Statistics of <i>I</i>_<i>TOT</i> the total number of COVID-19 cases per thousand inhabitants on 10 January 2022 since the beginning of the pandemic.

Four variables are considered: the unweighted number, the numbers weighted by the proportion of aged people (wA), by the number of hospital beds per inhabitant (wB) or both (w). The coefficient of variation cv = σ/μ is calculated for each variable (with σ the standard deviation and μ the mean). The variation coefficients of the weighted numbers are then compared to the unweighted ones. Statistics are provided for four ensembles (Europe, America, Asia and Africa), and for the whole world.</p

Dynamical regimes of the models.

The models dynamics was investigated considering 100 000 integration time steps of 0.1 day each (corresponding to a duration of 24 years). Metastable is mentioned when the integration could be checked on 20 000 time steps (∼6 years) only. P1, P2 and P5 refer to period cycles of period one, two and five, respectively. Toroidal chaos refers to chaotic attractors structured around, and bounded by, a toroidal structure (see [48] for details). (PDF)</p

Attractors (differential phase portraits in (<i>I</i>₁, <i>I</i>₂) projection) obtained by tuning the models parameters (or from alternative models) for cases or deaths dynamics of four countries.

In each case, the original phase portrait is also reported (in black line) for comparison. The two chaotic attractors presented in (A) for the cases dynamics in Egypt were obtained from Eq. (3) in S3 Appendix with κ = 1.25 (in magenta) and κ = 1.7 (in green). The three non chaotic attractors presented in (B) for the dynamics of cases in Algeria were obtained from Eq. (1) in S3 Appendix with (κ1, κ2) = (1., 0.75) (in magenta), (κ1, κ2) = (1., 1.2) (in blue), and (κ1, κ2) = (1.7, 1.) (in brown, with its transient in cyan). The attractors for the dynamics of cases in Kenya (C) were obtained from Eq. (6) in S3 Appendix with κ1 = κ2 = 1. (in green), (κ1, κ2) = (1.08, 0.9901) (in red), (κ1, κ2) = (1, 0.9882353) (in blue). The two attractors (in magenta and green) presented for cases dynamics in Ghana (D) were both obtained from equations (5) in S3 Appendix with κ = 0.85, but from different initial conditions revealing to a situation of bistability; The attractor presented in (E) was obtained from Eq. (13) in S3 Appendix, presenting one term less than Eq. (5). The two chaotic attractors presented in (F) for the dynamics of deaths in Algeria were obtained from Eq. (14) (in green) and (23) (in magenta), both in S3 Appendix, and correspond to two different epidemiological situations.</p

Estimating the contact number.

The number of contact per person and per day is an important parameter because this number plays a key role in the transmission of a virus during an epidemic. Here, a reformulation of the equations of an epidemic is used to reconstruct β(t) the number of contact per person and per day from I(t) the number of new cases per day and V(t) the number of vaccination at time t. All the details of these reformulation are provided in the present Appendix. To test its validy, the approach is applied to the dynamics of a 7-compartment model in S2 Appendix in order to show that, although based on a simple formulation, this formulation can apply to dynamics of higher complexity. (PDF)</p

First return maps of models for deaths.

Maps were reconstructed for (A) Algeria deaths model with κ1 = κ2 = 1. (in black) and κ1 = 1. and κ2 = 1.2 (in green); (B) Cameroon deaths model; (C) Egypt deaths model; (D) Namibia deaths model; and (E) Zimbabwe deaths model. Corresponding equations and initial conditions are provided in S3 Appendix (Section 2). (JPEG)</p

Differential phase portraits in (<i>D</i>₁, <i>D</i>₂) projection of the deaths dynamics.

Observations (black lines) and models (green lines). (JPEG)</p

Average number of contact estimated per person per day.

Average number of contact per person per day (green lines) reconstructed from the daily time series of new cases from 23 January 2020 (day 23) to 20 October 2021 (day 650) for seventeen countries (panels are organised geographically on the figure). For each country, an ensemble of 54 estimations is presented, obtained by varying the exposure duration (from 4.5 to 5.5 days), the sickness duration (from 4.5 to 5.5 days), the distribution shape (three types) and the percentage of asymptomatic (from 25% to 35%). The (reversed) index of stringency 100 − iox(t) [in %] (red line), from the University of Oxford is also provided. When available, the percentage of residential duration (100 + iRD(t) in orange) and retail-recreation time (100 + iRR(t) in purple) are also plotted (in comparison to the median of the period 3 January to 6 February 2020 which represents 100%).</p

Phase portraits for the dynamics of deaths.

Original (in black) and modelled (in green) differential phase portraits, in (D1, D2) projection, for the dynamics of COVID-19 deaths, for 8 African countries. Original phase portrait is reconstructed for the period 22 January 2020 to 21 June 2021 where solid lines were used for modelling, dashed lines for validation. (JPEG)</p

Statistical and geographical distributions of COVID-19 in the world.

Boxplots and geography of the numbers of COVID-19 cases (A and C) and deaths (B and D) per thousand inhabitants before health system correction is applied, and of the numbers of COVID-19 cases (E and G) and deaths (F and H) per thousand inhabitants of 60 years old and over, with health system correction based on the number of hospital beds per inhabitant applied. Geographical distributions resulting from these two factors separately are presented in S1 Fig. Results are presented for each continent separately (boxplots) and for the whole world (maps). Only countries with more than 1 million inhabitants are considered in the boxplots analysis where the rectangles correspond to the 25% and 75% centiles, and the extreme values to the minimum and maximum of the distributions; the central bar within the box corresponds to the median. Map created using Plotly and Mapbox. https://plotly.com/python/map-configuration.</p

Time series of deaths.

Observed and pre-processed time series of COVID-19 deaths per million inhabitants from January 1st 2020 (day 1) for seventeen African countries. The thin lines correspond to the original observations uncorrected (in blue) or corrected (in black) from the health system bias. Pre-processed time series are provided in thick lines with the one sigma error bar associated with it (in red). (TIFF)</p