Abstract
A simplified method to compute \(R_t\), the effective reproduction number, is presented. The method relates the value of \(R_t\) to the estimation of the doubling time performed with a local exponential fit. The condition \(R_t=1\) corresponds to a growth rate equal to zero or equivalently an infinite doubling time. Different assumptions on the probability distribution of the generation time are considered. A simple analytical solution is presented in case the generation time follows a gamma distribution.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
The effective reproduction number \(R_t\) is one of the main parameters that controls the evolution of an infection. It recently gained importance during the COVID19 pandemic outbreak and is used as one of the indicators to determine restrictive measures such as regional or national lockdowns.
Different algorithms for its computation are available [1,2,3,4,5], some of which are very CPU intensive.
Implementations are also available as software packages [6] for a number of algorithms, and results are presented on websites [7,8,9] with regular updates.
CPUeffective algorithms offer the advantage that estimates can be derived in real time as soon as new data are published. Often, results of simplified algorithms don’t differ too much from the results of more accurate methods, in particular due to the limited quality of input data.
The following proposes a simplified approach to the estimate of \(R_t\) based on the determination of the doubling time, or equivalently the growth rate. The growth rate can be simply obtained as the slope parameter from a linear interpolation, in a certain interval of time, of the logarithms of the daily number of infected persons.
2 The effective reproduction number, \(R_t\)
We assume \(I_t\) is the number of infected persons at the time t, measured as number of days from a conventional beginning of the epidemic, defined as \(t=0\).
Each contagious person can infect other people during his infection period. We assume that a person that got infected at a day d will infect, on average, a certain number of other persons that become infectious at the day \(t>d\) with a discrete probability distribution \(w_{s}\), with \(s=td\). The newly infected people, on turn, may infect more people with the same mechanism. \(s=td\) is defined as the generation time in literature and corresponds to the time interval between infectorinfected pair.
The probability distribution \(w_{s}\) is normalized to unity:
In practice, after a sufficiently large amount of time, i.e.: for a sufficiently large value of s, \(w_{s}\) becomes negligible. An estimate of \(w_s\) from Italian infection data, unfortunately from a limited number of cases, is published in [10] where \(w_s\) is approximated with a gamma distribution.
At a time t, the expected number of infected persons, \({\mathbb {E}}[I_{t}]\) can be determined from \(I_{d}\), \(d=0, \ldots , t1\), according to [3], as:
or, equivalently, defining \(s=td\), as:
The simplest assumption on \(w_s\) is a constant generation time g, which is equivalent \(w_s = \delta _{gs}\) where \(\delta _{gs}\) is a Kronecker delta, i.e.: \(w_g=1\) and \(w_s=0\) for \(s\ne g\). In this case, Eq. 3 becomes:
For COVID19, the average generation time, defined as the mean value of a gamma distribution fitted to the Italian data, is \(g=6.7 \pm 1.9\) days [10]. The Robert Koch Institute (RKI) takes instead for Germany the value \(g=4\) that gives a very simple estimate \({\hat{R}}_t\) of \(R_t\) [4]^{Footnote 1}:
or the smoother ratio of the moving averages over g days:
Usually, the moving average over few days does not sufficiently smooth the distribution of the number of daily infected cases \(I_t\). In particular, the lower number of swab tests taken during the weekend causes a “ripple” structure that requires a further smoothing to be applied to the input data before evaluating Eq. 6.
Figure 1 shows the number of daily confirmed cases, \(I_t\) for Italy according to public COVID19 Italian data from the Italian Dipartimento di Protezione Civile. The large dispersion of data is clearly visible, in particular around the more stable moving average over 7 days, also shown in the figure.
3 Relation between \(R_t\) and doubling time
Another indicator of the growth of the epidemic is the doubling time \(\tau _2\) defined as the time required to double the number of infected persons, assuming an exponential growth.
