Downloaded by guest on September 26, 2021 www.pnas.org/cgi/doi/10.1073/pnas.1811115115 modeling computational dynamics correct the data. characterize oper- from to can used number be reproduction aimed ationally instantaneous methodologies the that pop- estimating evidence the at provide of structure we contact Finally, (clustered) ulation. the be by can and explained it changes Rather, simply behavioral determinants. to epidemiological due exogenous not other conceptual is any the picture from provide theoretical departure classical This not evolution. the does epidemic untenable the it that is of understanding and number find populations, reproduction realistic We basic in the events. of concept record- classical individual-level time, generation explicitly all and We ing number data. reproduction epidemiological the empirical measure influenza on of history calibrated natural the and process for transmission accounting networks Dutch these and an on infer Italian simulate the we then of We subset Here, con- populations. a multiplex data. of two representative real networks data tact in sociodemographic detailed found highly theoretical hardly clear-cut from A is and epidemic. however, an primary of picture, charac- between dynamics the the for interval of allows terization number (the reproduction the time cases), secondary generation Along models. the mixing stratified infected analytical with clear and typical a homogeneous a finds in population by expression susceptible generated fully as a cases in definition secondary individual classical corner- of conceptual Its number the . 2018) the 27, of mathematical June one review of for is (received stones number 2018 16, reproduction October basic approved and The NJ, Princeton, University, Princeton Levin, A. Simon by Edited Spain; Zaragoza, 50018 and Zaragoza, Spain; of Zaragoza, University China; 50009 Systems, of Zaragoza, Complex Republic 02115; People’s of MA Sichuan, Physics Boston, 611731, and University, Chengdu Biocomputation Northeastern China, Systems, of Socio-Technical Technology and and Biological Science Electronic of University Center, a Liu Quan-Hui networks in contact number data-driven reproduction epidemic the of Measurability R definition the for Furthermore, of account patterns. to interaction (9) networks complex contact more heterogeneous mod- to stratified and to (8) extended concept els been The has 7). number (4, of reproduction spreads the infection timeline of the of which (e.g., patterns in contact population the system the and ), pathogen–host the inside the replication pathogen contact), transmission biological of the a probability by (e.g., given pathogen determined the are of time characteristics outbreak generation epidemic and an number of occurrence encom- the they cornerstones (R as for models, the condition epidemic the are basic pass quantities of and understanding These infector our the 6). of of (5, time fully infectees infection a the her/his time between in generation interval period the time is infectious the other the The of (4). course population susceptible entire indi- the infectious typical over a by vidual generated cases number of secondary of concepts. reproduction understanding number fundamental basic our two the to is approach, tied One modeling generally is of models type epidemic the Indepen- (1–3). of response dent and preparedness epidemic to support M number reproduction e cecsCne,Uiest fEetoi cec n ehooyo hn,Cegu613,Scun epesRpbi fChina; of Republic People’s Sichuan, 611731, Chengdu China, of Technology and Science Electronic of University Center, Sciences Web (t R 0 ) > 0 ie,teaeaenme fscnaycssgnrtdby generated cases secondary of number average the (i.e., a engnrlzdb h fetv erdcinnumber reproduction effective the by generalized been has ae r nraigyrcgie srlvn quantitative relevant as dis- recognized infectious increasingly of are models eases computational and athematical n h pdmcduln ie ohtereproduction the Both time. doubling epidemic the and 1) a,b,c ac Ajelli Marco , | eeaintime generation | netosdiseases infectious c,d g S onain 02 ui,Italy Turin, 10126 Foundation, ISI let Aleta Alberto , | R utpe networks multiplex 0 hc steaverage the is which , e,f tfn Merler Stefano , | Tg , 1073/pnas.1811115115/-/DCSupplemental. y at online information supporting contains article This 1 oeiaie ies . C BY-NC-ND) (CC 4.0 NoDerivatives License distributed under is article access open This Submission.y Direct research, PNAS a is designed article This A.V. interest.y of and conflict no Y.M., declare authors paper.y S.M., The the wrote A.A., and data, M.A., analyzed research, performed Q.