# Estimation of Daily Reproduction Numbers during the COVID-19 Outbreak

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{0}is calculated to quantify the contagiousness level of an infectious disease. (2) Methods: We provide the equation of the discrete dynamics of the epidemic’s growth and obtain an estimation of the daily reproduction numbers by using a deconvolution technique on a series of new COVID-19 cases. (3) Results: We provide both simulation results and estimations for several countries and waves of the COVID-19 outbreak. (4) Discussion: We discuss the role of noise on the stability of the epidemic’s dynamics. (5) Conclusions: We consider the possibility of improving the estimation of the distribution of daily reproduction numbers during the contagiousness period by taking into account the heterogeneity due to several host age classes.

## 1. Introduction

#### 1.1. Overview and Literature Review

_{0}(also called the average reproductive rate), a classical epidemiologic parameter that describes the transmissibility of an infectious disease and is equal to the number of susceptible individuals that an infectious individual can transmit the disease to during his contagiousness period. For contagious diseases, the transmissibility is not a biological constant: it is affected by numerous factors, including endogenous factors, such as the concentration of the virus in aerosols emitted by the patient (variable during his contagiousness period), and exogenous factors, such as geo-climatic, demographic, socio-behavioral and economic factors governing pathogen transmission (variable during the outbreak’s history) [4,5,6,7,8].

_{0}might change seasonally, but these factor variations are not significant if a very short period of time is considered. R

_{0}depends also on endogenous factors such as the viral load of the infectious individuals during their contagiousness period, and the variations in this viral load [9,10,11,12,13,14,15] must be considered in both theoretical and applied studies on the COVID-19 outbreak, in which the authors estimate a unique reproduction number R

_{0}linked to the Malthusian growth parameter of the exponential phase of the epidemic, during which R

_{0}is greater than 1 (Figure 1). The corresponding model has been examined in depth, because it is useful and important for various applications, but the distribution of the daily reproduction number R

_{j}at day j of an individual’s contagiousness period is rarely considered within a stochastic framework [16,17,18,19,20].

_{0}can be calculated from the evolutionary entropy defined by L. Demetrius as the Kolmogorov–Sinaï entropy of the corresponding random process [18], which measures the stability of the invariant measure, dividing the population into the subpopulations S (individuals susceptible to but not yet infected with the disease), I (infectious individuals) and R (individuals who have recovered from the disease and now have immunity to it). In the deterministic case, R

_{0}corresponds to the Malthusian parameter quantifying its exponential growth, and the stability of the asymptotic steady state depends on the subdominant eigenvalue [19,20].

#### 1.2. Calculation of R_{0}

_{0}, which correspond to the individual-level modeling and to the population-level modeling. At the individual level, if we suppose the susceptible population size constant (hypothesis valid during the exponential phase of an epidemic), the daily reproduction rates of an individual are typically non-constant over his contagiousness period, and the calculations we present in the following define a new method for estimating R

_{0}, as the sum of the daily reproduction rates. This new approach allows us to have a clearer view on the respective influence on the transmission rate by endogenous factors (depending on the level of immunologic defenses of an individual) and exogenous factors (depending on environmental conditions).

## 2. Materials and Methods

_{ikj}of people infected at day j by a given infectious individual i during the kth day of his period of contagiousness of length r. By summing up the number of new infectious individuals X

_{j−k}present on day j − k where started their contagiousness, we find that the number of new infected people on day j is equal to:

_{j}= Σ

_{k=1,r}Σ

_{i=1}X

_{j−k}R

_{ikj}

_{ikj}is the same, equal to R

_{k}, for all individuals I and day j, then depends only on day k. Then, we have:

_{j}= Σ

_{k=1,r}R

_{k}X

_{j−k}

#### 2.1. The Contagion Mechanism from a First Infectious Case Zero

#### 2.2. The Biphasic Pattern of the Virulence Curve of Coronaviruses

#### 2.3. Relationships between Markovian and ODE SIR Approaches

#### 2.3.1. First Method for Obtaining the SIR Equation from a Deterministic Discrete Mechanism

_{j}the number of new infected cases at day j (j ≥ 1), and R

_{k}(k = 1, …, r) the daily reproduction number at day k of the contagiousness period of length r for all infectious individuals. Then, we have obtained Equation (2) by supposing that the contagiousness behaviour is the same for all the infectious individuals:

_{j}= ∑

_{k=1,r}R

_{k}X

_{j−k},

_{j−k}new infected at day j − k give R

_{k}X

_{j−k}new infected on day j, throughout a period of contagiousness of r days, the R

_{k}’s being possibly different or zero. For example, if r = 3, for the number X

_{5}of new cases at day 5, equation X

_{5}= R

_{1}X

_{4}+ R

_{2}X

_{3}+ R

_{3}X

_{2}means that new cases at day 4 have contributed to new cases at day 5 with the term R

_{1}X

_{4}, R

_{1}being the reproduction number at first day of contagiousness of new infected individuals at day 4.

_{j}, …, X

_{j−r−1}) and R = (R

_{1}, …, R

_{r}) are r-dimensional vectors and M is the following r-r matrix:

_{0}= 1 and r = 5 (estimated length of the contagiousness period for COVID-19 [12,13,14,15,16,17,18,19,20,21]), we obtain:

_{5}= R

_{1}

^{5}+ 4R

_{1}

^{3}R

_{2}+ 3R

_{1}

^{2}R

_{3}+ 3R

_{1}R

_{2}

^{2}+ 2R

_{2}R

_{3}+ 2R

_{1}R

_{4}+ R

_{5}

_{j}’s, equal to the X

_{j}’s without their trend, by considering the length of the interval on which the auto-correlation function remains more than a certain threshold, e.g., 0.1 [4]. For example, by assuming r = 3, if R

_{1}= a, R

_{2}= b and R

_{3}= c, we obtain:

_{1}and R

_{2}are equal, respectively, to a and b, and if a = b = R/2, c = 0, then, X

_{5}behaves like:

_{5}= R

^{5}/32 + R

^{4}/4 + 3R

^{3}/8

_{j}}

_{i=1,∞}is the Fibonacci sequence, and more generally, for R > 0, the generalized Fibonacci sequence. Let us suppose now that b = c = 0 and a depends on the day j: a

_{j}= > C(j), where C(j) represents the number of susceptible individuals, which can be met by one contagious individual at day j. If infected individuals (supposed to all be contagious) at day j are denoted by I

_{j}, we have:

_{j}= ∆I

_{j}/∆j = (I

_{j+1}− I

_{j})/(j + 1 − j) = νC(j)I

_{j}

_{j}infectious individuals present at day j are located on a part of the sphere of centered on the first infectious 0 obtained by widening its radius (Figure 2). Then, we can consider that the function C(j) increases, then saturates due to the fact that an infectious individual can meet only a limited number of susceptible individuals as the sphere grows. We can propose for C(j) the functional form C(j) = S(j)/(c + S(j)), where S(j) is the number of susceptible individuals at day j. Then, we can write the following equation, taking into account the mortality rate µ:

_{j}= ∆I

_{j}/∆j = νC(j)I

_{j}− µI

_{j}= νI

_{j}S(j)/(c + S(j)) − µI

_{j}

#### 2.3.2. Second Method for Obtaining the SIR Equation from a Stochastic Discrete Mechanism

+ P(S(t) = k − 1, I(t) = N − k + 1) [f(k − 1) + ρ(N − k + 1)]dt

− P(S(t) = k+1, I(t) = N − k − 1) [µ(k + 1) + ν(k + 1) (N − k − 1)]dt

_{k}(t) denotes Probability({S(t) = k, I(t) = N − k}):

_{k}(t)/d = [P(S(t + dt) = k, I(t + dt) = N − k) − P(S(t) = k, I(t) = N − k)]/dt

= − P(S(t) = k, I(t) = N − k) [µk + νk (N − k)−fk-ρ(N − k)]

+ P(S(t) = k − 1, I(t) = N − k + 1) [f(k − 1) + ρ(N − k + 1)]

− P(S(t) = k + 1, I(t) = N − k − 1) [µ(k + 1) + ν(k + 1)(N − k − 1)],

_{k}(t)/dt = −[µk + νk(N − k)−fk − ρ(N − k)]P

_{k}(t) + [f(k − 1) + ρ(N − k + 1)]P

_{k−1}(t) − [µ(k + 1) + ν(k + 1)(N − k1)]P

_{k+1}(t)

