . [22] while for influenza it was reported as R = 1 .
7, resulting in ascaling factor of 1 . [20] Hospital-related transmission rates are calculated by scaling equivalent non-hospital rateswith data from our clinical consultant in Italy. Specifically, we use the fact that there wasa 7.2% increase in infection among hospital employees in a given week in Italy compared toa 3.7% increase in the general population. Thus, we use a ratio of these percentages as ourscaling factor to multiply a base rate of choice. The base rate for a hospital employee is theworkplace rate, and for an agent hospitalized as a non-COVID-19 patient, it is the householdrate. Other hospital rates are set relative to these following personal communication withthe clinical consultant in Italy.Hospitalization duration in the model is derived from the literature, [22] and linearly scaledby a factor of 0 .
39 according to the data in the paper by Richardson et al. [49] which are spe-cific to the geographic region considered in this work. Specifically, the study by Richardsonet al. [49] provides actual hospitalization duration in New York City, though without distin-guishing between ICU and non-ICU treatment, and relative lengths of these two. Hence, weuse the ratio of the total hospital treatment duration reported in Richardson et al. [49] andFerguson et al. [22] to obtain locally realistic hospitalization periods.
To demonstrate our platform’s viability, we simulated the spread of COVID-19 from February22 nd to July 14 th , 2020, from the onset of the epidemic to phase three of the reopening. Inparticular, this window includes the period in which all the schools in the town of NewRochelle were closed (March 13 th , 2020), followed by the restrictive state-wide lockdownduring which only essential businesses, such as grocery stores, were allowed to operate.In our calibration, we used officially reported data on the total number of detected infec-tions, number of people currently infected with the disease, and the total number of fatalities.From the cumulative number of cases and mortality, we extracted the number of new casesand deaths reported each week. To calibrate upon this dataset, we varied the initial numberof infected agents, the percentage of tested population, the reduction in workplace trans-mission rates during the lockdown and its subsequent increase during the reopening periods,and age-dependent fractions of asymptomatic agents as is summarized in Section S2 of the12upporting Information. Parameters were manually initialized, while the testing percentageswere later refined using simplex optimization in MATLAB via the fminsearch function.Testing prevalence was set to vary with time, mimicking the actual testing practices in theregion. [43] In other words, the fraction of infected individuals who could be tested was time-dependent. To match it, we used the data on newly confirmed cases every week during thesimulation period, computed from total detected cases. We performed 100 realizations of thesimulation, randomly selecting a fixed number of initially infected agents each time. All theparameters used in the simulations are listed in Section S2 of the Supporting Information.The computational performance of the model is summarized in Section S4 of the SupportingInformation; the implementation is fairly efficient and approachable for general use, with 600steps (150 days) of the simulation taking less than 30s on a standard laptop.Figures 4 and 5 show the results of the validation. Figure 4 compares the model outputwith real data along with five different metrics: i) the total number of cases, ii) the number ofnew cases, iii) the weekly average of active cases, iv) the total number of deaths, and v) thenumber of deaths in a week. The total number of cases was calculated as the number of agentswho tested positive, including false positives and those who died without treatment. Thenumber of new weekly cases was calculated as the weekly increase in the total number of cases.Working with weekly averages facilitates comparisons by filtering out spurious oscillationsfrom uneven reporting and data collection by the authorities. In our model, the number ofactive cases was computed as the number of agents undergoing treatment confirmed positiveor false positive. These were compared directly to the reported weekly average. The numberof deaths includes both treated and undetected, untreated agents, assuming that COVID-19would be confirmed in individuals who have died, regardless of their testing status. Thenumber of deaths per week was computed as the weekly rise in total fatalities.A comparison of the total number of cases shows good agreement between the model andreal data obtained from official outlets of Westchester County. [42]
Similarly, the number ofnew cases is well predicted by our model. Looking closely, however, the model has a smoothprogression of the disease compared to a sudden high number of initial cases in the real data.We note that our model imparts a simplistic scenario of the testing practice, whereby anaggressive contact tracing followed the town’s initial case detection. [60]
This likely resultedin a large difference in the number of initial cases in our model versus the real data. Simulatingthis particular scenario is currently outside the scope of our model.In the case of weekly averages of active cases, the model reasonably matches real datatrends. At the same time, the mean value predicted by the model is slightly lower than thereported values, likely due to longer recovery times of COVID-19 patients than utilized inthe model. We note that compared to the total number of reported cases, which provideinformation about new infections, the number of active cases also includes the process ofrecovery. Regarding the number of deaths, the reported values were obtained for the entireWestchester County and scaled down to New Rochelle, proportionally to its population. Heretoo, we find a close agreement between simulated and real data.Figure 5 compares the total number of tests and positivity in our simulations with theavailable data from local testing practices. Similarly to the number of deaths, this data wasreported at the county level and was scaled down to match the population of New Rochellefor a meaningful comparison. Our results indicate reasonable agreement in the early phases ofthe epidemic with discrepancies later on. The lower number of total tests in our model is due13o the rule used for testing, which is based on the total number of exposed and symptomaticindividuals. In contrast, in reality, testing was ramped up to include the general population.The only susceptible agents who can be tested in the model, in its present incarnation, arethose exhibiting COVID-19-like symptoms. The fraction of these individuals is low andchosen upfront in the model. This trend is also visible in the positivity values, wherebywe find an inflated positivity in our simulations by a factor of four compared to the realdata, again due to limited testing in our model. This difference shows that negative testingoutcomes do not affect the general number of cases, as evidenced by the model agreement onthe number of cases and deaths.Figure 6 shows the number of agents who were undergoing a given treatment type. Thenumber of agents isolated at home comprised individuals who were waiting for a test ortest results. The prevalence of each type of treatment qualitatively matches the generaldistribution of cases. Home isolated individuals constituted the bulk of infected agents,followed by hospitalized individuals, and finally, a few hospitalized in ICUs.According to the New York State Department of Health, the New Rochelle hospital has211 general and 12 ICU beds at its disposal under normal circumstances. [39]
In all realizationsof our model, the number of hospitalized agents was always below the reported normal bedvolume. In the model, the ICU demand was on average within standard hospital capacities,but in many simulations exceeded it two or even threefold. Given the expansion of hospitalcapacity as a response to COVID-19 [61] and in the absence of reliable data, we consider thisagreement reasonable.