-H.L., contributions: Author of intervention role the 17, performed disentangling 16, the methods, (10, these population of for the Unfortunately, impact in 37). changes the behavioral to and strategies due the mostly for are developed were methods of (35, statistical analysis data reason, real-world epidemic this by the For backed of 36). not “typi- often growth oversimplification evidence early a an growing exponential is classic of this, the to representative that addition shows In necessarily individual. not Indeed, infectious cal” is 32). which (31, values outbreak, estimated show the and R in limitations have differences methods marked both However, . fscnaycssgnrtdb h ne aeo h outbreak the of case index the by (R generated cases secondary workplaces) of of schools, estimation households, the for (25–34), accounting pop- human (e.g., the of ulations structure actual the resembling closely compute models oper- entailing to an models way although in (18–24), ational structure number household reproduction and the community structures of of definition contact estimat- the value in local communities) the the families, ing of extended have importance households, studies (e.g., the several in However, on used 11). widely cautioned are (10, and approaches models homogeneous predictive in epidemic the of (10). population susceptible fully a time at individual infectious an owo orsodnesol eadesd mi:[email protected] Email: addressed. be should correspondence whom To h ntnaeu erdcinrt sago ecitrof descriptor good a dynamics. of transmission as use the rate operational a reproduction the and instantaneous show phase the not for We growth number. is exponential that, reproduction it steady basic show a networks, define contact we to realistic possible approach, on microsimulation populations. spreading Dutch epidemics a and Italian using the describ- of By network pattern multiplex sociodemo- contact a the high-quality generate ing we on here number Based data, reproduction graphic situations. basic frame- the real-world as modeling in such homogeneous quantities, classic of and issues the work raised of has adequacy data epidemiological the real of analysis The Significance 0 index R 0 index d d 0 ai Moreno Yamir , rn ese onain 82 rno Italy; Trento, 38123 Foundation, Kessler Bruno and n/rteaayi ftegot aeo h simulated the of rate growth the of analysis the and/or ) scmue rmteaayi fteidxcs fan of case index the of analysis the from computed is Tg R (t r ahmtclywl endi h al stages early the in defined well mathematically are f eateto hoeia hsc,Uiest of University Physics, Theoretical of Department suigta h aitoso hsquantity this of variations the that assuming ), R 0 e,f,g 1–7.Tertclwr a explored has work Theoretical (12–17). n lsadoVespignani Alessandro and , R R 0 . y 0 ssillcig o oecomplex more For lacking. still is anyrle ntedrc count direct the on relied mainly raieCmosAttribution-NonCommercial- Commons Creative t ,tu eaigtehptei of hypothesis the relaxing thus ), c aoaoyfrteMdln of Modeling the for Laboratory www.pnas.org/lookup/suppl/doi:10. 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POPULATION BIOPHYSICS AND BIOLOGY COMPUTATIONAL BIOLOGY connectivity patterns from interventions and behavioral changes in the definition of the reproduction number is challenging due to the lack of microscale data on human contact patterns for large fractions of the population. In summary, a clear operational way to compute R is still lacking, and no predictive modeling studies have thus far focused on the harder task of analyzing the dynamics of the reproduction number over time. Here, we study the very definition and measurability of R0, R(t), and Tg with data-driven microsimulation models of the infection transmission process based on a highly detailed syn- thetic population of agents, where we can have complete access to microlevel data (e.g., keeping track of all interactions of the simulated agents and the entire transmission chain) (25–30). Namely, we consider an infection transmission model where the pathogen spreads on two high-resolution synthetic multiplex contact networks inferred from real-world sociodemographic data. We simulated the progression of a simple influenza trans- mission model accounting for the natural history of the and calibrated to match empirical epidemiological data on the inferred multiplex networks. We measure basic quantities, such as R0, R(t), and Tg, and show that the heterogeneity and cluster- ing of human contact patterns (e.g., contacts between household members, classmates, work colleagues) determine a nontrivial variability of R(t) and Tg(t). This evidence, contrasted with the results from null models, highlights the challenges in measuring R0 and Tg in real-world systems. This raises questions about the adequacy of their definitions given the complex