Given n consecutive counts of infected people, \(I_{tn+1}, \ldots , I_t\), the following function model can interpolate the n counts:
or, equivalently:
The growth rate \(\lambda \) is related to the doubling time \(\tau _2\) by:
Estimates of A and \(\lambda \), or equivalently \(\tau _2\), can be determined with a numerical fit procedure. In particular, the exponential fit can be conveniently implemented as a linear regression on \(\log {I_t}\).
Assuming \(R_t=R\) constant during the considered time interval, the evolution model in Eq. 4 represents an exponential growth. In a time period formed by a number of days n which is an integer multiple of g: \(n=N g\), we have:
where \(h=tn+1\), or:
Changing the base from R to e gives:
Comparing with Eq. 7, considering that \(t=h+n+1\), and \(A\,e^{\lambda t} = A\,e^{\lambda n}e^{\lambda (h+1)}\), we have:
hence the estimate \({\hat{R}}\) of R is:
where \({\hat{\lambda }}\) and \({\hat{\tau }}_2\) are the estimates of \(\lambda \) and \({\tau }_2\), respectively.
4 Simplified algorithm
We studied the progression of the COVID19 pandemic in Italy, considering the data published on daily basis by Italian Dipartimento di Protezione Civile [13]. For each day t, we perform an exponential fit to the n last days’ counts, \(I_{tn+1}\), \( \ldots ,\) \(I_t\). We determine an estimate \({\hat{\tau }}_2\) of the doubling time \(\tau _2\), or an equivalent estimate \({\hat{\lambda }}\) of the growth rate \(\lambda \), from a fit of the model in Eq. 7 or Eq. 8. Then, assuming a reasonable estimate g of the average generation time, we estimate \(R_t\) according to Eq. 14 as:
There are some advantages of Eq. 15 compared to the simplified model from Eq. 6:

Equation 15 can also be applied in case g, the average generation time, is not an integer, while Eq. 6 must approximate g to the nearest integer.

The exponential fit better follows an exponential growth in the considered time interval, as it is the case when \(R_t\) is a constant, with respect to a moving average.
At the cost of a modest increase in the computing time, yet maintaining very good speed, we consider the method proposed here to be more flexible and reliable compared to the method adopted in [4]. Moreover, the data smoothing can be tuned by including a sufficient number of points in the fit. In this way, no preliminary smoothing of the data is needed before the application of the algorithm.
In the following sections, we will introduce extensions of Eq. 15 that allow a more precise determination of \({\hat{R}}_t\) than with the simplified assumption that \(w_s=\delta _{gs}\), i.e.: s is constant and equal to g.
5 Uncertainty estimate
Given Eq. 15, the uncertainty on \({\hat{R}}_t\) is determined by the uncertainties on \({\hat{\lambda }}\) (or \(\tau _2\)) and the uncertainty on g. Namely, if \(\sigma _{{\hat{\lambda }}}\) and \(\sigma _g\) are the uncertainties on \({\hat{\lambda }}\) and g, respectively, within a Gaussian error approximation, the variance of \(R_t\) is given by:
The error on \({\hat{R}}_t\) is:
The uncertainty on \({\hat{\lambda }}\) derives from the exponential fit procedure, while the uncertainty on g depends on how well the probability distribution of the generation time \(w_s\) is known. From [10], the estimate of \(w_s\) and its average g for COVID19 in Italy is known from a limited number of cases.
In particular, when \({\hat{\lambda }}=0\) (infinite doubling time), which corresponds to \({\hat{R}}_t=1\), \(\sigma _g\) doesn’t contribute to the \({\hat{R}}_t\) uncertainty. This means that an imperfect assumption on g does not affect the condition \({\hat{R}}_t=1\) which is important to determine the turning point of infection, from growing to receding, or vice versa.
The uncertainty computed in Eq. 17 does not take into account the systematic uncertainty due to the assumed approximation that the generation time s is constant, and equal to g. Moreover, the assumption of Gaussian uncertainties may not hold for an asymmetric distribution.