To demonstrate the value of our platform, we performed a comparative analysis of differentvaccination strategies. Specifically, we evaluated the effect of vaccinating only high-riskgroups of individuals, hospital, school, or retirement home employees, or retirement homeresidents and compare the results to a random immunization across the entire population.The time period of this prospective study was aligned with the first wave of the epidemic,making the previously calibrated model the basis for the prediction. None of the parameterswere changed in this study with respect to the earlier validation. The only differences inthis vaccination study were the absence of school closures and any form of lockdown. In thiscontext, the study also investigated the consequences of leaving schools and non-essentialbusinesses open throughout the first wave of the epidemic upon the availability of a vaccine.Vaccination was implemented in the simulation on March 2 nd , simultaneously with thebeginning of testing in New Rochelle. All the vaccines were distributed simultaneously, grant-ing full protection against the disease. By then, some of the agents were already infected andwere therefore excluded from vaccination. Susceptible agents with COVID-19-like symptomswere not vaccinated either, in an attempt to maintain an approximately fixed number ofsuch agents in the simulation. We performed six sets of simulations with vaccinations of:i) hospital employees only, ii) school employees only, iii) retirement home employees only,iv) retirement home residents only, v) randomly selected fraction of the population, withthe same size as the number of hospital employees, and vi) about a quarter of the town,corresponding to ten times the number of hospital employees.14igure 7 shows the predictions from these six “what-if” scenarios. The importance ofclosures is evident, with numbers of infections and fatalities exceeding reality many times.The vaccination of hospital employees resulted in only minor differences compared with thevaccination of an equivalent number of individuals among the general population. Simi-lar observations can be made about targeted immunization of other vulnerable groups ofagents. Significant differences only occur in mortality when vaccinating the elderly residentsof retirement homes. Although both, targeted and random, approaches had some effect onCOVID-19 spread, massive immunization was the only truly impactful strategy. This findingis consistent with “herd immunity” predictions where effective containment of COVID-19can only be achieved with the large majority of the population acquiring immunity. [62] Until widespread vaccination efforts are underway, maintaining a balance between safetyand normalcy during the current COVID-19 crisis requires the use of non-pharmaceuticalprevention measures as well as efficient detection strategies. The large number of testingstrategies, unknowns, and high levels of uncertainties of this epidemic calls for the principleduse of predictive computational models, potentially informing policy-making with respect towidespread vaccination efforts.In this work, we proposed a high-resolution ABM of COVID-19, developed for granu-lar simulations of a small city or town, where each individual is explicitly modeled. Weintroduced several elements of novelty with respect to state of the art on ABM, includingi) different testing strategies in hospitals and drive-throughs; ii) time variations in testingprevalence; iii) multiple types of treatment, from home isolation to hospitalization in anICU; iv) the presence of susceptible agents who have COVID-19-like symptoms due to otherinfections; v) explicit modeling of employees of hospitals, schools, and retirement homes;vi) school and business closures and reopenings; vii) comprehensive model calibration withofficially reported data; and viii) incorporation of expert knowledge from the field.We applied our model to the US town of New Rochelle, where one of the first COVID-19outbreaks in the country took place. Using an in-house, detailed database of building loca-tions, public and residential, and Census data, we created a geographically and statisticallyaccurate representation of the town and its population. We demonstrated the possibility ofaccurately capturing the first wave of the COVID-19 epidemic in the town.As New Rochelle is a representative US small town, we believe that our validated modelcan serve as an analysis platform for numerous similar towns across the entire country,many currently facing the COVID-19 crisis. To illustrate the model’s value in analyzingprospective “what-if” questions, we performed an immunization study in which we evaluatedseveral vaccination strategies of future importance. In particular, we compared the impact ofvaccination of select group of vulnerable individuals, including school employees, retirementhome employees and residents, and the totality of the two thousand hospital employees in thetown, a randomly selected group of two thousand individuals, and twenty thousand randomlyselected individuals out of the eighty thousand people living in New Rochelle. Our resultssuggest that prioritizing vaccination of high-risk individuals has a marginal effect on the countof COVID-19 deaths. Predictably, a much more significant improvement is registered when15 quarter of the town is vaccinated. Importantly, the benefits of the restrictive measuresin place during the first wave greatly surpass those from any of these selective vaccinationscenarios.While undoubtedly useful, our model bears several limitations. First, the model lacksexplicit agent mobility and random contacts, which manifest in a faster decline of the epidemicnear the end of the simulation. The original model by Ferguson et al. [19, 20] , serving as thebasis for our own, had an additional term to the model disease spreading through randomcontacts in the community. However, these contacts were based on commute and travel dataat the level of the entire country, which is not directly applicable to the problem at hand of asmall town. Simulating truly random interactions using a contact network approach similarto Hinch et al. [24] may offer an alternative, which will be part of our future work. Alongthese lines, the impact of local travel and commute can further be included in the model byintegrating traffic flow simulations. [63–65]
In addition to mobility and random contacts, our model does not include testing thegeneral population, leading to possible under-detection of cases in later reopening stages.While the model allows for testing uninfected individuals with symptoms, massive communitytesting is needed to align its outcomes with reality in later phases of the epidemic. Testing ofgeneral population along with random interactions is also expected to highlight the effects ofdifferent testing strategies already encoded within the model. Combined with contact tracingoptions, testing of general population is part of the next step in our platform’s development.Another limitation of our approach is in the modeling of hospitals in terms of theirworkforce and capacity. Specifically, we assume that infections of hospital employees donot trigger changes in the treatment of hospital patients and that hospitals have infinitebeds and ICUs. Finally, we do not explicitly account for the use of personal protectiveequipment (PPEs), such as face coverings, and social distancing of agents. While thesemeasures are included indirectly through reduction of disease transmission during the lock-down and reopening phases, the ability of specific agents to protect themselves from thecontagion would improve the granularity of the model and add a further realistic element,at the expenses of the computational burden. Advantageous impact of PPEs and distancingcan be introduced to both susceptible and infected agents in a similar way to agent’s currentinfectiousness variability, and this will be one of our goals in the nearest future.Despite these limitations, our model matched real data very closely as the epidemicprogressed through its initial stages. This correspondence allowed us to prospect and analyzealternative scenarios for COVID-19, in which vaccination was accessible right at the onset ofthe first wave. Beyond the timely study of vaccination strategies, our model can be adaptedto explore a range of pressing problems that are ahead of us by interested users who candirectly modify our open-source platform. For example, the model can be swiftly adapted todescribe the concurrent spread of influenza with COVID-19, which is expected to exacerbatethe impact of second and third waves. Likewise, the model can provide clear and quantitativesupport to the long-debated recommendations regarding the need to avoid large gatheringsand always use masks.