temporal dynamics of R(t) and Tg(t), both of which highly deviate from the classical theory. Our study suggests that epidemic inflection points, often ascribed to behavioral changes or control strategies, could also be explained by the natural contact structure of the population. Finally, we show that Bayesian analyses of the instan- Fig. 1. Model structure. (A) Visualization of the multiplex network repre- taneous reproduction number over time represent valuable tools senting a subsample of 10,000 individuals of the synthetic population. Note for understanding complex epidemic dynamics, provided that that the community layer is a complete graph, although not all edges are reliable estimates of Tg are available. visible for the sake of readability of the illustration. (B) Degree distributions in the school, household, and workplace layers. (C) Schematic represen- Results tation of the infection transmission model along with examples of the We use detailed sociodemographic data to generate two multi- computation of individual reproduction number and generation time for plex networks describing the contact patterns of about 500, 000 the simulated transmission chains. I, infectious; R, removed; S, susceptible. agents, each representative of a subset of the Italian and Dutch populations (Materials and Methods discusses the methodology). The effective contacts through which the infection can spread removal rate [such that an infectious individual spends an expo- are determined by the copresence of two individuals in the same nential amount of time (the removal time or ) settings. This effectively defines a weighted multiplex network in the infectious stage before recovering]. We keep the transmis- (38–40) made by four layers representing the network of contacts sion scheme as simple as possible (e.g., avoiding the introduction between household members, schoolmates, work colleagues, and of other classes, such as a latent compartment, the distinction casual encounters in the general community (Fig. 1A) (15, 26, between symptomatic and asymptomatic individuals, hospital- 34, 41, 42). Each node in the household layer represents one ized individuals, and so on) to avoid confounding effects. Of individual of the real population and is linked only to the other course, more refined models are needed to answer complex nodes representing members of her/his own household. A second questions, such as the impact of control strategies. layer represents contacts in school (i.e., every node represents We simulate the transmission dynamics as a stochastic process one student or teacher and has contact only with other individu- for each specific individual, each with her/his own characteris- als attending/working in the same school). A third layer accounts tics (e.g., age, individual infectiousness, membership to a specific for contacts in workplaces, and a fourth layer encodes contacts in household or school, and so on), and accounting for the cluster- the community, where we assume a complete network [i.e., each ing of contacts typical of human populations. For instance, an individual has a certain (low) probability of infecting any other agent can transmit the infection in a given school only if she/he individual of the population]. The four layers are characterized studies or works there. In each layer, the infection transmission by remarkably different degree distributions (Fig. 1B). This rep- between nodes is calibrated in such a way that the fractions of resentation of links between individuals readily highlights the cases in the four layers are in agreement with literature values typical strong clustering of human populations, where individuals (namely, 30% of all influenza are linked to transmis- tend to meet the same set of contacts (e.g., household members, sion occurring in the household setting, 18% of all influenza schoolmates, colleagues) on a regular basis (43–45). infections are linked to transmission occurring in schools, 19% The influenza-like transmission dynamics are defined through of all influenza infections are linked to transmission occurring a susceptible, infectious, removed (SIR) scheme (Fig. 1C). Ess- in workplaces, and 33% of all influenza infections are linked to entially, susceptible individuals can acquire the infection through transmission occurring in the community) (26, 27, 33). Moreover, contacts with infectious individuals, and as soon as they are we set these layer-specific transmission rates such that the repro- index infected, they proceed to the infectious stage. Infectious indi- duction number of the index case (R0 ) is 1.3 [in agreement viduals then move to the removed compartment according to a with typical values reported for influenza in the literature (46)].