^{k}and summing over k, we obtain the characteristic function of the random variable S. If births do not compensate deaths, we have:

+ P(S(t) = k − 1, I(t) = j + 1) [f(k − 1) + ρ(j + 1)]dt

− P(S(t) = k + 1, I(t) = j − 1) [µ(k + 1) + ν(k + 1)(j − 1)]dt

or, if f = µ, dE(S)/dt ≈ E(I) [−νE(S) + ρ],

## 3. Results

#### 3.1. Distribution of the Daily Reproduction Numbers R_{j}’s along the Contagiousness Period of an Individual. A Theoretical Example Where They Are Supposed to Be Constant during the Epidemics

_{0}denotes the basic reproduction number (or average transmission rate) in a givenpopulation, we can estimate the distribution V (whose coefficients are denoted V

_{j}= R

_{j}/R

_{o}) of the daily reproduction numbers R

_{j}along the contagious period of an individual, by remarking that the number X

_{j}of new infectious cases at day j is equal to X

_{j}= I

_{j}− I

_{j−1}, where I

_{j}is the cumulated number of infectious at day j, and verifies the convolution equation (equivalent to Equation (2)):

^{−1}X

_{1}, M

_{2}, … Following Equation (4), we put the values of X

_{i}’s in the two matrices below, with r = 3 for two periods, the first from day 1 to day 3 and the second from day 4 to day 6.

_{1}and M

_{2}can be calculated from the R

_{j}’s as:

_{2}is given by:

_{j}’s on a contagiousness period of 3 days, are equal to:

_{ij}= X

_{7−(i+j)}gives the R

_{j}’s from Equation (16), hence allows the calculation of X

_{j}= Σ

_{k=1,3}R

_{k}X

_{j−k}.

^{−1}and verifies: R = M

^{−1}X, where X = (X

_{6}, X

_{5}, X

_{4}), with X

_{1}= 1, X

_{2}= 2, X

_{3}= 5, X

_{4}= 14, X

_{5}= 37, X

_{6}= 98 and we obtain:

_{j}’s:

_{1}= −49/2 + 37 − 21/2 = 2

_{2}= 98 − 111 + 14 = 1

_{3}= −147/2 + 37 + 77 = 2

_{1}by the matrix M

_{2}.

#### 3.2. Distribution of the Daily Reproduction Numbers R_{j}’s When They Are Supposed to Be Random

_{j}’s, e.g., by randomly choosing their values following a uniform distribution on the three intervals: [2 − a, 2 + a], [1 − a/2, 1 + a/2] and [2 − a, 2 + a] (for having a U-shape behavior), with increasing values of a, from 0.1 to 1, in order to see when the deconvolution would give negative resulting R

_{j}’s, with conservation of the average of their sum R

_{0}, if the random choice of the values of the R

_{j}’s at each generation is repeated, following the stochastic version of Equation (2): X

_{j}= Σ

_{k=1,r}(R

_{k}+ ε

_{k}) X

_{j−k}, where r is the contagiousness period duration and ε

_{k}is a noise perturbing R

_{k}, whose distribution is chosen uniform on the interval [0, 2a] for k = 1,3, and [0, a] for k = 2. This choice is arbitrary, and the main reason of the randomization is to show that the deconvolution can give negative results for R

_{k}’s, as those observed for increasing values of a, from 0.1 to 1, with explicit calculations for three consecutive periods, from day 1 to day 3, from day 4 to day 6, and from day 7 to day 9.

_{j}’s, we can calculate a matrix M

_{1}corresponding to Equation (3). Its inversion into the matrix M

_{1}

^{−1}makes it possible to solve the problem of deconvolution of Equation (2)—that is to say, to obtain new R

_{j}’s as a function of the observed X

_{k}’s. We can then calculate a new matrix M

_{2}from these new Rj’s and thus continue during an epidemic the estimation of the daily reproduction numbers R

_{j}’s from the successive matrices M

_{1}, M

_{2}, …, and observed X

_{k}’s.

- 1.
- For a = 0.1, let us randomly and uniformly choose the initial distribution of the daily reproduction numbers R
_{1}in the interval [1.9, 2.1], R_{2}in [0.95, 1.05] and R_{3}in [1.9, 2.1] as R_{1}= 2.1, R_{2}= 0.95, R_{3}= 2.1. Then, the transition matrix M_{1}is equal to:

_{6}= 113.491, X

_{5}= 41.7391, X

_{4}= 15.351, resulting R

_{j}’s are: R

_{1}= 2.1, R

_{2}= 0.95, R

_{3}= 2.1.

_{j}’s are chosen as: R

_{1}= 2, R

_{2}= 0.95, R

_{3}= 1.9 and we have:

_{7}= 2X

_{6}+ 0.95X

_{5}+ 1.9X

_{4}= 226.982 + 39.652 + 29.17 = 295.8

_{8}= 2X

_{7}+ 0.95X

_{6}+ 1.9X

_{5}= 591.6 + 107.816 + 79.304 = 778.72

_{2}and M

_{2}

^{−1}:

_{j}’s equal: R

_{1}= 2.0279, R

_{2}= 7.6158, R

_{3}= −16.426.

_{j}’s are: R

_{1}= 2, R

_{2}= 1.05, R

_{3}= 1.9 and we have:

_{9}= 2X

_{8}+ 1.05X

_{7}+ 1.9X

_{6}= 1557.44 + 310.59 + 215.63 = 2083.66

_{10}= 2X

_{9}+ 1.05X

_{8}+ 1.9X

_{7}= 4167.32 + 817.656 + 562.02 = 5546.996

_{9}and X

_{10}, we obtain the matrices M

_{3}and M

_{3}

^{−1}:

_{j}’s equal: R

_{1}= 2.486, R

_{2}= −2.33, R

_{3}= 7.38769.

- 2.
- For a = 1, let us choose the initial R
_{1}in [1, 3], R_{2}in [0.5, 1.5] and R_{3}in [1, 3], e.g., R_{1}= 1, R_{2}= 1.355 and R_{3}= 1.1. Then, the transition matrix M_{1}is equal to:

_{6}= 18.209, X

_{5}= 9.101, X

_{4}= 4.81, X

_{3}= 2.355, X

_{2}= 1, X

_{1}= 1, and by deconvoluting, we obtain the resulting R

_{j}’s equal to: R

_{1}= 1, R

_{2}= 1.355, R

_{3}= 1.1, i.e., the exact initial distribution.

_{j}’s: R

_{1}= 1, R

_{2}= 1, R

_{3}= 1. That gives a new matrix M

_{2}, with new X

_{7}and X

_{8}calculated from the new initial R

_{j}’s, by using the former values of X

_{6}, …, X

_{2}:

_{7}= X

_{6}+ X

_{5}+ X

_{4}= 18.209 + 9.101 + 4.81 = 32.12

_{8}= X

_{7}+ X

_{6}+ X

_{5}= 32.12 + 18.209 + 9.101 = 59.43

_{j}’s equal: R

_{1}= 2.90, R

_{2}= 5.4888, R

_{3}= −14.696.

_{9}and X

_{10}using new initial R

_{j}’s: R

_{1}= 3.0, R

_{2}= 0.5, R

_{3}= 2.9:

_{9}= 3X

_{8}+ 0.5X

_{7}+ 2.9X

_{6}= 178.29 + 16.06 + 52.81 = 247.16

_{10}= 3X

_{9}+ 0.5X

_{8}+ 2.9X

_{7}= 741.48 + 29.715 + 93.148 = 864.343

_{j}’s equal: R

_{1}= 3.66898, R

_{2}= −33.857, R

_{3}= 61.32.

_{j}’s distributions, for a = 0.1 and a = 1 and until time 20. These simulations show a great sensitivity to noise, but a qualitative conservation of their U-shaped distribution along the contagiousness period of individuals. More precisely, because of the presence of noise on the Rj’s, we cannot always obtain positive values from the data for the Rj’s by applying the deconvolution, which explains the presence of negative values in empirical examples, as in the theoretical noised examples. A way to solve this problem could be to suppose that noise is stationary during all of the growth period of a wave, then calculate the Rj’s for all running time windows of length equal to the contagiousness duration and then obtain the mean of the Rj’s corresponding to these windows. As this stationary hypothesis is not widely accepted, we prefer to keep negative values and focus on the shape of the distribution of the Rj’s.

