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A New Framework and Software to Estimate Time-Varying Reproduction Numbers During Epidemics
Open accessAbstract
The quantification of transmissibility during epidemics is essential to designing and adjusting public health responses. Transmissibility can be measured by the reproduction number R, the average number of secondary cases caused by an infected individual. Several methods have been proposed to estimate R over the course of an epidemic; however, they are usually difficult to implement for people without a strong background in statistical modeling. Here, we present a ready-to-use tool for estimating R from incidence time series, which is implemented in popular software including Microsoft Excel (Microsoft Corporation, Redmond, Washington). This tool produces novel, statistically robust analytical estimates of R and incorporates uncertainty in the distribution of the serial interval (the time between the onset of symptoms in a primary case and the onset of symptoms in secondary cases). We applied the method to 5 historical outbreaks; the resulting estimates of R are consistent with those presented in the literature. This tool should help epidemiologists quantify temporal changes in the transmission intensity of future epidemics by using surveillance data.
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Citations
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Cleary, E., Atuhaire, F., Sorichetta, A., Ruktanonchai, N., Ruktanonchai, C., Cunningham, A., Pasqui, M., Schiavina, M., Melchiorri, M., Bondarenko, M., et al. (2025) · PLOS Global Public Health · doi:10.1371/journal.pgph.0003431 · Cited by 3
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Asymmetric limits on timely interventions from noisy epidemic data
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INTEGRAL MODEL OF COVID-19 SPREAD AND MITIGATION IN UK: IDENTIFICATION OF TRANSMISSION RATE
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Self-citationA method to estimate the serial interval distribution under partially-sampled data
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“For the local student population, R eff was estimated from the epidemic curve as described by Cori et al . with a one-day sliding window 17 .”
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“We can estimate the daily number of cases by employing a Poisson process, following the renewal in Equation [20,72]: Here, ω denotes the serial-interval distribution, representing the time between successive cases in a transmission chain (i.e., the interval between infection and subsequent…”
MethodSoucy, J.-P. R., Sturrock, S. L., Berry, I., Westwood, D. J., Daneman, N., Fisman, D. N., MacFadden, D. R., & Brown, K. A. (2024) · PeerJ · doi:10.7717/peerj.17455 · Cited by 3
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InfluentialTransmissibility of the variant of concern for SARS-CoV-2 in six regions
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SARS-CoV-2 effective breeding number estimation in Vitória de Santo Antão/PE, Brazil
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Seposo, X., Ng, C. F. S., & Madaniyazi, L. (2021) · Atmosphere · doi:10.3390/atmos12040513 · Cited by 3
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BackgroundReal-Time Projections of SARS-CoV-2 B.1.1.7 Variant in a University Setting, Texas, USA
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“We estimate the daily reproduction number (Rt) using the EpiEstim package (10) applied to the test-level corrected case counts.”
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A Daily Measure of the SARS-CoV-2 Effective Reproduction Number for all Countries
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“Districts with less than 12 cumulative cases in the study period were excluded from the analysis, and we discarded R t estimates obtained before 12 cumulative cases were reported [7].”
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“Both the simplest methods of estimating time-varying reproduction numbers, which use the Robert Koch Institute rule [3, 4], and more complex methods based on modern mathematical models and fi lters [5 – 9] use only time series data on registered cases.”
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Paredes, M. I., Liang, C., Suen, S., Holloway, I. W., Garrigues, J. M., Green, N. M., Bedford, T., Müller, N. F., & Osmundson, J. (2026) · Nature Communications · doi:10.1038/s41467-026-71993-w · Cited by 2
“…however, both have limitations: understanding the interplay of local transmission and viral importations is dif fi cult via case counts alone 11,31 , limiting the power of the microsimulation to capture these dynamics; phylodynamics works to understand shared ancestry as it relates to…”
Grané, C. M., Manso, M. D., Manica, M., Sacco, C., Menegale, F., Bella, A., Riccardo, F., Liverani, A., Pesaresi, A., Mattei, G., et al. (2026) · Eurosurveillance · doi:10.2807/1560-7917.es.2026.31.27.2600035 · Cited by 2
“We leveraged the estimated generation time to compute the net reproduction number (R t ) for the main transmission foci, by applying the renewal equation to the time series of symptomatic cases by date of symptom onset, using a Markov Chain Monte Carlo (MCMC) approach [5,6,20].”