2 of 6 | www.pnas.org/cgi/doi/10.1073/pnas.1811115115 Liu et al. Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 r ftemlilxntokhlsi nesadn h origin the understanding in helps network multiplex the of ers model (6). homogeneous time generation the the of to by analysis predicted the is 2 and (Fig. what from theory the differs classic before This the right 2C). shortening (Fig. marked peak more epidemic (≈ a average d with estimated 2.67 value), is The oretical epidemic simulations). infectious whole our the the in of over d duration generation 3 the the (i.e., than that period shorter find considerably We genera- model. is estimated data-driven time the the of in In analysis time analogous 2.1). tion an section show Appendix, we (SI 2C, simulations Fig. performed the simulation, in single of tern each increase of time early outcome the the that sup- highlighting of is analysis result our this model, by the ported of realizations 50,000 over averaged of favor in indicator that mental repro- implying the phase, of constant of dynamics a growth temporal average number The an number. higher to heavy-tailed duction leading in a thus observed with already (51)], degree can effect networks individuals average [an model of contacts the adequate infection of data-driven of variation the the the by in by induced explained found partly epidemiology pattern be mathematical also The classical by (4). predicted peak theory epidemic as the before 2A) declines (Fig. rapidly then and phase demic ntehmgnosmdl hc ak h yia structures typical the lacks populations, which human model, of homogeneous the in from ing i tal. et Liu R indi- each time by at infected of generated vidual infections track secondary keeping of by number microsimulations exact the the from computed be can reproduc- basic the of definition connectivity proper patterns. realistic by a characterized populations find in relation number to tion the difficult through is con- defined in it number be that, reproduction basic readily suggests the in where This can models time 2A). SIR simple over (Fig. to decrease trast phase monotonically epidemic and early rate slowly growth the epidemic sharp to in the about expected is where decrease result theory, is a classic marked Such the 2A). a (Fig. with peak contrast by epidemic non- the followed a before d occurs find 20 phase we epidemic initial where the the over model, of trend data-driven increasing constant the an in behavior: nearly monotonous case a the at not is exponentially increases num- cases rate the new models, homogeneous of In associated ber epidemic. the the and of time of rate function growth a as Time. inci- infections) the new are (of Generation dence models epidemic in and investigated generally Number quantities Reproduction synthetic the Effective to refer text configura- the time) Italy. in for over population reported fixed Results are models. (edges tion quenched and in rewired) are (details model model 1.2–1.5 homogeneous with data-driven the models the from null complexity alternative to homogeneous of of the level set increasing a with studied an Along also we probability. model, fixed contact SIR in same are and the identical are with individuals indi- all the where at level, model vidual SIR mixing report homogeneous also a of we of implementation text, analysis the In corresponding ). 1.1 the section Appendix, (SI model sion d 3 to time removal the 48). fix 47, we (15, generality, of loss without Finally, (t lsrlo ttetasiso rcs ntedfeetlay- different the in process transmission the at look closer A h al fetv erdcinnme n eeaintime generation and number reproduction effective daily The In ) r nrae vrtm nteerypaeo h pdmc start- epidemic, the of phase early the in time over increases erpr l ftedtiso h transmis- the of details the of all report we Appendix, SI ,weetelnt fteifciu eidcorresponds period infectious the of length the where C), 5 9 0 uigteerypaeo h pdmc This epidemic. the of phase early the during 50) 49, (5, .Nl oesicueanae egsaeconstantly are (edges annealed include models Null ). R 0 index 1 = . 