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Tobias strategies immunity herd of COVID-19 impracticality the reveal dynamics Transmission www.pnas.org/cgi/doi/10.1073/pnas.2008087117 dmSho fEooy nvriyo eri,Ahn,G,30602; GA, Athens, Georgia, of University Ecology, of School Odum ral paig w itntapoce ocnrligthe controlling to approaches distinct two speaking, Broadly rm ooaiu SR-o-)() OI-9is COVID-19 (1), (SARS-CoV-2) syn- respiratory 2 acute coronavirus severe coronavirus, drome novel a by aused c eateto netosDsae,Uiest fGoga tes A 30602 GA, Athens, Georgia, of University Diseases, Infectious of Department a,b,1 | ahmtclmodeling mathematical n emnRohani Pejman and | yaia systems dynamical a,b,c b etrfrteEooyo netosDsae,Uiest fGoga tes GA, Athens, Georgia, of University Diseases, Infectious of Ecology the for Center doi:10.1073/pnas.2008087117/-/DCSupplemental e’ xml n tep oaheehr muiyi the the in immunity available, Given herd Swe- is achieve (10). following to vaccine relaxation considered attempt (16). a country their have and to until example for den’s appears wait push government a long UK and to potentially (14) mea- costs led distancing States the economic social have United with severe associated the sures The pressures [including (15)]. societal clear acute Kingdom less United coun- strate- is and other and appears transmission, intention Meanwhile, and sustained (13). gic experience strategies, immunity to mitigation herd continue for for tries aiming child be poster to the Swe- is Italy, (12). and den strategies Spain, suppression implemented Korea, successfully South have China, including countries, tiple dis- (social measures control and of vary. intensities durations necessary types the although same self-isolation), and the require tancing approaches of both rollout practice, with- In the even (11). immunity impossible, distancing) herd social is diminish, achieve out transmission to to local pool able sustained potentially susceptible (whereby population the the trans- allows with halting) overwhelm strategy than not this (rather does rate reducing mission, burden By growth disease (3). the ensure systems healthcare reduce to halt epidemic to ultimately the aims to of as mitigation aims hereafter transmission, suppression to or local (referred While manage impacts (10). to health “mitigation”) aims negative approach the to second reintroduction mitigate The subsequent preventing resurgence. to prevent shift then will focus is ulse etme 2 2020. 22, September published First at online information supporting contains article This 1 under distributed BY).y is (CC article access open This Submission.y Direct PNAS a is article This interest.y competing no declare authors paper.y The the wrote T.S.B. P.R. research; and T.S.B. performed and T.S.B. data; research; analyzed designed P.R. and T.S.B. contributions: Author owo orsodnemyb drse.Eal [email protected] Email: addressed. be may correspondence whom To lgclfcos uhfietnn fsca itnigrenders distancing impractical. social extended strategy of this an fine-tuning Such for epidemi- factors. quantified manner poorly ological to adaptive sensitivity care- acute an despite be period, must in levels manipulated Intervention error. capacity fully for hospital room overwhelming simulating little achiev- without leaves that study immunity finds Our herd Kingdom United crisis. ing the the in spread to SARS-CoV-2 resolution presents tantalizing COVID-19, of a while impacts spread, health to negative epidemic the serious the mitigating allowing by have population a immu- which in herd nity Establishing measures repercussions. economic distancing and societal impos- social to resort drastic to had have ing pandemic the at of edge countries leading outbreaks, the COVID-19 escalating with Confronted Significance ttetm fwiig rnsi niec aasgetmul- suggest data incidence in trends writing, of time the At PNAS | coe 3 2020 13, October raieCmosAtiuinLcne4.0 License Attribution Commons Creative | . y o.117 vol. https://www.pnas.org/lookup/suppl/ | o 41 no. | 25897–25903

