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David the second in outbreaks plague of Acceleration l niet(i a es rdrc pemnchuman-to- (pneumonic direct or www.pnas.org/cgi/doi/10.1073/pnas.2004904117 the fleas) primar- to rat was contribute (via transmission can indirect plague inference ily whether this (26). purpose concerning how this debate consider for previously then developed We have we method- that exploiting ology processes, transmission underlying the about of rate growth initial the in in increase outbreaks. change fourfold a striking Renaissance namely, the a outbreaks of period, course revealing plague the over (London), of dynamics transmission location span plague same 300-y a the from in exploring information by of epidemiological) investiga- and studies sources demographic, genetic genomic (historical, complement traditional from important we more Here identify patterns alone. to (Mod- ecological tions challenging strains or it makes human evolutionary which extant (London (21), genomes to core Death similar the Black similar Plague): the remarkably ern of is victims from 1348) 25). isolated 24, complete (21, strain fragments nearly The DNA reconstructing persistent from and genomes other pathogen sequencing and Researchers discov- plague by (23). Yersin of plague history pathogens after evolutionary bubonic century the to reconstructed 19th link have the bacterium’s in the proposed ered by as caused were 22), (21, plague historical of agent, that lished source causative the a of remains potential also bioterror pestis Plague the Yersinia some (14–18). to in due world occur concern the pandemic to third of continuing the outbreaks parts in now with evolution are Plague), and have We and ecology (Modern (6–13). 3–5), the disease (1, of infectious Asia studies of and theoretical Europe many in inspired centuries) pandemic behavior, 19th second and to the with (14th demography associated human outbreaks those on especially effects long-lasting and P observed COVID-19 the driving plague in density, change population climate rat as with acceleration. or such inconsistent factors, human roles are ecological potential or century and the discuss demographic observed 14th We of the the transmission. in that (pneumonic) direct rates suggest growth parameters slow trans- order-of-magnitude plague of nevertheless, of epidemic; estimates mode given epidemic the not any in the infer do mission to culminated data that information available estimate that enough Currently we provide fourfold. 1665, increased later of rate the Plague growth and Great epi- the the 1348 later with Between of that (“accelerated”). Death shows faster Black records significantly to these 14th spread of the demics from Analysis of Kingdom, centuries. sequence United 2020) a 25, 17th London, March of review in for patterns epidemics (received temporal 2020 plague 19, the August reveal approved records and FL, Historical Gainesville, Florida, of University Singer, H. Burton by Edited Canada; and 3R4, Canada; V8W 4K1, BC L8S Victoria, ON Victoria, Hamilton, University, of University Statistics, & Mathematics Canada; 4K1, L8S ON eateto ahmtc ttsis catrUiest,Hmlo,O 8 K,Canada; 4K1, L8S ON Hamilton, University, McMaster Statistics, & Mathematics of Department eqatf hscag ihu aigayassumptions any making without change this quantify We estab- definitively have paleogenomics in advances Recent es h it etr 1 ) hs vnshv a dramatic had have events at These since 2). (1, populations century sixth human the afflicted least have epidemics lague | London 1,20). 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J.D., D.J.D.E., and D.J.D.E., J.M., research; D.J.D.E., research; designed tools; performed analytic B.M.B. new B.M.B. and and J.D., J.D., H.P., J.M., J.M., D.J.D.E., contributions: Author ). Methods and (Materials epidemic an the after from and rate for before death adjust baseline years also a we estimating epi- plague; by plague to mortality due During nonplague likely (31). were deaths deaths total most of demics, 80% are use which cover Cummins, We to by mortality. estimated online published on deaths of information counts provide weekly registers parish don Registers. Parish 1593; of epidemic the LBoM for the (except available, Appendix). records SI a When reliable as 26). most use (6, our will are incidence we plague which epidemic, for centuries. mortality each proxy about 17th for specifically plague and information to each 16th us attributable during the give cause in thus by LBoM