Note: In this section we use the shorthand

The effective reproduction number

Typically, we do not know who infects whom. Therefore, the straightforward idea of averaging the total number of secondary infections caused by sufficiently many infected individuals is not practically feasible.

The estimation of

In reality, however, infected individuals are infectious for several days, and on each of those infectious days a different number of secondary infections may occur.

In order to arrive at an estimation procedure for

First, let us denote as

If every *fraction* of infections that occur on the **assume** that the fractions **serial interval**. (In our implementation of EpiEstim we assume that the distribution of the serial interval is given by a discretized Gamma distribution with mean 4.46 and standard deviation 2.63, based on estimates of AGES.)

Next, let

If we allow

Given that, one would attribute a decrease in the infections **today**, at day *today*, on day *today*, one would rather update the above formula again to

(Given such definition of **effective** reproduction number, which is written explicitly as R_{eff}.)
The updated formula also allows an easier estimation of

At its core, this is already the fundamental formula on which many estimation procedures of

In order to capture this, the mathematical model considers

Based on this, one uses a *Bayesian model* to infer *London School of Hygiene & Tropical Medicine* (LSHTM) / *epiforecasts.io* .

In Bayesian statistics, observed data (e.g., case numbers) are used to assess the plausibility of different values of a parameter. As a results, a so-called *posterior distribution* of the parameter of interest – namely *prior distribution* of the parameter, which summarizes our previous knowledge about its value.
If a model contains further parameters, which are not fully known, then one can treat those similarly with their own prior distributions. Essentially this means that for the *estimate* of

In the software package *EpiEstim* one can do this for the serial interval.
The method used by *epiforecasts.io* is also based on a *Bayesian estimator*, however, it also considers additional sources of uncertainty such as the **reporting delay**, i.e. the time duration between infection and onset of symptoms.

A credible interval is a region in which the parameter falls with a given probability according to its posterios distribution. Here, we show the 50% and 90% credible intervals for the method developed by LSHTM / *epiforecasts.io*.

We are primarily interested in whether the disease is spreading more rapidly or slowing down, i.e., whether **hypothesis test** (which is conceptually simpler than the estimator described in the previous section).
This hypothesis test is the basis for the *plausibility* assessment of

If we assume that

This model allows us to calculate the probability *smaller * yields

Therefore, if

The value of