POPULATION BIOLOGY The consequences of failure to either adequately mitigate Kingdom (Fig. 1A), and ultimately infect approximately 77% of or suppress COVID-19 are potentially catastrophic. Due to the population (Fig. 1B). Using data on the age-specific fatality the many uncertainties surrounding SARS-CoV-2 transmission, rate of COVID-19 (22), our results show around 350,000 fatal- evolution, and immunity, public health decision makers are pre- ities among individuals aged over 60 y, and around 60,000 aged sented with an unenviable task. To help inform control policies below 60 y (Fig. 1C). While we caution that our model makes a under uncertainty, mathematical modeling can assist in evalu- number of simplifying assumptions [e.g., no spatial dependence ating the viability of mitigation and suppression as objectives in transmission (11)], the total of fatalities is comparable to pre- (17), by simulating the impacts of control strategies on viral dictions made in other UK modeling studies (e.g., within the 95% transmission, hospital burden, fatalities, and population-level prediction interval of ref. 20). immunity. Sustained social distancing by older individuals (assumed to Recent studies have modeled impacts of both mitigation and result in a 90% reduction in contacts with individuals under 25 y suppression strategies, including for China (18), low-income old, a 70% reduction with 25- to 59-y-olds, and a 50% reduction countries (19), and the United Kingdom (20, 21). Crucially, we between one another), and moderately effective self-isolation are not aware of any systematic studies that focus on 1) walking by symptomatic individuals (at 20% efficacy) results in a shal- the tightrope of achieving herd immunity without overburden- lower epidemic curve (Fig. 1D) and a much smaller outbreak size ing healthcare systems and 2) the control effort (e.g., reduction among individuals aged 60+ y (Fig. 1E). The attendant mortality in contacts) required for successful mitigation relative to sup- burden among 60+-y-old individuals is also substantially reduced pression. These two knowledge gaps motivate our study. We use (to 62,000), with a smaller reduction in fatalities in those aged an age-stratified disease transmission model, taking the United < 60 y (to 43,000; Fig. 1F). Kingdom as an example, to simulate SARS-CoV-2 spread con- The addition of school (and university) closures, correspond- trolled by individual self-isolation and widespread social distanc- ing to a 70% reduction in contacts among school-aged individ- ing. We simulated various levels of self-isolation effectiveness uals and a 20% reduction with 25- to 59-y-olds, dramatically and three distinct types of social-distancing measures: 1) school reduces the rate of epidemic growth (Fig. 1G), although such (including university) closures, 2) work and social place closures, levels of control are insufficient to suppress the epidemic (i.e., and 3) effective isolation by older individuals. the number of daily cases still rises after implementation). The Our modeling confirms that suppression is possible with plau- premature reopening of schools after 100 d (while the virus is sible levels of social distancing and self-isolation, consistent still circulating) triggers a second wave of infection, with only a with experience in multiple countries. Our research does not, moderately reduced peak in daily new cases, largely eroding any however, support attempting to mitigate COVID-19 with the additional gains made (23). The final proportion of the popu- aim of building herd immunity. Achieving herd immunity while lation exposed (Fig. 1H) and the number of fatalities (Fig. 1I) simultaneously maintaining hospital burden at manageable lev- are largely unaltered compared to if schools had not been closed els requires adaptive fine-tuning of mitigation efforts, in the (compare Fig. 1 E and F). face of imperfect epidemiological intelligence—something that Our modeling indicates that, if sustained, such control mea- is impractical. sures can lead to the suppression of COVID-19 in the United Kingdom by reducing R0 to <1 (Fig. 2 A and B). The effective- Results ness of self-isolation by symptomatic individuals (i.e., its impact In the absence of any intervention measures, our modeling on reducing overall transmission) is a product of two factors. suggests SARS-CoV-2 will spread rapidly through the United Firstly, the proportion of infections generated while the primary

A D G

B E H

C F I

Fig. 1. Simulated examples of SARS-CoV-2 spread in the United Kingdom using an age-structured Susceptible Exposed Infectious Recovered (SEIR) model under different control scenarios. (A) Daily new cases assuming no intervention measures enacted. (B) Cumulative proportion of the population exposed to SARS-CoV-2 over the course of the epidemic. (C) Cumulative fatalities assuming fixed age-specific case fatality rates (see Materials and Methods). (D–F) Same as A–C, but assuming that specific control measures are introduced when daily cases reached 10,000 (day 57): Older individuals social distance, and symptomatic individuals self-isolate (at 20% effectiveness). (G–I) Same as D–F but, in addition, schools close on day 57. Reopening of schools after 100 d results in a resurgence.