epidemics The deaths plague of known the counts of weekly provide 28) (27, Mortality. of Bills London reconstructed be data can of sources disease three extent 1). analyzed and (Fig. have the We death in documents. unusual of extant from is patterns Kingdom, which United to London, of city com- The biological further the for Data highlight potential to the that limit is best that present the inferences. uncertainties this at and beyond done plexities that the be but in implausible transmission can is pneumonic century strictly that 14th argue We human). catrAcetDACnr,Dprmn fAtrplg,McMaster Anthropology, of Department Centre, DNA Ancient McMaster * owo orsodnemyb drse.Eal [email protected] Email: addressed. be may correspondence whom To tan of strains h oegnm saot8%o h ulgnm,sae ewe l sequenced all between shared genome, full the of 80% about is genome core The lgeeieissra uhfse nte1t etr than century century. 17th 14th of the the that in in reveals thousands faster much Kingdom, spread of United epidemics plague Analysis London, from bacterium dynamics. records the archival pat- transmission the overall was identify of that cannot epidemics terns analyses indicate these such but remains all pestis, Yersinia skeletal of studies of agent Genetic causative numbers periods. Renaissance modest and of Medieval the through- out population Europe’s devastated plague of Epidemics Significance .pestis. Y. b eateto ilg,MMse nvriy Hamilton, University, McMaster Biology, of Department a,b,c fe 58(e.2,p 4 n e.3) Lon- 30), ref. and 54, p. 29, (ref. 1538 After n ejmnM Bolker M. Benjamin and , h odnBlso otlt (LBoM) Mortality of Bills London The . y raieCmosAttribution-NonCommercial- Commons Creative . y https://www.pnas.org/lookup/suppl/ NSLts Articles Latest PNAS d a,b,c eatetof Department | f9 of 1

POPULATION BIOLOGY Fig. 1. (Top Left) Part of a will proved in the PCC, dated 18 December 1644 (34). (Top Center) A parish register page from St Giles without Cripplegate, August 1665 (38). Image credit: Wellcome Collection, licensed under CC BY 4.0.(Top Right) One of the LBoM, for the week beginning 26 September 1665 (photo by Claire Lees, taken in the Guildhall Library, City of London). (Bottom) Mortality in London, United Kingdom, 1340 to 1380 and 1540 to 1680, aggregated 4-weekly, plotted on a log scale. The three distinct sources of data (SI Appendix, Table S2) are last wills and testaments of Londoners whose wills were probated in the Court of Husting (14th century) or the PCC (16th to 17th centuries), weekly aggregations of burials listed in extant parish registers (29), and weekly plague deaths listed in the LBoM (27). Major plague years are highlighted in yellow. Aside from these and various minor plague years, there are notably unusual patterns during an epidemic of sweating sickness in 1551 (ref. 29, p. 70), the influenza epidemic of 1557 to 1559 (ref. 29, p. 70) (which coincided with the end of the reign of Bloody Mary I and the ascension of Elizabeth I in 1558 [indicated by a crown icon]), and the absence of a monarch during the Interregnum from 30 January 1649 to 29 May 1660.

Wills and Testaments. Before 1538, no direct tabulations of mor- use online data from the Prerogative Court of Canterbury (PCC) tality for London are available. However, we do have detailed (34) for the plague epidemics of the 16th and 17th century. information on last wills and testaments, which can be used as Plague epidemics in London are mentioned in historical docu- proxies for mortality (29). In particular, we have digitized and ments from the 15th century (ref. 27, chap. IV), but during this tabulated daily counts of wills recorded in the Court of Husting period, wills recorded in the Courts of Husting and Canterbury (32) for the 14th century; these give us data for the plague epi- were too sparse to enable identification of epidemics. demics of 1348, 1361, 1368, and 1375 [monthly counts for three Two dates are associated with each will: the date on which of these epidemics were previously published by Cohn (33)]. We it was written and the date on which it was probated (i.e.,