3 .I contrast, In 2B). (Fig. 2.1 about of peak a to t R .W n that find We 1C). (Fig. simulations the in (t ) R snal osati h al epi- early the in constant nearly is (t R lhuhw eotteresults the report we Although ). R (t 0 ) oe t enn safunda- a as meaning its loses and 11% Tg R sections Appendix, SI hre hntethe- the than shorter (t R (t ) R (t ) sacmo pat- common a is 0 ) o stochastic a for + 1 = osntshow not does h first The rTg (5), Tg he,wt n ne aeadtossetbemmes f at If, members. susceptible of two household and a case the time consider index of us one illustration with let simple three, households, a in provide effect house- To saturation for around data (26). real fluctuating analyzing transmission by average reported hold value an the to 3 B)—with close d, (Fig. found 2.6 also layers We layer, other the contacts. household all across susceptible the of uniform in depletion more that, the sim- trend by be decreasing led general to a ply follows tends thus, and layer Indeed, nodes different 3A). household (Fig. the layers community in and household the in 2.2 infec- section of Appendix, as degree (SI well average the as that nodes found tious we Specifically, 3). of (Fig. deviations the of infectious the of duration the of dotted value horizontal constant The the model. period. represent data-driven the line of of distribution gray realizations density single the the shows in area colored (C model. The mean data-driven the the represent of realizations lines single of distribution the density in the obtained shows area colored The models. neous apn ntesho ae.However, layer. school the in happens which time, Mean over (B) trend. infections linear influenza a new follow of not incidence does epi- daily time the mean at are to the aligned Results corresponds of are model. Results which of model. data-driven peak, each distribution the demic of realizations of density 50,000 realizations on the single based shows the in area obtained colored The models. rate, growth epidemic tial 2. Fig. C A B t h ne aeifcseatyoeo h w susceptibles, two the of one exactly infects case index the , endiyexponen- daily Mean (A) indicators. epidemiological Fundamental R r Tg(t R vrtm ftedt-rvnadhomogeneous and data-driven the of time over , (t (t ,ada es osm xet h same the extent, some to least at and ), ) fdt-rvnadhmgnosmodels. homogeneous and data-driven of ) ) and edt eki h oklc layer workplace the in peak to tend Tg Tg srmral hre hnin than shorter remarkably is (t t R(t ) = fdt-rvnadhomoge- and data-driven of ) rmtecascltheory classical the from 0. R NSLts Articles Latest PNAS (t Inset ) eeal decreases generally Tg(t hw h logarithm the shows ausobtained values ) h three The ) r R(t (t values ) | values ) R f6 of 3 (t )

POPULATION BIOPHYSICS AND BIOLOGY COMPUTATIONAL BIOLOGY A by using values estimated for Italy as reported in the literature (55). We also consider prepandemic immunity by age in the pop- ulation according to serological data (55). is not considered, as vaccination started only during the tail of the pan- demic and had a very limited uptake in Italy (vaccination cover- age < 1%) (56). As in the previous section, the model has four unknown parameters: the four layer-specific transmission rates. They are calibrated through a Bayesian Markov chain Monte Carlo (MCMC) approach on seroprevalence data by age collected in Italy before and after the 2009 H1N1 influenza (55). Model details are provided in SI Appendix, section 1.1. The calibrated model is able to well capture the seropositive rates by age at the end of the pandemic (Fig. 4A). The estimated growth rate from the influenza-like illness (ILI) cases reported in Italy over the course of the 2009 H1N1 influenza pandemic clearly shows an increasing trend during the early phase of the epidemic followed by a sharp drop about 3 weeks before the epidemic peak (Fig. 4B). The trend observed in the data is consis- B tent with that obtained in model simulations (Fig. 4C). Fitting a

A

B

Fig. 3. Layer-specific patterns. (A) Mean R(t) for the data-driven model in the four layers. The colored area shows the density distribution of R(t) values obtained in the single realizations. (B) Mean Tg(t) for the data-driven model in the four layers. The colored area shows the density distribution of Tg(t) values obtained in the single realizations. C then at time t + 1, the index case has to compete with the other infectious individual to transmit the infection; the resulting gen- eration time is shorter, simply because she/he cannot infect any other household member, although she/he may still be infectious. A similar argument was used in ref. 52 to explain why observed generation times are shorter than the infectious period. This evi- dence calls for considering within household competition effects when providing empirical estimates of the generation time (or ) from household studies. Saturation effects are also responsible for shortening Tg in the other transmission set- tings, with the exception of the general community (Fig. 3B). All of the observed patterns of R(t) and Tg(t) are robust with Fig. 4. The 2009 H1N1 influenza pandemic in Italy. (A) Seroprevalence rates respect to changes in the sociodemographic structure of the pop- by age as observed in a serosurvey conducted at the end of the 2009 H1N1 