6 Effect of finite width in the \(w_s\) distribution
The deviation of \(w_s\) from the hypothesis of a constant generation time \(s=g\) may be approximately estimated in the continuum approximation. Equation 3 for a continuous time variable t may be rewritten as:
where \(\rho (t)\) and i(t) are the continuum equivalent of R and \(I_t\), respectively.
The normalization condition is:
If s is a constant equal to g, we have \(w(s)=\delta (sg)\), where \(\delta \) is a Dirac’s delta function. Hence:
Assuming an exponential growth \(i(t) = A\,e^{\lambda t}\), one has:
which gives the continuous version of Eq. 15, where \(\rho (t)=\rho \) is a constant:
Assuming, instead, that w(s) deviates from the Dirac’s delta assumption and has average value g and standard deviation \(\sigma \), we may write Eq. 18 applying a series expansion of \(i(ts)\) around \(s=g\):
We assume that \(w(s)\simeq 0\) for \(s>t\), so that the integration can be extended from 0 to \(\infty \) instead of 0 to t.
After the integration, in the first term the normalization condition of w(d) can be applied. The second term vanishes, and in the third term the definition of standard deviation \(\sigma \) of w(s) can be applied. Equation 23 becomes:
If we assume again \(i(t) = A\,e^{\lambda t}\), hence \(i^{\prime \prime }(t) =A\,\lambda ^2\,e^{\lambda t}\), Eq. 24, becomes:
The term \(A\,e^{\lambda t}\) simplifies. If \(\lambda ^2\sigma ^2 \ll 1\), we may write, approximately:
hence:
Equation 27 has already been reported in [12]. This result implies the width of the distribution \(w_s\) has the effect to replace g in Eq. 22 with an “effective” generation time \(g^{\mathrm {eff}}\) that is somewhat smaller than the true average value and depends on \(\lambda \) according to:
In order to take into account more details of the distribution, more terms may be added to Eq. 23. Those would add a dependency of \(\rho \) on the higher moments: skewness, kurtosis and possibly more, if required by the desired accuracy. Those cases are not considered in the present work.
7 “Exact” solution
If we assume, as in the previous section, that i(t) is an exponential function, or at least that it can be approximated to an exponential function within a time interval that is at least as wide as the time range in which w(s) is not negligible, \(\rho (t)\) can be computed “exactly,” and is constant within that interval.
If we assume \(i(t) = A\, e^{\lambda t}\), Eq. 18 becomes:
Simplifying the term \(A\, e^{\lambda t}\), as in the previous cases, \(\rho (t)\) can be computed as:
If w(s) is negligible for values of \(s>t\), we can extend the integration from 0 to \(\infty \), and \(\rho (t)=\rho \) does not depend on t:
This result is also reported in [12].
Note that if \(\lambda =0\), Eq. 31 becomes:
The normalization of w(s) implies \(\rho =1\), regardless of the details of the probability distribution w(s).
8 The case of a gamma distribution
In [10], w(s) is approximated to a gamma distribution:
where \(\kappa \) and \(\theta \), the shape and scale parameters, are determined with a fit to the Italian data. Equation 31 becomes:
where the integration can be performed analytically:
With some simplification of the \(\varGamma \) functions, the result is:
The above equation is valid for \(1/\theta<\lambda < \infty \). Again, \(\lambda =0\) corresponds to \(\rho =1\) for any values of \(\kappa \) and \(\theta \), as demonstrated in general in the previous section.
9 \(R_t\) and \(\tau _2\) as indicators of the epidemic evolution
\(R_t\) is often used as indicator of the epidemic evolution. As we have seen, there is a very close relation between the Effective Reproduction Number and doubling time. The estimate of the doubling time \(\tau _2\) can be determine directly from the number of infected people, while \(R_t\) also requires an estimate of the average generation time g, which propagates an extra uncertainty with respect to the estimate of \(\tau _2\).
