Future Evolution of COVID-19 Pandemic in North Carolina: Can We Flatten the Curve?
Omar El Housni, Mika Sumida, Paat Rusmevichientong, Huseyin Topaloglu, Serhan Ziya
FFuture Evolution of COVID-19 Pandemic in North Carolina:Can We Flatten the Curve?
Omar El Housni ∗ , Mika Sumida , Paat Rusmevichientong , Huseyin Topaloglu , Serhan Ziya Department of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027 School of Operations Research and Information Engineering, Cornell Tech, New York, NY 10044 Marshall School of Business, University of Southern California, Los Angeles, CA 90089 Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, NC 27599
July 10, 2020
Abstract
On June 24th, Governor Cooper announced that North Carolina will not be moving into Phase3 of its reopening process at least until July 17th. Given the recent increases in daily positivecases and hospitalizations, this decision was not surprising. However, given the political andeconomic pressures which are forcing the state to reopen, it is not clear what actions will helpNorth Carolina to avoid the worst. We use a compartmentalized model to study the effects ofsocial distancing measures and testing capacity combined with contact tracing on the evolutionof the pandemic in North Carolina until the end of the year. We find that going back torestrictions that were in place during Phase 1 will slow down the spread but if the state wantsto continue to reopen or at least remain in Phase 2 or Phase 3 it needs to significantly expand itstesting and contact tracing capacity. Even under our best-case scenario of high contact tracingeffectiveness, the number of contact tracers the state currently employs is inadequate.
1. Background
This report provides a summary of our analysis into how social distancing measures along withtesting and contact tracing capacity will likely impact the evolution of COVID-19 pandemic withinthe state of North Carolina until the end of 2020. This analysis is based on a model, which wasfirst introduced in [4], a report on the findings of a similar study conducted for New York City. Forthe convenience of the reader, this paper is prepared as a self-contained document that includesa complete description of the mathematical model and therefore parts of it which pertain to themathematical model in general overlap with [4].The first COVID-19 case in North Carolina was identified on March 3rd and the state ofemergency was declared on March 10th. On March 30th, statewide stay-at-home order was put inplace and non-essential businesses were ordered closed. These orders stayed in place until May 8th.Starting with that date, the state entered Phase 1 of its reopening process. On May 22nd, Phase2 took effect with a scheduled end date of June 26th. However, on June 24th, Governor Cooperannounced that this date has been pushed three weeks into the future, to July 17th. It is not clearwhether the state will indeed move to Phase 3 on that date or it will be postponed again.Over the last few weeks, North Carolina has experienced significant increases in the numberof new cases and hospitalizations as the state has continued to lift restrictions. At the sametime, pressures that have forced the officials and other administrators to open up the economy(economical, political, or otherwise) appear to be still there and it is not clear what course of actionwill help balance the competing priorities. Our objective is to investigate the impact of differentsocial distancing measures the state might choose to put in place in the future, whether increasesin state’s testing capacity might help in mitigating the impact of reopening on hospitalizationsand deaths, and what resource demands will be needed for contact tracing. In the following, we ∗ Email addresses of the authors are [email protected], [email protected], [email protected],[email protected], and [email protected]. a r X i v : . [ q - b i o . P E ] J u l ummarize our main findings. For a broad description of our mathematical model, see Section 7.For details, see the Appendix.
2. Move Forward to Phase 3, Stay in Phase 2, or Go Back to Phase 1?
We do not know the exact current testing capacity in North Carolina, however, the data indicatethat the maximum number of tests performed on a single day is between 25,000 and 30,000.Therefore, we chose a testing capacity of 30,000 per day as the base case. We also do not know whatthe testing capacity will be in the future but to get a sense of the impact of the test availability oncontrolling the spread of the disease, we considered 60,000 per day as an alternative scenario. Thetop and bottom panels of Figure 1 respectively provide the trajectory of projected daily deathsunder the testing capacity of 30,000 per day and 60,000 per day. Each data series in the two panelscorresponds to a different level of social distancing that will be practiced starting with July 17th.We consider three possibilities: moving to Phase 3, staying in Phase 2, or going back to Phase 1 allassumed to be in place until the end of the year. It is important to note that we do not expect thestate to stick with any one of these three choices until the end of the year, however, comparing theprojection under each scenario provides some insights into what kind of impact each one of thesechoices will have on the disease’s spread.As we can see from the blue curve in the top panel of the figure, with a testing capacity of30,000 tests per day, moving to Phase 3 will likely be disastrous with a major peak that will arrivesome time in early November and a projected number of deaths of approximately 160,000 by theend of the year. (Labels on the right end of each panel give the total number of deaths by the end of2020.) In fact, such a major peak would mean that hospitals would be overwhelmed to such a levelthat they may not be able to provide their regular level of care and the mortality rates might endup being more than what the data to date suggest and our analysis assumes. If the state choosesto stay in Phase 2, there will surely be fewer deaths, however, we predict that there will still be asignificant increase in daily deaths and hospitalizations compared with current levels with a peakin December.From the bottom panel, we can see that having more tests would definitely help resulting infewer deaths but even a capacity of 60,000 tests per day will not be sufficient to prevent a majorwave of hospitalizations and deaths in Fall. The figure suggests that what will help in reining inthe spread of the disease is going back to the restrictions that were in place in Phase 1 regardlessof whether the testing capacity is 30,000 or 60,000. It appears that increasing testing capacity (atleast to 60,000) will not help mitigate the impact of relaxing social distancing measures.
3. A Closer Look into Testing Capacity and Social Distancing: What Might Help?
The discussion in the previous section suggests that even if the state does not move into Phase 3and stays in Phase 2 until the end of the year, hospitalizations and deaths will soar over the nextfew months even if the daily testing capacity increases to 60,000. Moving back to Phase 1 wouldhelp but given the highly likely economic impact of taking such an action it might be helpful toconsider alternatives. In particular, it is of interest to investigate what levels of testing capacitymight be sufficient for different levels of social distancing that might possibly be experienced in thecoming months as a result of different policies that might be employed.Figure 2 shows the trade-offs between the total number of deaths by the end of the year andthe level of social distancing to be practiced post July 17th, under different testing capacities. Thehorizontal axis shows the level of social distancing, expressed as a percentage relaxation in thesocial distancing norms in reference to what was experienced before Phase 1 under the stay-at-20,000 tests per day60,000 tests per dayFigure 1: Trajectory of the daily deaths under testing capacities of 30,000 and 60,000 per day.3igure 2: Trade-offs between the deaths by the end of 2020 and the social distancing norms.home order. Thus, 0% corresponds to stay-at-home, whereas 100% is full relaxation, correspondingto life before the pandemic with no restrictions in place. Social distancing levels that correspondto Phases 1, 2, and 3 are also marked on the x-axis so that the reader can get a sense of roughlywhat different social distancing levels correspond to. (Social distancing levels in Phase 1, 2, and3 respectively correspond to 25%, 50%, and 75%, which is in line with social distancing estimatesin [6].) The vertical axis shows the total number of projected deaths by the end of 2020. Eachdata series corresponds to a different level of testing capacity. Our analysis assumes that newtesting capacity and social distancing norms become effective on July 17th. In computing theseprojections, we use the reported number of tests performed on each day up to June 15th. BetweenJune 15 and July 17 we assume 30,000 tests are conducted daily.Current discussions and political and economical climate in the state suggest that going backto Phase 1 is not being considered as a serious possibility at least in the near future. In fact,with planned opening of the schools and colleges in late summer, the state is on path to furtheropening. This suggests that, it would make sense to pay more attention to the right half of Figure2, the part that corresponds to relaxation levels that are larger than 50%, i.e., either staying atPhase 2 or moving further to Phase 3 and beyond. There, we can observe that having more testsavailable everyday would make a significant difference. To be clear, having more tests cannot be asubstitute for social distancing. As we can see from the figure, even under the daily testing capacityof 200,000, the total number of deaths by the end of the year would be very large. Social distancingappears to be the only way to keep the spread under control. Still, it is important to note thateven if more tests will likely not alter the dynamics of the disease’s spread it will at least help in4eeping the numbers smaller.