#### 3.3. Distribution of the Daily Reproduction Numbers R_{j}’s. The Real Example of France

_{e}calculated between 20–25 October 2020 just before the second lockdown in France [28,29]. As the second wave of the epidemic is still in its exponential phase, it is more convenient (i) to consider the distribution of the marginal daily reproduction numbers and (ii) to calculate its entropy and simulate the epidemic dynamics using a Markovian model [4]. By using the daily new infected cases given in [30], we can calculate, as in Section 3.1, the inverse matrix M

^{−1}for the period from 20 to 25 October 2020 (exponential phase of the second wave), by choosing 3 days for the duration of contagiousness period and the following raw data for new infected cases: 20,468 for 20 October, then 26,676, 41,622, 42,032, 45,422 and 52,010 for 25 October. Then, for France between 15 February and 27 October 2020, we obtain the daily reproduction numbers given in Figure 3 with a U-shape as observed for influenza viruses.

_{j}’s, i.e., the vector (R

_{1}, R

_{2}, R

_{3}):

_{0}≈ 1.174, a value close to that calculated directly (Figure 3), giving V = (0.7, 0.085, 0.215), with a maximal daily reproduction number the first day of the contagiousness period. The entropy H of V is equal to:

_{k=1,r}V

_{k}Log(V

_{k}) = 0.25 + 0.21 + 0.33 = 0.79.

#### 3.4. Calculation of the Rj’s for Different Countries

#### 3.4.1. Chile

^{−1}for the period from 1 to 12 November 2020 (endemic phase), by choosing 6 days for the duration of the contagiousness period and the following 7-day moving average data for the new infected cases (Figure 4): 1400 for 1 November, then 1370, 1382, 1359, 1362, 1405, 1389, 1385, 1384, 1387, 1394 and 1408 for 12 November.

_{0}≈ 1.011, a value close to that calculated directly, with a maximal daily reproduction number the last day of the contagiousness period. Due to the negativity of R

_{1}, we cannot derive the distribution V and therefore calculate its entropy. As entropy is an indicator of non-uniformity, an alternative could be to calculate it by shifting values of Rj’s upwards by the value of their minimum.

_{j}’s.

#### 3.4.2. Russia

^{−1}for the period from 30 September to 5 October 2020 (exponential phase of the second wave), by choosing 3 days for the duration of the contagiousness period and the following raw data for new infected cases (Figure 5): 7721 for 30 September, then 8056, 8371, 8704, 9081, 9473 for 5 October.

_{1}= 298.905742490698404 – 250.588190354656833 − 47.155939991836672 = 1.161612144205

_{2}= −261.405343820026889−43.80070773806865 + 304.209011826875904 = −0.997039731220

_{3}= −51.322175958486764 + 317.385344255498631 – 265.15495733786368 = 0.90821095914

_{0}≈ 1.073, a value close to that calculated directly, with a maximal daily reproduction number the first day of the contagiousness period. Due to the negativity of R

_{2}, we cannot derive the distribution V and therefore calculate its entropy. The period studied corresponds to a local slow increase of new infected cases at the start of the second wave in Russia, which looks like a staircase succession of slightly inclined 4-day plateaus followed by a step: at the beginning of October, in Russia, new tightened restrictions (but avoiding lockdown) appeared [31], which could explain the change of the value of the slope observed in the new daily cases [30].

#### 3.4.3. Nigeria

^{−1}for the period from 5 November to 10 November (endemic phase), by choosing 3 days for the duration of the contagiousness period and the following raw data for the new infected cases (Figure 6): 141 for 5 November, then 149, 133, 161, 164, and 166 for 10 November.

_{0}≈ 1.129, value close to that calculated directly, with a maximal daily reproduction number the last day of the contagiousness period. The distribution V equals (0.143, 0.342, 0.515) and its entropy H is equal to:

_{k=1,r}V

_{k}Log(V

_{k}) = 0.29 + 0.37 + 0.34 = 1.

#### 3.5. Weekly Patterns in Daily Infected Cases

## 4. Discussion

- -
- In the virus transmitter, the transition between the mechanisms of innate (the first defense barrier) and adaptive (the second barrier) immunity may explain a transient decrease in the emission of the pathogenic agent during the phase of contagiousness [15],
- -
- -
- In the recipient of the virus, individual or public policies of prevention, protection, eviction or vaccination, which evolve according to the epidemic severity and the awareness of individuals and socio-political forces, can change the sensitivity of the susceptible individuals [32].

_{j}. In a future work, we will compare our results with those obtained by deconvolutions on contagiousness durations between 3 and 12 days in order to obtain possibly more realistic values for the R

_{j}’s, and hence, have perhaps a double explanation for the 7 days periodicity, both sociological and biological. Before this future work, we have extended our study using a duration r = 3 of contagiousness to r = 7. The results are given in Appendix B: they show the same existence of identical variations of U-shape type but they specify the values of R

_{j}’s, more often positive and of more realistic magnitude, while keeping a sum approximately equal to R

_{0}.

_{0}, even time-dependent [25]. In particular, they found that this distribution was generally not uniform, which we have confirmed here by showing many cases where we observe the biphasic form of the virulence already observed in respiratory viruses, such as influenza. The entropy of the distribution makes it possible to evaluate the intensity of its corresponding U-shape. This entropy is high if the daily reproduction numbers are uniform, and it is low if the contagiousness is concentrated over one or two days. If some Rj are negative, it is still possible to calculate this uniformity index, by shifting their distribution by a translation equal to the inverse of the negative minimum value.

_{j}= L X

_{j−1},

_{j}is the vector of the new cases living at day j and L is the Leslie matrix given by:

_{j}= 1 − μ

_{j}≤ 1, ∀ i = 1, …, r, is the recovering probability between days j and j + 1.

^{2}distance to the stationary infection age pyramid P = lim

_{j}X

_{j}/Σ

_{i=j,j−r+1}X

_{i}is related to |λ − λ′|, the modulus of the difference between the dominant and sub-dominant eigenvalues of L, namely λ = e

^{R}and λ′, where R is the Malthusian growth rate and P is the left eigenvector of L corresponding to λ. The dynamical stability for the distance (or symmetrized divergence) of Kullback–Leibler to P considered as stationary distribution is related to the population entropy H [26,27,28,29,30,31,32], which is defined if l

_{j}= ∏

_{i=1,j−1}b

_{i}and p

_{j}= l

_{j}R

_{j}/λ

^{j}, as follows:

_{j=1,r}p

_{j}Log(p

_{j})/Σ

_{j=1,r j}p

_{j}

- -
- The hypothesis of spatio-temporal stationarity of the daily reproduction numbers is no longer valid in the case of rapid geo-climatic changes, such as sudden temperature rises, which decrease the virulence of SARS CoV-2 [4], or mutations affecting its transmissibility.
- -
- The still approximate knowledge of the duration r of the period of contagiousness necessitates a more in-depth study at variable durations, by retaining the value of r, which makes all of the daily reproduction numbers positive.
- -
- The choice of uniform random fluctuations of the daily reproduction numbers is based on arguments of simplicity. A more precise study would undoubtedly lead to a unimodal law varying throughout the contagious period, the average of which following a U-shaped curve, of the type observed in the literature on a few real patients [10,54,55,56,57,58].

## 5. Conclusions and Perspectives

_{0}. The systematic use of R

_{0}simplifies the decision-making process by policymakers, advised by public health authorities, but it is too much of a caricature to account for the biology behind the viral spread. We have observed in the COVID-19 outbreak that it was non-constant during an epidemic wave due to exogenous and endogenous factors influencing both the duration of the contagiousness period and the daily transmission rate during this phase [54,55,56]. Then, the first challenge concerns the estimation of the mean duration of the infectious period for infected patients. As for the transmission rate, realistic assumptions made it possible to obtain an upper limit to this duration [45], mainly due to the lack of viral load data in large patient cohorts (see Figure A1 in Appendix A from [57,58,59]), in order to better guide the individual quarantine measures decided by the authorities in charge of public health. This upper bound also makes it possible to obtain a lower bound for the percentage of unreported infected patients, which gives an idea of the quality of the census of cases of infected patients, which is the second challenge facing specialists of contagious diseases. The third challenge is the estimation of the daily reproduction number over the contagiousness period, which was precisely the topic of the present paper. A fourth interesting challenge for this community is the extension of the methods developed in the present paper to the contagious non-infectious diseases (i.e., without causal infectious agent), such as social contagious diseases [59,60,61], the best example being that of the pandemic linked to obesity, for which many concepts and modelling methods remain available.