Supporting information
The database and the code for generating the town’s population are available at https://github.com/Dynamical-Systems-Laboratory/NR-ABM-population . The ABM code is16vailable at https://github.com/Dynamical-Systems-Laboratory/ABM-COVID-DSL . Author contributions
Conceptualization - AT, SB, ZPJ, AR, MP; methodology - AT, LZ, SB, ZPJ, AR, MP;software - AT; validation - AT; formal analysis - AT, BB, LZ; investigation - all the authors;resources - MP; data curation - JH; writing – original draft preparation - AT, BB, LZ; writing– review and editing - JH, SB, EC, ZPJ, AR, MP; visualization - AT; supervision - SB, EC,ZPJ, AR, MP; project administration - MP; funding acquisition - SB, ZPJ, AR, MP.
Acknowledgements
We would like to acknowledge Kyle C Payen for collecting and post-processing the of-ficially reported COVID-19 data, Malav Thakore for help with writing and verifying themanuscript draft, and Anna Sawulska for help with designing the graphical abstract. Thiswork was partially supported by National Science Foundation (CMMI-1561134, CMMI-2027990, and CMMI-2027988), Compagnia di San Paolo, MAECI (“Mac2Mic”), the Euro-pean Research Council (ERC-CoG-771687), and the Netherlands Organisation for ScientificResearch (NWO-vidi-14134).
Conflict of Interest
The authors declare no conflict of interest.
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Immunity , , 737.[63] E. Frias-Martinez, G. Williamson, V. Frias-Martinez, In . , 57–64.[64] H. Barbosa, M. Barthelemy, G. Ghoshal, C. R. James, M. Lenormand, T. Louail,R. Menezes, J. J. Ramasco, F. Simini, M. Tomasini, Phys. Rep , , 1.[65] H. Zheng, Y. Son, Y. Chiu, L. Head, Y. Feng, H. Xi, S. Kim, M. Hickman, A primerfor agent-based simulation and modeling in transportation applications (FHWA-HRT-13-054), Technical report, Federal Highway Administration, .22igure 1: Map of New Rochelle, NY, which highlights the residential and public buildingsincluded in the database. 23igure 2: Select characteristics of the created (virtual) households: a) sizes of residentialbuildings across town; b) percentage of households of a given size (Census data in brack-ets); c) distribution of employed members per family (Census data in brackets); and d) agedistribution of the population. 24igure 3: Schematic representation of modeled agent states and their possible transitions.Agent in the model can be in one of the following states: susceptible ( S ); exposed ( E );symptomatic ( Sy ); removed - dead ( D ); removed - healthy/recovered ( R ); Agents in differentstates can undergo testing in a test car ( T C ), or a hospital ( T Hs ) after which they can betreated through home isolation ( I Hm ), normal hospitalization ( H N ), or hospitalization in anintensive care unit, ICU ( H ICU ). In addition to symptomatic agents, exposed agents andagents who have COVID-19-like symptoms but are not infected can also be tested. Exceptfor the symptomatic agents, all positive test results, including false positives, will lead tohome isolation. 25igure 4: Comparison of the modeled COVID-19 epidemic and officially reported data: a)The cumulative number of infections; b) New infections detected within a week; c) Activecases averaged over each week; d) The total number of deaths; e) Number of deaths in eachweek. The grey lines represent each of the simulation’s 100 realizations, the blue line is theaverage value, and black circles are the reported data.Figure 5: Comparison of modeled and reported testing practices: a) The total number ofperformed tests; b) Fraction of positive test results, including false positives: the grey linesrepresent each of 100 realizations of the simulation, the blue line is the average value, anddashed black line is the reported data. 26igure 6: The number of agents undergoing each of the three treatment types at differenttime points in the simulation. The grey lines represent 100 realizations of the simulation; theblue line is the average value. 27igure 7: These figures show the spread of COVID-19 epidemics across a range of “what-if”scenarios, when no lockdowns or closures were set in place, and vaccinations were admin-istered selectively. Only means are reported for visual clarity; also, officially reported data(black circles) are presented for reference. In the scenario where the only vaccinated agentswere ones from specific high-risk groups the vaccination covered on average 2,201 hospitalemployees ( ± ± ± ± upporting Information for:High-resolution agent-based modeling of COVID-19 spreading in asmall town Agnieszka
Truszkowska , Brandon
Behring , Jalil
Hasanyan , Lorenzo
Zino , Sachit
Butail , Emanuele
Caroppo , , Zhong-Ping
Jiang , Alessandro
Rizzo , , and Maurizio
Porfiri , , Department of Mechanical and
Aerospace
Engineering and
Department of Biomedical
Engineering,
New
York
University
Tandon
School of Engineering,
Brooklyn NY USA Faculty of Science and
Engineering,
University of Groningen, AG Groningen,
Netherlands Department of Mechanical
Engineering,
Northern
Illinois
University,
DeKalb IL USA Mental
Health
Department,
Local
Health
Unit
ROMA Rome,
Italy University
Research
Center
He.R.A.,
Universit`a
Cattolica del
Sacro
Cuore,
Rome,
Italy Department of Electrical and
Computer
Engineering,
New
York
University
Tandon
School of Engineering,
Jay
Street,
Brooklyn NY USA Department of Electronics and
Telecommunications,
Politecnico di Torino,
Turin,
Italy Office of Innovation,
New
York
University
Tandon
School of Engineering,
Brooklyn NY USA Department of Biomedical
Engineering,
New
York
University
Tandon
School of Engineering,
Brooklyn NY USA Center for
Urban
Science and
Progress,
Tandon
School of Engineering,
New
York
University,
Jay
Street,