4. Testing Capacity and the Effectiveness of Contact Tracing
One important point to highlight about our model is that it assumes that there is a contact tracingpolicy in place. More specifically, it assumes that close contacts of individuals who test positive areidentified and they are asked to self-isolate, which helps in reducing new infections in the population.In fact, this is the main reason why in our model more testing helps. Without contact tracing, thebenefit of having more tests beyond a certain level would be minimal. Our analysis makes certainassumptions about the effectiveness of contact tracing but there is in fact significant uncertaintyaround this issue. To partially address this uncertainty, we consider alternative scenarios in regardsto contact tracing effectiveness and investigate the changes in our projections. (As we explain inthe Appendix, because we are not aware of any studies on contact tracing in the United States,we used a study conducted in China to come up with a rough approximation for a parameter thatcaptures the contact tracing effectiveness.)We control the effectiveness of contact tracing via the parameter CTEP, which stands for
Contact Tracing Effectiveness Parameter . For details on CTEP, we refer the reader to theAppendix, where likOfBeingInfected is used for CTEP, but it might be useful for the readerto know that CTEP is basically an indicator of how successful contact tracing is in identifyingasymptomatic infected individuals over the non-infected ones. If CTEP = 1, this means thatcontact tracing is of no use at all and higher values of CTEP indicate higher likelihood of identifyinginfected people or more effective contact tracing.Our analysis above assumed CTEP = 5 based on our rough approximation for the parameter.Figure 3 considers two additional cases, one with CTEP = 2 and the other with CTEP = 10 allunder the assumption that the state will continue to operate in Phase 2 until the end of the year.Figure 3: Impact of Contact Tracing Effectiveness on Projections for the Daily Number of Deaths.Not surprisingly, when contact tracing is more effective, number of deaths decline considerably.Nevertheless, even when CTEP = 10, we project a very large number of deaths and hospitalizations5y the end of the year with a major peak in late Fall. However, it is important to remind the readerthat, in fact, we do not know how effective contact tracing is or will be in the future and it is possiblethat even setting CTEP to 10 might be underestimating the actual effectiveness. We do hope thatis the case as that would suggest that, given the substantial decreases in the projected numberof deaths with increased contact tracing efficiency and perhaps combined with increased testingcapacity, the state might be able to keep the spread somewhat under control. The critical questionin that case would be whether the state has the sufficient resources for contact tracing. This iswhat we investigate next.
5. Projected Resource Needs for Contact Tracing
There are mainly two components of contact tracing. First, close contacts of individuals who testpositive are identified and tracers contact these individuals informing them about the possibilityof their having been infected and giving them directions as to what they need to do over a periodof roughly two weeks. Second, during this period, when these close contacts need to self-isolate(unless they are tested and result comes negative), tracers periodically contact them to check upon their conditions and ensure that they are self-isolating. Figure 4 shows our daily projections fornew cases, who need to be interviewed to determine their close contacts while Figure 5 shows ourprojections for the number of individuals who need follow-up calls from contact tracers.Figure 4: Projections of Daily Number of New Positive Cases.Again, we can see from the figures that contact tracing resource needs peak some time in Fallregardless of how effective tracing is, however, the effectiveness has a significant impact on thenumber of tracers that would be needed. For the mid-range effectiveness scenario, where CTEP =5, the maximum value for the number of daily new positive cases is 7,121 whereas the maximum6igure 5: Projections of Daily Number of Isolated Individuals who Need Follow-up.value for the number of isolated individuals who need follow-up is 127,617. According to theestimates by the Fitzhugh Mullan Institute for Health Workforce Equity of the George WashingtonUniversity, a single contact tracer, assuming an 8-hr workday, would be able to interview six indexcases (individuals who test positive), or make 12 initial contact notifications, or make 32 follow-upcalls [5]. This would then mean that North Carolina would need roughly 7,550 contact tracers.(This number is very close to 7,700, the current estimate by the Fitzhugh Mullan Institute). Ourestimate would be roughly 11,630 for CTEP = 2, and 2,860 for CTEP = 10. According to recentreports [14], North Carolina has roughly 1,500 contact tracers meaning that even under our moreoptimistic scenario of high contact tracing effectiveness, the state’s tracing resources would fallshort of meeting the demand.
6. Discussion
Our main objective in this work has not been to make precise predictions for key metrics likenumber of deaths, number of hospitalizations etc. but rather to understand the dynamics of thespread of the disease, how these dynamics would depend on the state’s testing and contact tracingcapacity, and provide insights into whether the existing capacity is sufficient for the state to reopenwithout overwhelming its healthcare resources. Therefore, the reader should take the precisevalues of the estimates we report in this paper with a grain of salt but give credence and paymore attention to the insights we generated through their analysis collectively. Unfortunately, thisanalysis strongly suggests that even if North Carolina does not move to Phase 3 on July 17th,without further restrictions, the number of new cases, hospitalizations and deaths will continue toincrease substantially peaking some time in Fall.7here are several ways that this grim prediction may not become reality. One possibility isfor the state to reimpose some of the restrictions that were in place in Phase 1 of the reopening.Alternatively, of course, the state might continue with is reopening plan but take other actions thatwould significantly limit person-to-person transmission such as enforcing face covering requirementsmore strictly. The goal here should be to bring down the transmission rate at least to the levelsachieved under Phase 1. The current economic and political climate in North Carolina suggeststhat reimposing restrictions is not very likely and it is not clear whether strict requirements forface coverings could keep new infections down as the state keeps opening.If the state continues with its reopening plan with no further restrictions, there appears to beonly one way the spread of the disease can be kept under control and that is through extensivetesting combined with effective contact tracing. There is some uncertainty as to how effectivecontact tracing will be but it is safe to say that North Carolina is significantly understaffed to keepup with the workload that will be generated for contact tracers over the next few months. Therefore,it is essential for the state to quickly hire new contact tracers to keep up with the impending jumpin their workload and ensure that contact tracing is effective in identifying infected individuals andprevent them from infecting others in the community. It is also important to understand that testingcapacity and contact tracing capacity are directly tied to each other and any meaningful responseshould include increasing both in some proportional manner. Beyond certain levels of testingcapacity, further increases will not bring much benefit if contact tracers are already overwhelmed.Similarly, hiring more contact tracers will be meaningless unless the state continues to test atcertain numbers everyday.Without further action from the state, one possibility that could alter our predictions is that,with the increased public awareness around how the disease spreads, the infection rates for differentphases might end up being lower than what we assumed in our analysis based on the mobility data.It is also possible that over the next few months, the disease’s spread might follow a course thatis quite different from what we observed so far. This may not necessarily be a result of some formof mutation in the virus but could simply be a result of the changes in the demographics of thepopulation who is infected by the virus over time. In fact, our analysis of the most recent datafrom North Carolina suggests that there appears to be some drop in the overall mortality rate andthis appears to be a national trend. Recent news articles reported that median age of individualswho test positive has dropped substantially since March [13] and such a change in the infectedpopulation would reduce hospitalization and mortality rates. (Interestingly, in North Carolina,even though we observe a drop in mortality rates, hospitalization rate appears to not have changedin any significant manner.) If this trend continues, it is possible that the scale of the problem we willface will be smaller than what we expect based on our analysis. However, it is important to keep inmind that younger individuals who tend to be mostly asymptomatic and are more socially active,can spread the virus more easily to different segments of the population, which might increase thehospitalization and mortality rates back up again. In any case, it would be prudent to not put ourhopes on possibilities that are beyond our control but take actions that would help even under theworst circumstances.