ulation (i.e., we simulated the infection spreading in both the influenza pandemic in Italy (55) and as estimated by the calibrated model. Italian and Dutch synthetic populations), influenza transmission (B) Epidemic growth rate over time r(t) as estimated from the weekly inci- intensity (measured in terms of Rindex), and the distribution of the dence of new ILI cases in Italy over the course of the 2009 H1N1 influenza removal time (we tested exponential and gamma distributions). pandemic and the best-fitting linear model from week 35 to week 41 in 2009 These analyses are reported in SI Appendix, sections 2.4–2.6. (scale on the left axis). Weekly incidence of new ILI cases in Italy over the course of the 2009 H1N1 pandemic (scale on the right axis). Data are avail- The 2009 H1N1 Influenza Pandemic in Italy. To test the robustness able at the ISS Influnet website (old.iss.it/flue/). Note that, over the period from week 35 to week 51 in 2009, schools were regularly open in Italy. (C) of the results in a more realistic epidemic transmission model, Temporal pattern of the mean weekly exponential epidemic growth rate (r) we used the data-driven modeling framework to model the 2009 resulting from the analysis of the data-driven model calibrated on the 2009 influenza pandemic in Italy. One of the characteristic signatures H1N1 influenza seroprevalence data. The colored area shows the density dis- of the 2009 H1N1 pandemic was the presence of a differential tribution of r(t) values obtained in the single realizations of the data-driven susceptibility by age (34, 53, 54); this is included in the model model.

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Contact Data-Driven Methods transmission and Materials epidemic the eventu- of dynamics. may temporal understanding and that potential our assumptions mislead untenable ally to resorting evidence without provide we of networks, estimates contact that realistic on the gen- for time the framework and eration number theoretical reproduction a lack- the providing of still not computation/definition is are theory we While the unified in ing. a number in theo- reproduction (19–24), the of models understanding define lot to nonhomogeneous done of a been Although has window work scenarios. retical epidemic a realistic open of analysis to numer- experiments silico” “in of ical advantage takes here presented analysis The of series time Conclusion (10). reliable underreporting a and of noise by availability affected the markedly generation on not the cases and of distribution 5) the (Fig. of estimates time knowledge obtained the the on of goodness depends The dynamics. possible epidemic is the it of that estimation the shows operationalize values to actual and estimated the between are realizations in (other realization shown model compari- stochastic one the for chain show we (11, 5, R approaches Fig. Bayesian In between using Methods). son by and cases (Materials new 17) of effec- number estimating its by daily of time terms over in number epidemic reproduction possible an tive still of approximate is dynamics an the it characterize However, as to dynamics. even epidemic the adequacy of its descriptor and populations behavior. realistic observed the of root the at are not structure models, populations, null human multiplex by of captured typical the effect clustering that strong the suggesting and model, data-driven the ( 2.7 models degree-preserving configuration as layer-preserving such con- models, and null human alternative of of Furthermore, analysis features. clustering the country-specific and than structure different rather patterns the in tact to consistent due observed be the are that to patterns suggesting 2.4), seem course section Appendix, results the (SI Our countries over epidemic. time an generation as of and such number indicators, reproduction epidemiological the fundamental of stan- results the between heterogene- alter dard contacts colleagues) (e.g., the work classmates, interactions how members, human household of highlight clustering clearly and ity results simulation Our Discussion 2.3). modeling section homogeneous devia- Appendix, the (SI noticeable from framework model the data-driven confirms the pandemic of tions influenza 2009 the on pat- of networks. dynamics these contact the of human for terms of In explanation structure intrinsic plausible the on a and based exist terns provide may results other explanations our for alternative (60), observed Although been (57–59). also countries has epidemic 0.064 the is of realizations stochastic (SE all regression 0.064 from (95% linear of incidence a case coefficient with obtained of estimated value model an mean time the in