Measurability of the epidemic reproduction number in data-driven contact networks Quan-Hui Liua,b,c, Marco Ajellic,d, Alberto Aletae,f, Stefano Merlerd, Yamir Morenoe,f,g, and Alessandro Vespignanic,g,1

aWeb Sciences Center, University of Electronic Science and Technology of China, Chengdu 611731, Sichuan, People’s Republic of China; bBig Data Research Center, University of Electronic Science and Technology of China, Chengdu 611731, Sichuan, People’s Republic of China; cLaboratory for the Modeling of Biological and Socio-Technical Systems, Northeastern University, Boston, MA 02115; dBruno Kessler Foundation, 38123 Trento, Italy; eInstitute for Biocomputation and Physics of Complex Systems, University of Zaragoza, 50018 Zaragoza, Spain; fDepartment of Theoretical Physics, University of Zaragoza, 50009 Zaragoza, Spain; and gISI Foundation, 10126 Turin, Italy

Edited by Simon A. Levin, Princeton University, Princeton, NJ, and approved October 16, 2018 (received for review June 27, 2018)

The basic reproduction number is one of the conceptual corner- an infectious individual at time t), thus relaxing the hypothesis of stones of mathematical epidemiology. Its classical definition as a fully susceptible population (10).

the number of secondary cases generated by a typical infected R0 and Tg are mathematically well defined in the early stages individual in a fully susceptible population finds a clear analytical of the epidemic in homogeneous models and are widely used in expression in homogeneous and stratified mixing models. Along predictive approaches (10, 11). However, several studies have with the generation time (the interval between primary and cautioned on the importance of the local contact structures secondary cases), the reproduction number allows for the charac- (e.g., households, extended families, communities) in estimat- terization of the dynamics of an epidemic. A clear-cut theoretical ing the value of R0 (12–17). Theoretical work has explored picture, however, is hardly found in real data. Here, we infer the definition of the reproduction number in models entailing from highly detailed sociodemographic data two multiplex con- community and household structure (18–24), although an oper- tact networks representative of a subset of the Italian and Dutch ational way to compute R0 is still lacking. For more complex populations. We then simulate an infection transmission process models closely resembling the actual structure of the human pop- on these networks accounting for the natural history of influenza ulations (e.g., accounting for households, schools, workplaces) and calibrated on empirical epidemiological data. We explicitly (25–34), the estimation of R0 mainly relied on the direct count measure the reproduction number and generation time, record- of secondary cases generated by the index case of the outbreak ing all individual-level transmission events. We find that the index (R0 ) and/or the analysis of the growth rate of the simulated classical concept of the basic reproduction number is untenable epidemics. However, both methods have limitations and show in realistic populations, and it does not provide any conceptual marked differences in the estimated values (31, 32). Indeed, understanding of the epidemic evolution. This departure from the index R0 is computed from the analysis of the index case of an classical theoretical picture is not due to behavioral changes and outbreak, which is not necessarily representative of a “typi- other exogenous epidemiological determinants. Rather, it can be cal” infectious individual. In addition to this, growing evidence simply explained by the (clustered) contact structure of the pop- shows that the classic exponential early growth of the epidemic ulation. Finally, we provide evidence that methodologies aimed is an oversimplification often not backed by real-world data (35, at estimating the instantaneous reproduction number can oper- 36). For this reason, statistical methods were developed for the ationally be used to characterize the correct epidemic dynamics analysis of R(t), assuming that the variations of this quantity from incidence data. are mostly due to the impact of the performed intervention strategies and behavioral changes in the population (10, 16, 17, computational modeling | infectious diseases | multiplex networks | 37). Unfortunately, for these methods, disentangling the role of reproduction number | generation time Significance athematical and computational models of infectious dis- Meases are increasingly recognized as relevant quantitative The analysis of real epidemiological data has raised issues of support to epidemic preparedness and response (1–3). Indepen- the adequacy of the classic homogeneous modeling frame- dent of the type of modeling approach, our understanding of work and quantities, such as the basic reproduction number epidemic models is generally tied to two fundamental concepts. in real-world situations. Based on high-quality sociodemo- One is the basic reproduction number R0, which is the average graphic data, here we generate a multiplex network describ- number of secondary cases generated by a typical infectious indi- ing the contact pattern of the Italian and Dutch populations. vidual over the entire course of the infectious period in a fully By using a microsimulation approach, we show that, for susceptible population (4). The other is the generation time Tg, epidemics spreading on realistic contact networks, it is not the time interval between the infection time of the infector and possible to define a steady exponential growth phase and a her/his infectees (5, 6). These quantities are the cornerstones basic reproduction number. We show the operational use of of our understanding of basic epidemic models, as they encom- the instantaneous reproduction rate as a good descriptor of pass the condition for the occurrence of an epidemic outbreak the transmission dynamics. (R0 > 1) and the epidemic doubling time. Both the reproduction number and generation time are determined by the biological Author contributions: Q.-H.L., M.A., A.A., S.M., Y.M., and A.V. designed research, characteristics of the pathogen (e.g., probability of transmission performed research, analyzed data, and wrote the paper.y given a contact), the pathogen–host system (e.g., timeline of The authors declare no conflict of interest.y pathogen replication inside the host), and the contact patterns of This article is a PNAS Direct Submission.y the population in which the infection spreads (4, 7). The concept This open access article is distributed under Creative Commons Attribution-NonCommercial- of the reproduction number has been extended to stratified mod- NoDerivatives License 4.0 (CC BY-NC-ND).y els (8) and to heterogeneous contact networks (9) to account for 1To whom correspondence should be addressed. Email: [email protected] more complex interaction patterns. Furthermore, the definition This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. of R0 has been generalized by the effective reproduction number 1073/pnas.1811115115/-/DCSupplemental.y R(t) (i.e., the average number of secondary cases generated by Published online November 21, 2018.