2 of 9 | www.pnas.org/cgi/doi/10.1073/pnas.2004904117 Earn et al. Downloaded by guest on September 27, 2021 Downloaded by guest on September 27, 2021 i.2. Fig. to by LBoM appear and registers counts dif- parish will and in recorded times, analysis mortality mortality lagged peak cross-correlation the have epidemic on the in written between Based for evident wills ferences data 2). of patterns the (Fig. numbers epidemic but too records the plague noise, is that (SI the the wills verify follow PCC from (37) outbreaks of signal three data number a later an the extract of 1593, are to segments and any small written 1563 on sampled date In relying poorly Appendix). by avoid or rates—and wills mortality questionable of for counts proxy asser- our adequate aggregated test to that particular, results—in tion cross-check to us allows (primarily property owned the who biased 2) strongly others times males). and is and 10 wills registers) about merchants wrote parish are toward who popu- on 1 individuals the based Fig. of of counts in subpopulation proportion the counts smaller than will much (the smaller a sampled 1) is that lation mind in keep disease with (36). can patterns symptoms hence epidemic influenza century (and for 21st searches predict internet been of as have will just fear period, to a incidence), long likely the which were a with dates on by will correlated death date epidemics, testator’s The severe the during incidence. but precede plague may for written proxy was a as val pro- Appendix, (SI were (35) most epidemics between written date; were century S2). lag they probate Fig. 14th after irregular the months with and several and associated bated death variable wills the the a of short of is date over inappropriate there (unknown) mortality are the as these of dates series scales by time probate long-term time died detailed inferring mortality, have reconstructing for annual for to well Tra- in works known by testator). approach trends are wills this the testators of While of counts dates. as death annual date the used probate after have demographers court ditionally, the by accepted pi,1Jl,ad1Otbro ahya.Weswt eocut r hw ln h otmeg ftegah(rsn nal1t etr epidemics, century 14th all in (present graph 1 the show were of lines wills edge dashed comparison, al. bottom Vertical visual et the Earn For counts). along 1. daily shown Fig. original are of the counts legend on the zero based and with were text Weeks year. main wills each the to in of (fits 1665). described October and observations are 1625, 1 mortality ) S2 and of Table July, Appendix , frequency 1 (SI the April, sources match data to The weekly 3). aggregated Fig. in plotted and 1 o h eidsne14,teeitneo utpesources multiple of existence the 1540, since period the For must we incidence, or mortality for proxies as wills using When inter- time given a in written wills of counts the use we Instead, bevdtm eis(ons n hnmnlgclmdlfis(lines; fits model phenomenological and (points) series time Observed ipa h aadrn h ao lgeeieiso ierrte hnlgrtmcscales. logarithmic than rather linear on epidemics plague major the during data the display S6, and S5 Figs. Appendix , SI htyedteetmtdgot ae lse nTable in (listed rates growth estimated the yield that ) Methods and Materials h vrl ifrnebteneoh,w hnueteestimates the use then we epochs, summarize between To difference CIs. overall profile the likelihood with along detail), first more see (using for overconfidence; curve a avoid death of cumulative to rate the differences growth to initial fitted the function of natural logistic estimate likelihood or maximum characteristics the disease the of (26). knowledge history any out are initial we R the which consider observa- rate forecasting, primarily from growth to we process Furthermore, primarily attempting]. separate applies not (39) that noise models exhortation influ- al.’s tion epidemic the et use [King estimate rate, processes to or growth mechanistic variability various size, of of (starting ence sources phe- series different simple time separate use size; short that total to to parameters and five fitted of chose of number be we total small can a a Thus, at with century week). models peaked epi- 14th nomenological 1375 per each the and written for 1368 during wills of series especially epidemics noisy, time the and data—the (e.g., short our are constrained of are demic limitations we rates, epidemics the growth plague epidemic by estimate during to us series allows Rates. time mortality Growth high-resolution Epidemic of Estimation sin- Approach a 1679. in May occurred 5 6 LBoM of are than the week in plague more the recorded from during deaths never plague deaths 1666, last 77 The after week. of gle LBoM total of the a end in 1666: the of recorded in since all Plague London in Great were occurred the in these have only epidemics 1666; discussed plague are and No and 1563 magnitude smaller between much London in Appendix). demics (SI lag growth this epidemic to about robust be inferences to our appear wk; rates 5 to 3 approximately 0 eetmt h pdmcgot aefrec pdmcas epidemic each for rate growth epidemic the estimate We epi- plague other 18 were there that indicate LBoM The cnb siae eibyfo otlt aaaoewith- alone data mortality from reliably estimated be —can r hc—nietebscrpouto number reproduction basic the which—unlike , n ontatmtto attempt not do and ) Methods and Materials hl u olcinof collection our While NSLts Articles Latest PNAS aeil n Methods and Materials IAppendix. 