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POPULATION BIOLOGY (10)] and the disease has not been successfully suppressed, then early. An idealized optimal strategy for achieving social distanc- much of the benefit of their closure is lost (Fig. 3 E–H) due to ing, calculated using a two-age group (<60 y old and ≥60 y a resurgent second wave. In this scenario, the principle effect old) SEIR model, is shown in Fig. 5 A and B. We assumed that of school and workplace closures is in delaying the peak, buying individuals ≥60 y old were able to perfectly self-isolate, while more time for preparations (Fig. 3G). herd immunity is built up in the <60-y age group. The epidemic It has been suggested that children might have reduced sus- in <60-y-old individuals is initially allowed to grow unimpeded ceptibility to infection with SARS-CoV-2 (25). We repeated our until the hospital burden caused by the number of sick individu- analysis of suppression and mitigation assuming 50% reduction als reaches the hospital capacity. At this point, social distancing in susceptibility of individuals under 20 y old, with transmission among those younger than 60 y commences (Fig. 5B), at a level rates among adults increased to ensure the same reproductive that ensures the epidemic growth stalls (i.e., Reff = 1). To achieve number, R0 = 2.3. As might be expected, there is a decreased herd immunity in minimal time, the rate of new infections impact of school closures; however, this is compensated by the must then remain at the maximum afforded by the healthcare increased impact of social distancing by over-20-y-olds, result- capacity. This requires reducing social distancing (i.e., increasing ing in little net change in our findings if both work and school contacts) at the exact rate to balance out reductions in trans- closures occur (SI Appendix, Figs. S1 and S2). mission stemming from the decreasing susceptible population To summarize, aiming to build herd immunity to SARS- (Fig. 5B)—too quickly and the epidemic grows above hospital CoV-2 in a population while mitigating the burden on hospitals capacity; too slowly and the epidemic with die out without reach- requires initially reducing the reproductive number to ensure ing herd immunity. Over time, social distancing in <60-y-olds available hospital capacity is not exceeded (Fig. 4A). If social dis- drops to zero, after which there are no longer enough suscep- tancing is maintained at a fixed level, hospital capacity needs to tible <60-y-old individuals for the epidemic to be sustained. be much larger than presently available to achieve herd immunity New infections continue but decline in frequency (meanwhile, without exceeding capacity; otherwise, the final outbreak size further decreasing the effective reproductive number), before will be insufficient to achieve herd immunity (Fig. 4B). Relax- reaching zero. ing social distancing measures linearly is also insufficient (SI While, at this point, the effective reproductive number is below Appendix, Fig. S3). 1 without any social distancing in the <60-y age group, full Instead, achieving herd immunity without exceeding hospital herd immunity has not necessarily been achieved (Fig. 5C). If capacity requires that social distancing is implemented nonlin- ≥60-year-olds cease self-isolating, then the increased contact rates cause the effective reproductive number to increase, and may take it above 1. Achieving herd immunity requires that, A while ≥60-y-olds are isolated, the additional new infections after Reff < 1 sufficiently further reduce Reff (the “overshoot”) such that, when ≥ 60-y-olds cease self-isolation, it remains below 1. The amount of overshoot increases with the hospital capacity (Fig. 5C). In addition to increasing the prospects of achieving herd immunity (and also its robustness; see also SI Appendix, Fig. S3C), the necessary duration of social distancing is inversely proportional to hospital capacity (Fig. 5D). Our modeling required estimates for the case hospitalization probability (22) and mean hospitalization stay (24). We explored sensitivity of the duration of social distancing to these two param- eters, and also performed a comparison with a rough calculation B of the duration based on estimates of seroprevalance at the end of April (26) (SI Appendix, Fig. S4). Our estimated necessary social distancing duration of 12 mo is close to the estimate from serology of 7 mo to 11 mo, especially considering the uncertainty in the case hospitalization probability (22). Throughout our analysis, we used R0 = 2.3, broadly in line with most early estimates; however, some estimates were higher. We repeated our analysis using R0 = 5.7 (27), finding that, as might be expected, the range of intervention parameters that led to successful suppression was reduced (SI Appendix, Fig. S5). However, our main finding was reinforced: Higher values of R0 narrow the window for successful mitigation (SI Appendix, Fig. S6). Discussion Fig. 4. Summary of prospects for achieving herd immunity. (A) The peak hospital burden (shown on a log scale) is highly sensitive to the reproductive Various governments have entertained the idea of achieving number. There is a narrow window of reproductive number values (shaded) herd immunity through natural infection as a means of ending where either 1) the number of COVID-19 cases requiring hospitalization the long-term threat of COVID-19. This has provoked alarm does not overwhelm hospital capacity (modeled at the average hospital in sections of the public health community (16, 28). Our work burden for April, 17,800 beds) or 2) circulation is suppressed. This window confirms that this alarm is well founded. depends subtly on the exact age-specific social distancing configuration; Attempting to achieve herd immunity while simultaneously however, all strategies studied fall between the two curves shown. (B) None mitigating the impact of COVID-19 on hospital burden is an of the simulated control scenarios shown in Figs. 2 and 3 achieved herd extremely challenging task. In order to ensure the hospital bur- immunity while also keeping cases below hospital capacity. For the parame- ters considered, a hospital capacity in excess of 300,000 is required for this to den does not exceed levels comparable with that of the United be possible—almost 3 times the total UK NHS hospital beds (around 125,000 Kingdom in April 2020, R0 needs to be reduced from its initial beds; see Materials and Methods), and around 15 times the average hospital value (assumed to be R0 = 2.3) to about 1.2. Suppression is pos- burden of April. sible if R0 is reduced below 1. Due to the fine margins (in terms

