Table 1.
Abbreviations and mathematical symbols used throughout the study.
Fig 1.
German expected normalised AMRs (exNAMRs) of the lumped cohort of the youngest (ages 0–29 years), the thirteen older 5-year age cohorts, and the lumped cohort of the eldest (ages 95+).
Plotted are the curves fitted to the observed values of one specific CW in each year, with the observations themselves not shown for clarity of the figure; all observed (year-specific) AMR values were normalised to the CW’s value in 2000; for each CW, an exponential two-parameter function (Eq (1)) was fitted through the respective (twenty) observed normalised AMR (NAMR) data points from 2000 through 2019, as depicted by a line with a CW-specific colour ranging from blue (CW01) through violet (CW52); for calculating prognoses (exAMR, exAMC, NEAMR), each CW’s fitted exNAMR function was extrapolated beyond 2019. Distinguishing females and males, the same is plotted in Fig A.1 and Fig A.2, respectively.
Fig 2.
For the fifteen age cohorts resolving the complete age range, German (Npop ≈ 83.6 million in 2024 [18]) weekly observed (magenta lines) all-cause mortality counts (AMCs) and expected (model-estimated), corresponding values (exAMCs; black lines) from 2000 through 2024; an exAMC is the product of its correspondingly fitted exAMR value and the year-specifically observed number of cohort members (demographics; see e.g. [4, Fig 3]).
Distinguishing females and males, the same is plotted in Fig B.3 and Fig B.4, respectively.
Fig 3.
German weekly normalised excess AMRs (NEAMRs; black lines) from 2000 through 2024, for the same fifteen age cohorts as in Fig 2; normalisation: weekly excess all-cause mortality count (EAMC: difference between its AMC and corresponding expected AMC (exAMC); both: see Fig 2) divided by its exAMC; NEAMR values exceeding (red) or dropping below (green) the 95% CI indicated by a spot. Distinguishing females and males, the same is plotted in Fig C.5 and Fig C.6, respectively.
Fig 4.
Top: Yearly AMCs (i.e. observed: black dots) and exAMCs (i.e. expected: green triangles enclosed by 95% CIs), respectively, for the years 2000-2024 in Germany for comparison, exAMCs estimated by our original model version [4] (yearly instead of weekly NAMR resolution, 10-year instead of 5-year age cohorts: orange squares enclosed by 90% CIs); model population in 2024: Npop = 83.6 million [18]; Various WHO predictions (red symbols) are shown for comparison, see [4, Fig 4] for a more detailed explanation. middle, left: yearly values of exAMCs and EAMCs (UM if negative); middle, right:
the same for just the ‘flu seasons’ (fls2,year & fls1,year + 1 : CW40-subsCW20); 95%-CI significance of EAMC indicated by (**); bottom: seasonal percentage NEAMR values of the 5-year-resolved age cohorts since 2020 through 2024.
Fig 5.
German seasonal NEAMRs from 2000 through 2024, for the same fifteen age cohorts as in Figs 2, 3; NEAMR values given in % as in Figs 3, 6, however, time-resolved more coarsely for just three seasons constituting a full year (A): a tile is plotted for each of the three seasons of any year (‘flu season 1’, fls1: CW04-CW20; ‘summer season’, sus: CW21-CW39; ‘flu season 2’, fls2: CW40-subsCW03), which codes by colour the percentage NEAMR values averaged in time over the respective season (reddish: over-mortality, i.e. positive NEAMR; greenish: UM, i.e. negative NEAMR), now indicating significance by either an asterisk (NEAMR exceeding 95% CI) or a circle (NEAMR dropping below 95% CI); B: two distributions of the relative frequencies (histograms) of NEAMR values are plotted in one sub-panel, of which each reflects the data of one age cohort: one distribution for the years 2000-2019 (grey), and one for 2020-2024 (magenta); their arithmetic mean values are symbolised by solid vertical lines; the significance of the difference in mean values is indicated by star symbols: p < 0.05 (one), p < 0.01 (two), p < 0.001 (three); distinguishing females and males, the same is plotted in Fig D.7 and Fig D.8, respectively.
Fig 6.
German weekly NEAMRs (black lines) from 2020 through 2024, however, with coarser age resolution (see Sect 2.2.5) of only seven instead of the fifteen cohorts applied in Figs 2, 3; 95% CI is the grey shaded area; NEAMR values exceeding (red) or dropping below (green) the 95% CI are indicated by a spot; abscissa: CW#; ordinates: left for NEAMR values (given in %; see Fig 3), right for all others (normalised incidence values). Additionally shown: (i; orange lines; ‘normalised mRNA-I incidences’, mRNA-I#) weekly number of mRNA-I, normalised to the cohort’s size, and multiplied by 25 for better visibility, i.e. adopted to the ordinate range of (ii), distinguishing the first (mRNA-I1), second (mRNA-I2), third (mRNA-I3), and fourth (mRNA-I4) mRNA-I. (ii; violet, magenta, cyan, sky blue and dark blue; ‘normalised PCR-incidences’, PCR+) weekly number of positive PCR tests, normalised to the number of PCR tests conducted as well as to (times) the cohort’s size, the numbers distinguishing ‘alpha’ (magenta), ‘delta’ (cyan), ‘omicron’ (sky blue), and ‘others’ (dark blue), as well as the total number not distinguishing SARS-CoV-2 variants (violet).
Distinguishing females and males, the same is plotted in Fig E.9 and Fig E.10, respectively.
Fig 7.
In Germany, during the time interval CW04-CW42,2021, and for seven age cohorts, the (Pearson) coefficients of cross-correlating their respective weekly NEAMR time course with several (time-lagged) normalised incidence signals (regarding normalisation, see caption of Fig 6) are plotted: with the normalised weekly numbers of positive PCR tests PCR+(V) of two SARS-CoV-2 variants V‘alpha’, ‘delta’}, and with the normalised weekly number of mRNA-I of the first (mRNA-I1), and second (mRNA-I2) mRNA-I, as well as with two products by week of incidences; maximum and minimum coefficient values to the right of a sub-panel; distinguishing females and males, the same is plotted in Fig F.11 and Fig F.12, respectively.
Fig 8.
In Germany, during the time interval CW30,2021-CW03,2022, and for seven age cohorts, the (Pearson) coefficients of cross-correlating their respective weekly NEAMR time course with several (time-lagged) normalised incidence signals (regarding normalisation, see caption of Fig 6) are plotted: with the normalised weekly numbers of positive PCR tests PCR+(V) of the SARS-CoV-2 variant V=‘delta’, and with the normalised weekly number of mRNA-I of the first (mRNA-I1), second (mRNA-I2), and third (mRNA-I3) mRNA-I, as well as with one product by week of incidences; maximum and minimum coefficient values to the right of a sub-panel; distinguishing females and males, the same is plotted in Fig G.13 and Fig G.14, respectively.