Brooklyn, NY USACorrespondence should be addressed to: [email protected] In this section, we detail the analytical expressions used for computation of COVID-19 infectiousness functionsin our model, as outlined in Section 3.2 in the main article.The probability that an agent i who is susceptible at discrete time t will be infected at the following time-step of the simulation depends on the contagion risk Λ i ( t ), which consists of the sum of the infectiousnessfunctions of all the locations that the agent is associated with. Following the notation used in the mainarticle, we denote by f q ( i ) the location of type q associated with agent i , where f q ( i ) = ∅ if no location oftype q is associated with agent i . The expression for the infection risk is represented by the following equation(that is, Equation (2) in the main article):Λ i ( t ) := λ H ,f H ( i ) ( t ) + λ W ,f W ( i ) ( t ) + λ S ,f S i ( t ) + λ Rh ,f Rh ( i ) ( t ) + λ Hsp ,f Hsp ( i ) ( t ) , (S1)where λ H ,f H ( i ) ( t ) is the risk associated with the household in which agent i lives, λ W ,f W ( i ) ( t ) is the riskassociated with agent i ’s workplace, λ S ,f S ( i ) ( t ) is the risk associated with agent i ’s school, λ Rh ,f Rh ( i ) ( t ) isthe risk associated with agent i ’s retirement home, and λ Hsp ,f Hsp ( i ) ( t ) is the risk associated with agent i ’shospital. According to the agent type and their health state, only some of the terms in Equation (S1) maybe different from zero, since an agent may not be associated with a location of type q . Table S1 lists, for eachtype and health state of agents, the locations that are associated to the agent and give a nonzero contributionto their corresponding Equation (S1).The infectiousness function of a location ‘ of type q at time t is defined by Equation (3) of the mainarticle, that is, λ q,‘ ( t ) := 1 n α q ‘ n ‘ X k =1 ( E k ρ k β q,k + Sy k ψ ‘ c k ρ k β q,k ) . (S2)Therein, specific elements may be kept or omitted, depending on location characteristics. Equation (S2)can be further detailed for different types of locations. Equations (S3)–(S7) refer to household, workplaces,schools, retirement homes, and hospitals, respectively. In these equations, to discriminate among differentcontributions to the formation of the infectiousness functions by agents belonging to different categories, weuse indicator functions denoted as E k and Sy k , that indicate a generic category and the function takes valueone if the agent k belongs to that category, and zero otherwise. All the indicator functions used are listed inTable S2.In all the Equations (S3)–(S7), ρ k ≥ c k > n ‘ is the number of agents at that location. Household.
The infectiousness function of a household ‘ consists of two separate contributions ofuntreated agents that live in the household and agents that are home isolated in the household, denoted bysubscripts Ut and Ih, respectively, yielding the following expression: λ H ,‘ = 1 n α‘ n ‘ X k =1 ( E k, Ut ρ k β H , Ut + E k, Ih ρ k β H , Ih + Sy k, Ut c k ρ k β H , Ut + Sy k, Ih c k ρ k β H , Ih ) , (S3)where α is the size scaling parameter, β h, Ut is the transmission rate of untreated agents, and β h, Ih is thetransmission rate of agents who are undergoing home isolation, as part of the testing procedure or subsequenttreatment. Workplaces.
For a generic workplace ‘ , that is, one that is not a school, a retirement home, or a hospital,the infectiousness function is defined as λ W ,‘ = 1 n ‘ n ‘ X k =1 ( E k, W ρ k β W + Sy k, W c k ρ k ψ W β W ) , (S4)where β W is the transmission rate of the workplace and ψ W is the absenteeism correction. The latter denotesthe portion of agents who will still be present at their workplace, regardless of having COVID-19 symptoms.For example, ψ W = 0 .
20 means that 20% of symptomatic agents will still go to work. If an individual i is2 ype of susceptible agent λ H λ W λ S1 λ S2 λ Rh λ Hsp
Hospital employee • •
Hospital employee attending school s • • • Non-COVID-19 patient in a hospital • Agent with COVID-19-like symptoms at the time oftesting • Home isolated retirement home resident with COVID-19-like symptoms • Home isolated agent with COVID-19-like symptoms • Agent with COVID-19-like symptoms who works • •
Agent with COVID-19-like symptoms who works at aschool s • • Agent with COVID-19-like symptoms who works at aschool s s s s • • • Agent with COVID-19-like symptoms who works at aretirement home and attends school s • • • Resident of a retirement home • Employee of a retirement home • •
Employee of a retirement home attending school s • • • School employee at school s • • School employee at school s s s s • • • Agent who works at a general workplace • •
Agent who works at a general workplace and attendsschool s • • • Table S1: Contributions to the formation of the infectiousness functions for each agent type and healthstate. The dots indicate the terms that are not zero for each agent category. Subscripts to the functions λ correspond to: H - household, W - workplace, S - school, S1, S2 — potentially different schools for schoolemployees if they are both working at a school and attending one, Rh - retirement home, and Hsp - hospital.All agents reside in either a household or a retirement home, unless they are hospital patients treated for acondition different than COVID-19 (e.g., undergoing a surgery). Agents who have COVID-19-like symptomsbut are not infected are in home isolation when waiting for a scheduled test, the test results, and if testedfalse positive. 3 ndicator Description E k, Ut Exposed agent in a household who is untreated E k, Ih Exposed agent in a household who is home isolated Sy k, Ut Symptomatic agent in a household who is untreated Sy k, Ih Symptomatic agent in a household who is home isolated E k, W Exposed agent at a workplace Sy k, W Symptomatic agent at a workplace E k, Emp
Exposed agent who is a school employee E k, St Exposed student Sy k, Emp
Symptomatic agent who is a school employee Sy k, St Symptomatic student E k, RhEmp
Exposed retirement home employee E k, RhUt
Exposed retirement home resident who is untreated E k, RhIh
Exposed retirement home resident who is home isolated Sy k, RhEmp
Symptomatic retirement home employee Sy k, RhUt
Symptomatic retirement home resident who is not treated Sy k, RhIh
Symptomatic retirement home resident who is home isolated E k, HspEmp
Exposed hospital employee E k, Pt Exposed patient previously admitted with a condition different than COVID-19 E k, Tst
Exposed agent getting tested at that time-step Sy k, Pt Symptomatic patient previously admitted with a condition different than COVID-19 Sy k, Hn Symptomatic agent routinely hospitalized with COVID-19 Sy k, ICU
Symptomatic agent hospitalized in an ICU with COVID-19 Sy k, Tst
Symptomatic agent getting tested at that time-stepTable S2: Summary of indicator functions used in Equation (S3)–Equation (S7). The value of each indicatoris 1 if the agent falls into the category in the description, and 0 otherwise. An indicator of 1 means that thisagent will be included in the summation, contributing to the infectiousness function of the location.associated with a workplace that is either a school, a retirement home, or a hospital, then the term λ W ,i ( t )in the expression of λ i ( t ) in Equation S1 is computed according to the corresponding formula for a school, aretirement home, or a hospital in Equations (S5)–(S7), detailed in the following. Schools.