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POPULATION BIOLOGY The mean latent and infectious periods were set to 1/ρ = 3 and 1/γ = A 3 d, respectively, consistent with various estimates of the serial interval (34, 35) and incubation period (4, 36, 37), assuming infectiousness starts 1 d to 2 d before symptoms develop. Both latent and infectious periods were assumed to be gamma dis- tributed and modeled using the method of stages (11, 38, 39), by dividing the exposed (Ei) and infectious (Ii) compartments for each age group into P4 (k) P4 (k) four subcompartments, Ei = k=1 Ei and Ii = k=1 Ii , where the super- script labels the subcompartment. The transmission dynamics for the age groups were governed by a system of ordinary differential equations,

dSi = −λi(t)Si, [1] dt 1 dEi (1) = λi(t)Si − 4ρE , [2] dt i dEk i = 4ρE(k−1) − 4ρE(k) for k = 2, 3, 4, [3] dt i i dI1 i = 4ρE(4) − 4γI(1), [4] dt i i dIk i = 4γI(k−1) − 4γI(k) for k = 2, 3, 4, [5] B dt i i X Ij(t) λi(t) = β ci,j(t) . [6] N j j

The transmission rate β was tuned using the next-generation matrix (40) to give a value of R0 = 2.3, consistent with estimates (34, 35). Simulations were initialized with one initial introduction in a fully susceptible popula- tion (Si = Ni). The resulting doubling time was observed to be about 3 d, broadly consistent with early observations from the United Kingdom.