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POPULATION BIOLOGY for each epidemic as observations in a linear mixed model includ- Results ing fixed effects of epoch (early/14th century vs. late/16th to 17th The fitted models do as good a job as one could hope for centuries) and data source, and a random effect of outbreak year in capturing the initial patterns of epidemic growth evident in (see below), where each estimate was weighted by its certainty. the noisy will counts and agree very well with the patterns of Finally, to robustly quantify the statistical significance of the growth in the mortality time series that are available in the later difference between epochs, we ran a permutation test. We com- epoch (Fig. 2). puted the null distribution of the between-epoch difference in The growth rates for the 14th century epidemics are lower growth rate by scrambling the estimates and associated CIs than those for the 16th to 17th centuries (Fig. 3); they may randomly across epochs. For each of the 126 possible permuta- also be more variable than in the later period, although some tions, we refitted the mixed model and extracted the estimated of the appearance of variability is due to larger uncertainties among-epoch difference in growth rate. (presumably due to the relatively poor sampling in the early epoch). The mixed model quantifies the dramatic increase in Estimating R0 and Attack Rate. Our analysis focuses on the epi- growth rate (“acceleration”) of epidemics between epochs. The demic growth rate r, rather than the more commonly investi- growth rates increased fourfold in the late vs. early epochs: r gated basic reproduction number R0 [the expected number of increased 3.9× [95% CI (1.5×, 10.3×)] (Materials and Methods). secondary cases caused by a primary case in a completely sus- The permutation test results support the statistical significance ceptible population (7)], because estimates of R0 depend on of this difference; the observed order of outbreaks yields the information about the life history of the pathogen as well as on most extreme difference in growth rates out of all 126 possible the epidemic curve itself (40). However, if we can estimate R0, orderings, giving a two-tailed P value of 2/126 = 0.016. then we can predict epidemic outcomes that we are unable to predict from r alone; then, by comparing observed with theoret- Discussion ical outcomes under different scenarios, we can make inferences Plague epidemics in London appear to have been faster in the about modes of transmission. 16th to 17th centuries than in the 14th. Our analysis shows that In particular, if the host population is homogeneously mixed, this difference is very likely real, rather than driven by changes in we can use R0 to estimate the expected final size Z (the propor- the types of documentary evidence that are available; in particu- tion of the population infected by the end of an epidemic, also lar, we have shown that inferences from will counts are consistent called the attack rate) (6, 41). The observed attack rate based on with those from death counts when both are available, in the 17th mortality records depends in turn on the fraction of infected indi- century. viduals who die from disease. We can thus compare theoretical Why the later plague epidemics were faster than the earlier expectations of observed attack rates under different transmis- ones is not clear. Nevertheless, to provide some context, we con- sion modes with historical information about observed attack sider below a number of mechanisms that could have contributed rates (see Implications of Growth Rate Estimates for Transmission to increases in epidemic growth rates. We then consider what Mode). our estimated epidemic growth rates suggest about the mode of We can use the distribution of the generation interval (the transmission during the various London plague epidemics. times elapsed between the moment a host is infected and the times at which they infect