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POPULATION BIOPHYSICS AND BIOLOGY COMPUTATIONAL BIOLOGY Finally, without loss of generality, we fix the removal time to 3 d A (15, 47, 48). In SI Appendix, we report all of the details of the transmis- sion model (SI Appendix, section 1.1). In the text, we also report the corresponding analysis of R(t) and Tg(t) for a stochastic implementation of a homogeneous mixing SIR model at the indi- vidual level, where all individuals are identical and are in contact with the same fixed probability. Along with the homogeneous SIR model, we also studied a set of alternative null models with an increasing level of complexity from the homogeneous model to the data-driven model (details are in SI Appendix, sections 1.2–1.5). Null models include annealed (edges are constantly rewired) and quenched (edges are fixed over time) configura- tion models. Results reported in the text refer to the synthetic B population for Italy.

Effective Reproduction Number and Generation Time. The first quantities generally investigated in epidemic models are the inci- dence (of new infections) as a function of time and the associated growth rate of the epidemic. In homogeneous models, the num- ber of new cases increases exponentially at a nearly constant rate r (5, 49, 50) during the early phase of the epidemic. This is not the case in the data-driven model, where we find a non- monotonous behavior: an increasing trend over the initial phase of the epidemic occurs followed by a marked decrease about 20 d before the epidemic peak (Fig. 2A). Such a result is in sharp C contrast with the classic theory, where the epidemic growth rate is expected to slowly and monotonically decrease over time in the early epidemic phase (Fig. 2A). This suggests that, in con- trast to simple SIR models where the basic reproduction number can readily be defined through the relation R0 = 1 + rTg (5), it is difficult to find a proper definition of the basic reproduc- tion number in populations characterized by realistic connectivity patterns. The daily effective reproduction number and generation time can be computed from the microsimulations by keeping track of the exact number of secondary infections generated by each indi- vidual infected at time t in the simulations (Fig. 1C). We find that Fig. 2. Fundamental epidemiological indicators. (A) Mean daily exponen- R(t) increases over time in the early phase of the epidemic, start- tial epidemic growth rate, r, over time of the data-driven and homogeneous ing from Rindex = 1.3 to a peak of about 2.1 (Fig. 2B). In contrast, models. The colored area shows the density distribution of r(t) values 0 obtained in the single realizations of the data-driven model. Results are in the homogeneous model, which lacks the typical structures based on 50,000 realizations of each model. Results are aligned at the epi- of human populations, R(t) is nearly constant in the early epi- demic peak, which corresponds to time t = 0. Inset shows the logarithm demic phase and then rapidly declines before the epidemic peak of the mean daily incidence of new influenza infections over time, which (Fig. 2A) as predicted by classical mathematical epidemiology does not follow a linear trend. (B) Mean R(t) of data-driven and homoge- theory (4). The pattern found in the data-driven model can neous models. The colored area shows the density distribution of R(t) values also be partly explained by the variation of the average degree obtained in the single realizations of the data-driven model. (C) The three induced by the infection