while over results 0.033), rate of data growth ILI estimated the the in to model regression linear (t u ueia td usin h esrblt of measurability the questions study numerical Our ,sosmrel ifrn eair hntoeehbtdby exhibited those than behaviors different markedly shows ), ) eutn rmtemcoiuaindt ftetransmission the of data microsimulation the from resulting CI, .Teoealgo agreement good overall The 2.8). section Appendix, SI .0–.0) nices nteiiilgot rate growth initial the in increase An −0.207–0.305). ~ G ecntu hrceieec layer each characterize thus can We α. R = (t (G ) R H , sifre rmtetm eiso ae and cases of series time the from inferred as (t G S ) , G a eue ocaatrz h epidemic the characterize to used be can W , G h oe osdr egtdmultiplex weighted a considers model The R C ,where ), (t ) and R H (t Tg , ) S , (t opoiepoetosof projections provide to W h oe calibrated model the ), and , section Appendix, SI G C α ersn house- represent R = E α (t (V stestof set the as ) , E rmthe from α ythe by ) R 0 in V ae rmdy1to 1 day from cases time at number duction where i.5. Fig. prahue nrf.1 n 7 easm httediynme fnew of number daily the equation: that following assume we 17, and cases 11 refs. in used approach of Estimation individuals. between γ contact step involve time not at that, does Given removed to infectious from by and contact given per probability is infectious) to that susceptible probability Specifically, the step individual. susceptible, time contact susceptible at a a that, requires and given infectious individual to infectious susceptible an between from sus- transition states: states The between three removed. transitions following (i of the possible: types of Two are one removed. model in and SIR infectious, be The ceptible, can model. SIR individuals an that as assumes process transmission influenza the simulate Model. Transmission Epidemic The population. have i we above, introduced notation the Following network. Network. Mixing Homogeneous nodes between link a matrix adjacency layer-specific distribution. and value time generation the of knowledge precise more a of estimate the of goodness The simulation. model selected of (C simulation. as model selected same the The of (B) d. 3 of of distribution average realization. an with model distributed stochastic exponentially one for infections new of C B A and orcvra iestep time at recover to Tg j vrtm sdrvdfo h nlsso h rnmsinte fthe of tree transmission the of analysis the from derived as time over C where , φ (t siainof Estimation ttime at ) stegnrto iedsrbto and distribution time generation the is R w Tg (t rmssetbet netosad(ii and infectious to susceptible from ) = rmTasiso vnsTm Series. 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hsmdlasmsasnl ul connected fully single a assumes model This si h netossae thsaprobability a has it state, infectious the in is α R(t N j j hr elements where , nec fteitoue ewrs we networks, introduced the of each On h aeas same The ) sifciu n t egbrnode neighbor its and infectious is infects stettlnme fidvdasi the in individuals of number total the is R(t ) α X s=1 β t sifre rmtediyincidence daily the from inferred as ) and = φ pw (s)C L i a (i.e., fteosre iesre of series time observed the of ij α where , (t = NSLts Articles Latest PNAS A − otherwise. 0 R(t i u sn h distribution the using but s) hne t ttsfrom status its changes steefcierepro- effective the is ) a ! ij α , p rmifciu to infectious from ) = Tg olwn h same the Following stetransmission the is R(t w sasmdt be to assumed is α a nrae with increases ) ij > = fteeis there if 0 w A o each for | u the but f6 of 5 i is

POPULATION BIOPHYSICS AND BIOLOGY COMPUTATIONAL BIOLOGY T t ! ACKNOWLEDGMENTS. Q.-H.L. acknowledges support from Program of the Y X L = P C(t), R(t) φ(s)C(t − s) , China Scholarships Council Grant 201606070059. M.A. and A.V. acknowl- edge the support of NIH Grant MIDAS-U54GM111274. A.A. acknowledges t=1 s=1 the support of the Formacion´ Personal Investigador Doctoral Fellowship from Ministerio de Econom´ıa y Competitividad (MINECO) and its mobil- ity scheme. Y.M. acknowledges partial support from the Government of where here, P(k, λ) is the probability mass function of a Poisson distribu- Aragon,´ Spain through a grant (to the group F´ısica Estad´ıstica y No Lineal) tion (i.e., the probability of observing k events if these events occur with as well as MINECO and Fondo Europeo de Desarrollo Regional Funds Grant a known rate λ). The posterior distribution of R(t) is then explored using FIS2017-87519-P. The funders had no role in study design, data collection MCMC sampling. and analysis, or preparation of the manuscript.

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