_{j}’s, that is to say, R

_{0}or the effective R

_{e}. For an individual health use, it is important to know the existence of a minimum of the R

_{j}’s, which generally corresponds to a temporary clinical improvement, after the partial success of the innate immune defenses. This makes it possible to prevent the patient from continuing to respect absolute isolation and therapeutic measures, even if a transient improvement occurs; otherwise, they risk, as in the flu, a bacterial pulmonary superinfection (a frequent cause of death in the case of COVID-19). On the theoretical level, the interest of the proposed method is its generic character: it can be applied to all contagious diseases, within the very general framework of Equation (1), which makes no assumption about the spatial heterogeneity or the longitudinal constancy of the daily reproduction numbers. The deconvolution of Equation (1) poses a new theoretical problem when it is offered in this context, and our future research will propose new avenues of research in this field.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

- 1.
- Beginning of the pandemic in France from 21 February 2020 to 9 March 2020

**21 February**2, 4, 19, 18, 39, 27, 56, 20, 67, 126, 209, 269, 236, 185

**9 March**

^{−1}X = $\left[\begin{array}{c}\begin{array}{c}0.239\\ 0.052\\ -0.783\end{array}\\ -0.295\\ 1.189\\ 3.060\\ 3.122\end{array}\right]$ and we can represent the evolution of X

_{j}’s on Figure A2.

**Figure A2.**Values of the daily reproduction numbers R

_{j}along the period of contagiousness of length 7 days.

- 2.
- Exponential phase in France from 25 October 2020 to 7 November 2020

**7 November**83,334, 58,581, 56,292, 39,880, 35,912, 51,104, 45,258, 33,447, 46,185, 44,705, 34,194, 31,360, 25,123, 48,808 25

**October**

_{j}’s equal to 1.11, close to the effective reproduction number R

_{e}= 1.13 [28].

**Figure A3.**Values of the daily reproduction numbers R

_{j}along the period of contagiousness of length 7 days.

- 3.
- Beginning of the pandemic in the USA from 21 February 2020 to 5 March 2020

**21 February**20, 0, 0, 18, 4, 3, 0, 3, 5, 7, 25, 24, 34, 63

**5 March**

_{j}’s equal to 2.72, less than the effective reproduction number R

_{e}= 3.27 [28].

**Figure A4.**Values of the daily reproduction numbers R

_{j}along the period of contagiousness of length 7 days.

- 4.
- USA exponential phase from 1 November 2020 to 4 November 2020

**N 14**163,961, 183,792, 167,665, 150,535, 159,565, 120,924, 108,248, 135,385, 136,292, 129,663, 113,709, 105,745, 86,030, 75,285

**N 1**

_{j}’s equal to 1.35, close to the effective reproduction number R

_{e}= 1.24 [28].

**Figure A5.**Values of the daily reproduction numbers R

_{j}along the period of contagiousness of length 7 days.

- 5.
- Beginning of the pandemic in the UK from 23 February 2020 to 7 March 2020

**23 February**4, 0, 0, 0, 3, 4, 3, 12, 3, 11, 33, 26, 43, 41

**7 March**

_{j}’s equal to 9.88, higher than the effective reproduction number R

_{e}= 2.95 [28].

**Figure A6.**Values of the daily reproduction numbers R

_{j}along the period of contagiousness of length 7 days.

- 6.
- UK exponential phase from 17 October 2020 to 30 October 2020

**30 October**24,350, 23,014, 24,646, 22,833, 20,843, 19,746, 22,961, 20,484, 21,195, 26,624, 21,282, 18,761, 16,943, 16,133

**17 October**

_{j}’s equal to 1.07, close to the effective reproduction number R

_{e}= 1.06 [28].

**Figure A7.**Values of the daily reproduction numbers R

_{j}along the period of contagiousness of length 7 days.

## Appendix C

_{j}’s twice, for 12.12 ± 6 expected with 0.95 confidence (p < 10

^{−12}), and 189 times, a U-shape evolution for all countries and waves (397), for 99.3 ± 9 expected with 0.95 confidence (p < 10

^{−24}). Hence, the U-shape is the most frequent evolution of daily R

_{j}’s, which confirms the comparison with the behavior of seasonal influenza (see Section 2.2).

**Table A1.**Calculation of the daily R

_{j}’s and shape of their distribution for 194 countries and for the two first waves.

All Countries | First Wave | Second Wave | ||||||
---|---|---|---|---|---|---|---|---|