The infectiousness function of a school ‘ accounts for both presences of the employees, denotedwith the subscript Emp, and students, denoted with the subscript St, according to: λ S ,‘ = 1 n ‘ n ‘ X k =1 ( E k, Emp ρ k β S , Emp + E k, St ρ k β S , St + Sy k, Emp c k ρ k ψ Emp β S , Emp + Sy k, St c k ρ k ψ St ,‘ β S , St ) . (S5)An employee contributes to the school infectiousness function with a transmission rate of β S , Emp , whilea student with a transmission rate β S , St . The two categories of agents may, in principle, have differentabsenteeism corrections, ψ Emp and ψ St ,‘ , respectively. Note that the absenteeism correction for students, ψ St ,‘ , depends on the location ‘ , since it can vary across different school types: daycare, primary school,middle school, high school, and college. Retirement homes.
In a retirement home, ‘ , the infection may spread through employees, denotedwith the subscript Emp, and residents, denoted with the subscript Rh. Similarly to a household, residentscan be either untreated, Ut, or home isolated, Ih. The infectiousness function has the following expression: λ Rh ,‘ = 1 n ‘ n ‘ X k =1 ( E k, RhEmp ρ k β Rh , Emp + E k, RhUt ρ k β Rh , RhUt + E k, RhIh ρ k β Rh , RhIh + Sy k, RhEmp c k ρ k ψ Rh , Emp β Rh , Emp + Sy k, RhUt c k ρ k β Rh , RhUt + Sy k, RhIh c k ρ k β Rh , RhIh ) , (S6)4here employees’ transmission rate is β Rh , Emp while the transmission rates of an untreated and a home-isolated resident are β Rh , RhUt and β Rh , RhIh , respectively; ψ Rh , Emp is the absenteeism correction for employees.
Hospitals.
Computation of the infectiousness function of a hospital ‘ should account for several types ofagents: i) hospital employees (denoted by the subscript Emp), ii) patients who were admitted with a conditionother than COVID-19 and got infected while hospitalized (subscript Pt), iii) patients who are routinelyhospitalized with COVID-19 (subscript Hn), iv) patients hospitalized with COVID-19 in an intensive careunit, ICU (subscript ICU), and v) infected agents who are getting tested for COVID-19 in the hospital(subscript Tst). Contributions of all these agents yield the following expression: λ Hsp ,‘ = 1 n ‘ n ‘ X k =1 ( E k, HspEmp ρ k β Hsp , Emp + E k, Pt ρ k β Hsp , Pt + E k, Tst ρ k β Hsp , Tst + Sy k, Pt c k ρ k β Hsp , Pt + Sy k, Hn c k ρ k β Hsp , Ih + Sy k, ICU c k ρ k β Hsp , ICU + Sy k, Tst c k ρ k β Hsp , Tst ) . (S7)Equation (S7) is derived upon a number of model assumptions. First, hospital employees do not come to workonce they develop disease symptoms. This is modeled through the lack of symptomatic contribution and anyabsenteeism correction for hospital employees. Thus, a hospital employee will contribute to λ Hsp ,‘ only asan exposed agent, with a transmission rate of β Hsp , Emp . Such complete absenteeism holds even before thepopulation develops awareness of the disease, reflecting specific hospital conditions as a workplace. However,before the COVID-19 detection period starts, a symptomatic hospital employee may still contribute to theinfectiousness function in a school if they are attending one. A patient admitted with a condition otherthan COVID-19 can contribute both as exposed and as symptomatic agents, with a transmission rate of β Hsp , Pt . After disease awareness and testing begin, hospital employees and patients are always tested upondeveloping symptoms. While all agents who are being tested, including the symptomatic hospital employees,are placed in home isolation during the testing procedure, hospital patients remain in the hospital, but theirstatus and transmission rate change to an agent hospitalized with COVID-19. Furthermore, agents whoare routinely hospitalized with a COVID-19 diagnosis can only be symptomatic and are characterized by atransmission rate of β Hsp , Ih . The same holds for agents hospitalized in an ICU, having a transmission rate of β Hsp , ICU . Finally, any agent tested in the hospital contributes to the contagion at the test’s time step witha transmission rate of β Hsp , Tst . 5
Here, we list all the parameters on which our model relies and reproduce the results in Sections 4 and 5 ofthe main article.Parameters are briefly discussed in Section 3.9 of the main article and are here presented grouped intofour categories: COVID-19 transmission dynamics parameters (Table S3), testing parameters (Table S4),parameters related to closures and reopenings (Table S5), and other parameters (Table S6). Additionally,Table S7 outlines the treatment and mortality statistics, Table S8 shows the age-dependent portion of agentswho will not develop symptoms during the course of the disease, and Table S9 records the fractions of exposedand symptomatic agents tested as a function of time. References are included, and the assumptions madeare clearly stated. The distributions of symptoms onset to death and hospitalization from [1] were digitizedusing the free software WebPlotDigitizer [2]. 6 arameter Value References
Severity correction, c k ρ k Gamma distribution withmean 1, shape parameter0.25, scale parameter 4 [5]Household transmission rate - un-treated, β H , Ut , time − − [6] scaled by 1 . β H , Ih , time − − AssumptionHousehold size scaling parameter, α H β W , time − − [6] scaled by 1 . ψ w β Rh , Emp , time − − AssumptionRetirement home employee absen-teeism correction, ψ Rh , Emp β Rh , RhUt , time − − AssumptionRetirement home resident transmis-sion rate - home isolated, β Rh , RhIh − AssumptionSchool student transmission rate, β S , St , time − − [6] scaled by 1 . β S , Emp , time − − AssumptionSchool employee absenteeism correc-tion, ψ Emp , S ψ S , St β Hsp , Emp , time − − Estimated based on datafrom a clinical consultantTransmission rate of hospital pa-tients with a condition different thanCOVID-19, β Hsp , Pt , time − − Estimated based on datafrom a clinical consultantTransmission rate of hospitalizedagents, β Hsp , Ih , time − − Estimated based on datafrom a clinical consultantTransmission rate of ICU hospital-ized agents, β Hsp , ICU , time − − Estimated based on datafrom a clinical consultantTransmission rate of infected hospi-tal visitors being tested for COVID-19, β Hsp , Tst , time − − Estimated based on datafrom a clinical consultantTable S3: COVID-19 transmission parameters. Assumed values were based on discussion with Clinicalconsultant. 7 arameter Value References
Probability of an agent getting anegative test result, P (negative) 0.89 [7]Fraction of susceptible agents withCOVID-19-like symptoms 0.0483 [8]Probability agent is tested in a hos-pital, P ( T Hsp ) 0.60 Data from a clinicalconsultantProbability of a false negative test, P (false negative) 0.05 Data from a clinicalconsultantProbability of a false positive test, P (false positive) 0.05 Data from a clinicalconsultantWait time for the test, time 2 .