7. Outline of the Mathematical Model
Our model follows the standard susceptible-infected-recovered paradigm with compartmentscapturing individuals classified along the dimensions of infected, noninfected, and recovered, aswell as symptomatic-isolated, asymptomatic-isolated, and asymptomatic-nonisolated. Once anindividual is tested positive, a certain number of individuals who are expected to have been in close8ontact with the positive individual are transferred to a certain isolated compartment. In this way,we build a contact tracing mechanism to ensure that those who have been in contact with a positiveindividual reduce their contact with the rest of the population. Running our model with a startingdate of March 2, we get a trajectory of the pandemic that is in reasonably close agreement withthe actual trajectory in North Carolina so far. In the appendix, we give a discussion of our modeland provide comparisons of its output with the actual trajectory of the pandemic.
References [1] Q. Bi, Y. Wu, S. Mei, C. Ye, X. Z. Zou, Z. Zhang, X. Liu, L. Wei, S. A. Truelove, T. Zhang, W. Gao,C. Cheng, X. Tang, X. Wu, Y. Wu, B. Sun, S. Huang, Y. Sun, J. Zhang, T. Ma, J. Lessler, and T. Fen.Epidemiology and transmission of covid-19 in 391 cases and 1286 of their close contacts in shenzhen,china: a retrospective cohort study, 2020. https://doi.org/10.1016/S1473-3099(20)30287-5 .[2] CDC. Discontinuation of isolation for persons with covid-19 not in healthcare settings, 2020.[3] CDC. Past seasons estimated influenza disease burden, 2020.[4] O. El Housni, M. Sumida, P. Rusmevichientong, H. Topaloglu, and S. Ziya. Can testing ease socialdistancing measures? future evolution of covid-19 in nyc, 2020. https://arxiv.org/pdf/2005.14700.pdf .[5] F. M. I. for Health Workforce Equity. Contact tracing workforce estimator, 2020. .[6] IHME. Covid-19 projections (north carolina), 2020. http://covid19.healthdata.org/united-states-of-america/north-carolina .[7] Johns Hopkins Coronavirus Resource Center. Hubei timeline, 2020. https://coronavirus.jhu.edu/data/hubei-timeline .[8] Joseph T. Wu, Kathy Leung, Mary Bushman, Nishant Kishore, Rene Niehus, Pablo M. de Salazar,Benjamin J. Cowling, Marc Lipsitch, and Gabriel M. Leung. Estimating clinical severity of covid-19from the transmission dynamics in wuhan, china, 2020. https://doi.org/10.1038/s41591-020-0822 .[9] Q. Li, X. Guan, P. Wu, X. Wang, L. Zhou, Y. Tong, R. Ren, K. Leung, E. Lau, J. Y. Wong, X. Xing,N. Xiang, Y. Wu, C. Li, Q. Chen, D. Li, T. Liu, J. Zhao, M. Li, and Z. Feng. Early transmission dynamicsin Wuhan, China, of novel coronavirus-infected pneumonia. New England Journal of Medicine, 2020.[10] Mizumoto Kenji, Kagaya Katsushi, Zarebski Alexander, Chowell Gerardo. Estimating the asymptomaticproportion of coronavirus disease 2019 (covid-19) cases on board the diamond princess cruise ship,yokohama, japan, 2020. Eurosurveillance, 2020.[11] Natalie Linton, Tetsuro Kobayashi, Yichi Yang, Katsuma Hayashi, Andrei R. Akhmetzhanov, Sung-mok Jung, Baoyin Yuan, Ryo Kinoshita, Hiroshi Nishiura. Epidemiological characteristics of novelcoronavirus infection: A statistical analysis of publicly available case data, 2020. .[12] NBC Boston. Gov. baker signs order requiring mass. residents to wear masks in public, 2020. .[13] New York Times. As virus surges, younger people account for “disturbing” number of cases, 2020. .[14] News and Observer. North carolina now has 1,500 covid-19 contact tracers. study says state needs7,100, 2020. .[15] NYC Department of Health and Mental Hygiene. Nyc coronavirus (covid-19) data, 2020.[16] Reuters. Coronavirus clue? most cases aboard u.s. aircraft carrier are symptom-free, 2020.[17] Steven Sanche, Yen Ting Lin, Chonggang Xu, Ethan Romero-Severson, Nick Hengartner, Ruian Ke.High contagiousness and rapid spread of severe acute respiratory syndrome coronavirus. EmergingInfectious Diseases, 2020. https://doi.org/10.3201/eid2607.200282 .[18] Wang, Dawei and Hu, Bo and Hu, Chang and Zhu, Fangfang and Liu, Xing and Zhang, Jing and ang, Binbin and Xiang, Hui and Cheng, Zhenshun and Xiong, Yong and Zhao, Yan and Li, Yirongand Wang, Xinghuan and Peng, Zhiyong. Clinical characteristics of 138 hospitalized patients with 2019novel coronavirus-infected pneumonia in Wuhan, China. JAMA, 2020. ppendix The appendix is organized as follows. In Appendix A, we discuss our model at a high level. Thissection avoids any technical details but it is useful for seeing how our model works, whatcompartments it uses, and how the different compartments interact with each other. In Appendix B,we provide the full mathematical details of our model. In Appendix C, we elaborate on theparameters of our model and how they are estimated. We borrow some of the parameters from theliterature and derive others based off of problem primitives. We explain how we arrive at the derivedmodel parameters. In Appendix D, we compare the output of our model with the trajectory of thepandemic in North Carolina until early May and demonstrate that our model does a reasonablygood job of predicting the trajectory that has been observed so far.