others) to compute R0 from r (40). Possible Causes of Acceleration. What factors could have caused If we know only the mean generation interval Tg, we can plague epidemics in London in the 16th and 17th centuries to still approximate R0 ≈ rTg + 1, by assuming that the genera- accelerate relative to the earlier plagues in the 14th century? tion intervals are exponentially distributed; this approximation Pathogen or host evolution. Faster growth could be a conse- underestimates R0 in the typical case where the generation inter- quence of the pathogen evolving to infect hosts more effectively val distribution is less variable than an exponential distribution or to remain infectious for longer. However, such evolution with the same mean. may be challenging to demonstrate, particularly in light of the Beyond the information on generation interval that we use to strong genetic similarity between the Second Pandemic and derive R0 from r, the observed attack rate depends on addi- Modern Plague strains (21). Evolution of host resistance has also tional characteristics of an epidemic. If the population does been suggested as a cause of changes in plague dynamics (11), not mix homogeneously, the final size will typically decrease although we have no particular mechanisms in mind for how (e.g., figure 6 in ref. 42). Since the case fatality proportion the evolution of resistance over the course of centuries could (CFP; the proportion of plague cases who die) is not 100%, accelerate epidemics. the observed attack rate is less than the final size. Observed Shifts in transmission mode. Modes of transmission may have attack rates also decrease if some individuals are infected with- differed between epochs, which could account for differences out showing symptoms (asymptomatic ) (43) or without in epidemic growth rates. The reservoir host of Y. pestis in symptoms identified as plague (subclinical infections). Finally, London was rat populations where it was primarily spread by the observed attack rate depends on the initial proportion of the flea vectors such as Xenopsylla cheopis (ref. 14, p. 54). This population susceptible, which will be lower if some individuals transmission mode gives rise to bubonic plague in humans, have previously acquired immunity after prior nonfatal plague as a consequence of zoonotic spillover; epidemic growth rate infections. is driven by spread among rats and fleas, with human infec- We can use independent estimates of the generation inter- tion and mortality as a secondary consequence. In contrast, val to estimate R0 (and hence the theoretical attack rate) for when Y. pestis infection in humans spreads to the lungs, London epidemics under different transmission modes. Given it gives rise to pneumonic plague, which can then spread the limited information that we have, we consider the additional directly from human to human like other severe respiratory (extreme) assumptions that 1) the population was homoge- infections. It has also been more recently hypothesized that neously mixed, 2) the population was initially 100% susceptible, human ectoparasites (including lice and human fleas) could 3) there were no asymptomatic infections, and 4) all infected have supported indirect plague transmission without involving individuals died (i.e., the CFP was 100%). All of these assump- rats (13, 45). tions increase the observed attack rate (as detected from wills or Pneumonic plague probably has shorter generation intervals deaths), so we can calculate an upper bound on the total mortal- than bubonic plague (44, 46) and is often thought to spread much ity we would expect to observe in an epidemic with a particular faster at the population level (44); one potential explanation for transmission mode. our observations is that 14th century epidemics were primarily

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(the occurred centuries Age 17th Ice tle to the 16th between and significantly 14th changed climates European (54). triangle Northern disease host–parasite–environment the of side third even occur change. 