Modeling Nonpharmaceutical Interventions. Two types of nonpharmaceuti- cal intervention were modeled: 1) self-isolation by symptomatic infectious individuals and 2) mass social distancing by differing age groups (Fig. 6). The Fig. 6. Modeling the impact of nonpharmaceutical interventions on disease effectiveness of self-isolation of symptomatic individuals is dependent on transmission. (A) Different social distancing measures (e.g., school closures, the product of two factors: 1) the proportion of infections that occur due to work and social place closures, and older individuals distancing) reduce con- symptomatic individuals (excluding both presymptomatic and asymptomatic tact rates between individuals in different ways. To reduce the complexity transmission), ps, and 2) the observance rate of social isolation among symp- of the model, and to understand their differential impacts, we assume that tomatic individuals, k. The fractional reduction of contacts between age individuals are only affected by one of these measures, dependent on their groups i and j due to social distancing is given by qi,j. age. Individuals aged 0 y to 24 y are affected by school closures (included Both of these interventions take the form of modifications to the contact in this are university closures). School closures were assumed to result in matrix between infectious and susceptible individuals, a 70% reduction in contacts among school-aged individuals (qYY ) and 20% reduction in their contacts with individuals aged 25 y to 59 y (qYA). Work and eci,j = (1 − kps)(1 − qi,j)ci,j. [7] social place closures were assumed to reduce contacts among adults (qAA) by 50%. Finally, older individuals distancing reduced contacts by 60+-y-old indi- This expression for eci,j is inserted in place of ci,j in Eq. 6. viduals with 0- to 24-y-olds by 90% (q ), with 25- to 59-y-olds by 70% (q ), Age groups in the model are divided into whether they are young (Y; YO AO and among one another by 50% (q ). The effectiveness of symptomatic corresponding to 0- to 24-y-olds and age groups i = 1 to 4), adults (A; 25- OO individuals self-isolating is dependent on two factors: 1) the observance by to 59-y-olds, age groups i = 5 to 11) and older (O; 60+-y-olds, age groups symptomatic individuals, k, and 2) the proportion of transmission due to i = 12 to 15). The reduction in contacts due to social distancing, qi,j, is then individuals who are symptomatic, .( ) We modeled five distinct combina- determined by which of these three categories the contacter and contactee ps B tions of social distancing measures, assuming that older individuals’ social fall into, given by the block matrix distancing will always be prioritized.   qYY qYA qYO q = qYA qAA qAO. [8] qYO qAO qOO Pre–COVID-19 hospital beds and occupancy rates in the UK National Health Systems (NHS) were taken from the most recent (autumn 2019) We assume school closures reduce contact rates between young individuals published numbers for Northern Ireland, Wales, Scotland, and England. P by a factor of qYY = 0.7 and between young people and adults by qYA = 0.2. Hospital burden was calculated as H = d i hiΛiSi. Although there is Social distancing among adults (e.g., due to workplace closures and reduc- a lag between exposure and hospitalization, this has no bearing on our tion in social events) was modeled as a reduction of qAA = 0.5. Social mathematical results. distancing of older individuals was represented by qYO = 0.9, qAO = 0.7, and qOO = 0.5. For simulations with social distancing fatigue, qYY , qYA, and qAA Optimal Strategy to Achieve Herd Immunity. We used a two-age group were modeled as linearly decreasing from these initial values to 0 over the model (<60-y-olds and ≥60-y-olds) to demonstrate an optimal social periods indicated in SI Appendix, Fig. S3. distancing strategy while shielding ≥60-y-old individuals. To preserve the data-informed contact structure, we made the approximation that Estimating Hospital Burden and Case Fatalities. Age-specific hospitalization the age-specific incidence was constant for age groups <60 y and probabilities, hi, and fatality rates were taken from point estimates cal- similarly for ≥60 y. The two-age group contact matrix is then given culated in a study of cases in Wuhan, China (22). Due to differences in by a weighted sum of the POLYMOD-derived contact matrix, c(2) = P P a,b the final age group of our model (70+ y) and those of the Wuhan study i∈a j∈b(Ni/Nb)ci,j, where a and b index age groups {< 60, ≥ 60}. Aside (70- to 79-y-olds and 80+-y-olds), the hospitalization and fatality rates for from the contact matrix, the model is identical in structure to that given 70+-y-old individuals were calculated by summing the estimated 70- to 79- in Eqs. 1–6. y-old and 80+-y-old rates weighted by their relative UK population sizes. If ≥60-y-old individuals are shielded (contact rates reduced to zero), the Based on data from the United Kingdom, we assumed the average duration <60-y-old susceptible population is depleted at the fastest rate if the epi- of hospitalization with COVID-19, d, was 7 d (24). demic is allowed to grow unimpeded until the hospital burden (as defined

