Epidemiological measures for assessing the dynamics of the SARS-CoV-2-outbreak: Simulation study about bias by incomplete case-detection

During the SARS-CoV-2 outbreak, several epidemiological measures, such as cumulative case-counts (CCC), incidence rates, effective reproduction numbers (Reff) and doubling times, have been used to inform the general public and to justify interventions such as lockdown. It has been very likely that not all infectious people have been identified during the course of the epidemic, which lead to incomplete case-detection. We compare CCC, incidence rates, Reff and doubling times in the presence of incomplete case-detection. For this, an infection-age-structured SIR model is used to simulate a SARS-CoV-2 outbreak followed by a lockdown in a hypothetical population. Different scenarios about temporal variations in case-detection are applied to the four measures during outbreak and lockdown. The biases resulting from incomplete case-detection on the four measures are compared in terms of relative errors. CCC is most prone to bias by incomplete case-detection in all of our settings. Reff is the least biased measure. The possibly biased CCC may lead to erroneous conclusions in cross-country comparisons. With a view to future reporting about this or other epidemics, we recommend including and placing an emphasis on Reff in those epidemiological measures used for informing the general public and policy makers.


Infection-age structured SIR model
As for SARS-CoV-2 there is evidence that probability of making an effective contact between an infector and a susceptible subject depends on the infector's time since infection [He20], we use the infection-age structured SIR model. Migration, fertility and mortality of non-diseased people plays a minor role in the simulated period of 60 days. Thus, demography of the background host is ignored. Similar to the conventional SIR model (without infection age), the population is partitioned into three states, the susceptible state, the infected and the removed state. The initial letters of the three states give the model's name SIR. The removed state comprises people recovered and deceased from the infected state. The numbers of the people in the susceptible and the removed states at time t are denoted by S(t) and R(t), respectively. Furthermore, let i(t, τ ) denote the density of infected people at time t and duration τ since infection (i.e., the infection age), such that the number I(t) of infected at t is (1) The transmission rate of the infected with infection age τ at time t is β(t, τ ) and the removal rate from the infectious stage is γ(τ ). The rate γ comprises mortality as well as remission.
The SIR model and the rates β and γ controlling the transitions between the states is shown in Figure (  We can formulate the following model equations for the infection-age SIR model [Ina17]: The incidence rate λ in Eq. (2) is given by and is usually called force of infection [Ina17].
Detailed discussion of Equations (2) -(4) with initial conditions (5) -(8) can be found in [Ina17, Chapter 5.3]. In [Ina17, Chapter 5.5] we also find that the age-structured SIR model is a generalization of the frequently used SEIR model.
Using the definition the effective reproduction number R eff (t) is given by These are the incidence densities at the grid points located on and above the diagonal of the grid (on and above the dashed line in Figure 2).

Cumulative case counts
The cumulative case count CCC(t) up to time t is the sum of incident cases until day t: Accordingly, the observed cumulative case count, CCC (o) (t) is defined as

Incidence rate
As usual, the incidence rate is defined as the number of incident cases F t over the population at risk. The size of population at risk is the number of susceptibles S t .
Compared to the number S(t) defined via Equations (2) -(4) an initial conditions (5) -(8), S t is an daily average of S(t). Similarly, the observed incidence rate is defined via

Effective reproduction number R eff
We use the Fraser-method to estimate the instantaneous reproduction number R eff .

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Obviously, the estimation method for R eff (t) uses data from the J = 9 preceding days t−9, t−8, . . . , t, the earliest day with an available estimate for R eff (t) is day t = 10. The corresponding estimates of R eff (t) based on the daily numbers F t are shown as solid black line in Figure 5. For comparison, the time-continuous values R eff (t) calculated by numerically integrating Eq. (10) are drawn as dashed blue line.

Doubling times
As described in the main text, the doubling time ∆ at time t is defined by For estimating ∆(t), the cumulative case count CCC(t) at day t is modelled by fitting a linear regression line to the logarithmized case counts of the J = 9 previous days, i.e., to the points t − j, log CCC(t − j) , j = 0, . . . , J.
Assumed the associated regression line at day t reads as a t + b t × t, then an easy calculation shows that the doubling time ∆(t) is given by ∆(t) = log(2) bt .
As the estimation method for ∆(t) uses data from the J = 9 preceding days, the earliest day with an estimate is t = 10.
4 Choice of the parameters for mimicking the

SARS-CoV-2 pandemic
We use the parameters as shown in Table 1 to mimic the spread of the virus in the hypothetical population. The transmission rate β is assumed to be a product of two factors β τ and β t . Figure 1 of the main text shows the factors β t and β τ . β Transmission assumed to be a product of two rate functions β(t, τ ) = β t (t) × β τ (τ ) β t factor of β mimics a lockdown, chosen such that [Lav20] depending on t R eff drops from > 1 to < 1 (cf. left part of Figure 2 in the main text) β τ factor of β follows a Gamma distribution with [He20] depending on τ shape 2 and rate .25 ⇒ modal value of 4 and mean 8 (see right part of Figure 2 in the main text) γ Removal rate Asymptotics similar to β τ [He20] 5 Source code Source code for running the simulation in the free statistical software R (including solving the age-structured SIR model) can be found in the open public repository Zenodo under DOI 10.5281/zenodo.4750942 [Bri21].