A. Model at a High Level
In our model, we capture the two benefits of testing, which are isolating the specific individualtested positive and tracing the contacts of the positive individual. In particular, our modeldistinguishes the infected individuals who know that they are infected from the infected individualswho do not know whether they are infected. Once a person is tested positive, this personis an infected individual who knows that she is infected. Such a person minimally infectsothers. Moreover, considering the individuals who do not know whether they are infected, weclassify them as symptomatic-isolated, asymptomatic-isolated, or asymptomatic-nonisolated. Oncean individual is tested positive, a certain number of people around this person are classifiedas asymptomatic-isolated people, so these individuals reduce their contact with the generalpopulation, preventing them from infecting others to some extent.Our model follows the standard susceptible-infected-recovered modeling paradigm. In Figure 6,we show the compartments in our model. The boxes represent the compartments. The arrowsrepresent the possible flows between the compartments. The three compartments on the right ofthe figure capture the individuals who know that they are infected. In particular, the compartmentlabeled “KI” corresponds to the infected individuals who know that they are infected and not (yet)hospitalized. The compartment labeled “H” captures the infected individuals who are hospitalized.The individuals in the latter compartment also know that they are infected. The compartmentlabeled “KR” captures the recovered individuals who knew that they were infected. Perhapsoptimistically, in our model, the individuals in all of these three compartments minimally infectothers, since they are aware that they were infected.The nine compartments on the left side of the figure capture the individuals who do not knowwhether they are infected. Some of these individuals will, in actuality, be infected, some willbe noninfected, and some will even have recovered without knowing they had been infected. Weclassify the individuals who do not know whether they are infected along two dimensions. Inthe first dimension, any such individual could be symptomatic-isolated, asymptomatic-isolated, orasymptomatic-nonisolated. Asymptomatic-nonisolated individuals do not show symptoms of thedisease and do not make an effort to reduce their contact. Asymptomatic-isolated individuals makea conscious effort to reduce their contact, mainly because they have been in touch with a personwho tested positive. Symptomatic-isolated individuals show symptoms but do not know whetherthey are infected. Their symptoms may be due to other diseases.Once again, perhaps optimistically, in our model, all symptomatic individuals isolate themselves,so we do not have a compartment for symptomatic-nonisolated people. Symptomatic-isolatedand asymptomatic-isolated individuals infect others with a smaller rate, when compared with11 Si I Ai KI Known
Infected H Hospitalization
Unknown
Non-Infected Unknown
Recovered Unknown
InfectedSymptomatic IsolatedAsymptomatic IsolatedAsymptomatic Non-isolated KR Known
Recovery D Death I An R Si R Ai R An N Si N Ai N An Figure 6: Compartments used in our model.asymptomatic-nonisolated people. When stricter social distancing norms are enforced, allindividuals will reduce their contact with the general population, but we follow the assumptionthat asymptomatic-nonisolated individuals will still maintain a higher contact rate with the rest ofthe population than asymptomatic-isolated and symptomatic-isolated individuals.Considering the individuals who do not know whether they are infected, we classify them alonganother dimension. In this second dimension, an individual who does not know whether she isinfected could be infected, noninfected, or recovered. Thus, considering the nine compartments onthe left side of Figure 6, the compartment labeled “ I An ,” for example, captures the infected andasymptomatic-nonisolated individuals who do not know whether they are infected. Note that theseindividuals are infected in actuality, but they do not know that they are infected and they maintaina high contract rate with the rest of the population, being nonisolated. When we perform tests ona segment of a population, the proportion of positive tests is given by the fraction of the infectedindividuals in the segment relative to the size of the whole segment. For example, using the labelof a compartment to also denote the number of individuals in the compartment, since there are N An + R An + I An asymptomatic-nonisolated individuals, if we test T asymptomatic-nonisolatedindividuals, then the number of positive tests is T × I An N An + R An + I An . So far, we indicated two reasons for our model to be optimistic. In addition, our model assumesthat close contacts of every positive case are traced and these close contacts remain in isolationfor 14 days or until they are tested negative. In other words, our model implicitly assumes thatthe bottleneck is the testing capacity, not contact tracing capacity. It is important to note thatgrowing the testing capacity in the state beyond a certain level without growing the contact tracingcapacity will not be of much benefit, so any decision regarding how much to grow the contact tracingcapacity should be directly informed by the plans to grow the testing capacity.12 . Mathematical Description of the System Dynamics
In this section, we provide a detailed description of the system dynamics of our model. The stateof the system at each time period t is described by a vector (cid:0) I t Si , I t Ai , I t An , R t Si , R t Ai , R t An , N t Si , N t Ai , N t An , D t , H t , KI t , KR t (cid:1) , where the variables I t Si , I t Ai , I t An , R t Si , R t Ai , R t An , N t Si , N t Ai , N t An denote the states associated with thenine compartments on the left side of Figure 6. These nine compartments correspond to individualswhose COVID-19 status is unknown. The variable D t denotes the number of individuals who diein period t , and the variable H t denotes the number of individuals who are hospitalized in period t .The variable KI t captures the number of individuals who have been confirmed to have COVID-19and are currently infected in period t . The variable KR t captures the number of individuals whohave been confirmed to have COVID-19 and recovered by period t . Description of Sub-compartments:
For our state variables, we use regular font to denotescalars and bold font to denote vectors. We use vector notation when a compartment has multiplesub-compartments representing different subgroups of the population. Here are the descriptions ofthe sub-compartments. • Sub-compartments within the I Si compartment: The I Si compartment has 3 sub-compartments:(a) those who will recover naturally without requiring any hospitalization, (b) those who willrequire hospitalization, and (c) those who will die without access to a COVID-19 diagnostic test.Thus, I t Si = (cid:0) I t Si (recovered) , I t Si (hospitalized) , I t Si (death) (cid:1) . We use the notation ¯ I t Si = I t Si (recovered) + I t Si (hospitalized) + I t Si (death) to denote the totalnumber of individuals in the I Si compartment. The trick we use here is that when an individualis infected, we immediately decide whether this person will recover, will be hospitalized, or willdie. We keep the identity of the individual accordingly throughout the simulation. • Sub-compartments within the I Ai compartment: The I Ai compartment has 2 sub-compartments:(a) those who will never develop COVID-19 symptoms, and (b) those who will show COVID-19symptoms but are currently pre-symptomatic. Thus, I t Ai = (cid:0) I t Ai (recovered) , I t Ai (show symptom) (cid:1) , and we let ¯ I t Ai = I t Ai (recovered) + I t Ai (show symptom) denote the total number of individualsin the I Ai compartment. • Sub-compartments within the I An compartments: Similar to the I Ai compartment, the I An compartment has 2 sub-compartments: (a) those who will never develop COVID-19 symptoms,and (b) those who will show COVID-19 symptoms but are currently pre-symptomatic. Thus, I t An = (cid:0) I t An (recovered) , I t An (show symptom) (cid:1) , and as before, we let ¯ I t An = I t An (recovered) + I t An (show symptom) denote the total number ofindividuals in the I An compartment. • Sub-compartments within the H compartment: The H compartment consists of two types ofindividuals: (a) those who will die and (b) those who will eventually recover. Thus, we have H t = (cid:0) H t (die) , H t (recovered) (cid:1) . 13 Sub-compartments within the KI compartment: The KI compartment consists of two typesof individuals: (a) those who will require hospitalization and (b) those who will recover withoutvisiting a hospital. Therefore, we have that KI t = (cid:0) KI t (hospitalized) , KI t (recovered) (cid:1) . Impact of Testing:
