Fig 1.
Disease intervals and reproduction.
(A) Disease and transmission intervals. Schematic diagram showing the events infection, symptom onset and registration (denoted by T) in the timeline of a hypothetical infection pair. The time period between two consecutive disease transmissions is called the generation interval, here written as Δgen. The time period between the symptom onsets in an infection pair is called the serial intervalΔser. We further denote the time period between registration of two consecutive cases as the case intervalΔcase. Further intervals shown are the incubation delayΔinc, the registration delayΔreg and the forward and backward reporting offsets ,
. (B) Visual outline for the quantification of reproductive dynamics based on time series of reported cases. Infectious load and activity can be derived from reported cases using the statistical distributions of the reporting offsets. The obtained characterization of epidemic progression is only a surrogate for actual reproduction dynamics. (C) Probability densities of the backward and forward reporting offset distributions and the case interval distribution inferred from data. All interval distributions were calculated for different parameter settings; here, the weekday of case registration is encoded in the lightness of the color. (D) Transformation between different statistical models for the reporting offset intervals. We investigate our model and the resulting dispersion indicator (EffDI) under gradual transformation of the probability densities to analyze potential impacts of their characteristic features (S9 Text). Here, the continuous transformation—implemented by means of a transition parameter in the interval [0, 1] and displayed according to continuous color ramps—of the forward and backward reporting offset into a degenerate distribution and the case interval is portrayed.
Fig 2.
Varying regularity of reported case numbers.
Normalized segments of daily reported case numbers [43] during low-prevalence (top row) and high prevalence (bottom row) periods in different countries (columns). Normalization was achieved by dividing the daily case numbers by their mean value during the respective three-week period. During the respective high prevalence periods, we observe a high degree of regularity and weekly seasonal patterns, whereas in low-prevalence phases, the segments seem to be more erratic.
Fig 3.
Quantification of effective aggregate dispersion.
(A) Number of reported cases in Austria [43] and infectious load and activity according to Eq (3). The low- and a high-incidence phases from Fig 2 are indicated with vertical lines. (B) Plausibility diagram for the dispersion parameter κt. The transition between the plausible and implausible regime (e.g. p = 0.9) is abrupt. (C) EffDI results by the transformed level set line (green) (κt(p = 0.9))−1/2. It is a quantifier for dispersion and corresponds to the coefficient of variation in the underlying statistical model Eq (6). The progression of dispersion aligns with the progression of a socio-demographic heterogeneity measure (orange). Socio-demographic heterogeneity is measured via the total variation distance between infectious load and infectious activity. (D) Resulting case reproduction number.
Fig 4.
EffDI evaluated for different countries.
Reported case numbers [43] are shown in gray. In the case of Switzerland and the UK, certain dates are highlighted to illustrate the effect of irregularities or changes in the reporting regime. a: After September 19, 2020, Switzerland only reported aggregate case numbers on weekdays (cf. Fig 2). b, c, d, e, f, h: Missing aggregate case numbers in Switzerland on several days in 2021. g: Missing entries on a Friday and the following Monday in Switzerland during September 2021. i, j: Reporting of negative aggregate case numbers on two days in April and May 2021 in the UK.
Fig 5.
Heterogeneity of infectious activity and in reproduction.
Media coverage about certain SSEs in Austria is indicated by vertical dashed lines and the identifiers C1-C8 (see main text). (A) Crude number of reported cases in Austria; estimate for the time series of infections of subsequently reported cases (infectious activity); and estimated time series of the currently infectious population (infectious load). (B-D) For the social dimensions sex, age, and geographic region, the difference of the distribution of the estimate newly infected population (infectious activity) to the distribution of the total population is visualized. High values (red) indicate over-representation of infectious activity and low values (blue) indicate under-representation. (E) Quantification of the heterogeneity of infectious activity using the total variation distance measure. To increase lucidity, a smoothed version of the resulting time series is shown. (F) Quantification of the heterogeneity in reproduction; if the distance measure is small, spread is confined to specific social strata; if the distance is large infections shift to previously unaffected social compartments.