53). that Environmental to (46, dynamics epidemic attempting single population a without complex during manner, truly quantitative the (crude) model in a are but in effects estimate, these them of to end rat magnitudes rats difficult The higher dying hosts. extremely addition, departing rat fleas In susceptible that on (52). likely up density more it flea, make and densities cer- rat, almost However, affect conditions) epidemics. living tainly pneumonic (and of population human speed from have in the changes only spillover on would effect by (51). density direct houses driven a population pest are human in epidemics, epidemics contacts by rat–flea their plague century and bubonic 17th sick S1), the Since the Fig. in of and exacerbated isolation S1 were the Table increases Appendix, 14th density (SI the and (48–50) from centuries enormously 17th increased pop- to London and in size sup- density population community Human ulation ecological humans. concern the here of studied rest the on porting not and humans of ecology changes. demographic and Ecological at made be can mode present. transmission below about consider inferences we but any hypothesis, whether this are reject epidemics we or confirm data, century to available unable currently 17th With to pneumonic. primarily 16th were while transmitted, bubonically estimates and All (44). centuries) times). epidemics 17th doubling plague associated to pneumonic the century 16th lists 20th late, also from vs. estimated (which that 1 century Table as (R 14th in same number (early, listed the reproductive was are epoch intrinsic (40) CIs implied each distribution and the for interval show generation scales rate the vertical growth that additional assumption average the interpretation, overall aid of To source. ) Methods data and (Materials estimates mixed-model siae nta rwhrts(r rates growth initial Estimated .pestis Y. .pestis Y. eas h nydt vial o h epochs the for available data only the because seteeycmlctd(7.W ou on focus We (47). complicated extremely is h xenlevrnetrpeet the represents environment external The n 5 rfieCso oaihi cl,frec fteeieissoni i.2 eodadtidpnl show panels third and Second 2. 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POPULATION BIOLOGY Table 1. Maximum likelihood estimates (MLEs) of the initial exponential growth rate (r) and doubling time ((log 2)/r) with their 95% CIs, obtained from the time series shown in Fig. 2 (see Materials and Methods) Assuming pneumonic plague

2 Source Epidemic Growth rate (1/y) Doubling time (days) R0,P Attack rate ZP R Husting wills 1348 4.5 (3.4, 6.2) 55.6 (41.1, 74.1) 1.06 (1.04, 1.09) 0.11 (0.08, 0.16) 0.439 Husting wills 1361 9.2 (4.9, 15.4) 27.5 (16.4, 51.6) 1.14 (1.07, 1.25) 0.23 (0.12, 0.37) 0.693 Husting wills 1368 8.4 (3.3, 21.7) 30.2 (11.7, 77.2) 1.12 (1.04, 1.38) 0.21 (0.08, 0.49) 0.333 Husting wills 1375 20.3 (6.0, 60.3) 12.5 (4.2, 42.2) 1.35 (1.08, 2.41) 0.47 (0.15, 0.88) 0.669 London bills 1563 21.8 (20.1, 23.7) 11.6 (10.7, 12.6) 1.38 (1.34, 1.42) 0.49 (0.46, 0.52) 0.961 London parish 1563 20.6 (18.9, 22.5) 12.3 (11.2, 13.4) 1.35 (1.32, 1.39) 0.47 (0.44, 0.51) 0.978 London parish 1593 15.9 (14.2, 17.3) 15.9 (14.6, 17.8) 1.26 (1.23, 1.29) 0.38 (0.35, 0.41) 0.986 Canterbury wills 1603 14.0 (8.3, 23.8) 18.0 (10.6, 30.4) 1.23 (1.12, 1.42) 0.34 (0.21, 0.53) 0.853 London bills 1603 29.1 (26.5, 31.8) 8.7 (8.0, 9.5) 1.54 (1.48, 1.60) 0.61 (0.57, 0.64) 0.927 London parish 1603 19.9 (18.2, 21.8) 12.7 (11.6, 13.9) 1.34 (1.30, 1.38) 0.46 (0.43, 0.49) 0.988 Canterbury wills 1625 27.1 (11.4, 53.5) 9.3 (4.7, 22.3) 1.49 (1.18, 2.19) 0.58 (0.28, 0.84) 0.900 London bills 1625 27.3 (25.9, 28.4) 9.3 (8.9, 9.8) 1.50 (1.46, 1.52) 0.58 (0.56, 0.60) 0.996 London parish 1625 23.0 (22.3, 23.7) 11.0 (10.7, 11.3) 1.40 (1.39, 1.42) 0.51 (0.50, 0.53) 0.999 Canterbury wills 1665 35.8 (17.6, 47.5) 7.1 (5.3, 14.4) 1.69 (1.29, 2.01) 0.69 (0.42, 0.80) 0.842 London bills 1665 22.2 (21.0, 23.4) 11.4 (10.8, 12.0) 1.39 (1.36, 1.41) 0.50 (0.48, 0.52) 0.984 London parish 1665 23.2 (20.6, 26.0) 10.9 (9.7, 12.3) 1.41 (1.35, 1.47) 0.52 (0.47, 0.56) 0.987 Husting wills Early 5.9 (3.5, 9.8) 43.2 (25.7, 72.5) 1.08 (1.04, 1.15) 0.15 (0.08, 0.25) Canterbury wills Late 23.0 (11.7, 45.0) 11.0 (5.6, 21.5) 1.40 (1.18, 1.94) 0.51 (0.29, 0.78) London bills Late 23.9 (21.3, 26.7) 10.6 (9.5, 11.9) 1.42 (1.37, 1.48) 0.53 (0.49, 0.57) London parish Late 20.4 (18.3, 22.8) 12.4 (11.1, 13.8) 1.35 (1.31, 1.40) 0.47 (0.43, 0.51)

The goodness of fit measure (R2) is the proportional reduction in the mean squared error, i.e., 1 − d2/σ2 where d2 is the model mean squared error and σ2 is the (population) variance of the data; predictions and observations are aggregated to weekly before computing the R2 for wills, for consistency among data sources. The implied basic reproduction numbers (R0,P) and attack rates (ZP), assuming the generation interval distribution for Modern pneumonic plague (44), are also shown. All MLEs and associated CIs are shown in Fig. 3.