We assume that the diagnostic test is 100% accurate and the result isobtained instantaneously. However when we perform the test, we cannot differentiate betweenunknown infected, unknown non-infected, and recovered people. We can only test people based ontheir observable characteristics. At the beginning of each period, we test T Si symptomatic-isolatedindividuals, T Ai asymptomatic-isolated individuals, and T An asymptomatic-nonisolated individuals.We assume that the tests are administered at random within each of these populations. Let π Si , π Ai ,and π An denote the fraction of symptomatic isolated, asymptomatic isolated, and asymptomaticnon-isolated individuals, respectively, who receive diagnostic tests. Then, π Si = min (cid:26) T t Si ¯ I t Si − I t Si (death) + R t Si + N t Si , (cid:27) ,π Ai = min (cid:26) T t Ai ¯ I t Ai + R t Ai + N t Ai , (cid:27) ,π An = min (cid:26) T t An ¯ I t An + R t An + N t An , (cid:27) . The above fractions drive the dynamics across compartments, which are described below basedon the events that can occur in our model. In the fraction π Si , we assume that the infectedsymptomatic-isolated individuals from the death sub-compartment are inaccessible for testing. Description of Dynamics Between Compartments:
Our description below is organizedby the compartments. For each compartment, we describe the inflows, outflows, and the new stateat time period t + 1. • Infected Symptomatic-Isolated Compartment ( I Si ): As noted earlier, the I Si compartmenthas 3 subgroups of individuals: (a) those who will recover naturally without requiring anyhospitalization, (b) those who require hospitalization, and (c) those who die without accessto a COVID-19 diagnostic test. Thus, I t Si = (cid:0) I t Si (recovered) , I t Si (hospitalized) , I t Si (death) (cid:1) , and ¯ I t Si = I t Si (recovered) + I t Si (hospitalized) + I t Si (death) denotes the total number of individualsacross the three sub-compartments.Inflow: The inflow to the I Si compartment consists of 3 sources given by: – Non-infected symptomatic-isolated ( N Si ) individuals who did not get tested and were newlyinfected during period t . The number of such individuals is given by β (cid:96) × (cid:16) ¯ I t An (1 − π An ) + ¯ I t Ai (1 − π Ai ) + ¯ I t Si (1 − π Si ) (cid:17) × N t Si (1 − π Si ) , where ¯ I t An (1 − π An ) + ¯ I t Ai (1 − π Ai ) + ¯ I t Si (1 − π Si ) represents the infected population who remainsuntested, and β (cid:96) represents the effective contact rate among isolated individuals. See SectionC for more details. 14 Infected asymptomatic-nonisolated ( I An ) individuals who become symptomatic. The totalinflow of such individuals is given by I t An (show symptom) infToSympTime , where infToSympTime represents the average number of days until an infected person displayssymptoms. In our model, we set infToSympTime to be 5 days; see Section C for more details. – Infected asymptomatic-isolated ( I Ai ) individuals who become symptomatic. Using a similarlogic as above, the total amount of such inflow is given by I t Ai (show symptom) infToSympTime .We assume that a fraction hospOutOfSympFrac , currently set at 20%, of the total inflow to the I Si compartment will require hospitalization, and a fraction deathOutOfSympFrac (2%) will diewithout being tested. The remaining 1 − hospOutOfSympFrac − deathOutOfSympFrac = 78%will recover at home. Justifications for these fractions are given in Section C.Outflow: There are four destinations for the outflow from the I Si compartment. – Individuals in the “recovered” sub-compartment, I t Si (recovered), will move to the recoveredasymptomatic-nonisolated ( R An ) compartment at the rate of 1 / sympToRecoveryTime , where sympToRecoveryTime represents the average time for a symptomatic individual to recover fromthe disease. Currently, sympToRecoveryTime is set at 14 days. – Individuals in the “hospitalized” sub-compartment, I t Si (hospitalized), will move to the H compartment at the rate of 1 / sympToHospTime , where sympToHospTime (5 days) representsthe average number of days from symptom onset until hospitalization. – Individuals in the “death” sub-compartment, I t Si (death), will move to the death compartment( D ) at the rate of 1 / sympToDeathTime , where sympToDeathTime (14 days) denotes the averagetime from symptoms onset to death. – Finally, a fraction π Si in every sub-compartment will move to the known infected( KI ) compartment due to testing.Update Equations: Let recoverOutOfSympFrac = 1 − hospOutOfSympFrac − deathOutOfSympFrac .We have the following update equations. I t +1 Si (recovered)= I t Si (recovered) × (1 − π Si ) × (cid:18) − sympToRecoveryTime (cid:19) + recoverOutOfSympFrac × (cid:34) β (cid:96) × (cid:16) ¯ I t An (1 − π An ) + ¯ I t Ai (1 − π Ai ) + ¯ I t Si (1 − π Si ) (cid:17) × N t Si (1 − π Si )+ I t An (show symptom)(1 − π An ) infToSympTime + I t Ai (show symptom)(1 − π Ai ) infToSympTime (cid:35) ,I t +1 Si (hospitalized)= I t Si (hospitalized) × (1 − π Si ) × (cid:18) − sympToHospTime (cid:19) hospOutOfSympFrac × (cid:34) β (cid:96) × (cid:16) ¯ I t An (1 − π An ) + ¯ I t Ai (1 − π Ai ) + ¯ I t Si (1 − π Si ) (cid:17) × N t Si (1 − π Si )+ I t An (show symptom)(1 − π An ) infToSympTime + I t Ai (show symptom)(1 − π Ai ) infToSympTime (cid:35) ,I t +1 Si (death)= I t Si (death) × (1 − π Si ) × (cid:18) − sympToDeathTime (cid:19) + deathOutOfSympFrac × (cid:34) β (cid:96) × (cid:16) ¯ I t An (1 − π An ) + ¯ I t Ai (1 − π Ai ) + ¯ I t Si (1 − π Si ) (cid:17) × N t Si (1 − π Si )+ I t An (show symptom)(1 − π An ) infToSympTime + I t Ai (show symptom)(1 − π Ai ) infToSympTime (cid:35) . • Infected Asymptomatic-Isolated Compartment ( I Ai ) : We have two sub-compartmentswith I t Ai = (cid:0) I t Ai (recovered) , I t Ai (show symptom) (cid:1) , and ¯ I t Ai = I t Ai (recovered) + I t Ai (show symptom)denotes the total number of individuals across the two sub-compartments.Inflow: The inflow to the I Ai compartment has two sources: – Non-infected asymptomatic-isolated ( N Ai ) individuals who were not tested and became newlyinfected during period t . The number of such individuals is given by β (cid:96) × (cid:16) ¯ I t An (1 − π An ) + ¯ I t Ai (1 − π Ai ) + ¯ I t Si (1 − π Si ) (cid:17) × N t Ai (1 − π Ai ) , where β (cid:96) denotes the effective contact rate for isolated individuals. – Infected asymptomatic-nonisolated ( I An ) individuals who were not tested but were identifiedas a close contact of a positive case. These individuals voluntarily self isolate and reduce theircontact rate with others. The number of such individuals is given by contactPerPosCase × ¯ I t Si π Si + ¯ I t Ai π Ai + ¯ I t An π An ¯ H × likOfBeingInfected × ¯ I t An (1 − π An ) , where contactPerPosCase is the average number of contacts per positive case, currently setat 4, whereas ¯ I t Si π Si + ¯ I t Ai π Ai + ¯ I t An π An denotes the number of positive cases identified that timeperiod and ¯ H is given by ¯ H = (cid:0) likOfBeingInfected × ¯ I t An (1 − π An ) (cid:1) + ( R t An + R t Si π Si + R t Ai π Ai ) + ( N t An + N t Si π Si + N t Ai π Ai ) , where the parameter likOfBeingInfected measures the likelihood that a contacted individualis in the infected population as compared to the non-infected and recovered populations. Weset likOfBeingInfected at 5.We assume that a fraction symptomFrac , currently set at 50%, of the total inflow to the I Ai compartment