No | Country Name |
R_{0} |
R_{j’}s
| U-Shape |
R_{0} |
R_{j}’s
| U-Shape | |

1 | AFGHANISTAN | 0.65 | 0.17; 0.09; 0.39 | YES | 0.04 | −1.38; −0.36; 1.78 | INCR | |

2 | ALGERIA | 1.25 | 3.93; −6.21; 3.53 | YES | 0.91 | 1.28; −1.06; 0.69 | YES | |

3 | ARUBA | 5.46 | 10.31; −39.32; 34.47 | YES | 1.10 | 1.54; −1.60; 1.16 | YES | |

4 | ANDORRA | 1.36 | 1.00; 0.79; −0.43 | DECR | 0.12 | 4.34; −1.63; −2.59 | DECR | |

5 | ANGOLA | 0.63 | 0.33; 1.42; −1.12 | INV | 1.70 | 9.22; −1.58; −5.94 | DECR | |

6 | ANTIGUA | 1.92 | 0.00; 1.25; 0.67 | INV | 2.13 | −0.40; 1.33; 1.20 | INV | |

7 | ALBANIA | 0.96 | 0.48; 0.50; −0.02 | INV | 0.66 | 1.98; −0.56; −0.76 | DECR | |

8 | ARGENTINA | 0.73 | 0.57; −1.28; 1.44 | YES | 0.36 | 1.27; 0.75; −1.66 | DECR | |

9 | ARMENIA | 4.43 | 17.99; −36.99; 23.43 | YES | 0.86 | 1.41; −0.97; 0.42 | YES | |

10 | AUSTRALIA | 2.79 | −1.02; 3.47; 0.34 | YES | 1.50 | −0.88; 0.68; 1.70 | INCR | |

11 | AUSTRIA | 1.17 | −1.78; −0.05; 3.00 | INCR | 2.08 | 0.62; −3.55; 5.01 | YES | |

12 | AZERBAIJAN | 1.16 | 1.23; −1.32; 1.25 | YES | 0.37 | 10.36; −6.45; −3.54 | YES | |

13 | BAHAMAS | 0.57 | −0.13; −0.98; 1.68 | YES | 1.22 | 0.22; −0.86; 1.86 | YES | |

14 | BAHRAIN | 1.10 | −0.74; 0.28; 1.56 | INCR | 1.14 | 1.98; −2.69; 1.85 | YES | |

15 | BANGLADESH | 1.04 | 2.37; −2.97; 1.64 | YES | 0.99 | 0.86; −0.69; 0.82 | YES | |

16 | BARBADOS | 1.86 | 0.86; −0.64; 1.64 | YES | 1.14 | 0.22; −0.81; 1.73 | YES | |

17 | BELARUS | 1.57 | −2.37; −4.58; 8.52 | YES | 1.07 | −0.33; 0.24; 1.16 | INCR | |

18 | BELGIUM | 0.43 | 11.66; −15.63; 4.41 | YES | 2.23 | 1.17; −2.39; 3.45 | YES | |

19 | BELIZE | 0.99 | 0.80; 0.42; −0.23 | DECR | 0.51 | 1.77; −0.21; −1.05 | DECR | |

20 | BENIN | 0.85 | 0.81; 0.47; −0.43 | DECR | 0.85 | 1.17; 0.22; −0.54 | DECR | |

21 | BHUTAN | 15.00 | 14.00; 15.00; −14.00 | INV | 1.08 | 0.80; 0.57; −0.29 | DECR | |

22 | BOLIVIA | 2.17 | 8.47; −1.17; −5.13 | DECR | 1.61 | 0.96; −0.30; 0.95 | YES | |

23 | BOSNIA | 0.09 | −1.06; −1.05; 2.20 | INCR | 1.56 | −0.57; −0.51; 2.64 | INCR | |

24 | BOTSWANA | 28.47 | 0.22; 0.00; 28.25 | YES | 28.43 | 0.22; −0.05; 28.26 | YES | |

25 | BRAZIL | 0.77 | 0.31; 1.08; −0.62 | INV | 0.46 | 1.21; 0.16; −0.91 | DECR | |

26 | BRUNEI | 1.08 | 0.10; −0.15; 1.13 | YES | 1.00 | 1.00; −1.00; 1.00 | YES | |

27 | BULGARIA | 5.06 | 14.73; −66.02; 56.35 | YES | 0.75 | 1.34; −0.98; 0.39 | YES | |

28 | BURKINA FASO | 1.08 | 0.72; −0.34; 0.70 | YES | 0.94 | 0.31; 0.24; 0.39 | YES | |

29 | BURUNDI | 1.33 | 1.33; −0.67; 0.67 | YES | 2.18 | 0.53; 1.80; −0.15 | INV | |

30 | CABO VERDE | 0.82 | −0.08; −0.26; 1.16 | YES | 0.19 | 0.56; 1.37; −1.74 | INV | |

31 | CAMBODIA | 0.34 | 0.08; 0.25; 0.01 | INV | 0.27 | 0.06; 0.15; 0.06 | INV | |

32 | CAMEROON | 2.17 | 2.36; 1.25; −1.44 | DECR | 2.48 | 0.50; −0.25; 2.23 | YES | |

33 | CANADA | 1.10 | −0.55; −0.72; 2.37 | YES | 0.44 | 2.36; −0.44; −1.48 | DECR | |

34 | CAR | 1.66 | −0.07; 0.64; 1.09 | INCR | 0.33 | 0.44; −0.22; 0.11 | YES | |

35 | CHAD | 1.19 | 0.77; −1.15; 1.57 | YES | 0.77 | 1.19; 0.25; −0.67 | DECR | |

36 | CHILE | 1.00 | 0.72; 0.17; 0.11 | DECR | 1.64 | 0.37; −4.45; 5.72 | YES | |

37 | CHINA | 1.10 | 0.90; −0.49; 0.69 | YES | 0.87 | 1.16; 0.60; −0.89 | DECR | |

38 | COLUMBIA | 1.00 | 1.75; −0.86; 0.11 | YES | 1.47 | −1.14; 3.08; −0.47 | INV | |

39 | COMOROS | 3.75 | 0.00; −2.75; 6.5 | YES | 1.65 | −0.58; 1.24; 0.99 | INV | |

40 | CONGO DEM | 0.03 | −0.37; −0.39; 0.79 | YES | 0.88 | 0.66; 0.74; −0.52 | INV | |

41 | CONGO REP | 0.92 | 0.92; 0.92; −0.92 | DECR | 0.39 | −0.12; 0.19; 0.32 | INCR | |

42 | COSTA RICA | 0.50 | −2.79; −3.84; 7.13 | YES | 1.26 | 1.21; −0.85; 0.90 | YES | |

43 | COTE D’VOIRE | 1.18 | −0.49; −0.63; 2.30 | YES | 2.09 | 4.32; −7.09; 4.86 | YES | |

44 | CROTIA | 0.75 | 0.53; 0.79; −0.57 | INV | 0.57 | 0.68; −0.64; 0.53 | YES | |

45 | CUBA | 0.48 | −37.25; 16.17; 21.56 | INCR | 0.78 | 0.34; −0.73; 1.17 | YES | |

46 | CURACAO | 0.50 | 3.00; −1.00; −1.50 | DECR | 4.19 | 1.93; −4.01; 6.27 | YES | |

47 | CYPRUS | 0.69 | 0.27; 2.49; −2.07 | INV | 0.45 | −0.42; 1.76; −0.89 | INV | |

48 | CZECH | 0.16 | −0.16; 3.88; −3.56 | INV | 0.88 | 1.88; −1.41; 0.41 | YES | |

49 | DENMARK | 0.80 | −0.11; 0.41; 0.50 | INCR | 0.64 | −0.03; 4.65; −3.98 | INV | |

50 | DJIBOUTI | 0.17 | 1.23; 0.24; −1.30 | DECR | 0.36 | 0.64; 0.41; −0.69 | DECR | |

51 | DOMINICAN | 1.02 | 1.05; −0.31; 0.28 | YES | 1.57 | 0.32; −0.06; 1.31 | YES | |

52 | DOMINICA | 7.75 | 2.00; −4.00; 9.75 | YES | 0.67 | −0.36; 0.72; 0.31 | INV | |

53 | ECUADOR | 1.46 | −0.47; 1.06; 0.87 | INV | 1.14 | 0.73; −0.14; 0.55 | YES | |

54 | EGYPT | 0.84 | 0.30; 0.37; 0.17 | INV | 0.51 | 11.99; −3.76; −7.72 | DECR | |

55 | EL SALVADOR | 1.70 | −0.20; 0.59; 1.31 | INCR | 0.66 | −0.76; −14.49; 15.91 | YES | |

56 | EQUITORIAL G. | 0.38 | 0.85; −0.20; −0.27 | DECR | 1.48 | 0.81; −0.66; 1.33 | YES | |

57 | ERITREA | 1.18 | 1.44; −0.05; −0.21 | DECR | 0.80 | 1.02; 0.20; −0.42 | DECR | |

58 | ESTONIA | 0.87 | 1.96; 0.82; −1.91 | DECR | 3.04 | −0.70; −1.80; 5.54 | YES | |

59 | ESWATINI | 0.94 | 1.41; −1.42; 0.95 | YES | 0.71 | −0.02; 1.52; −0.79 | INV | |

60 | ETHIOPIA | 0.80 | −0.56; −1.45; 2.81 | YES | 1.24 | 0.34; 0.13; 0.77 | YES | |

61 | FIJI | 2.00 | 0.00; 1.00; 1.00 | INCR | 0.50 | 0.75; −0.50; 0.25 | YES | |

62 | FINLAND | 1.14 | 0.91; −0.42; 0.65 | YES | 2.41 | 0.56; −2.38; 4.23 | YES | |

63 | FRANCE | 1.17 | 0.82; 0.10; 0.25 | YES | 2.17 | 0.88; −0.86; 2.15 | YES | |

64 | GABON | 0.97 | 0.20; 0.47; 0.30 | INV | 0.19 | −0.51; 0.00; 0.70 | INCR | |

65 | GAMBIA | 0.83 | −0.25; 0.43; 0.65 | INCR | 0.37 | −0.38; 0.00; 0.75 | INCR | |

66 | GEORGIA | 1.23 | 0.16; 0.43; 0.64 | INCR | 0.79 | 1.52; −0.49; −0.24 | YES | |