discussed in Approach would lower the total mortality). Total maximum attack rate of 51% [CI (27%, 81%)], compared to mortality of 15% is implausibly low in comparison to typical esti- the estimated ≈ 20% of the population who died in each of mates for 14th century plague epidemics [e.g., “[b]etween the the late-epoch epidemics (ref. 31, p. 4). Given that the CFP years 1346 and 1352, [plague] caused the death of ... one-third for (Modern) pneumonic plague is nearly 100% (20, 67–69), of the world’s population at that time” (ref. 19, p. 163)] and cer- reconciling the predicted attack rates with observed mortality tainly inconsistent with the careful and comprehensive analysis of would require strong effects of heterogeneity, a large frac- Creighton for the 1348 to 1349 epidemic in London specifically tion of individuals resistant to plague due to prior nonfatal [“the mortality was about one-half the population” (ref. 27, p. plague infections, a large proportion of asymptomatic or sub- 128)]. It is therefore unlikely that 14th century plague epidemics clinical infections, or a decrease in transmission rate as the in London were primarily pneumonic. plague spread (perhaps due to behavioral avoidance of trans- In the early epoch, growth rates appear to be consistent mission). The effects of heterogeneity and behavioral avoid- with bubonic plague. What if we assume bubonic transmis- ance may indeed have been stronger in the late epoch, as sion? Unfortunately, the expected total human mortality from a result of concentration of the wealthy in central London a bubonic plague epidemic depends on several poorly known and a tendency for them to flee London during plague epi- aspects of rat ecology. For the generation interval Tg for bubonic demics (ref. 31, p. 4). While we still consider direct (pneumonic) plague we use a rough estimate of 18 d (≈ 0.05 y) based on transmission in late-epoch London plague epidemics unlikely the flea incubation period of 9 to 26 d (46), which dominates because of the strong effects that would be needed to reduce the time scales of other processes involved in rat–flea plague the expected mortality rates to the observed levels, we cannot dynamics (see SI Appendix for further detail). In combination rule it out. with a growth rate estimate of r ≈ 6/y, we obtain R0,B ≈ 1.3 In the late epoch (as in the early epoch), growth rates appear and ZB ≈ 42%. However, this estimate of the final size gives the to be consistent with bubonic plague. Similar calculations with fraction of rats, not humans, infected over the course of the epi- the estimated r for the late epoch and the approximate gen- demic. Translating this rat final size to the final size of the human eration interval for bubonic plague give estimated R0,B = 2.1 epidemic requires us to quantify the amount of spillover from [CI (1.5, 3.5)] and predicted rat attack rates of ZB = 83% [CI rats to humans. In particular, we need to know the rat-to-human (61%, 96%)]. The human ZB would be identical, if we were to ratio and the number of infected humans per infected rat, which make the same assumptions as above. This attack rate is even can both be crudely estimated as 1:1 (53) (although these ratios higher than the 51% predicted for pneumonic plague, but the will vary during the course of plague epidemics). Given an esti- CFP for bubonic plague was probably only 30 to 60% in the mated CFP of 70 to 80% for early-epoch plagues (66), our (very 16th to 17th centuries (e.g., refs. 19 and 69), which could bring rough) mortality predictions are reasonably consistent with the the total mortality roughly in line with the observed ≈ 20%. observed human mortality rates. The lower CFP of bubonic plague would also lead to a build- In the late epoch, primarily pneumonic transmission is unlikely but up in the population of individuals who had survived previous cannot be ruled out. Now suppose that the late-epoch plagues plague epidemics and retained immunity (reducing the suscep- were pneumonic with a generation interval similar to that of tible population available to be attacked), which in combination Modern pneumonic plague. The implied reproduction num- with heterogeneity and asymptomatic infections makes bubonic ber (Fig. 3) would be R0,P = 1.4 [CI (1.2, 2.1)], which yields a transmission plausible for the late epoch.

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The allowed changes imagined later. have large have years emerged not acterize of have could hundreds that liv- used 1665 series be People or might y. 1348 records 300 in these than London more in spanning ing records handwritten of thousands of composed databases paper—machine-readable this flea and diffi- rat conclu- about are period. uncertainties Definite this plague) substantial during epoch. ecology the (bubonic early of transmission because the have cult indirect in to that about mode unlikely concluded sions primary and was the transmission plague) been of (pneumonic mode transmission the direct to respect with currently is acceleration substantial this unclear. of esti- of cause history on on natural The rely or infection. transmission not solely of do mode based the estimates about early our assumptions is particular, invaded the rates—in first conclusion than growth Death mated This Black faster the populations. much when century grew of human 14th plagues centuries the major of 17th The plagues and 1666). to 16th (1348 the pandemic second dur- the Kingdom, United ing London, in epidemics plague recorded the Conclusion. for parameters disease (79–81). of COVID-19 estimates incorrect lead or overconfident can to transmission—that presymptomatic of and importance the asymptomatic are assessing and there dynamics, and disease for perfect, while accounting means intervals generation no estimating issues—including by number many rate are still reproduction data growth basic modern convert course, the to Of of about easier estimates it information into more make estimates yield improved will should addition, intervals and generation In streams data estimates. modern consequently accurate are to outbreaks apply plague to reported of easier growth newly used par- initial have of we In estimate techniques counts to paper. The deaths. daily this and the hospitalizations, report in of cases, analyzed countries that have many exceed we far ticular, data data plague COVID-19 of historical quantity The (79). and discourse health quality public COVID-19 the of elements tant 78). 