of individuals with a higher number lines represent the mean Tg(t) of data-driven and homogeneous models. of adequate contacts [an effect already observed in heavy-tailed The colored area shows the density distribution of Tg(t) values obtained networks (51)], thus leading to an average growth of the repro- in the single realizations of the data-driven model. The horizontal dotted gray line represent the constant value of the duration of the infectious duction number. The temporal dynamics of R(t) does not show period. a constant phase, implying that R0 loses its meaning as a funda- mental indicator in favor of R(t). Although we report the results averaged over 50,000 realizations of the model, this result is sup- of the deviations of R(t) and Tg(t) from the classical theory ported by our analysis of the outcome of each single simulation, (Fig. 3). Specifically, we found that the average degree of infec- highlighting that the early time increase of R(t) is a common pat- tious nodes as well as R(t) tend to peak in the workplace layer tern in the performed simulations (SI Appendix, section 2.1). In (SI Appendix, section 2.2), and at least to some extent, the same Fig. 2C, we show an analogous analysis of the estimated genera- happens in the school layer. However, R(t) generally decreases tion time in the data-driven model. We find that the generation in the household and community layers (Fig. 3A). Indeed, R(t) time is considerably shorter than the duration of the infectious in the household layer tends to be more uniform across the period (i.e., 3 d in our simulations). The estimated average Tg different nodes and thus, follows a general decreasing trend sim- over the whole epidemic is 2.67 d (≈ 11% shorter than the the- ply led by the depletion of susceptible contacts. We also found oretical value), with a more marked shortening right before the that, in the household layer, Tg is remarkably shorter than in epidemic peak (Fig. 2C). This differs from what is predicted by all other layers (Fig. 3B)—with an average fluctuating around the classic theory and the analysis of the homogeneous model 2.6 d, close to the value reported by analyzing real data for house- (Fig. 2C), where the length of the infectious period corresponds hold transmission (26). To provide a simple illustration of the to the generation time (6). saturation effect in households, let us consider a household of A closer look at the transmission process in the different lay- three, with one index case and two susceptible members. If, at ers of the multiplex network helps in understanding the origin time t, the index case infects exactly one of the two susceptibles,

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POPULATION BIOPHYSICS AND BIOLOGY COMPUTATIONAL BIOLOGY linear regression model to the estimated growth rate over time A in the ILI data results in an estimated coefficient of 0.064 (SE of 0.033), while the mean value obtained with a linear regression model of case incidence from all stochastic realizations is 0.064 (95% CI, −0.207–0.305). An increase in the initial growth rate of the epidemic has also been observed for other diseases and countries (57–59). Although alternative explanations may exist (60), our results provide a plausible explanation for these pat- terns based on the intrinsic structure of human contact networks. In terms of the dynamics of R(t) and Tg(t), the model calibrated on the 2009 influenza pandemic confirms the noticeable devia- B tions of the data-driven model from the homogeneous modeling framework (SI Appendix, section 2.3).

