computation

Article
Estimation of Daily Reproduction Numbers during the
COVID-19 Outbreak
Jacques Demongeot 1, * , Kayode Oshinubi 1 , Mustapha Rachdi 1 , Hervé Seligmann 1,2 , Florence Thuderoz 1
and Jules Waku 3

1 Laboratory AGEIS EA 7407, Team Tools for e-Gnosis Medical & Labcom CNRS/UGA/OrangeLabs
Telecom4Health, Faculty of Medicine, University Grenoble Alpes (UGA), 38700 La Tronche, France;
Kayode.Oshinubi@univ-grenoble-alpes.fr (K.O.); Mustapha.Rachdi@univ-grenoble-alpes.fr (M.R.);
varanuseremius@gmail.com (H.S.); florence.thuderoz@gmail.com (F.T.)
2 The National Natural History Collections, The Hebrew University of Jerusalem, Jerusalem 91404, Israel
3 UMMISCO UMI IRD 209 & LIRIMA, University of Yaoundé I, P.O. Box 337, Yaoundé 999108, Cameroon;
jules.waku@gmail.com
* Correspondence: Jacques.Demongeot@univ-grenoble-alpes.fr

Abstract: (1) Background: The estimation of daily reproduction numbers throughout the contagious-
ness period is rarely considered, and only their sum R0 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 de-
convolution 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.
Citation: Demongeot, J.; Oshinubi,
K.; Rachdi, M.; Seligmann, H.;
Keywords: daily reproduction number; COVID-19 outbreak; discrete epidemic growth equation;
Thuderoz, F.; Waku, J. Estimation of
discrete deconvolution; COVID-19 in several countries
Daily Reproduction Numbers during
the COVID-19 Outbreak. Computation
2021, 9, 109. https://doi.org/
10.3390/computation9100109
1. Introduction
Academic Editor: Simone Brogi
1.1. Overview and Literature Review
Following the severe acute respiratory syndrome outbreak caused by coronavirus
Received: 22 September 2021 SARS CoV-1 in 2002 [1] and the Middle East Respiratory Syndrome outbreak caused
Accepted: 8 October 2021 by coronavirus MERS-CoV in 2012 [2], the COVID-19 disease caused by coronavirus
Published: 18 October 2021 SARS CoV-2 is the third coronavirus outbreak to occur in the past two decades. Human
coronaviruses, including 229E, OC43, NL63 and HKU1, are a group of viruses that cause a
Publisher’s Note: MDPI stays neutral significant percentage of all common colds in humans [3]. SARS CoV-2 can be transmitted
with regard to jurisdictional claims in from person to person by respiratory droplets and through contact and fomites. Therefore,
published maps and institutional affil- the severity of disease symptoms, such as cough and sputum, and their viral load, are often
iations. the most important factors in the virus’s ability to spread, and these factors can change
rapidly within only a few days during an individual’s period of contagiousness. This
ability to spread is quantified by the basic reproduction number R0 (also called the average
reproductive rate), a classical epidemiologic parameter that describes the transmissibility of
Copyright: © 2021 by the authors. an infectious disease and is equal to the number of susceptible individuals that an infectious
Licensee MDPI, Basel, Switzerland. individual can transmit the disease to during his contagiousness period. For contagious
This article is an open access article diseases, the transmissibility is not a biological constant: it is affected by numerous factors,
distributed under the terms and including endogenous factors, such as the concentration of the virus in aerosols emitted
conditions of the Creative Commons by the patient (variable during his contagiousness period), and exogenous factors, such
Attribution (CC BY) license (https:// as geo-climatic, demographic, socio-behavioral and economic factors governing pathogen
creativecommons.org/licenses/by/
transmission (variable during the outbreak’s history) [4–8].
4.0/).

Computation 2021, 9, 109. https://doi.org/10.3390/computation9100109 https://www.mdpi.com/journal/computation

Computation 2021, 9, 109 2 of 31

Due to these exogenous factors, R0 might change seasonally, but these factor variations
are not significant if a very short period of time is considered. R0 depends also on endoge-
nous factors such as the viral load of the infectious individuals during their contagiousness
period, and the variations in this viral load [9–15] must be considered in both theoretical
and applied studies on the COVID-19 outbreak, in which the authors estimate a unique
reproduction number R0 linked to the Malthusian growth parameter of the exponential
phase of the epidemic, during which R0 is greater than 1 (Figure 1). The corresponding
model has been examined in depth, because it is useful and important for various applica-
tions, but the distribution of the daily reproduction number Rj at day j of an individual’s
contagiousness period is rarely considered within a stochastic framework [16–20].

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 R0 = 3,
with a deterministic dynamic (left) and a stochastic one (right), with standard deviation σ of the uniform distribution on an
interval centered on R0 and with a random variable time interval i between infectious generations (after [16]).