67 | GERMANY | 0.73 | 0.15; −1.04; 1.62 | YES | 0.79 | 1.15; −0.56; 0.20 | YES | |

68 | GHANA | 1.48 | 0.55; 0.70; 0.23 | INV | 0.62 | 0.13; −0.81; 1.30 | YES | |

69 | GREECE | 0.71 | 0.33; −0.27; 0.65 | YES | 0.71 | 0.95; 0.28; −0.52 | DECR | |

70 | GRENADA | 14.00 | −5.00; 3.00; 16.00 | INCR | 0.10 | −0.15; 0.00; 0.25 | INCR | |

71 | GUADELOUPE | 1.35 | 0.00; 0.76; 0.59 | INV | 1.35 | 0.00; 0.76; 0.59 | YES | |

72 | GUATEMALA | 0.25 | 2.01; −0.70; −1.06 | YES | 0.27 | 1.19; −0.11; −0.81 | DECR | |

73 | GUIANA FRENCH | 0.88 | 1.30; −0.38; −0.04 | YES | 0.43 | 0.99; 0.27; −0.83 | DECR | |

74 | GUINEA | 0.46 | 0.65; −0.56; 0.37 | YES | 1.68 | 0.21; 0.68; 0.79 | INCR | |

75 | GUINEA BISSAU | 1.14 | 0.06; 1.59; −0.51 | INV | 4.20 | −0.11; 0.04; 4.27 | INCR | |

76 | GUYANA | 2.38 | −3.45; −0.20; 6.03 | INCR | 4.23 | −0.53; 0.58; 4.18 | INCR | |

77 | HAITI | 0.60 | 0.30; −0.13; 0.43 | YES | 0.61 | 0.32; 0.42; −0.13 | INV | |

78 | HONDURAS | 0.57 | −2.94; 3.12; 0.39 | INV | 1.64 | 0.13; 0.54; 0.97 | INCR | |

79 | HONGKONG | 0.04 | 0.95; −0.69; −0.22 | YES | 0.24 | 2.50; −8.79; 6.53 | YES | |

80 | HUNGARY | 0.90 | 0.66; −0.12; 0.36 | YES | 1.93 | 1.91; −2.72; 2.74 | YES | |

81 | ICELAND | 2.28 | −0.85; 3.93; −0.80 | INV | 0.66 | 0.84; 0.22; −0.40 | NO | |

82 | INDIA | 0.98 | 1.82; 0.53; −1.37 | DECR | 0.96 | 1.08; −0.57; 0.45 | YES | |

83 | INDONESIA | 0.95 | 0.67; 0.88; −0.60 | INV | 0.99 | 1.06; −0.03; −0.03 | YES | |

84 | IRAN | 1.04 | 1.73; −0.67; −0.02 | YES | 0.90 | 6.62; −6.62; 0.90 | YES | |

85 | IRAQ | 0.77 | 0.15; −0.35; 0.96 | YES | 0.96 | 0.77; −0.40; 0.59 | YES | |

86 | IRELAND | 2.16 | −2.83; −5.64; 10.63 | YES | 1.12 | 1.12; −0.39; 0.39 | YES | |

87 | ISRAEL | 0.21 | −1.39; 1.08; 0.52 | INV | 1.16 | −0.16; 0.44; 0.88 | INCR | |

88 | ITALY | 1.04 | 2.24; −1.85; 0.65 | YES | 3.69 | 1.65; −7.89; 9.93 | YES | |

89 | JAMAICA | 0.43 | 0.13; 0.06; 0.24 | YES | 2.47 | −0.34; 2.06; 0.75 | INV | |

90 | JAPAN | 1.02 | 0.69; 0.88; −0.55 | INV | 1.16 | 0.61; 0.42; 0.13 | DECR | |

91 | JORDAN | 2.53 | 10.82; −18.20; 9.91 | YES | 0.93 | 1.28; 0.57; −0.92 | DECR | |

92 | KAZAKHSTAN | 0.60 | 0.53; −5.45; 5.52 | YES | 2.06 | −0.05; 2.37; −1.26 | INV | |

93 | KENYA | 1.14 | 0.05; 0.65; 0.44 | INV | 1.18 | 0.47; 1.34; −0.63 | INV | |

94 | KOREA REP. | 1.00 | 0.12; 0.87; 0.01 | INV | 1.04 | 0.60; −0.03; 0.47 | YES | |

95 | KOSOVO | 1.02 | 1.00; 1.02; −1.00 | INV | 0.99 | 1.31; −0.29; −0.03 | YES | |

96 | KUWAIT | 0.88 | 0.5; −0.34; 0.67 | YES | 1.10 | 0.58; −0.84; 1.36 | YES | |

97 | KYRGYZSTAN | 0.17 | −0.73; 0.26; 1.64 | INCR | 1.05 | 0.28; −0.32; 1.09 | YES | |

98 | LAO PDR | 0.50 | 0.50; 0.50; −0.50 | DECR | 0.15 | 0.33; 0.74; −0.92 | INV | |

99 | LATVIA | 0.74 | 1.97; −0.76; −0.47 | YES | 0.50 | 0.40; −0.22; 0.32 | YES | |

100 | LEBANON | 1.03 | 0.57; 0.12; 0.34 | YES | 0.90 | 0.23; 0.06; 0.61 | YES | |

101 | LESOTHO | 7.08 | −2.86; 7.22; 2.72 | INV | 1.42 | 0.37; 1.51; −0.46 | INV | |

102 | LIBERIA | 0.31 | 0.18; −0.04; 0.17 | YES | 4.56 | 0.14; 4.61; −0.19 | INV | |

103 | LIBYA | 0.96 | 0.19; −0.71; 1.48 | YES | 0.79 | −0.42; 0.56; 0.65 | INCR | |

104 | LITHUANIA | 0.83 | 0.56; 0.11; 0.16 | YES | 2.49 | −0.90; −0.52; 3.91 | INCR | |

105 | LUXEMBOURG | 0.24 | −8.55; −3.75; 12.54 | INCR | 1.48 | 1.16; −0.91; 1.23 | YES | |

106 | MACAO | 0.29 | 1.14; 2.29; −3.14 | INV | - | - | - | |

107 | MADAGASCAR | 0.94 | 0.61; −0.16; 0.49 | YES | 0.75 | 0.38; −1.54; 1.91 | YES | |

108 | MALAWI | 1.12 | −0.23; 0.53; 0.82 | INCR | 6.46 | −0.41; 0.99; 5.88 | INCR | |

109 | MALAYSIA | 1.25 | 0.38; 2.79; −1.92 | INV | 1.30 | −0.57; 1.82; 0.05 | INV | |

110 | MALDIVES | 0.83 | 0.60; −0.53; 0.76 | YES | 1.05 | −0.27; 0.70; 0.62 | INV | |

111 | MALI | 0.64 | 0.59; 0.42; −0.37 | DECR | 7.78 | −2.64; −4.96; 15.38 | YES | |

112 | MALTA | 1.06 | 1.15; 0.24; −0.33 | DECR | 0.99 | −0.73; 1.81; −0.09 | INV | |

113 | MAURITANIA | 1.76 | −0.94; 0.29; 2.41 | INCR | 1.14 | 0.73; −0.41; 0.82 | YES | |

114 | MAURITIUS | 4.49 | −4.05; 0.36; 8.18 | INCR | 0.35 | 1.41; 0.53; −1.59 | DECR | |

115 | MAYOTTE | 5.46 | −9.46; −2.50; 17.42 | INCR | 1.05 | 0.72; −0.17; 0.50 | YES | |

116 | MEXICO | 0.86 | −1.39; 3.07; −0.82 | INV | 2.53 | −0.55; 0.10; 2.98 | INCR | |

117 | MOLDOVA | 1.03 | 2.73; −0.67; −1.03 | DECR | 0.36 | 1.27; 0.66; −1.57 | DECR | |

118 | MONACO | 3.15 | 0.52; −1.93; 4.56 | YES | 0.54 | 1.02; −0.12; −0.36 | DECR | |

119 | MONGOLIA | 10.25 | 1.25; 19.25; −10.25 | INV | 0.68 | 0.91; 0.25; −0.48 | DECR | |

120 | MONTENEGRO | 1.37 | 2.94; −3.90; 2.33 | YES | 0.66 | 2.36; 0.26; −1.96 | DECR | |

121 | MOROCCO | 0.90 | 0.36; 1.41; −0.87 | INV | 0.95 | 0.95; −0.15; 0.15 | YES | |

122 | MOZAMBIQUE | 0.72 | 0.92; 0.001; −0.20 | DECR | 0.70 | 2.46; −2.45; 0.69 | YES | |

123 | MYANMAR | 1.12 | −0.75; 1.07; 0.80 | INV | 1.15 | −1.36; −2.17; 4.68 | YES | |

124 | NAMIBIA | 0.68 | 1.37; −1.82; 1.13 | YES | 1.22 | −0.26; 0.95; 0.53 | INV | |

125 | NEPAL | 0.74 | 0.35; 0.76; −0.37 | INV | 0.78 | 0.11; 0.58; 0.09 | INV | |

126 | NETHERLAND | 1.19 | 0.11; 0.11; 0.97 | YES | 1.04 | 1.05; −0.99; 0.98 | YES | |

127 | NEW CALEDONIA | 5.00 | −2.00; 2.00; 5.00 | YES | 1.00 | 1.00; −1.00; 1.00 | YES | |

128 | NEW ZEALAND | 0.74 | 2.30; −3.40; 1.84 | YES | 0.72 | −0.52; 0.43; 0.81 | INCR | |

129 | NICARAGUA | 0.97 | −0.03; 0.97; 0.03 | INV | 1.02 | 0.86; 0.14; 0.02 | DECR | |

130 | NIGER | 0.63 | 0.28; −0.12; 0.47 | YES | 2.21 | −0.14; 0.39; 1.96 | INCR | |

131 | NIGERIA | 1.13 | 0.16; 0.39; 0.58 | INCR | 1.02 | 1.38; −0.65; 0.29 | YES | |

132 | MACEDONIA | 0.74 | 1.83; −1.16; 0.07 | YES | 0.74 | 1.26; −0.10; −0.42 | DECR | |

