The effective reproduction number R_{eff} is a fundamental epidemiological quantity. It describes the spread of an infectious disease by answering the question: How many people does an infected person infect on average?
If R_{eff} is bigger than _{eff} is smaller than _{eff} = 2, the number of new infections doubles in every disease cycle and therefore grows exponentially.

R_{eff} is not fixed over time, but depends on several factors.
On the one hand, there are factors which we cannot actively influence (e.g., properties of the virus itself). On the other hand, it depends on factors such as how many people interact in confined spaces, which can be influenced by mitigation measures.

In order to monitor the efficacy of such mitigation measures, it is helpful to understand how R_{eff} evolves.

Unfortunately, R_{eff} cannot be directly assessed, since we typically do *not* know who infected whom. Instead, one tries to infer R_{eff} from the series of case numbers. This procedure is based on a **mathematical model**, which tries to encapsulate our knowledge about the **dynamics** of the spread of infectious diseases.

An internationally recognized, frequently used base model is explained in the section Mathematical background. This model and the resulting estimation procedure was developed by Cori et al. (2013) and is implemented in the software package EpiEstim. This method is used to compute the estimates of R_{eff} which are shown on the current numbers page.
The estimation procedure depends on various input parameters, which we explain in detail here.
The credible intervals shown on the index page are based on an extension of EpiEstim which was developed by the London School of Hygiene & Tropical Medicine (*LSHTM*), cf. epiforecasts.io.

_{eff} is assumed to be constant.

To account for random and systematic variability in the number of new cases (e.g., on the weekend there are typically fewer cases), *EpiEstim* makes the assumption that R_{eff} has remained constant over the previous _{eff} for the current date. This provides an estimate of R_{eff} for the current date that can be roughly considered as an "average" of R_{eff} over

The size of _{eff}. For example, a small τ has the advantage that changes in R_{eff} can be identified more quickly. In contrast, if τ is larger, the estimate is more robust with respect to random fluctuations in the number of new cases. In our graphs, we present the estimates for τ=13 (this choice of parameter is for instance used by AGES) as well as estimates for τ=7 (as used, e.g. in Cori et al. (2013)).

These opposing effects are best illustrated via example: One can simulate case numbers from the model on which *EpiEstim* is based after providing R_{eff} as an input parameter. The following graph shows a series of virtual case numbers that has been generated accordingly.
The "virtual epidemic" has three phases. In the beginning, R_{eff}=2.2, then R_{eff}=0.6, and ultimately R_{eff}=1.3.

Based on this simulated number of cases, we can use *EpiEstim* to estimate the (actually known) value of R_{eff}. In this scenario, we have nearly perfect conditions for the *EpiEstim* procedure, since all model assumptions of *EpiEstim* are fulfilled, except one: at 11.10 and 25.10 as well as some days thereafter the model assumption which says that the actual R_{eff} has been constant across the last _{eff} series and the corresponding estimate by *EpiEstim* (including credible intervals). Jumps in the actual R_{eff} turn into "ramps" in the estimates. Furthermore, the actual value of R_{eff} is correctly estimated after a delay of _{eff} correctly when

R (EpiEstim, τ=7) | |

R (EpiEstim, τ=13) | |

R (specified) |

In the real world several reasons contribute to a time delay in the estimation of R_{eff}. After someone is infected, it takes several days until symptoms emerge. It takes additional time until they are tested and till the testing result is digitally recorded. Hence, if one tries to attribute a case to the day when the infection actually occurred, it has to be moved backwards by several days. Accounting for these delays, our estimates of R_{eff} are assigned to the date 10 days before the last date for which data has been used to compute the estimate on the main page.

Additionally, since the estimation procedure averages across *last figure of the previous section*, this would mean that the estimated values of R_{eff} move closer to the actual values, overall. **But**, this would be accompanied by the unwanted effect that the estimate of R_{eff} starts decreasing **before** the true R_{eff} drops. Therefore this additional shift is not incorporated in our estimates of R_{eff}.

We reiterate that the time offset has to be considered when comparing different estimation procedures. For instance, it is also common to associate the estimates with the **last** day for which data has been used to calculate the esitmate. This would lead to significantly different plots.

The 90% credible interval for R_{eff} gives the range in which 90% of the *plausible* values for R_{eff} lie (analogous for the 50% credible interval).

*EpiEstim*, as well as the extended method from *epiforecasts.io*, use *Bayes estimation* for R_{eff}: Different possible values of R_{eff} are weighted depending on how plausible they are given the current development of the case numbers. A more detailed explanation can be found in the Mathematical background.

For the interpretation of such estimates and credible intervals, it is crucial to notice that only **uncertainties which were considered in the model** are taken into account.

*EpiEstim* assumes that the serial interval is known accurately and that all infected individuals are equally infectious. Furthermore, it is assumed that all infected individuals test positively and that it is known exactly at which day the infection event occured. Additionally, R_{eff} is considered to have been constant for a stretch of _{eff} is *accurately inferred*. For the parameters we used and the recorded infection cases, the corresponding credible interval is typically very small (i.e., single-digit percentage values for the 95% credible interval); however, the model assumptions are apparently not satisfied and one expects that the error of the model should be much larger than the credible interval of the Bayes estimator. Therefore, the credible interval of the *EpiEstim* estimation procedure could give a *highly unrealistic* measure of the uncertainties actually present and is **not** displayed in our graphics.

In contrast, the implementation of the *LSTHM* group (*epiforecasts.io*) considers model uncertainties in several additional ways: variation in the length of the serial interval is accounted for, the date of the infection event is considered *stochastic*, and the assumption of R_{eff} being constant over the last

We want to stress that further uncertainties exist which are not accounted for. One can assume that many infected individuals (e.g., asymptomatic carriers) are not tested. In addition, uncertainties exist as to how infectious different individuals are and to which extent the behavior of individuals affects their infectiousness (e.g., "super-spreaders"). The last argument also questions the validity of the model assumption that the number of newly infected individuals per day is Poisson distributed (cf. the section Mathematical background).