will develop symptoms, and the remaining 1 − symptomFrac = 50% will not developany symptoms.Outflow: There are three destinations for the outflow from the I Ai compartment.16 The individuals in the “recovered” sub-compartment, I t Ai (recovered), will move to the R An compartment at the rate of 1 / asympToRecoveryTime . The parameter asympToRecoveryTime represents the average time that an asymptomatic infected person self-isolates after beingidentified as a close contact. We currently set this value to 10 days. – The “show symptom” sub-compartment, I t Ai (show symptom), will move to the I Si compartment at the rate of 1 / infToSympTime , where infToSympTime (5 days) is the averagetime until symptom onset. – A fraction π Ai of individuals who are tested will move to the known infected( KI ) compartment.Update Equations: We have the following update equations. I t +1 Ai (recovered)= I t Ai (recovered) × (1 − π Ai ) × (cid:18) − asympToRecoveryTime (cid:19) + (1 − symptomFrac ) × (cid:34) β (cid:96) × (cid:16) ¯ I t An (1 − π An ) + ¯ I t Ai (1 − π Ai ) + ¯ I t Si (1 − π Si ) (cid:17) × N t Ai (1 − π Ai )+ contactPerPosCase × ¯ I t Si π Si + ¯ I t Ai π Ai + ¯ I t An π An ¯ H × likOfBeingInfected × ¯ I t An (1 − π An ) (cid:21) ,I t +1 Ai (show symptom)= I t Ai (show symptom) × (1 − π Ai ) × (cid:18) − infToSympTime (cid:19) + symptomFrac × (cid:34) β (cid:96) × (cid:16) ¯ I t An (1 − π An ) + ¯ I t Ai (1 − π Ai ) + ¯ I t Si (1 − π Si ) (cid:17) × N t Ai (1 − π Ai )+ contactPerPosCase × ¯ I t Si π Si + ¯ I t Ai π Ai + ¯ I t An π An ¯ H × likOfBeingInfected × ¯ I t An (1 − π An ) (cid:21) . • Infected Asymptomatic-Nonisolated Compartment ( I An ) : We have two sub-compartments, I t An = (cid:0) I t An (recovered) , I t An (show symptom) (cid:1) , and we denote the total numberof individuals across the two sub-compartments as ¯ I t An = I t An (recovered) + I t An (show symptom).Inflow: The inflow to the I An compartment comes from non-infected asymptomatic-nonisolated N An individuals who become newly infected during period t . The number of such individuals isgiven by (cid:16) β h ¯ I t An (1 − π An ) + β (cid:96) ¯ I t Ai (1 − π Ai ) + β (cid:96) ¯ I t Si (1 − π Si ) (cid:17) × N t An (1 − π An ) , where β h represents the effective contact rate among nonisolated individuals. See Section C formore details. We assume that a fraction symptomFrac , currently set at 50%, of the total inflowto the I An compartment will develop symptom, and the remaining 1 − symptomFrac = 50% willnot develop any symptoms.Outflow: The outflow from the I An compartment has four destinations. – The individuals in the “recovered” sub-compartment, I t An (recovered), will move to the R An compartment at the rate of 1 / asympToRecoveryTime .17 The individuals in the “show symptom” sub-compartment, I t An (show symptom), will move tothe I Si compartment at the rate of 1 / infToSympTime . – Some individuals who were not tested will reduce their contacts and move to the I Ai compartment. The number of such individuals is contactPerPosCase × ¯ I t Si π Si + ¯ I t Ai π Ai + ¯ I t An π An ¯ H × likOfBeingInfected × ¯ I t An (1 − π An ) . – A fraction π An of individuals are tested and moved to the known infected KI compartment.Update Equations: We have the following update equations. I t +1 An (recovered)= I t An (recovered) × (1 − π An ) × (cid:18) − asympToRecoveryTime − contactPerPosCase × ¯ I t Si π Si + ¯ I t Ai π Ai + ¯ I t An π An ¯ H × likOfBeingInfected (cid:19) + (1 − symptomFrac ) × (cid:20)(cid:16) β h ¯ I t An (1 − π An ) + β (cid:96) ¯ I t Ai (1 − π Ai ) + β (cid:96) ¯ I t Si (1 − π Si ) (cid:17) × N t An (1 − π An ) (cid:21) ,I t +1 An (show symptom)= I t An (show symptom) × (1 − π An ) × (cid:18) − infToSympTime − contactPerPosCase × ¯ I t Si π Si + ¯ I t Ai π Ai + ¯ I t An π An ¯ H × likOfBeingInfected (cid:19) + symptomFrac × (cid:20)(cid:16) β h ¯ I t An (1 − π An ) + β (cid:96) ¯ I t Ai (1 − π Ai ) + β (cid:96) ¯ I t Si (1 − π Si ) (cid:17) × N t An (1 − π An ) (cid:21) . • Non-infected Symptomatic-Isolated Compartment ( N Si ) :Inflow: The inflow to the N Si compartment has two sources: – Non-infected asymptomatic-nonisolated ( N An ) individuals who did not get tested anddeveloped symptoms caused by another condition or disease such as the seasonal flu. Thereare N t An + N t Si π Si + N t Ai π Ai such individuals. The rate that an individual develops symptomsdue to a non-COVID-19 disease is given by the parameter nonCOVIDSymptRate , which wecurrently set at 1/1200; see Section C for more details on how we arrive at this number. – Non-infected asymptomatic-isolated ( N Ai ) individuals can also develop flu-like symptoms.There are N t Ai × (1 − π Ai ) such individuals.Outflow: The outflow from the N Si compartment has three destinations: – Individuals who are tested will return negative and move to the N An compartment. This causesthem to increase their contact with other individuals. – Some individuals will become infected from coming in contact with infected people. – Individuals leave self-quarantine at a rate of 1 / selfQuarTime , where selfQuarTime is theaverage time that an individual self-isolates. We currently set this value to 10 days.18pdate Equations: We have the following update equations. N t +1 Si = N t Si × (1 − π Si ) × (cid:18) − selfQuarTime − β (cid:96) × (cid:16) ¯ I t An (1 − π An ) + ¯ I t Ai (1 − π Ai ) + ¯ I t Si (1 − π Si ) (cid:17)(cid:19) + nonCOVIDSymptRate × (cid:0) N t Si π Si + N t Ai + N t An (cid:1) . • Non-infected Asymptomatic-Isolated Compartment ( N Ai ) :Inflow: The inflow to the N Ai compartment comes from individuals in the N An compartment whoreduce their exposure to other individuals after being identified as a close contact. The numberof such individuals is given by contactPerPosCase × ¯ I t Si π Si + ¯ I t Ai π Ai + ¯ I t An π An ¯ H × (cid:0) N t An + N t Si π Si + N t Ai π Ai (cid:1) , where contactPerPosCase is the average number of contacts per positive case, currently setat 4, and ¯ H denotes the total number of high-contact individuals; see the discussion in the I Ai compartment.Outflow: The outflow from the N Ai compartment has four destinations. – Individuals who are tested will test negative and move to the N An compartment, end theirself-isolation, and increase their contacts. – Untested individuals leave self-isolation at a rate of 1 / selfQuarTime . – Some individuals will become infected and move to the I Ai compartment. – Some individuals will develop flu-like symptoms but not COVID-19, at the rate of nonCOVIDSymptRate .Update Equations: We have the following update equations. N t +1 Ai = N t Ai × (1 − π Ai ) × (cid:18) − selfQuarTime − nonCOVIDSymptRate − β (cid:96) × (cid:16) ¯ I t An (1 − π An ) + ¯ I t Ai (1 − π Ai ) + ¯ I t Si (1 − π Si ) (cid:17)(cid:19) + contactPerPosCase × ¯ I t Si π Si + ¯ I t Ai π Ai + ¯ I t An π An ¯ H × (cid:0) N t An + N t Si π Si + N t Ai π Ai (cid:1) . • Non-infected Asymptomatic-Nonisolated Compartment ( N An ) :Inflow: The inflow to the N An compartment comes from three sources. First, individuals from the N Si compartment recover from non-COVID-19 symptoms at a rate of 1 selfQuarTime . Second,individuals from the N Ai compartment leave self-isolation and increase their contact with othersat a rate of 1 selfQuarTime . Third, individuals from the N Si and N Ai compartments who testnegative will also increase their contact rates.19utflow: The outflow from the N An compartment has three destinations. First, some individualsare identified as close contacts and move to the N Ai compartment, thereby reducing their contactlevels. Second, other individuals will develop flu-like symptoms independent of COVID and moveto the N Si compartment at the rate of nonCOVIDSymptRate . Third, some individuals will becomeinfected and move to the I An compartment.Update Equations: We have the following update equations. N t +1 An = (cid:0) N t An + N t Si π Si + N t Ai π Ai (cid:1) × (cid:32) − nonCOVIDSymptRate − contactPerPosCase × ¯ I t Si π Si + ¯ I t Ai π Ai + ¯ I t An π An ¯ H − (cid:16) β h ¯ I t An (1 − π An ) + β (cid:96) ¯ I t Ai (1 − π Ai ) + β (cid:96) ¯ I t Si (1 − π Si ) (cid:17)(cid:33) + N