500,000 (76, order impact studied annually of worldwide greater cause epidemics deaths which much epidemics, plague a and influenza had seasonal 77) historical than has it (76, the COVID-19 that influenza Nevertheless, for 1918 indicates here. than the 75) lower for (74, evidence than much IFR the lower this but substantially is about limitations, is COVID-19 date detection for to case (IFR) accumulated to SARS-CoV-2 ratio fatality due the infection by uncertain The caused (73). disease, virus COVID-19 of pandemic Context. COVID-19 72). ref. cf. 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POPULATION BIOLOGY Ma et al. (26) found that a delayed logistic curve, which adds an exponen- Outbreak Years and Outbreak Windows. We refer to the calendar year in tially distributed delay between infection and mortality, estimated growth which an epidemic begins as the outbreak year. For each epidemic, we rates from mortality curves with less bias. However, the delay between infec- define an outbreak window that is the maximum time interval to be con- tion and mortality in our data is extremely uncertain—it would represent sidered for fitting. For the 14th century epidemics, we defined the outbreak another parameter that would have to be estimated from noisy data. For window to be the full outbreak year, except for outbreak year 1348 when wills data, the delay could even be negative; i.e., wills could have been the epidemic spanned both 1348 and 1349 (so we took the outbreak win- written out of fear of future infection. Furthermore, because we found dow to be the full length of both calendar years). For the late-epoch little difference between the Richards and logistic fits—which Ma et al. epidemics, we defined the outbreak windows as sequences of consecutive (26) found to have opposite biases—growth rates estimated by these two weeks during which at least one plague death was listed in the LBoM (SI methods are likely to bracket the growth rate estimated by the delayed Appendix, Tables S6 and S7). logistic. Data and Code Availability. Our epigrowthfit R package for estimating initial Fitting Windows. How well a given model can fit a time series may depend growth rates, which includes all of the data used in this paper, is available on the time window used for fitting (88). In order to avoid choosing a in Github at https://github.com/davidearn/epigrowthfit. The data can also window separately for each individual epidemic, we considered window be obtained from the International Infectious Disease Data Archive (http:// selection criteria that could be applied uniformly to all epidemics with- iidda.mcmaster.ca). The additional code required to reproduce all of out first examining the data. In all cases we defined the end of the fitting our results is available in Github at https://github.com/davidearn/plague window to be one data point past the observed peak date (the date of max- growth. imum observed deaths). For wills and parish registers, we began the fitting window at the start of the outbreak window (see below); for LBoM data, ACKNOWLEDGMENTS. The plague mortality data were photographed we started immediately after the last observation before the peak for which and/or entered by Seth Earn, Kelly Hancock, Claire Lees, James McDonald, David Price, and Hannah Price. Valerie Hart facilitated research and photog- the mortality rate was ≤ 1/50 of the peak mortality rate. For two epidemics raphy at the Guildhall Library, City of London. Nonexponential comparison for which the data are particularly noisy (1593 and 1625), we were unable fits were conducted by David Champredon. Sang Woo Park assisted in find- to find any simple heuristic that would automatically choose a good fitting ing published estimates of auxiliary epidemic parameters. Alexandra Bushby window and resorted to choosing the fitting window by eye. assisted with compiling probate dates. Sigal Balshine, Ann Carmichael, and The fitting windows used for all our fits are listed in SI Appendix, David Price made helpful comments. Early stages of this research were Tables S6 and S7. Although we experimented with different methods for supported by a J. S. McDonnell Foundation Research Award to D.J.D.E. determining the fitting window (e.g., starting the window from the obser- (https://www.jsmf.org/grants/d.php?id=2006014). All of the authors were vation after the last local minimum before the peak or based on a threshold supported by Discovery grants from the Natural Sciences and Engineering Research Council of Canada. H.P. was supported by the Social Sciences and of 1/100 of the peak mortality rate), we emphasize that our choices were Humanities Research Council of Canada. The Shared Hierarchical Academic entirely determined by seeking good fits (judged by visual inspection) and Research Computing Network (SHARCnet; http://www.sharcnet.ca) provided not by the pattern of estimated initial growth rates. Moreover, our conclu- computational resources. D.J.D.E. dedicates this paper to Josephine Earn, sion that later plagues were much faster than the earlier plagues is robust who enthusiastically discussed the history of London (and plagues) with him to choices we made about window selection. over many years.

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