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COVID-19 the of course Lancet the influence measures mitigation country-based Nature ac 6 2020. 16, March patients. medRxiv:10.1101/2020.03.03.20028423 contacts. close 2020). their March of (27 1,286 and cases 391 of Med. J. Engl. N. Med. J. (2020). 1018 China. Wuhan, Med. J. Engl. R olfel ¨ pdmooyadtasiso fCVD1 nSeze hn:Analysis China: Shenzhen in COVID-19 of transmission and Epidemiology al., et eff u eeto fCVD1 ncide neryJnay22 nWhn China. Wuhan, in 2020 January early in children in COVID-19 of Detection al., et 3–3 (2020). 931–934 395, 35e8 (2020). e375–e385 5, oe ooaiu rmptet ihpemnai hn,2019. China, in pneumonia with patients from coronavirus novel A al., et 6–6 (2020). 465–469 581, 382 ASCV2vrlla nuprrsiaoyseieso infected of specimens respiratory upper in load viral SARS-CoV-2 al., et h feto oto taeist euesca iigo ucmsof outcomes on mixing social reduce to strategies control of effect The al., et actIfc.Dis. Infect. Lancet (t siae ftesvrt fcrnvrsdsae21:Amodel-based A 2019: disease coronavirus of severity the of Estimates al., et .Eg.J Med. J. Engl. N. rsmtmtctasiso fSR-o-—igpr,Jnay23– January SARS-CoV-2—Singapore, of transmission Presymptomatic al., et lnclfaue fptet netdwt 09nvlcrnvrsin coronavirus novel 2019 with infected patients of features Clinical al., et actIfc.Dis. Infect. Lancet .Eg.J Med. J. Engl. N. iooia seseto optlzdptet ihCOVID-2019. with patients hospitalized of assessment Virological al., et ) .Eg.J Med. J. Engl. N. OI-9 oad otoln fapandemic. a of controlling Towards COVID-19: al., et lnclcaatrsiso ooaiu ies 09i China. in 2019 disease coronavirus of characteristics Clinical al., et 7812 (2020). 1708–1720 = q(t Ofr nvriyPes 1992). Press, University (Oxford fet fnnpamcuia nevnin nCVD1 cases, COVID-19 on interventions non-pharmaceutical of Effects al., et 2–3 (2020). 727–733 328, Ipc fnnpamcuia nevnin NI)t reduce to (NPIs) interventions non-pharmaceutical of “Impact al., et γ Lancet ) h mato OI-9adsrtge o iiainand mitigation for strategies and COVID-19 of impact The al., et 3017 (2020). 1370–1371 382, MR(ob otl ky Rep.) Wkly. Mortal. (Morb. MMWR N = 0 /(β  1 0 9–0 (2020). 497–506 395, q(t c − 0,0 (2) 1717 (2020). 1177–1179 382, 1/R 3 (2020). e31 382, gives ) 7 (2020). e75 382, 69P7 (2020). P669–P677 20, oeigIfciu iessi uasadAnimals and Humans in Diseases Infectious Modeling S 3–3 (2020). 533–534 20, 0 0yod.Sbtttn Eq. Substituting <60-y-olds. (t eff u )i h fetv erdcienme with- number reproductive effective the is )) (t if ) otherwise. H H netosDsae fHmn:Dynamics Humans: of Diseases Infectious ≥ c tti on,cnatrtsare rates contact point, this At . H c and Science 1–1 (2020). 411–415 69, R eff u 1–2 (2020). 413–422 369, q(t t (t d o h parameters the For . ) ,sc htEqs. that such ), actPbi Health Public Lancet > 9 1, Lancet noEq. into actPublic Lancet 1015– 395, BMJ .Engl. N. 1 and 369, BMJ 2–5 [9] N. 5, 1 .Bet aafr rnmsindnmc eelteipatclt fCVD1 herd COVID-19 of impracticality the reveal sub-threshold dynamics Transmission for: and Data Brett, numbers T. 41. Reproduction of Watmough, management J. the for Driessche, den models Van Appropriate P. Keeling, 40. M. Rohani, P. Wearing, Changing models: H. epidemic in 39. periods infectious of distributions Realistic Lloyd, L. A. 38. Lauer A. S. 37. Linton M. N. 36. Zhang J. vaccination 35. booster One resurgence: Li pertussis Q. Combating 34. Rohani, P. Riolo, A. M. 33. Mossong J. 32. Cell de D. M. 31. Antia A. 30. transmis- the Projecting Lipsitch, M. Grad, H. Y. Goldstein, E. Tedijanto, C. Kissler, M. S. reckoning. COVID-19—A 29. Offline: Horton, R. 28. Sanche S. 27. May 28 pilot: survey infection (COVID-19) Coronavirus Statistics, National for Office 26. Stringhini S. 25. Docherty Mitigation B. Anderson, A. M. 24. R. Heesterbeek, H. Klinkenberg, D. Hollingsworth, D. T. 23. osdrd hr slmtdssetbedpeinpirt optlcapacity hospital to so prior and depletion reached, susceptible being limited is there considered, netosDsaeAetSuyGat5R01GM123007. Grant Study Agent Disease Infectious ACKNOWLEDGMENTS. (41). (10.5281/zenodo.400093) Availability. Data eoie 5Ags 2020. August http://doi.org/10.5281/zenodo.4000983. 25 Deposited Zenodo. 2.0.0). (Version strategies immunity transmission. disease (2002). of 29–48 models 180, compartmental for equilibria endemic diseases. infectious dynamics. and persistence of patterns (2020). of application. 577–582 and analysis Estimation cases: confirmed statistical reported publicly A truncation: right with data. case infections available publicly coronavirus novel 2019 study. modelling and descriptive A China: Dis. Infect. province, Hubei outside 2019 ease pneumonia. infected all. fit not does schedule diseases. infectious vaccines. period. postpandemic (2020). the through SARS-CoV-2 of dynamics sion 2. coronavirus syndrome 2020. June 3 Accessed conditionsanddiseases/bulletins/coronaviruscovid19infectionsurveypilot/28may2020. https://www.ons.gov.uk/peoplepopulationandcommunity/healthandsocialcare/ 2020. study. population-based medRxiv:10.1101/ (2020). A protocol. (SEROCoV-POP): characterisation Switzerland clinical WHO ISARIC 2020). April (28 the 2020.04.23.20076042 using 19 objectives. policy conflicting Balancing A: Biol. influenza Comput. pandemic for strategies oeaeadimnt nprussresurgence. pertussis on immunity and coverage al rnmsindnmc nWhn hn,o oe coronavirus– novel of China, Wuhan, in dynamics transmission Early al., et vligeieilg n rnmsindnmc fcrnvrsdis- coronavirus of dynamics transmission and epidemiology Evolving al., et LSBiol. PLoS eeoeet n ogvt fatbd eoyt iue and viruses to memory antibody of longevity and Heterogeneity al., et ihcnaiuns n ai pedo eeeauerespiratory acute severe of spread rapid and contagiousness High al., et 9–0 (2020). 793–802 20, h nuainpro fcrnvrsdsae21 CVD1)from (COVID-19) 2019 disease coronavirus of period incubation The al., et oilcnat n iigpten eeatt h pedof spread the to relevant patterns mixing and contacts Social al., et s .M apna,A .Kn,P oai h mato atvaccination past of impact The Rohani, P. King, A. A. Magpantay, M. F. es, ` eorvlneo niSR-o- g nioisi Geneva, in antibodies IgG anti-SARS-CoV-2 of Seroprevalence al., et nuainpro n te pdmooia hrceitc of characteristics epidemiological other and period Incubation al., et 1006(2011). e1001076 7, etrso 679hsiaie Kptet ihCOVID- with patients UK hospitalised 16,749 of Features al., et orecd aahv endpstdi Zenodo in deposited been have data code Source PNAS 2061(2018). e2006601 16, LSMed. PLoS Med. PLoS .Eg.J Med. J. Engl. N. hswr sfne yteNHtruhMdl of Models through NIH the by funded is work This mr.Ifc.Dis. Infect. Emerg. rc al cd c.U.S.A. Sci. Acad. Natl. Proc. t d | ≈ .Ci.Med. Clin. J. coe 3 2020 13, October 14(2005). e174 2, (2008). e74 5, dh H 0 c N 0 1910 (2020). 1199–1207 382, ho.Ppl Biol. Popul. Theor.

3 (2020). 538 9, 1 − 4017 (2020). 1470–1477 26, Lancet β c.Tas.Med. Transl. Sci. c γ 0,0 (2) | 3 (2020). 935 395, 42E7 (2015). E472–E477 112, ! o.117 vol. . 97 (2001). 59–71 60, Lancet n.Itr.Med. Intern. Ann. | Science aj78(2018). eaaj1748 10, o 41 no. P313–P319 396, 860–868 368, ah Biosci. Math. | Lancet 25903 PLoS [10] 172,

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