t Ai (1 − π Ai ) selfQuarTime + N t Si (1 − π Si ) selfQuarTime . • Recovered Symptomatic-Isolated Compartment ( R Si ) :Inflow: The inflow to the R Si compartment has two sources. Recovered asymptomatic-nonisolated( R An ) and recovered asymptomatic-isolated ( R Ai ) individuals, who did not get tested, can developflu-like symptoms independent of COVID-19, at the rate of nonCOVIDSymptRate .Outflow: The outflow from the R Si compartment occurs from individuals who recover from non-COVID-19 symptoms at the rate of 1 / selfQuarTime .Update Equations: We have the following update equations. R t +1 Si = R t Si × (1 − π Si ) × (cid:18) − selfQuarTime (cid:19) + nonCOVIDSymptRate × (cid:0) R t Si π Si + R t Ai + R t An (cid:1) . • Recovered Asymptomatic-Isolated Compartment ( R Ai ) :Inflow: The inflow to the R Ai compartment comes from individuals in the R An compartment whoreduce their exposure to other individuals after being identified as a close contact. The numberof such individuals is given by contactPerPosCase × ¯ I t Si π Si + ¯ I t Ai π Ai + ¯ I t An π An ¯ H × (cid:0) R t An + R t Si π Si + R t Ai π Ai (cid:1) , Outflow: The outflow from the R Ai compartment has two destinations. First, untested individualsleave isolation at the rate of 1 / selfQuarTime . Second, other individuals will develop symptomsindependent of COVID-19, at the rate of nonCOVIDSymptRate .20pdate Equations: We have the following update equations. R t +1 Ai = R t Ai × (1 − π Ai ) × (cid:18) − selfQuarTime − nonCOVIDSymptRate (cid:19) + contactPerPosCase × ¯ I t Si π Si + ¯ I t Ai π Ai + ¯ I t An π An ¯ H × (cid:0) R t An + R t Si π Si + R t Ai π Ai (cid:1) . • Recovered Asymptomatic-Nonisolated Compartment ( R An ) :Inflow: The inflow to the R An compartment has six sources: – Untested people from the I Si compartment who recovered naturally. – Untested people from the I Ai compartment who recovered naturally. – Untested people from the I An compartment who recovered naturally. – People from the R Ai and R Si compartments who test negative. – People from the R Ai compartment who leave self-quarantine. – People from the R Si compartment whose non-COVID-19 symptoms disappeared.Outflow: The outflow from the R An compartment occurs in two ways. First, some individualsare identified as close contacts and move to the R Ai compartment. Second, some individuals willdevelop symptoms independent of COVID-19, at the rate of nonCOVIDSymptRate .Update Equations: We have the following update equations. R t +1 An = (cid:0) R t An + R t Si π Si + R t Ai π Ai (cid:1) × (cid:32) − nonCOVIDSymptRate − contactPerPosCase × ¯ I t Si π Si + ¯ I t Ai π Ai + ¯ I t An π An ¯ H (cid:33) + R t Ai (1 − π Ai ) selfQuarTime + R t Si (1 − π Si ) selfQuarTime + ¯ I t Si (1 − π Si ) sympToRecoveryTime + ¯ I t Ai (1 − π Ai ) asympToRecoveryTime + ¯ I t An (1 − π An ) asympToRecoveryTime . • Known Infected Compartment ( KI ) : This compartment has two sub-compartments, with KI t = (cid:0) KI t (hospitalized) , H t (recovered) (cid:1) and we have the following dynamics.Inflow: There are three sources of inflow. The infected individuals in I Si , I Ai , and I An compartments who are tested. We assume that individuals from the I Si (hospitalized) sub-compartment flow into the KI t (hospitalized) sub-compartment, and that individuals from the I Si (recovered) sub-compartment flow into the KI t (recovered) sub-compartment. We assumethat hospOutOfSymptFrac (20%) of the individuals from the I Ai and I An compartments requirehospitalization, while the remaining 1 - hospOutOfSymptFrac = 80% will recover naturally.Outflow: Individuals who require hospitalization move to the H compartment at the rate of1 / infToHospTime , where infToHospTime represents the average time until an infected person21equires hospitalization. We currently set this to 5 days. Individuals who will recover move to the KR compartment at the rate of 1 / sympToRecoveryTime , where sympToRecoveryTime denotesthe average recovery time. We set this to 14 days.Update Equations: Let recoverOutOfSymptFrac = 1 − hospOutOfSymptFrac = 80%. Then, KI t +1 (recovered) = (cid:18) KI t (recovered) + I t Si (recovered) π Si + I t Ai (recovered) π Ai + I t An (recovered) π An + recoverOutOfSymptFrac × (cid:2) I t Ai (sympt . ) π Ai + I t An (sympt . ) π An (cid:3) (cid:19) × (cid:18) − sympToRecoveryTime (cid:19) ,KI t +1 (hospitalized) = (cid:18) KI t (hospitalized) + I t Si (hospitalized) π Si + hospOutOfSymptFrac × (cid:2) I t Ai (sympt . ) π Ai + I t An (sympt . ) π An (cid:3) (cid:19) × (cid:18) − infToHospTime (cid:19) . • Hospitalization Compartment ( H ) : This compartment has two sub-compartments, with H t = (cid:0) H t (die) , H t (recovered) (cid:1) .Inflow: There are two sources of inflow to the hospitalization compartment: the infectedindividuals from the I Si and KI compartments who require hospitalization. We assume that afraction deathFrac , currently set at 1/3, of the inflow to the hospitalization compartment willdie. We assume the remaining 1 − deathFrac = 2 / / hospToDeathTime , where hospToDeathTime is the average time between hospitalization and death. Individuals whowill recover move to the KR compartment at the rate of 1 / hospToRecoveryTime , where hospToRecoveryTime is the average time between hospitalization and recovery. We currentlyset both to 14 days.Update Equations: Let recFrac = 1 − deathFrac = 2 /
3. We have the following updateequations. H t +1 (die) = H t (die) × (cid:18) − hospToDeathTime (cid:19) + deathFracinfToHospTime × (cid:32) I t Si (hosp . ) × (1 − π Si ) + KI t (hosp . ) + I t Si (hosp . ) π Si + hospOutOfSymptFrac × (cid:2) I t Ai (sympt . ) π Ai + I t An (sympt . ) π An (cid:3) (cid:33) ,H t +1 (recovered) = H t (recovered) × (cid:18) − hospToRecoveryTime (cid:19) + recFracinfToHospTime × (cid:32) I t Si (hosp . ) × (1 − π Si ) + KI t (hosp . ) + I t Si (hosp . ) π Si hospOutOfSymptFrac × (cid:2) I t Ai (sympt . ) π Ai + I t An (sympt . ) π An (cid:3) (cid:33) . • Known Recovery Compartment ( KR ) :Update Equations: Let recoverOutOfSymptFrac = 1 − hospOutOfSymptFrac = 80%. We havethe following update equations. KR t +1 = KR t + H t (rec . ) hospToRecoveryTime + KI t (rec . ) sympToRecoveryTime + I t Si (rec.) π Si + I t Ai (rec.) π Ai + I t An (rec.) π An sympToRecoveryTime + recoverOutOfSymptFrac × (cid:104) I t Ai (sympt . ) π Ai + I t An (sympt . ) π An (cid:105) sympToRecoveryTime . • Death Compartment ( D ) :Update Equations: We have the following update equations. D t +1 = D t + H t (die) hospToDeathTime + I t Si (death) sympToDeathTime . C. Model Parameters and Their Estimated Values
In this section, we discuss how we obtain the estimates for different model parameters.
Estimating Flow Rates Between Compartments:
We assume an incubation period of 5 days [11]. After an individual develops symptoms, weassume it takes another 5 days for the individual to become hospitalized [18]. Specifically, weset infToSympTime = infToHospTime = sympToHospTime = 5.We set the recovery time for symptomatic individuals to be 14 days. In our model, we donot differentiate between recovery at home versus at the hospital and set sympToRecoveryTime = hospToRecoveryTime = 14. We set the time for a symptomatic individual to die to be 14 days.Similar to the recovery time, we do not differentiate between deaths at home versus at the hospitaland set both sympToDeathTime = hospToDeathTime = 14 [11].We assume that individuals who self-isolate after being identified as a close-contact or developsymptoms from a non-COVID-19 illness spend an average of 10 days reducing their contact withothers. While the WHO recommends a quarantine period of 14 days, the CDC gives a less stringentrecommendation of 10 days [2]. We set selfQuarTime = asympToRecoveryTime = 10.In our model, non-infected and recovered individuals develop symptoms due to non-COVID-19diseases at a rate of nonCOVIDSymptRate = . This rate is based on the assumption that 10%of the population is infected over the course of 4 months. This is roughly estimated using the flusymptom rate in past years as reported by the CDC [3]. We set N = 10 , , ,
966 individuals, corresponding to 0 .