Xia, F., Xiao, Y., Liu, P., Cheke, R., & Li, X. (2020) · Mathematical Biosciences & Engineering · doi:10.3934/mbe.2020274 · Cited by 2
Zhao, S., Lou, J., Cao, L., Zheng, H., Chong, K. C., Chen, Z., Chan, R. W. Y., Zee, B. C.-Y., Chan, P. K. S., & Wang, M. H. (2021) · BMC Infectious Diseases · doi:10.1186/s12879-021-06729-w · Cited by 2
Brinks, R., Küchenhoff, H., Timm, J., Kurth, T., & Hoyer, A. (2022) · PLoS ONE · doi:10.1371/journal.pone.0276311 · Cited by 2
Rasul, A., & Ibrahim, S. (2022) · Spatial Information Research · doi:10.1007/s41324-022-00497-8 · Cited by 2
“[15] framework was adopted in this study (Eq.”
BackgroundMethodAnalysis of reporting lag in daily data of COVID-19 in Japan
Kanatani, T., & Nakagawa, K. (2023) · Letters in Spatial and Resource Sciences · doi:10.1007/s12076-023-00334-y · Cited by 2
“By contrast, delay or lag is usually discussed in relation to the serial interval between infection and symptom onset, which is essential to estimate the epidemiological curve and the effective reproductive number; see, for example, Cori et al. (2013), Nishiura et al. (2020), Ali et al. (2020).”
BackgroundAdvancements in Defining and Estimating the Reproduction Number in Infectious Disease Epidemiology
Li, K., Wang, J., Xie, J., Rui, J., Abudunaibi, B., Wei, H., Liu, H., Zhang, S., Li, Q., Niu, Y., et al. (2023) · China CDC Weekly · doi:10.46234/ccdcw2023.158 · Cited by 2
Abry, P., Fort, G., Pascal, B., & Pustelnik, N. (2023) · European Signal Processing Conference (EUSIPCO) · doi:10.23919/eusipco58844.2023.10290048 · Cited by 2
“[1] proposed an epidemiology model designed to estimate the reproduction numbers R from the observed counts of new daily infections, denoted by At time t , Φ t is a weighted mean of the number of new infections over the past τ φ days, with weights φ ( · ) which model the delay between a primary…”
BackgroundInfluentialLu, M., Ali, M., Zhang, W., & Kumar, N. (2023) · Public Management Review · doi:10.1080/14719037.2023.2283125 · Cited by 4
“…t is the reproduction number at calendar time t ; E[ I t ] is the expected value for new infections at t ; I t (cid:0) s is the incidence at time step t-s ; and w s is a function to measure the risk of disease transmission, dependent on the time since an infection of the case s (Cori et al. 2013).”
McCabe, R., Danelian, G., Panovska‐Griffiths, J., & Donnelly, C. A. (2024) · Infectious Disease Modelling · doi:10.1016/j.idm.2024.01.011 · Cited by 2
“These parameters are traditionally estimated using case data as a proxy for the infection incidence curve (Cori, Ferguson, Fraser, & Cauchemez, 2013; Parag, 2021), but methods have also been developed to estimate these parameters from other sources, such as hospitalisations (Moore, Rosato, &…”
BackgroundMethodInfluentialExamining the Evolution and Drivers of COVID-19 Transmission Waves in Ghana, 2020-2022
Tamal, C. S., Ohene, S.-A., Okine, R. N. A., Adjabeng, M. J., Guyo, A. G., Osei-Sarpong, F., Avevor, P. M., Owusu-Asare, A. A., Asiedu‐Bekoe, F., Kwarteng, G., et al. (2024) · Journal of Biotechnology and Biomedicine · doi:10.26502/jbb.2642-91280129 · Cited by 2
Assessing the Time Evolution of COVID-19 Effective Reproduction Number in Brazil
da Silva, E. P., & Lima, A. M. N. (2024) · Anais da Academia Brasileira de Ciências · doi:10.1590/0001-3765202420221050 · Cited by 2
“Instead, the instantaneous effective reproduction number R(t) (Fraser, 2007; Cori et al., 2013; Flaxman et al., 2020; Gostic et al., 2020) provides a better picture of an ongoing epidemic.”
BackgroundInfluentialSARS-CoV-2 Transmission in Alberta, British Columbia, and Ontario, Canada, January 2020–January 2022
Kehoe, A. R. D., Mallhi, A. K., Barton, C. R., Martin, H. M., Turner, C. M., Hua, X., Kwok, K. O., Chowell, G., & Fung, I. C. (2024) · Emerging infectious diseases · doi:10.3201/eid3005.230482 · Cited by 2
“R t is estimated by calculating the ratio between the number of incident cases at time t and the total infectiousness of all infected persons at that same time ( 23 , 24 ).”