133 | NORWAY | 0.77 | −0.19; −0.61; 1.57 | YES | 2.13 | 6.02; −10.80; 6.91 | YES | |

134 | OMAN | 3.70 | 0.39; 0.12; 3.19 | YES | 9.80 | −16.87; 39.41; −12.74 | INV | |

135 | PAKISTAN | 1.22 | −0.61; 1.07; 0.76 | INV | 1.19 | 0.55; −0.11; 0.75 | YES | |

136 | PALESTINE | 0.96 | −0.18; −0.23; 1.37 | YES | 1.06 | −0.21; 0.18; 1.09 | INCR | |

137 | PANAMA | 0.96 | 0.16; 0.56; 0.24 | INV | 0.79 | 1.22; −0.16; −0.27 | DECR | |

138 | PAPAU NEW G. | 0.49 | 0.35; −1.96; 2.10 | YES | 0.88 | −0.39; 0.04; 1.23 | INCR | |

139 | PARAGUAY | 0.59 | −1.52; 1.90; 0.21 | INV | 1.20 | −3.20;3.06; 1.34 | INV | |

140 | PERU | 0.89 | 8.30; −2.47; −4.94 | DECR | 0.53 | 3.98; −4.72; 1.27 | YES | |

141 | PHILLIPPINES | 1.15 | 0.89; −0.08; 0.34 | YES | 1.54 | 0.07; 2.84; −1.37 | INV | |

142 | POLAND | 0.92 | 2.32; −1.89; 0.49 | YES | 1.31 | 1.71; −1.63; 1.23 | YES | |

143 | POLYNESIA | 0.66 | 0.22; 0.20; 0.24 | YES | 0.21 | −1.05; 1.09; 0.17 | INV | |

144 | PORTUGAL | 1.56 | −1.34; −8.29; 11.19 | YES | 3.89 | 1.13; −4.00; 6.76 | YES | |

145 | QATAR | 0.80 | −0.84; −1.99; 3.63 | YES | 1.03 | 0.62; 0.61; −0.20 | INV | |

146 | ROMANIA | 0.88 | 0.90; 0.06; −0.08 | DECR | 0.95 | 1.23; −0.48; 0.20 | YES | |

147 | RUSSIA | 1.07 | 1.16; −1.00; 0.91 | YES | 0.87 | 0.83; −5.77; 5.81 | YES | |

148 | RWANDA | 1.80 | 3.20; 2.20; −3.60 | DECR | 0.14 | 3.93; −2.75; −1.04 | YES | |

149 | SAO TOME | 1.44 | 0.44; 0.64; 0.36 | INV | 2.67 | 2.25; −3.45; 3.87 | YES | |

150 | SAN MARINO | 5.10 | 0.28; 1.14;3.68 | INCR | 0.26 | −0.05; 2.32; −2.01 | INV | |

151 | SAUDI ARABIA | 0.90 | −1.70; 2.94; −0.34 | INV | 0.98 | −1.05; 0.54; 1.49 | INCR | |

152 | SENEGAL | 0.72 | −0.19; 1.48; −0.57 | INV | 1.59 | 0.73; 0.23; 0.63 | YES | |

153 | SERBIA | 1.62 | −0.40; 0.47; 1.55 | INCR | 0.82 | 2.02; −0.94; −0.26 | YES | |

154 | SEYCHELLES | 0.48 | 0.30; 0.51; −0.33 | INV | 0.54 | 0.38; −0.19; 0.35 | YES | |

155 | SIERRA LEONE | 2.23 | −2.93; −0.80; 5.96 | INCR | 1.37 | 0.95; −1.25; 1.67 | YES | |

156 | SINGAPORE | 1.33 | 1.15; 0.51; −0.33 | DECR | 2.83 | 1.61; −2.44; 3.66 | YES | |

157 | SLOVAK | 0.99 | −2.67; 1.90; 1.76 | INV | 0.74 | 0.97; −0.73; 0.50 | YES | |

158 | SLOVENIA | 0.75 | 1.56; −0.71; −0.10 | DECR | 0.64 | 1.47; −0.47; −0.36 | YES | |

159 | SOMALIA | 1.18 | −0.16; 1.51; −0.17 | INV | 0.29 | 0.86; 0.57; −1.14 | DECR | |

160 | SOUTH AFRICA | 0.87 | 0.22; 0.73; −0.08 | INV | 1.49 | 0.20; −0.04; 1.33 | YES | |

161 | SOUTH SUDAN | 0.58 | 0.10; 0.16; 0.32 | INCR | 1.72 | 0.63; −0.63; 1.72 | YES | |

162 | SPAIN | 0.38 | −0.18; 0.27; 0.29 | INCR | 0.51 | 1.21; −0.86; 0.16 | YES | |

163 | SRI LANKA | 2.13 | 2.73; −0.75; 0.15 | YES | 0.79 | 0.42; 1.00; −0.63 | INV | |

164 | ST KITTS NEVIS | 2.00 | 0.00; 1.00; 1.00 | INCR | 1.07 | 0.25; 0.18; 0.64 | YES | |

165 | ST LUCIA | 1.13 | −0.53; −0.04; 1.70 | INCR | 1.00 | 1.00; −1.00; 1.00 | YES | |

166 | ST VINCENT | 0.04 | −0.29; 0.24; 0.10 | INV | 0.69 | −0.24; 0.35; 0.58 | INCR | |

167 | SUDAN | 0.36 | −1.46; 2.34; −0.52 | INV | 2.00 | 0.00; 2.00; 0.00 | INV | |

168 | SURINAME | 10.34 | 2.70; 18.77; −11.13 | INV | 1.63 | 2.95; −1.25; −0.07 | YES | |

169 | SWEDEN | 0.56 | 0.58; −1.20; 1.18 | YES | 1.21 | 0.67; −0.91; 1.45 | YES | |

170 | SWITZERLAND | 1.21 | 1.25; 0.13; −0.17 | DECR | 0.28 | 0.89; 1.18; −1.79 | INV | |

171 | SYRIA | 1.43 | 1.39; 4.13; −4.09 | INV | 0.18 | 0.31; −0.68; 0.55 | YES | |

172 | TAIWAN | 1.88 | −0.13; 1.38; 0.63 | INV | 0.66 | −5.21; 13.83; −7.96 | INV | |

173 | TAJIKISTAN | 1.02 | 0.71; −0.60; 0.91 | YES | 1.49 | 1.83; −0.17; −0.17 | YES | |

174 | TANZANIA | 0.91 | −1.50; 0.18; 2.23 | INCR | 1.89 | 3.42; 14.26; −15.79 | INV | |

175 | THAILAND | 0.69 | 0.42; 0.07; 0.20 | YES | 2.71 | −1.77; −0.75; 5.23 | INCR | |

176 | TIMOR LESTE | 5.00 | 1.00; 0.00; 4.00 | YES | 1.33 | 0.00; 1.00; 0.33 | INV | |

177 | TOGO | 0.08 | 6.05; −6.18; 0.21 | YES | 1.14 | 0.18; 0.09; 0.87 | YES | |

178 | TRINIDAD | 0.32 | −0.26; 1.46; −0.88 | INV | 0.55 | 0.26; 0.03; 0.26 | YES | |

179 | TUNISIA | 1.53 | 0.77; −0.04; 0.80 | YES | 2.77 | −3.21; −2.41; 8.39 | INCR | |

180 | TURKEY | 1.15 | −1.50; −1.13; 3.78 | INCR | 2.21 | 19.82; −47.90; 30.29 | YES | |

181 | UAE | 0.97 | 2.07; −1.11; 0.01 | YES | 1.15 | 1.25; −0.64; 0.54 | YES | |

182 | UGANDA | 0.95 | 0.87; −0.37; 0.45 | YES | 0.64 | 0.44; −0.06; 0.26 | YES | |

183 | UKRAINE | 0.96 | 1.35; −1.04; 0.65 | YES | 0.30 | 3.10; 1.07; −1.73 | DECR | |

184 | UK | 0.76 | −0.02; −0.76; 1.54 | YES | 1.03 | 0.43; 0.82; −0.22 | INV | |

185 | USA | 8.42 | 31.42; −99.18; 76.18 | YES | 0.49 | 3.32; −0.38; −2.45 | DECR | |

186 | URUGUAY | 0.63 | 0.71; 0.31; −0.39 | DECR | 1.03 | −0.23; 0.35; 0.91 | INCR | |

187 | UZBEKISTAN | 0.95 | 0.04; 0.10; 0.81 | INCR | 0.90 | −0.03; −0.39; 1.32 | YES | |

188 | VENEZUELA | 1.54 | 1.65; 2.95; −3.06 | INV | 0.82 | 1.09; −2.53; 2.26 | YES | |

189 | VIETNAM | 3.29 | −0.84; −0.39; 4.52 | YES | 1.43 | 0.76; −0.11; 0.78 | YES | |

190 | VIRGIN ISLANDS | 0.51 | 0.01; −0.06; 0.56 | YES | 0.33 | 0.44; −0.22; 0.11 | YES | |

191 | WEST GAZA | 1.00 | −1.00; −2.00; 4.00 | YES | 0.98 | 0.59; −0.11; 0.50 | YES | |

192 | YEMEN | 0.70 | −0.34; 0.17; 0.86 | INCR | 1.50 | 1.00; 0.00; 0.50 | YES | |

193 | ZAMBIA | 0.75 | 0.25; −0.13; 0.63 | YES | 1.12 | 1.11; −0.44; 0.45 | YES | |

194 | ZIMBABWE | 1.44 | 0.24; 0.60; 0.60 | INCR | 1.62 | 1.08; −1.12; 1.66 | YES |

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**Figure 1.**Spread of an epidemic disease from the first infectious “patient zero” (in red), located at the center of its influence sphere comprising the successive generations of infected individuals, for the same value of the reproduction number R