The main feature of \(R_t\) is the passage through the threshold value of one: \(R_t>1\) indicates a growing phase, while \(R_t<1\) indicates a receding phase of the epidemic. Those conditions are equivalent to \(\tau _2>0\) and \(\tau _2<0\), respectively, as evident form Eq. 15. In the case \({\hat{\lambda }}=0\), \({\hat{R}}_t\) is not affected by the uncertainty on the estimate of g.
For this reason, we consider \(\tau _2\), or equivalently \(\lambda \), a better indicator of the situation of the epidemic compared to \(R_t\), which may be of interest for other epidemiology purposes.
10 Results
Figure 2 shows \(R_t\), evaluated with the presented algorithm assuming a constant generation time, using the public Italian COVID19 data released by the Italian Dipartimento di Protezione Civile [13]. Different values of the average generation time g have been assumed, from 3 to 7 days.
The magnitude of the dependence of \(R_t\) on g gives also a clue about the uncertainty on \(R_t\) due to imperfect knowledge of g, which mainly affects the regions where \(R_t\) is significantly different from 1.
Figure 3 shows instead the evaluation performed with the three models discussed above:

1.
Eq. 15, assuming a constant generation time of \(g=6.7\) days;

2.
Eq. 28, assuming a mean value of 6.7 days and a standard deviation of 4.88 days;

3.
Eq. 36, assuming a gamma distribution having parameters \(\kappa =1.87\) and \(\theta =3.57\) days, respectively, as determined in [10].
Note that the mean of the gamma distribution is equal to the product \(\kappa \theta \).
All three methods give similar values for \(R_t\) close to 1, but exhibit some discrepancy at more extreme values. Compared to the “exact” solution that assumes a gamma distribution (Eq. 36), assuming a fixed generation time (Eq. 15) gives a result that is about 9% larger at the highest value and about 4% larger at the lowest value. Including the contribution of the standard deviation term (Eq. 28) gives a reduction of about 12% at the larges value and 3% at the lowest value. Using (Eq. 15) with a lower “effective” g may improve the agreement with the “exact” solution at higher values at the cost of a poorer agreement at lower values. This is effectively done in the implementation of the RKI algorithm.
Figure 4 shows the application of different algorithms to the official Italian COVID19 data published by the Italian Dipartimento di Protezione Civile [13]. The algorithm presented in this paper is noted as CovidStat and assumes a gamma distribution with the parameters reported above. It is compared with algorithms by Wallinga and Teunis [1], Bettencourt and Ribeiro [2], Cori et al. [3], and RKI [4]. Algorithms by Wallinga and Teunis and Cori et al. use the details of the probability distribution \(w_s\) and are here implemented assuming the same \(w_s\) as our algorithm. Bettencourt and Ribeiro uses a fixed time, that we have set to 7 days.
The method proposed here has been implemented with an exponential fit to the last 14 days. The RKI algorithm has been applied with generation time \(g=5\), since the original implementation with \(g=4\) showed significant discrepancy with respect to the other algorithms, consistently with what can be noted in Fig 3. A smoothing of the infection data with a SavitzkyGolay filter [14] using a time window of 15 days and a thirdorder polynomial was also applied to the infection data before applying the RKI algorithm.
The comparison of the proposed method with other algorithms shows a good agreement, considering the possible source of uncertainties and the intrinsic “ripple” structure of the data that may depend on the applied smoothing. In particular, agreement of our method is very good with the WallingaTeunis and with the Cori et al. algorithms. The agreement with the BettencourtRibeiro is also good, considering that it includes a “ripple” structure due to the data fluctuations. The agreement with the RKI method is also reasonable after the assumed constant generation time is “tuned,” with a residual disagreement for the cases where \(R_t<1\). This feature is consistent with what can be observed comparing the “exact” solution computed for the gamma distribution to the one computed assuming a fixed generation time “tuned” to the more convenient value \(g=5.5\), as shown in Fig 5.