Discussion Our simulation results clearly highlight how the heterogene- ity and clustering of human interactions (e.g., contacts between household members, classmates, work colleagues) alter the stan- dard results of fundamental epidemiological indicators, such as the reproduction number and generation time over the course C of an epidemic. Our results seem to be consistent in different countries (SI Appendix, section 2.4), suggesting that the observed patterns are due to the structure and clustering of human con- tact patterns rather than country-specific features. Furthermore, the analysis of alternative null models, such as degree-preserving and layer-preserving configuration models (SI Appendix, section 2.7), shows markedly different behaviors than those exhibited by the data-driven model, suggesting that the multiplex structure and the strong clustering effect typical of human populations, not Fig. 5. Estimation of R(t). (A) Daily R(t) as inferred from the daily incidence captured by null models, are at the root of the observed behavior. of new infections for one stochastic model realization. Tg is assumed to be Our numerical study questions the measurability of R0 in exponentially distributed with an average of 3 d. (B) The same as A but the realistic populations and its adequacy even as an approximate distribution of Tg has been derived from the analysis of the transmission tree descriptor of the epidemic dynamics. However, it is still possible of the selected model simulation. (C) The same as A but using the distribution to characterize the dynamics of an epidemic in terms of its effec- of Tg over time as derived from the analysis of the transmission tree of the R(t) selected model simulation. The goodness of the estimate of R(t) increases with tive reproduction number over time by estimating from the a more precise knowledge of the generation time value and distribution. daily number of new cases by using Bayesian approaches (11, 17) (Materials and Methods). In Fig. 5, we show the compari- α α son between R(t) as inferred from the time series of cases and layer-specific adjacency matrix A , where elements aij = wα > 0 if there is α R(t) resulting from the microsimulation data of the transmission a link between nodes i and j in layer α and aij = 0 otherwise. chain for one stochastic model realization (other realizations are shown in SI Appendix, section 2.8). The overall good agreement Homogeneous Mixing Network. This model assumes a single fully connected between the estimated and actual values shows that it is possible network. Following the notation introduced above, we have aij = w for each = / − to operationalize the estimation of R(t) to provide projections of i and j, where w 1 (N 1) and N is the total number of individuals in the population. the epidemic dynamics. The goodness of the obtained estimates depends on the knowledge of the distribution of the generation The Epidemic Transmission Model. On each of the introduced networks, we time (Fig. 5) and on the availability of a reliable time series of simulate the influenza transmission process as an SIR model. The SIR model cases not markedly affected by noise and underreporting (10). assumes that individuals can be in one of the following three states: sus- ceptible, infectious, and removed. Two types of transitions between states Conclusion are possible: (i) from susceptible to infectious and (ii) from infectious to The analysis presented here takes advantage of “in silico” numer- removed. The transition from susceptible to infectious requires a contact between an infectious individual and a susceptible individual. Specifically, ical experiments to open a window of understanding in the given that, at time step t, node j is infectious and its neighbor node i is analysis of realistic epidemic scenarios. Although a lot of theo- susceptible, the probability that j infects i (i.e., i changes its status from retical work has been done to define the reproduction number in susceptible to infectious) is given by β = pw, where p is the transmission nonhomogeneous models (19–24), a unified theory is still lack- probability per contact and w is the contact weight. However, the transition ing. While we are not providing a theoretical framework for the from infectious to removed does not involve contact between individuals. computation/definition of the reproduction number and the gen- Given that, at time step t, node i is in the infectious state, it has a probability eration time on realistic contact networks, we provide evidence γ to recover at time step t + 1. that estimates of R(t) can be used to characterize the epidemic without resorting to untenable assumptions that may eventu- Estimation of R (t ) from Transmission Events Time Series. Following the same approach used in refs. 11 and 17, we assume that the daily number of new ally mislead our understanding of the epidemic transmission cases C(t) at time t can be approximated by a Poisson according to the potential and temporal dynamics. following equation:

t ! X Materials and Methods C(t) ≈ Pois R(t) φ(s)C(t − s) , Data-Driven Contact Network. The model considers a weighted multiplex s=1 ~ network (38) G = (GH, GS, GW , GC ), where H, S, W, and C represent house- hold, school, workplace, and community layers, respectively. Let us define V where φ is the generation time distribution and R(t) is the effective repro- as the set of all nodes, which is common to all layers, and Eα as the set of duction number at time t. The likelihood L of the observed time series of edges in layer α. We can thus characterize each layer Gα = (V, Eα) by the cases from day 1 to T is thus given by