MethodLee, H.-W., Choi, H., Lee, H., Lee, S., & Kim, C. (2024) · Frontiers in Public Health · doi:10.3389/fpubh.2024.1362823 · Cited by 2
“In the context of COVID-19 policies, our current knowledge of how infections spread through transmission networks is primarily based on virtual data and theoretical models (14, 15), with evidence from actual data (16–18) being limitedly available.”
BackgroundMethodInfluentialStratified epidemic model using a latent marked Hawkes process
Lamprinakou, S., & Gandy, A. (2024) · Mathematical Biosciences · doi:10.1016/j.mbs.2024.109260 · Cited by 3
25 Models in the COVID-19 Pandemic
Imai, N., Baguelin, M., & Ferguson, N. M. (2024) · Springer eBooks · doi:10.1007/978-3-031-48408-7_37 · Cited by 2
Self-citationCalabrese, J. M., Schüler, L., Fu, X., Gawel, E., Zozmann, H., Bumberger, J., Quaas, M. F., Wolf, G., & Attinger, S. (2024) · Journal of The Royal Society Interface · doi:10.1098/rsif.2024.0301 · Cited by 3
“Then, for each day in a given time window ( 1 or t 2 ), we extract the model-fitted value of case numbers and estimate R t according to Cori et al. [51]. t Finally, we average R t values over all days in a given time window.”
Evolution of COVID-19 dynamics in Guangdong Province, China: an endemic-epidemic modeling study
Huang, Z., Lin, L., Li, X., Rong, Z., Hu, J., Zhao, J., Zeng, W., Zhu, Z., Li, Y., Huang, Y., et al. (2024) · Archives of Public Health · doi:10.1186/s13690-024-01406-1 · Cited by 2
“R t was defined as the mean number of secondary infections that were generated by a primary case of infection at time t [20].”
BackgroundMultiphasic stochastic epidemic models
Barmpounakis, P., & Demiris, N. (2024) · Journal of the Royal Statistical Society Series C (Applied Statistics) · doi:10.1093/jrsssc/qlae064 · Cited by 2
Meng, Y., Lin, Y., Xiong, W., Lau, E. H. Y., Ho, F. C. S., Wong, J. Y., Wu, P., Tsang, T. K., Cowling, B. J., & Yang, B. (2024) · The Journal of Infectious Diseases · doi:10.1093/infdis/jiae592 · Cited by 2
Quevedo, D. S., Domínguez, N. T., Perez, D. F., Cabrera-Polanía, M. A., Medina, J. D. S., Abril-Bermúdez, F. S., Moyano, D., Rios, D. S., Mayorga, M. A. G., Whittaker, C., et al. (2024) · BMC Global and Public Health · doi:10.1186/s44263-024-00105-x · Cited by 3
“We estimated the time-varying R ( t ) using the epidemiological R package EpiEstim [14] .”
Machine learning mathematical models for incidence estimation during pandemics
Fajardo-Fontiveros, O., Mattei, M., Burgio, G., Granell, C., Gómez, S., Arenas, À., Sales‐Pardo, M., & Guimerà, R. (2024) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1012687 · Cited by 5
Zhao, S., Guo, Z., Wang, K., Sun, S., Sun, D., Wang, W., He, D., Chong, K. C., Hao, Y., & Yeoh, E. (2025) · Bulletin of Mathematical Biology · doi:10.1007/s11538-025-01421-5 · Cited by 2
“…a Poisson-distributed secondary case distribution cannot incorporate heterogeneity in transmissibility, it was commonly adopted to construct the likelihood framework in time series analysis of infectious disease epidemiology owning to its simplicity (Wallinga and Teunis 2004; Cori et al. 2013).”
Yang, T., Du, X., Li, J., Tin, Z., Wang, Y., & Wang, L. (2025) · Epidemics · doi:10.1016/j.epidem.2025.100844 · Cited by 2
Nonparametric serial interval estimation with uniform mixtures
Gressani, O., & Hens, N. (2025) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1013338 · Cited by 2
“Originally developed for estimation of incubation period distributions, the methodology of [26] is available in an R software package [32] and associated routines are embedded in the EpiEstim package of [33] to estimate the serial interval [34].”