25 days Mid point of data froma clinical consultantTime from test to results, time 1 .
75 days Mid point of data froma clinical consultantTime the testing procedure startsfor a susceptible agents withCOVID-19-like symptoms −
10 + Γ ( x ), with Γ ( x ) avalue from Gamma distri-bution with a shape pa-rameter 25.4621, and scaleparameter 1.4301 AssumptionTable S4: Testing parameters. Parameter Value References
Start of the simulation February 22 nd , 2020 (day0) [9]Start of data collection March 2 nd , 2020 (day 9) A day before first con-firmed case [10]Start of testing March 2 nd , 2020 (day 9) A day before first con-firmed case [10]School closure March 13 th , 2020 (day 20) [11]State-wide lockdown March 22 nd , 2020 (day 29) [12]Reopening - phase I May 26 th , 2020 (day 94) [13]Reopening - phase II June 9 th , 2020 (day 108) [14]Reopening - phase III June 23 rd , 2020 (day 122) [15]Workplace transmission rate reduc-tion - lockdown, β w β W β W β W ψ W arameter Value References Latency period, τ E , time log-normal distribution with1.621 mean and 0.418 standarddeviation, days [16]Number of days between infectionand infectiousness, time 4.6 days [5]Probability of death in an ICU, P ( D | N, T ) 0.5 [5]Time average of probability asymptomatic agent will be tested, P ( T | Sy ) 0.636369 Average ofa calibratedparameterTime spent in a hospital if not ad-mitted to an ICU 3 days [5] and [17]Time before death spent in an ICU 2 days AssumptionTime spent in an ICU if recovering 4 days [5] and [17]Time spent in a hospital after anICU treatment if recovering 2 days [5] and [17]Time from the onset of symptoms torecovery 4.9 days [5]Time from the onset of symptoms tohospitalization Gamma distribution with shapeparameter 0.7696 and scale pa-rameter 3.4192 [1]Time from the onset of symptoms todeath Log-normal distribution with2.6696 mean and 0.4760 stan-dard deviation [1]Number of initially infected agents 22 CalibratedparameterTimestep, ∆ t Age group(years) % symptomaticcases requiringhospitalization % hospitalized casesrequiring ICU Infection FatalityRatio
Age group (years) Fraction imulation step Fraction of exposedagents tested Fraction of symptomaticagents tested × − × −
20 0 4.85 × −
30 4.35 × − × −
40 1.46 × − × − nd ,2020 as step 0. S3 Sociodemographic model
This section describes the steps for generating the population of New Rochelle, NY, from the in-house towndatabase ( https://docs.google.com/spreadsheets/d/1AJAMPvVfxbp5JBC80e0qkgCwemNF2wvC/edit ) and the US Census data. Unless otherwise specified, the Census data is from the 2018 5-yearestimates collected in March and April 2020. A brief overview of the steps and data sources is reported inSection 2 of the main article. Unless otherwise specified, all the described operations use the default roundingand floating-point numbers conversion rules of the Python version with which the program is executed.
S3.1 Generation of households
Households were assigned to the agents using the buildings collected in the database and Census data on thetotal number of households ( https://data.census.gov/cedsci/table?g=1600000US3650617&hidePreview=false&tid=ACSCP5Y2018.CP04&vintage=2017&cid=CP05 2013 001E&t=Housing%3AHousing%20Units ).The database distinguishes between single-family houses, townhouses, and multilevel/multi-unit buildings.We assume that one family house represents one household. The information on townhouses includes anestimate of the number of units a townhouse has, and we treat each unit as a single household. For multilevelbuildings, we estimate the number of households based on the number of floors registered in the database.To estimate the number of households in multilevel buildings, we use the following algorithm:1.
Identify all known housing units
Known housing units are single-family homes and townhouses with a clearly identified number of sub-units.2.
Estimate the number of units in apartment complexes
The number of units in a given complex, n u, Bld , can be approximated as n u, Bld = n Fl , Bld n u, Fl . (S8)Here, n Fl , Bld is the number of floors in the building and n u, Fl is the approximate number of units perfloor calculated as n u, Fl = (cid:22) ( N u, total − N u, known ) N Fl , total (cid:23) , (S9)where N u, total is the total number of units reported in the Census data, N u, known is the number ofknown units identified in Step 1, N Fl , total is the sum of all existing residential floors in New Rochelle,and b·c denotes rounding down to the lower integer.10. Redistribute all the remaining housing units into the multilevel buildings
This is done in a round-robin way — each building gets one unit in a circular fashion until no moreunits are left to distribute.