12684 | www.pnas.org/cgi/doi/10.1073/pnas.1811115115 Liu et al. Downloaded by guest on September 27, 2021 Downloaded by guest on September 27, 2021 5 acee ,e l 21)Rl fsca ewrsi hpn ies transmission disease shaping in and networks social structured of Role in (2011) al. (t) remedies. et and S, R Biases Cauchemez epidemics: number emerging 15. in Estimation reproduction (2018) GS The Tomba T, (2009) Britton G 14. Chowell T, Burr 13. 6 jliM ta.(05 h 04Eoavrsdsaeotra nPjhn iraLeone: Sierra Pujehun, in outbreak disease virus Ebola 2014 The (2015) al. et M, Ajelli 16. first The Africa: West in Cintr disease 12. virus Ebola (2014) Team Response respiratory Ebola acute WHO severe for 11. curves epidemic Different (2004) P Teunis J, Wallinga 10. 0 rsrC(07 siaigidvda n oshl erdcinnmesi an in numbers reproduction household and individual Estimating and transmission (2007) on C distribution household Fraser of 20. effect The (1995) K Dietz NG, models. Becker epidemic network 19. for Guinea challenges Eight in (2015) al. epidemic et Ebola L, Pellis the 18. of dynamics Spatiotemporal (2016) al. et M, Ajelli 17. 2 odti ,e l 20)Rpoutv ubr,eiei pedadcnrli a in control and spread epidemic numbers, Reproductive (2009) al. household et explicit with E, models Goldstein epidemic 22. Deterministic (2008) MJ Keeling T, House 21. 0 uael ,Ael ,Mre ,Fruo M acee 21)Mdlbsdcom- Model-based (2016) S Cauchemez NM, Ferguson influenza S, an Merler M, of Ajelli containment L, Fumanelli layered targeted 30. Modeling (2008) al. Italy: et in ME, influenza pandemic Halloran for measures 29. Mitigation (2008) al. et ML, pandemic. Atti influenza Degli an Ciofi mitigating 28. for Strategies (2006) al. et NM, source. Ferguson pandemic the influenza emerging 27. an at containing for influenza Strategies (2005) al. pandemic et NM, Containing Ferguson (2005) 26. al. et repro- IM, household Longini and rate 25. growth Epidemic (2011) C Fraser population NM, a in Ferguson dynamics L, disease Pellis of Calculation 24. (2010) MJ Keeling T, House JV, Ross 23. i tal. et Liu CCsampling. MCMC rate observing known of a probability the (i.e., tion here, where .Pso-aorsR atlaoC a ige ,Vsinn 21)Epidemic (2015) A Vespignani P, Mieghem Van com- C, the Castellano and definition R, the Pastor-Satorras On A 9. (1990) infections: JA viral Metz respiratory JAP, acute Heesterbeek of O, periods Diekmann Incubation 8. (2009) al. et repro- J, basic Lessler the relationship 7. of estimation the Model-consistent shape (2007) intervals J generation Heesterbeek M, How Roberts (2007) 6. M Lipsitch J, Wallinga Africa 5. West (1991) B the Anderson of RM, and modeling May RM, Mathematical Control Anderson (2015) Disease 4. DB George for S, Centers Riley JP, the Chretien from 3. Results (2016) al. et M, Biggerstaff 2. .Vbu ,e l 21)TeRPD bl oeatn hleg:Snhssadlessons and Synthesis challenge: forecasting ebola RAPIDD The (2018) al. et C, Viboud 1. uigacmuiyotra f20 11pnei influenza. pandemic H1N1 2009 USA of outbreak community a during arXiv:1803.01688. populations. nonstructured data. outbreak disease Eng from number reproductive effective the of mation pdmooyadipc finterventions. of impact and Epidemiology projections. forward and epidemic the 1495. of months 9 measures. control of impacts 516. similar reveal syndrome in diseases infectious networks. for complex in models processes in R0 ratio reproduction populations. basic heterogeneous the of putation infection. review. systematic emerging an of incidence the 803–816. from number duction numbers. reproductive and rates growth between Control and epidemic. Ebola Challenge. Season Influenza 2013–2014 16:357. the Predict Prevention’s mrigepidemic. emerging diseases. infectious highly of control 62. modeling computational A elimination: disease analysis. and vaccination for implications and learnt. omnt fhouseholds. of community structure. rhnieaayi fsho lsr oiisfrmtgtn nunaeieisand epidemics influenza mitigating for policies pandemics. closure school of analysis prehensive States. United the in pandemic scenarios. different considering model based individual An 442:448–452. Asia. Southeast in 309:1083–1087. workplaces. and schools households, 63:691–734. of communities in number duction households. of 6:261–282. 108:2825–2830. nAisA Castillo-Ch A, on-Arias ´ Epidemics M Med BMC ahBiosci Math LSCmu Biol Comput PLoS P Ofr nvPes Oxford). Press, Univ (Oxford (k .Tepseirdsrbto of distribution posterior The λ). LSOne PLoS , eLife stepoaiiyms ucino oso distribu- Poisson a of function mass probability the is λ) 22:13–21. Nature actIfc Dis Infect Lancet L 14:130. LSOne PLoS = 4:e09186. 213:29–39. Y t =1 T 5:e9666. 437:209–214. ahBiosci Math vzC etnor M ly L ak 20)Teesti- The (2009) H Banks AL, Lloyd LM, Bettencourt C, avez ´ P ahBoc Eng Biosci Math ahBiol Math J