Figure 6 shows the estimated growth rate \(\lambda \) and the corresponding \(R_t\) for Italy data. Estimates are done with an exponential fit over the last 14 days. For \(R_t\) the contribution to uncertainty due to the propagation of the statistical uncertainty on \(\lambda \) is, in most of the range, much smaller than the total uncertainty that also takes in to account the uncertainty on the parameters that model w(s), according to the estimate from [10]. This contribution to the total uncertainty is particularly large as the values of \(R_t\) depart from one. For \(R_t=1\), as noted before, the uncertainty contribution form the parameters that model w(s) is null. The magnitude of the total uncertainty is comparable with what is obtained from the algorithm by Cori et al. that tales into account the uncertainty on w(s).
11 Performances
We compared the CPU time required to run the five algorithms considered in this paper. The benchmarks ran on a dedicated cluster with 32 cores/64 threads on two AMD EPYC 7301 processors and 64GB RAM. The algorithm ran on a single thread avoiding any multithread implementation. The results are reported in Table 1.
The algorithm proposed in this paper outperforms all other algorithms, in particular when the number of cases is large, as for Itay and Lombardia. The comparison with the RKI algorithm is not very meaningful. RKI estimates \(R_t\) as the ratio of the number of infected persons last \(g=5\) days divided by the the number of infected persons in the previous g days, which takes a very small CPU time. Nonetheless, our implementation is largely dominated by the overhead introduced by the python module pandas [15] compared to numpy [16], which is faster, and is the one we use for the CovidStat algorithm. The choice was only dictated by convenience, and we didn’t consider any porting of our implementation of the RKI algorithm to numpy, that would outperform the CovidStat algorithm, because the gain would be negligible anyway.
We report in the CovidStat website [9] \(R_t\) estimates for Italy, for North, Center and South separately, for the 20 Italian regions, and for the autonomous provinces of Bolzano and Trento. On the aforementioned dedicated 64thread cluster, each geographic area running on a separate thread, the computation takes about 30 minutes for all five algorithms, including all the data management overhead.
In addition, we compute \(R_t\) for the 107 provinces and for about 30 countries. For those, we only compute the CovidStat \(R_t\) estimate in order to reduce the required computation time. This evaluation takes a negligible CPU compared with the other methods of computation of \(R_t\).
Updates are published on our website daily and are produced automatically, with no human intervention, as soon as the data from the Dipartimento della Protezione Civile are available.
12 Conclusion
A simplified method to determine an estimate of \(R_t\) based on a local exponential fit is presented. The method can be applied assuming a fixed generation time, including the contribution of the standard deviation of the generation time distribution, or assuming a functional form for the probability distribution of the generation time. If a gamma distribution is assumed, a simple analytic solution is reported. The method offers some advantages compared to the simplified method adopted by the Robert Koch Institut, yet preserving good computing performances that makes it suitable for a realtime evaluation.
Results of the method applied to the public Italian COVID19 data have been presented. The proposed method shows a good agreement with other, more complex, algorithms available in literature and implemented in public software packages.
We note a close relation between \(R_t\) and the doubling time of the number of infections \(\tau _2\), or equivalently the growth rate \(\lambda \). In particular, the condition \(R_t>1\) is equivalent to \(\tau _2>0\) or \(\lambda >0\). Since the determination of \(R_t\) is affected by additional uncertainty sources compared to \(\tau _2\), we consider \(\tau _2\) or \(\lambda \) to be a more sound and simpler indicator of the condition of growing or receding epidemic compared to \(R_t\), while \(R_t\) may have more importance in other contexts of epidemiological interest.
We publish in real time daily estimates of \(R_t\) as computed by our algorithm and by all the other algorithms quoted in this article for the cases in Italy and all the Italian regions under [9]. Daily values for the major world countries are also reported.