Disentangling Temporal Trends of Clade Ib Monkeypox Virus Transmission in Burundi
Jin, S., Asakura, T. R., Murayama, H., Jung, S.-M., Niyukuri, D., Nyandwi, J., Nkengurutse, L., Kamatari, O., Lim, J. T., Endo, A., et al. (2025) · The Journal of Infectious Diseases · doi:10.1093/infdis/jiaf475 · Cited by 2
“The largest eigenvalue of M t generated the instantaneous reproduction number, R t , which quantifies disease transmission potential at time t [10].”
The R = 1 threshold can misclassify epidemic stability
Parag, K. V., Santillana, M., Cori, A., & Obolski, U. (2026) · Communications Physics · doi:10.1038/s42005-026-02631-6 · Cited by 2
Self-citationASSESSING EPIDEMIC TRENDS IN REAL TIME WITH A SIMPLE RATIO PLOT
Yu, X. (2020) · American Journal of Epidemiology · doi:10.1093/aje/kwaa123 · Cited by 2
Toraih, E. A., Hussein, M. H., Elshazli, R. M., Fawzy, M. S., Houghton, A., Tatum, D. M., Killackey, M. T., Kandil, E. H., & Duchesne, J. C. (2020) · Annals of Surgery · doi:10.1097/sla.0000000000004590 · Cited by 2
“Estimation in some articles may be associated with significant uncertainty, especially early in the outbreak.(14) Therefore, we used the pooled summary for the mean and standard deviation of the serial interval reported in 14 studies, which was equivalent to 5.”
MethodTransitivity of COVID-19, based on parameters R_0 and R_((t)) in Pombos/PE municipality, Brazil
Ferreira, D. S. D. A., de Lima, F. M., de Freitas, J. R., Nascimento, G. I. L. A., Alves, D. A. N. D. S., Gomes, D. A., do Amaral, L. S., dos Santos, A. L. P., Júnior, A. F. D. S., Cunha, D. X., et al. (2020) · Research Society and Development · doi:10.33448/rsd-v9i10.8441 · Cited by 2
Inference under superspreading: Determinants of SARS‐CoV‐2 transmission in Germany
Schmidt, P. (2024) · Statistics in Medicine · doi:10.1002/sim.10046 · Cited by 2
“14,22 Thirdly, the model allows for dispersion Ψ of the offspring distribution following the seminal model of superspreading.”
BackgroundImpact of the COVID-19 Epidemic: Scenario in a Tropical Environment
Sharma, S., & Kumar, P. (2021) · Pure and Applied Geophysics · doi:10.1007/s00024-021-02793-0 · Cited by 2
“Cori et al. (2013) developed a ready-to-use tool for estimating R0 from incidence time series.”
Method
Metadata
- Authors
- Cori, A., Ferguson, N. M., Fraser, C., & Cauchemez, S.
- Affiliations
- Imperial College London (GB)
- University of Oxford (GB)
- Institut Pasteur (FR)
- Corresponding author
- Cori, Anne
- Venue
- American Journal of Epidemiology
- Publisher
- Oxford University Press (OUP)
- Published
- 2013-09-15
- Volume
- 178
- Issue
- 9
- Pages
- 1505-1512
- ISSN
- 0002-9262 (ISSN-L)
- 1476-6256
- PMC
- PMC3816335
- OpenAlex
- W2097446414
- Cited by
- 1,530
- References
- 45
- Citations by year
- 2026: 94 · 2025: 141 · 2024: 161 · 2023: 199 · 2022: 268 · 2021: 328 · 2020: 262 · 2019: 21 · 2018: 16 · 2017: 16 · 2016: 16 · 2015: 3 · 2014: 4 · 2013: 1
- FWCI
- 8.33
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- American Journal of Epidemiology (publishedVersion)
- PubMed Central (publishedVersion)
- PubMed (publishedVersion)
- Funders
- Royal Society
- Imperial College London
- Medical Research Council (MR/K010174/1, G0800596)
- National Institute of General Medical Sciences
- Topics
- Physical Sciences › Mathematics › Modeling and Simulation › COVID-19 epidemiological studies
- Physical Sciences › Mathematics › Applied Mathematics › Mathematical and Theoretical Epidemiology and Ecology Models
- Health Sciences › Medicine › Epidemiology › Data-Driven Disease Surveillance
- Keywords
- time-varying reproduction number, analytical estimation, transmission intensity, public health response
- MeSH terms
- Humans, Software, Time Factors, Virus Diseases, Models, Statistical, Incidence, Epidemics
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