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(2018) Y Moreno F, Rodrigues G, Arruda de Ferraz E, Cozzo 40. for strategies Mitigation (2006) CA Macken in IM, component Longini contacts unstructured K, the Kadau of TC, impact Germann The 32. (2008) S Merler M, Ajelli 31. I21-71-.Tefneshdn oei td ein aacollection data design, manuscript. study the in of Grant preparation role Funds or Regional no analysis, Desarrollo and had de of funders Europeo Fondo The Government and FIS2017-87519-P. the MINECO as from well as support partial acknowledges Arag Y.M. scheme. Econom ity de Formaci acknowledges Ministerio A.A. the acknowl- from MIDAS-U54GM111274. of A.V. Grant support and NIH M.A. the of support 201606070059. the Grant edge Council Scholarships China ACKNOWLEDGMENTS. admciflez dynamics. influenza pandemic bioRxiv:10.1101/247569. modelling. epidemic and descriptions Data Madagascar: USA. the in pandemic A/H1N1 influenza 2009 Viruses the Respir of phase early in diseases. infectious emerging of potential Surveill susceptibility. population HN)vrsi h ntdStates. United the in virus (H1N1) findings. diseases. communicable for times 223:24–31. generation observing on siderations networks. scale-free in outbreaks epidemic 92:178701. of spread hierarchical h erdcinnme rmeryeiei rwhrt fiflez (H1N1) A influenza of rate growth epidemic early 2009. from number reproduction The the measures: health public of effects Uganda. and the Congo and of Ebola cases of number reproductive basic households. in A influenza influenza. of interval systematic A influenza: literature. zoonotic the and of pandemic, review seasonal, for number reproduction Russia. in diseases infectious spread. diseases infectious of Biol analysis the Comput in PLoS data demographic from contacts social of diseases. infectious syndrome. outbreaks. disease infectious of phase ascending early the dynamics. growth subexponential early with Interface epidemics real-time of for number Implications duction Europe: in pandemic H1N1 2009 the modelling. of dynamics poral influenza. pandemic of spread the in praht td rnmsino nuna plcto ohueodlongitudinal household to Application influenza: of data. transmission study to approach pandemic. 2009 the during transmission influenza shaping Cham, (Springer, Complexity in Briefs Springer Switzerland). Properties, Structural and Formalism X Rev States. United the in influenza pandemic modeling. pandemic influenza n pi hog rn t h ru F group the (to grant a through Spain on, ´ ,e l 21)Mliae networks. Multilayer (2014) al. et M, a ¨ lm ,Bra ,Pso-aorsR epgaiA(04 eoiyand Velocity (2004) A Vespignani R, Pastor-Satorras A, Barrat M, elemy ttMed Stat ´ ho ilMdModel Med Biol Theor 3:041022. 15:19744. Science 13:20160659. Science LSCmu Biol Comput PLoS 3:267–276. 23:3469–3487. PNAS 324:1557–1561. 8:e1002673. 300:1966–1970. LSMed PLoS Epidemiology .HL cnwegsspotfo rga fthe of Program from support acknowledges Q.-H.L. 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