Data Availability Statement
This manuscript has associated data in a data repository. [Authors’ comment: Available at covid19.infn.it.]
Notes
We tested this algorithm over the \(R_t\) values computed by RKI for the German cases as reported in [11], and we found an excellent agreement with the data published by the RKI.
References
J. Wallinga, P. Teunis, Different epidemic curves for severe acute respiratory syndrome reveal similar impacts of control measures. Am. J. Epidemiol. 160(6), 509–516 (2004). https://doi.org/10.1093/aje/kwh255
L.M.A. Bettencourt, R.M. Ribeiro, Real time bayesian estimation of the epidemic potential of emerging infectious diseases. PLoS ONE 3(5), e2185 (2008). https://doi.org/10.1371/journal.pone.0002185
A. Cori, N.M. Ferguson, C. Fraser, S. Cauchemez, A new framework and software to estimate timevarying reproduction numbers during epidemics. Am. J. Epidemiol. 178(9), 1505–1512 (2013). https://doi.org/10.1093/aje/kwt133
Robert Koch Institut, Erläuterung der Schätzung der zeitlich variierenden Reproduktionszahl (2020), https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Projekte_RKI/RWertErlaeuterung.pdf
K. Systrom, The Metric We Need to Manage COVID19.\(R_t\): the effective reproduction number, (2020), http://systrom.com/blog/themetricweneedtomanagecovid19/
EpiEstim: Estimate Time Varying Reproduction Numbers from Epidemic Curves, https://cran.rproject.org/web/packages/EpiEstim/index.html
K. Systrom, \(R_t\), Effective Reproduction Number, https://rt.live/
G. Bonifazi, \(R_t\)COVID19 Italia, Numero effettivo di riproduzione del virus, https://rtitaly.live/
CovidStat INFN, https://covid19.infn.it/
D. Cereda et al., The early phase of the COVID19 outbreak in Lombardy, Italy,arXiv:2003.09320 (2020)
J. Wallinga, M. Lipsitch, How generation intervals shape the relationship between growth rates and reproductive numbers. Proc Biol Sci. 274(1609), 599–604 (2007). https://doi.org/10.1098/rspb.2006.3754
Dipartimento della Protezione Civile, Dati COVID19 Italia, https://github.com/pcmdpc/COVID19
A. Savitzky, M.J.E. Golay, Smoothing and Differentiation of Data by Simplified Least Squares Procedures. Anal. Chem. 1964 36(8), 1627–1639 (1964). https://doi.org/10.1021/ac60214a047
Pandas, https://pandas.pydata.org/
NumPy, https://numpy.org/
Acknowledgements
The present work has been done in the context of the INFN CovidStat project that produces an analysis of the public Italian COVID19 data. The results of the analysis are published and updated daily on the website covid19.infn.it. The project has been supported in various ways by a number of people from different INFN Units. In particular, we wish to thank, in alphabetic order: Stefano Antonelli (CNAF), Fabio Bredo (Padova Unit), Luca Carbone (MilanoBicocca Unit), Francesca Cuicchio (Communication Office), Mauro Dinardo (MilanoBicocca Unit), Paolo Dini (MilanoBicocca Unit), Rosario Esposito (Naples Unit), Stefano Longo (CNAF), and Stefano Zani (CNAF). We also wish to thank Prof. Domenico Ursino (Università Politecnica delle Marche) for his supportive contribution.
Funding
Open access funding provided by Università degli Studi di Napoli Federico II within the CRUICARE Agreement.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Bonifazi, G., Lista, L., Menasce, D. et al. A simplified estimate of the effective reproduction number \(R_t\) using its relation with the doubling time and application to Italian COVID19 data. Eur. Phys. J. Plus 136, 386 (2021). https://doi.org/10.1140/epjp/s13360021013396
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/s13360021013396