S3.2 Generation of public places
Our model distinguishes four types of public places: workplaces, schools, hospitals, and retirement homes.The database on those locations includes geographic coordinates, type of place, short description, Census-defined workplace category under which this location can be classified, and the number of employees. Wealso identify the number of patients in each hospital, the number of residents in each retirement home, andthe number of students in each school, along with the school type.Number of employees in workplaces other than hospitals are manually adjusted based on Census employ-ment data ( https://data.census.gov/cedsci/table?g=1600000US3650617&tid=ACSDP5Y2018.DP03&vintage=2018&layer=VT 2018 160 00 PY D1&cid=S2501 C01 001E&hidePreview=false&t=Class%20of%20Worker%3AEmployment%3AEmployment%20and%20Labor%20Force%20Status%3AIndustry%3AOccupation ) indifferent workplace categories. The number of employees in all categories, including hospitals, matches theCensus’s total employed population.The number of full-time teachers and students at a school is estimated by National Center for EducationStatistics ( https://nces.ed.gov/globallocator/index.asp?search=1&State=&city=&zipcode=&miles=&itemname=dav&sortby=name&School=1&PrivSchool=1&College=1&Library=1&CS=FB1D5F98 ). Astandalone preschool is considered a daycare, while for schools that combine more types (like primary throughhigh school), both the earliest and latest types are recorded for further processing. Specifically, if there are N s students in a school and the school combines N t educational levels (e.g., primary and middle school),each level will be nominally assigned N s /N t number of students, rounded down, towards the smaller integer.Hospital data includes the number of employees and the number of in-patients who, on average, reside inthe hospitals. This data is estimated from the records of the New York State Department of Health [18] andthe American Hospital Directory ( ).The number of residents in retirement homes is estimated based on the building’s size, supported by thewebsites of specific homes where possible. S3.3 Assignment of agents into households
After generating all the public and residential places, we use Census data, and the knowledge derived fromit, to create a population of agents and assign it to residential locations as follows:1.
Create agents in each age group
Number of agents in each Census-specified age group is calculated using Census data on populationstructure ( https://data.census.gov/cedsci/table?g=1600000US3650617&tid=ACSST5Y2018.S0101&vintage=2018&layer=VT 2018 160 00 PY D1&cid=S2501 C01 001E&hidePreview=false&t=Populations%20and%20People ) and the total number of agents. The operations round down to the lowerinteger and then use a round-robin distribution scheme to match the total population reported. Themaximum age is assumed to be 100 years old. When an agent is assigned a residence, the count ofagents in their age group is decreased.2.
Assign nursing/retirement home residents and hospital patients
Agents aged 75 and older are randomly distributed into retirement homes based on the estimatednumber of agents in each home. In the same way, agents of any age are randomly assigned as hospitalpatients.3.
Exclude vacant households
The Census data defines the percent of vacant households ( https://data.census.gov/cedsci/table?g=1600000US3650617&tid=ACSCP5Y2018.CP04&vintage=2018&layer=VT 2018 160 00 PY D1&cid=S2501 C01 001E&hidePreview=false&t=Housing%3AHousing%20Units ). The corresponding numberof households is excluded and do not have any agents assigned. This number is obtained by roundingdown to the lower integer. 11.
Assign the age of a householder
Each of the non-vacant households has a householder (head of the household) assigned to it. Thehouseholder has to be at least 18 years old, and the distribution follows the Census data on householderage ( https://data.census.gov/cedsci/table?g=1600000US3650617&tid=ACSDT5Y2018.B09021&vintage=2010&layer=VT 2018 160 00 PY D1&cid=S2501 C01 001E&hidePreview=false&t=Housing%3AHousing%20Units ). The data is first converted to percentages, after which the number ofhouseholders in each age group is calculated. The householder is randomly assigned a specific age inthe group and a household.5.
Select household size
Households are assigned sizes based on Census data on household size distribution ( https://data.census.gov/cedsci/table?g=1600000US3650617&tid=ACSST5Y2018.S2501&vintage=2018&layer=VT 2018 160 00 PY D1&cid=S1101 C01 001E&hidePreview=false&t=Housing ), following roundingdown to the lower integer, finished with a round-robin distribution to match the number of non-vacanthouseholds. Each householder is assigned a randomly chosen household size.6.
Assign remaining household members
Households with more than one member are classified as follows: • Two parents and children • Single parent and children • Family with no children (married couple) • Person living alone • Group of peopleThese categories are assigned using probabilities based on percentages from Census data on householdstructure ( https://data.census.gov/cedsci/table?g=1600000US3650617&tid=ACSST5Y2018.S1101&vintage=2018&layer=VT 2018 160 00 PY D1&cid=S0901 C01 001E&hidePreview=false ). Agesof members follow the knowledge base in [3]: there cannot be more than 15 years difference betweenspouses, and a parent has to be at least 18, and at most 43, years older than the child. Therefore,parents in the model are at most 60 years old, and children are considered as such while below 18 yearsold. During the entire process, ages are randomly chosen within given bounds, and if the agents are notavailable anymore in that age interval, an agent with a random, still available age is assigned to thathousehold. Agents remaining after the end of the distribution are randomly assigned to householdswith four or more members.For a household size of 2, it is first determined if the household is a family. If it is a family, they aredesignated as a married couple or a single parent with a child. Ages are assigned accordingly to theoutlined rules. Households with 3 members are assigned either to two parents and a child, one parentand two children, or a random group of adults. Households with four or more members, if not a randomgroup of adults, have one parent and three children or two parents and two children. These householdsare later randomly assigned to any remaining agents.Agents who are 60 years or older are assigned to households or retirement homes, following the Census-reported percentage of households with one or more members of that age range.