_{0}= 3, with a deterministic dynamic (

**left**) and a stochastic one (

**right**), with standard deviation σ of the uniform distribution on an interval centered on R

_{0}and with a random variable time interval i between infectious generations (after [16]).

**Figure 2.**Spread of an epidemic disease from a first infectious case 0 (located at its influence sphere centre) progressively infecting its neighbours in some regions (rectangles) on successive spheres.

**Figure 3.**

**Top**: estimation of the effective reproduction number R

_{e}’s for 20 October and the 25 October 2020 (in green, with their 95% confidence interval) [28,29].

**Bottom left**: daily new cases in France between 15 February and 27 October [30].

**Bottom right**: U-shape of the evolution of the daily R

_{j}’s along the 3-day contagiousness period of an individual.

**Figure 4.**

**Top**: estimation of the effective reproduction number R

_{e}’s for the 1 November and the 12 November 2020 (in green, with their 95% confidence interval) [28,29].

**Bottom left**: Daily new cases in Chile between 1 November and 12 November [30].

**Bottom right**: U-shape of the evolution of the daily R

_{j}’s along the infectious 6-day period of an individual.

**Figure 5.**

**Top**: estimation of the effective reproduction number R

_{e}’s for 30 September and the 5 October 2020 (in green, with their 95% confidence interval) [28,29].

**Bottom left**: Daily new cases in Russia between 15 February and 21 November [30].

**Bottom right**: U-shape of the evolution of the daily R

_{j}’s along the 3-day contagiousness period.

**Figure 6.**

**Top**: estimation of the effective reproduction number R

_{e}’s for 5 November and 10 November 2020 (in green, with their 95% confidence interval) [28,29].

**Bottom left**: Daily new cases in Nigeria between 15 February and 21 November [30].

**Bottom right**: increasing evolution of the daily R

_{j}’s along the 3-day contagiousness period of an individual.

**Figure 7.**Confirmed world daily new cases (from [30]) as a function of days since 26 February until 23 August 2020 + indicates Sundays, X indicates Mondays.

**Figure 9.**Smoothed confirmed world daily new cases [30], from Figure 7, as a function of days since 26 February 2020 until 23 August 2020 + indicates Sundays, X indicates Mondays. For each specific day j, the mean number of confirmed daily new cases calculated for days j − 1, j − 2, j, j + 1 and j + 2 is subtracted from the number for day j.

**Figure 10.**Smoothed confirmed world daily new cases [30] applied to z-scores from Figure 8, as a function of days since 26 February 2020 until 23 August 2020 + indicates Sundays, X indicates Mondays. Z-transformations are specific to each weekday. For specific day j, the mean number of confirmed new cases calculated for days j − 1, j − 2, j, j + 1, j + 2 is subtracted from the number for day j.

**Table 1.**Simulation results obtained for extreme noises a = 0.1 and a = 1, showing great variations of deconvoluted distribution of daily reproduction numbers X

_{j}’s and a qualitative conservation of their U-shaped distribution along contagiousness period.

a | Initial R_{j}’s | t | X_{t} | X_{t+1} | X_{t+2} | Resulting R’_{j}s | R_{0} | Distribution Shape, Sign R_{0} |
---|---|---|---|---|---|---|---|---|

0.1 | 2.1; 0.95; 2.1 | 4 | 15.35 | 31.74 | 113.5 | 2.1; 0.95; 2.1 | 5.15 | U-shape, positive |

2; 0.95; 1.9 | 6 | 113.5 | 295.8 | 778.7 | 2.03; 7.6; −16.4 | −6.77 | Inverted U-shape, negative | |

2; 1.06; 1.9 | 8 | 778.7 | 2083.7 | 5547 | 2.49; −2.33; 7.39 | 7.55 | U-shape, positive | |

1.9; 1.05; 1.9 | 10 | 5547 | 14,207 | 36,776 | 2.69; −16.7; 43.8 | 29.8 | U-shape, positive | |

1.9; 0.95; 1.9 | 12 | 36,776 | 93,910 | 240,359 | 2.92; 1.68; −6.7 | −2.1 | Decreased shape, negative | |

1.9; 1; 1.9 | 14 | 240,359 | 622,149 | 1,605,227 | 2.3; −4.83; 14.3 | 11.8 | U-shape, positive | |

2; 1.05; 1.9 | 16 | 1,605,227 | 4,331,630 | 11,561,153 | 2.76; 27; −70 | −40.2 | Inverted U-shape, negative | |

1.9; 1; 1.95 | 18 | 11,561,153 | 29,558,395 | 76,502,587 | 2.5; −6.48; 17.9 | 13.9 | U-shape, positive | |

2; 1; 2.1 | 20 | 76,502,587 | 2,076,519 | 556,226,772 | 2.67; −7.6; 19.7 | 14.8 | U-shape, positive | |

1 | 1; 1.355; 1.1 | 4 | 4.81 | 9.1 | 18.21 | 1; 1.355; 1.1 | 3.455 | Inverted U-shape, positive |

1; 1; 1 | 6 | 18.21 | 32.12 | 59.43 | 2.9; 5.49; −14.7 | −6.31 | Inverted U-shape, negative | |

3; 0.5; 2.9 | 8 | 59.43 | 247.16 | 864.34 | 3.7; −33.9; 61.3 | 31.1 | U-shape, positive | |

2.6; 0.7; 2.6 | 10 | 864.34 | 2574.82 | 7942 | 3; −1.79; 7.14 | 8.35 | U-shape, positive | |

2.5; 0.75; 1.5 | 12 | 7942.2 | 23,083.1 | 67,526.6 | 3.35; 2.54; −11.6 | −5.71 | Decreased shape, negative | |

2.4; 0.8; 2.4 | 14 | 67,526.6 | 199,590 | 588,437 | 2.58; −0.5; 4.8 | 6.88 | U-shape, positive | |

2; 1; 2 | 16 | 588,437 | 1,511,517 | 4,010,652 | 2.72; −1.08; 3.19 | 4.83 | U-shape, positive | |

2.3; 1.15; 2.3 | 18 | 4,010,652 | 12,316,150 | 36,415,885 | 2.88; −7.9; 21.7 | 16.7 | U-shape, positive | |

2.8; 0.6; 2 | 20 | 36,415,885 | 117,375,471 | 375,133,150 | 3.7; 4.1; −17 | −9.2 | Inverted U-shape, negative |

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**MDPI and ACS Style**

Demongeot, J.; Oshinubi, K.; Rachdi, M.; Seligmann, H.; Thuderoz, F.; Waku, J.
Estimation of Daily Reproduction Numbers during the COVID-19 Outbreak. *Computation* **2021**, *9*, 109.
https://doi.org/10.3390/computation9100109

**AMA Style**

Demongeot J, Oshinubi K, Rachdi M, Seligmann H, Thuderoz F, Waku J.
Estimation of Daily Reproduction Numbers during the COVID-19 Outbreak. *Computation*. 2021; 9(10):109.
https://doi.org/10.3390/computation9100109

**Chicago/Turabian Style**

Demongeot, Jacques, Kayode Oshinubi, Mustapha Rachdi, Hervé Seligmann, Florence Thuderoz, and Jules Waku.
2021. "Estimation of Daily Reproduction Numbers during the COVID-19 Outbreak" *Computation* 9, no. 10: 109.
https://doi.org/10.3390/computation9100109