A two-step procedure is used to estimate the Rt from data informing the SI and from data on the daily temporal evolution of incidence onset of cases (7). The first step uses data on known pairs of index (infector) and secondary (infectee) cases to estimate the SI distribution; the second step estimates the time-varying reproduction number jointly from disease onset time series and from the SI distribution fitted in the first step.
The distribution of serial intervals can be estimated during an ongoing outbreak using data from the list of censored lines by interval, i.e. the lower and upper limits of the time of symptom onset in index and secondary cases (8). For each pair, the number of days between the reported symptom onset date for the infector and the reported symptom onset date for the infectee is calculated (9). Negative values indicate that the infectee developed symptoms before the infector.
Maximum likelihood (ML) estimates and the Akaike information criterion (AIC) are used to evaluate widely used parametric candidate models for the SARS-CoV-2 serial interval distributions namely normal, lognormal, Weibull, and gamma. Since our SI data includes a considerable number of non-positive values, we fit the four distributions both to positive values (truncated) and to shifted data, in which 12 delays are added to each observation (9). However, caution against making assessments and projections based on the truncated data should be carefully explored and we do not believe there is cause for excluding the non-positive data.
At the beginning of an epidemic, when the whole population is susceptible (i.e. not immune), this number takes on a particular value denoted R0 and called the basic reproduction number (10).
The calculation of R0 is based on three underlying assumptions as follows:
Screening strategy in Tunisia is assumed to be constant,
Spatial structure is neglected,
Incidences used are those available since February 29, 2020 and until March 18, 2020 (date of the curfew) for R0 and until 5 May 2020 for temporal reproduction number.
During the outbreak, when the proportion of immunized persons becomes sufficiently large to slow the transmission of the virus (by an effect similar to a reduction in the number of individuals still susceptible), we speak about the effective, or temporal, reproduction number denoted Rt (11).
Analyses to estimate the reproduction number were conducted using the EpiEstim (7, 12, 13) package on the R statistical software (version 3.6.3) (14). This package is based on an approach that is motivated by the fact that in the situation where the epidemic under study would still be ongoing, and more particularly when it comes to evaluating the effectiveness of control measures, the total number of infections caused by the latest cases detected is not yet known. For EpiEstim package, the highlighted approach to temporal reproduction number leads to the instantaneous reproduction number, which is prospective: its calculation is based on the potential number of secondarily infected persons that a cohort of cases could have caused if the conditions of transmissibility had remained the same as at the time of their detection.
Let's denote the total cases by symptom onset arising at time-step by It (assuming total cases of local and imported). Following (6, 7), in which the time-dependent reproduction number, Rt, is defined as the ratio of the number of new infected cases at time t, It, and the total infection potential across all infected individuals at time t, Λt. If there is a single serial interval distribution ωs(s = 1,2,...), representing the probability of a secondary case arising a time period s after the primary case, each incident case that appeared at a previous time-step t-s contributes to the current infectiousness at a relative level given by ωs. Therefore conditional on ωs, Λt can be computed as follows:
Formally the EpiEstim package maximizes the likelihood of incidence data (seen as a Poisson count) observed over a time window of size τ ending at t. The assumption made here is that the reproduction number is constant over this time window [t-τ,t]. The estimation of the reproduction number at each time window, denoted Rt,τ, for the time interval [t-τ,t] verifies:
Thereby, an estimation of Rt is obtained given both the incidence and serial interval data, from which the mean and 95% intervals of Rt can be computed. The proposed formula may also be used for early detection of the effect of control measures to prevent the spread of the virus on the incidence of new cases.
For the next parts of the document, Rt is denoted R for simplicity. If R > 1, then one person infects more than one person, on average and the epidemic is growing. As the epidemic spreads, R decreases as an increasing proportion of the population becomes immune. When the threshold for group immunity is exceeded, R drops below 1, an epidemic peak is reached and the epidemic decreases. Public health control measures can also decrease R and thus reach epidemic peak before the threshold of population immunity is reached. Therefore, knowing the value of R at time t is essential to determine the status of the epidemic.
Moreover, the overall infectivity due to previously infected individuals of an outbreak at time t, denoted λt, is a relative measure of the current force of infection. It is calculated as the sum the previously infected individuals It, weighted by their infectivity at time t and is given by:
The critical parameter for these calculations is the distribution of SI. If λ is falling, then that’s good: if not, bad.