S3.4 Assignment of agents to workplaces and schools
After being distributed in households, agents older than 18 that are not hospital patients or in retirementhomes are distributed into workplaces. This operation is performed by assigning agents to workplaces ina sequential fashion, one workplace after another. The maximum working age is set to 75 years. Hospitalemployees are distributed in the same way except the maximum working age is lower and measures 65 years.Lastly, agents younger than 22 that are not hospital patients are distributed into schools according totheir age, • < −
10 - kindergarten and elementary • −
13 - middle school • −
17 - high school • −
21 - collegeFor each agent, the school that corresponds to their age is chosen randomly from the list of collectedschools. Agents are assigned to a school until it reaches its capacity. If a chosen school is already full, theagent is randomly placed in any other school suitable for their age. If there are no more schools to select,agents in daycare and college age are omitted, while others are added randomly to schools in their age group,exceeding nominal capacities.Since colleges are mostly attended by students from outside New Rochelle who are not counted in theCensus, we approximate college attendance instead of introducing outside agents into the model. Specifically,we select a fraction of the population that is 18 −
21 years old, assumed to pursue higher education, and thenevenly distribute it across all colleges in town. 13
The ABM code is publicly available through our GitHub repository, ABM-COVID-DSL ( https://github.com/Dynamical-Systems-Laboratory/ABM-COVID-DSL ). The code structure and compilation areexplained through an accompanying
README.md file. The documentation is created using Doxygen ( ) — details how to retrieve it are also in the
README . There is also a manualand usage examples. The code was extensively tested, with tests and scripts to run them automaticallystored in the repository.We used the exact simulation setup in Section 3 of the main article to estimate the time needed to runour code. The average time out of 100 realizations, each ran for 600 computational steps, was 27 .
84 s with astandard deviation of 1 .
95 s. This result shows the applicability of the code for purposes requiring numeroussimulations, such as comprehensive parametric studies. The computational time can be further reduced withless data collection.The time complexity of the code was measured with a synthetic population similar to the one used inthe publication. The number of agents was varied from 100 to 1,000,000. Simulations were run for 100computational steps with no data collection. The average time needed to complete the simulations for eachnumber of agents is indicated in Table S10. For the studied interval, the time complexity was very broadlylinear.Timing and complexity analyses were performed on a MacBook Pro laptop computer with a 2.9 GHzIntel Core i5 processor and 16 GB RAM. The compiler was Apple LLVM version 8.0.0 (clang-800.0.42.1)with C++11 standard and − O Number of agents Simulation time, s100 0.08 (0.03)1000 0.11 (0.01)10,000 0.57 (0.04)100,000 6.09 (0.45)1,000,000 162.55 (22.65)Table S10: Simulation time for different number of agents. Each data point represents an average out of 100realizations of the code. Standard deviation is indicated in parentheses.14 upplementary References [1] N. Linton, T. Kobayashi, Y. Yang, K. Hayashi, A. Akhmetzhanov, S. mok Jung, B. Yuan, R. Kinoshita,H. Nishiura,
J. Clin. Med. , , 2, 538.[2] A. Rohatgi, WebPlotDigitizer, https://automeris.io/WebPlotDigitizer , Last accessed online12/20/2020.[3] M. Ajelli, B. Gon¸calves, D. Balcan, V. Colizza, H. Hu, J. J. Ramasco, S. Merler, A. Vespignani, BMCInfect. Dis. , , 1, 190.[4] N. M. Ferguson, D. A. Cummings, S. Cauchemez, C. Fraser, S. Riley, A. Meeyai, S. Iamsirithaworn,D. S. Burke, Nature , , 7056, 209.[5] N. M. Ferguson, D. Laydon, G. Nedjati-Gilani, N. Imai, K. Ainslie, S. B. Marc Baguelin, A. Boonyasiri,Z. Cucunub´a, G. Cuomo-Dannenburg, A. Dighe, I. Dorigatti, H. Fu, K. Gaythorpe, W. Green, A. Hamlet,W. Hinsley, L. C. Okell, S. van Elsland, H. Thompson, R. Verity, E. Volz, H. Wang, Y. Wang, C. W.Patrick G.T. Walker, P. Winskill, C. Whittaker, C. A. Donnelly, S. Riley, A. C. Ghani, Impact of non-pharmaceutical interventions (NPIs) to reduce COVID-19 mortality and healthcare demand, Availableat https://doi.org/10.25561/77482 , Report of the Imperial College London, UK.[6] N. M. Ferguson, D. A. Cummings, C. Fraser, J. C. Cajka, P. C. Cooley, D. S. Burke, Nature , ,7101, 448.[7] Center for Disease Control and Prevention, Testing in the U.S., Available at , Last accessed online: June2020.[8] New York State Department of Health, Influenza activity, surveillance and reports, Available at , Last accessed online:12/20/2020.[9] The Evening Tribune, Coronavirus timeline in New York: Here’s how we got here and where we’reheaded, , Last accessed online 12/20/2020.[10] The official website of New Rochelle, NY, Coronavirus newsflash, , Last accessed online 12/20/2020.[11] CBS New York, Coronavirus update: New Rochelle’s 1-mile containment zone takes effect, district closesall schools, https://newyork.cbslocal.com/2020/03/12/coronavirus-new-rochelle-containment/ , Last accessed online 12/20/2020.[12] The official website of New York State, New York State on PAUSE, https://coronavirus.health.ny.gov/new-york-state-pause , Last accessed online 12/20/2020.[13] The official website of New Rochelle, NY, Coronavirus updates for businesses, , Last accessed online 12/20/2020.[14] The official website of New Rochelle, NY, Coronavirus updates for businesses, , Last accessed online 12/20/2020.[15] The official website of New Rochelle, NY, Coronavirus updates for businesses, , Last accessed online 12/20/2020.[16] S. A. Lauer, K. H. Grantz, Q. Bi, F. K. Jones, Q. Zheng, H. R. Meredith, A. S. Azman, N. G. Reich,J. Lessler, Ann. Intern. Med. , , 9, 577.[17] S. Richardson, J. S. Hirsch, M. Narasimhan, J. M. Crawford, T. McGinn, K. W. Davidson, the NorthwellCOVID-19 Research Consortium, JAMA . 1518] New York State Department of Health, NYS Health Profiles - Montefiore New Rochelle Hospital,Available at https://profiles.health.ny.gov/hospital/view/103001https://profiles.health.ny.gov/hospital/view/103001