3% of the23opulation, initially has symptoms due to the seasonal flu. This is calculated assuming that flusymptoms last for an average of 4 days.
Estimating Disease Dynamics:
We use sympOutOfInfFrac to denote the fraction of COVID-19 infected individuals who developsymptoms. Numerous reports have given different estimates for this fraction: from 82% on theDiamond Princess cruise ship [10] to roughly 40% on the USS Theodore Roosevelt [16]. We set sympOutOfInfFrac = 50%.The parameter hospOutOfSympFrac denotes the fraction of symptomatic individuals who needhospitalization. We set this value to be 20%. The parameter deathOutOfHospFrac denotes thefraction of hospitalized individuals who die. We set this value to be 33 . ¯3%. Both of these valuesare roughly estimated using historical hospitalization and death numbers as reported by New YorkCity [15]. Finally, we assume that some portion of the infected population die at home withouthospitalization. We denote this portion as deathOutOfInfFrac and set it to 2%. We estimate thisvalue using the number of probable deaths as reported by the City. Infection Dynamics and Estimating Contact Rates:
The basic reproductive number, R , captures the average number of new infections produced byeach infected individual. There are numerous recent studies estimating the value of R for COVID-19. These estimates vary widely, from anywhere between 2.2 [9] and 5.7 [17]. Furthermore, itis possible that R varies across different countries and cities due to climate, environmental, andsociological differences. As a result, we set R = 3 .
04 according to a tuning procedure usinghistorical hospitalization and death statistics as reported by North Carolina. We briefly discuss thedetails of this procedure in Section D.The rate of new infections is controlled by β , the contact rate between infected and non-infectedpopulations. In a standard compartmentalized model, the number of new infections during anytime period is captured by β × { } × { } . In our model, we have a similar term for every pair of infected and non-infected compartments. Weemploy two different contact rates β h and β (cid:96) , so we differentiate their usage according to whetherwe are considering isolating or non-isolating compartments.We set β h = R N × sympToRecoveryTime and use this as the contact rate between infected andnon-infected compartments that are both non-isolating. When at least one of the compartmentsis isolating or symptomatic, we assume that contact between these populations is reduced by onethird and use the contact rate β (cid:96) = × R N × sympToRecoveryTime .Beginning on March 27, we reduce both β h and β (cid:96) by a factor of 1 /
2. This reduction is due tosocial distancing measures associated with the NC stay-at-home order. We set this reduction to 1 / Dynamics of Contact Tracing:
The parameter contactPerPosCase denotes the average number of close contacts that are identifiedvia tracing for each COVID-19 positive individual. We assume that close contacts are always24dentified from within the non-isolating population. To estimate contactPerPosCase and theproportion of infected and non-infected individuals within the close-contact group, we use theresults of [1], a comprehensive study on contact tracing. This study reports the results of contacttracing in Shenzhen, China but the index cases are mostly travelers arriving from Hubei province.(To the best of our knowledge there are no detailed studies on contact tracing in the UnitedStates. Therefore, we use findings reported for the Hubei province in China and its capital Wuhanto estimate contactPerPosCase and to get a rough approximation for likOfBeingInfected asexplained in the following.)In this study, the authors identified 1,286 close contacts based on 292 COVID-positive cases.This suggests an average close contact group size of 1286 /
292 = 4 . contactPerPosCase to 4 in our model.The parameter likOfBeingInfected is a multiplier that captures how much more likely it is foran infected individual to be identified as a close contact as compared to a non-infected individualand therefore can be seen as a measure of the effectiveness of contact tracing. Specifically, likOfBeingInfected = P (close-contact | infected) P (close-contact | non-infected) . It would be reasonable to consider settings where likOfBeingInfected ≥ likOfBeingInfected , we use the following approach:We can show that likOfBeingInfected = P (infected | close-contact) P (non-infected | close-contact) ÷ P (infected) P (non-infected) . Thus, we need estimates for the proportion of infected versus non-infected individuals within thenon-isolating population as well as the proportion of infected versus non-infected individuals amongpeople identified as close contacts. Of the 1,286 close contacts studied in [1], 98 tested positive.Thus, we estimated the proportion of the infected to non-infected within the close contact groupto be P (infected | close-contact) /P (non-infected | close-contact) = 98 / . P (infected) /P (non-infected), the proportion of infected to non-infected inthe general non-isolating population, using findings of studies done on data collected from Wuhanaround the same time the study discussed in [1] was conducted. According to [8], in Wuhan, theprobability of death after developing symptoms was 1.4%. Using this estimate and the number ofdeaths in Wuhan, which is reported as 4,512 by the Johns Hopkins Coronavirus Resource Center [7],we can estimate the total number of infected individuals in Wuhan over the course of the pandemic(at least as of now) to be 322,286. In order to estimate the total number of infected individuals atsome random time between mid-January and mid-February, we divide this by two and use 161,143 asthe average number of infected individuals at a given time. This approximation implicitly assumesa linear build-up of new COVID-positive cases, which is not true for infection spread within apopulation, but nevertheless serves as a reasonable rough estimate. Then, using 11,000,000 forthe population of Wuhan, we estimate P (infected) /P (non-infected) to be 161 , / (11 , , − , . . / .
015 = 5 .
47. Clearly, this is a very rough approximation and it would not be reasonableto reach conclusions based on results only on this estimated value. Therefore, in our analysis weconsider scenarios that assume different values around 5.47. Specifically, in the main body of thepaper, we report results based on three settings with likOfBeingInfected set to 2, 5, and 10,respectively corresponding to increasing levels of contact tracing effectiveness.
D. Model Validation
We manually calibrated two parameters of the model to roughly match its output to the observedtrajectory of the pandemic up until April 15. These two parameters are the R value and theinitial number of infected asymptomatic-nonisolated individuals on March 2. We ended up withan R value of 3.04 and an initial infected asymptomatic-nonisolated population of 676. Thesestarting parameters are tuned based on hospitalization, death, and test data up to June 15. Weuse testing data from the NC Department of Health and Human Services. We assume that thetests are conducted daily on people in the order of symptomatic, isolated, and finally non-isolatedpeople.The plots in Figure 7 show the epidemic trajectory as predicted by our model. In particular,Figure 7a shows the total number of currently hospitalized, symptomatic infected, asymptomaticinfected and known infected people on each day as projected by the model. In Figure 7b, wecompare the cumulative number of deaths predicted by our model with the actual cumulativenumber of deaths. In Figure 7c, we plot the actual daily number of tests conducted and positivecases found in North Carolina. We use the same historical testing capacity in our model. We setthe testing capacity to 30,000 for future projections in this plot. Note that the daily number ofpositive tests in our model fluctuates because we use the actual number of tests performed in NorthCarolina until June 15 as the number of tests performed in our model. However, the number ofpositive cases identified in our model emerges from the internal behavior of our model. The fitbetween the model and the actual trajectory in terms of positive cases is reasonably good. Figure7d shows the daily number of active hospitalizations reported by North Carolina against the numberof hospitalizations projected by our model. 26 a) Population overview(b) Cumulative deaths(c) Daily new cases(d) Daily active hospitalizationsa) Population overview(b) Cumulative deaths(c) Daily new cases(d) Daily active hospitalizations