Epidemological modelling has come into focus due to the Corona Crisis. The Basic Reproduction Number (R0) is a core indicator, defined as the number of secondary infections that arise from a typical primary case in a completely susceptible population. In emperical work it may be more convenient to work with The Effective Reproduksjon Number (\(R_{eff}\) or simply \(R\)). This is defined as the number of secondary infections that arise from a typical primary case.
How can this number be calculated?
The time between the symptom onset of the primary and secondary case is usually called generation interval, but sometimes serial interval or generation time. This can be empirically observed in detailed outbreak studies and it’s distribution estimated. Wallinga and Teunis(2004) found that the generation intervals observed during the SARS-2 outbreak in Singapore in 2003 to follow a Weibull distribution with a shape parameter α and a scale parameter β, with values corresponding to a mean generation interval of 8.4 days and a standard deviation of 3.8 days Obadia et al (2012) apply a gamma distribution with mean 2.6 and standard deviation 1.0 for the German 1918 Spanish Flu outbreak.
For an outbreak with a new virus such as Corona/Covid-19 the precise values of parameters of the generation interval distribution will not be known (i.e. before detailed tracking studies have been carried out). However, if we can assume the new virus to have similar distribution characteristics to previously known cases, then the efficient reproduction number R can be calculated from the growth rate of the confirmed cases in the new outbreak.
Wallinga and Lipsitch(2007) (WL) derive the relationship between the outbreak growth rate \(r\) and the reproduction number \(R\) from the Lotka-Euler equation and the moment generating function of the generation interval distribution. WL discusses varous theoretical generation distributions, but most useful is their empirical treatment. This approach is in more detail described in Wallinga and Teunis(2004)(WT). WT show that the relative likelihood \(p_{ij}\) that case \(_{i}\) has been infected by case \(_{j}\), given their difference in time of symptom onset \(t_{i} – t_{j}\) , can be expressed in terms of the probability distribution for the generation interval. This property makes analysis of outbreaks possible even if detailed generation studies have not yet been performed.
The further presentation here is split into 4 sections as follows:
There exists a R implementation in the R0 Package on cran r, described in Obadia(2012). Here we implement the methods by means of Julia code. Bernd Blasius Github Repository contains data and Julia code for more complete epidemologic (SEIR) modelling of the corona outbreak. Peter Turchin Github Repository contains another R SIRD-model for Corona Analysis.
Links and pdfs:
Wallinga and Teunis(2004)) pdf
Wallinga and Lipsitch(2007) pdf
Obadia et al(2012) R0 Package (Article) pdf
Blasius Julia Github Repository