We therefore defined a partial reproduction number for each day of an individual’s

contagiousness period, and, assuming initially that this number was the same for all
individuals, we obtained the evolution equation for the number of new daily cases in a
population. Assuming that the distribution of partial reproduction numbers (referred to as
daily for simplicity) was subject to fluctuations, we calculated the consequences for their
estimation, and we estimated them for a large number of countries, taking a duration of
contagiousness of 3 followed by 7 days.
When this distribution is considered, it is possible to calculate its entropy as a parame-
ter quantifying its uniformity and to simulate the dynamics of the infectious disease either
using a Markovian model such as that defined in Delbrück’s approach [17] or a classical
discrete or ODE SIR deterministic model. In the Markovian case, R0 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, R0 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 R0
In epidemiology, there are essentially two broad ways to calculate R0 , which cor-
respond 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
Computation 2021, 9, 109 3 of 31

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 R0 , 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

The methodology chosen starts from an attempt to reconstruct an epidemic dynamic
from the knowledge of the number Rikj 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 Xj−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:

Xj = Σk=1,r Σi=1 Xj−k Rikj (1)

We will assume in the following that Rikj is the same, equal to Rk , for all individuals I
and day j, then depends only on day k. Then, we have:

Xj = Σk=1,r Rk Xj−k (2)

The convolution Equation (2) is the basis of our modelling of the epidemic dynamics.

2.1. The Contagion Mechanism from a First Infectious Case Zero

Let us suppose that the secondary infected individuals are recruited from the centre
of the sphere of influence of an infectious case zero and that the next infected individuals
remain on a sphere centred on case 0, by just widening its radius on day 2. Therefore, the
susceptible individuals C(j), which each infectious on day j − 1 can recruit, are on a part of
the sphere of influence of case 0 reached at day j (rectangles on Figure 2).

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.

2.2. The Biphasic Pattern of the Virulence Curve of Coronaviruses

Mostly, the clinical course of patients with seasonal influenza shows a biphasic oc-
currence of symptoms with two distinct peaks. Patients have a classic influenza disease
followed by an improvement period and a recurrence of the symptoms [11]. The influenza
RNA virus shedding (the time during which a person might be contagious to another
person) increases sharply one half to one day after infection, peaks on day 2 and persists
for an average total duration of 4.5 days, between 3 and 6 days, which explains why we
will choose in the following contagiousness duration these extreme values, i.e., either 3
or 6 days, depending on the positivity of the estimated daily reproduction numbers. It is
common to consider this biphasic evolution of influenza clinically: after incubation of one
day, there is a high fever (39–40 ◦ C), then a drop in temperature before rising, hence the
Computation 2021, 9, 109 4 of 31

term “V” fever. The other symptoms, such as coughing, often also have this improvement
on the second day of the flu attack: after a first feverish rise (39–39.5 ◦ C), the temperature
drops to 38 ◦ C on the second day, then rises before disappearing on the 5th day, the fever
being accompanied by respiratory signs (coughing, sneezing, clear rhinorrhea, etc.). By
looking at the shape of virulence curves observed in coronavirus patients [12–16], we often
see this biphasic pattern.

2.3. Relationships between Markovian and ODE SIR Approaches

In the following, we suppose that the susceptible population size remains constant,
which constitutes a hypothesis valid during the exponential phase of epidemic waves.
The Markovian stochastic and ODE deterministic approaches are linked by a common
background consisting of the birth and death process approach used in the kinetics of
molecular reactions by Delbrück [17], then in dynamical systems theory by numerous
authors [18–23], namely in modelling of the epidemic spread in exponential growth. In the
ODE approach, the Malthusian parameter is the dominant eigenvalue, and the equivalent
in the Markovian approach is the Kolmogorov–Sinai entropy (called evolutionary entropy
in [24–26]).

2.3.1. First Method for Obtaining the SIR Equation from a Deterministic
Discrete Mechanism
Let us suppose the model is deterministic and denote by Xj the number of new
infected cases at day j (j ≥ 1), and Rk (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:
Xj = ∑k=1,r Rk Xj−k , with Xj-k=0, if k>j
which says that the Xj−k new infected at day j − k give Rk Xj−k new infected on day
j, throughout a period of contagiousness of r days, the Rk ’s being possibly different
or zero. For example, if r = 3, for the number X5 of new cases at day 5, equation
X5 = R1 X4 + R2 X3 + R3 X2 means that new cases at day 4 have contributed to new cases at
day 5 with the term R1 X4 , R1 being the reproduction number at first day of contagiousness
of new infected individuals at day 4.
In matrix form, we obtain:
X = MR, (3)
where X = (Xj , . . . , Xj−r−1 ) and R = (R1 , . . . , Rr ) are r-dimensional vectors and M is the
following r-r matrix:
 
Xj−1, Xj−2, . . . , Xj − r
M =  Xj−k−1, Xj−k−2, . . . , Xj − k − r  (4)
Xj−r Xj−r−1, . . . , Xj−2r+1

It is easy to show that, if X0 = 1 and r = 5 (estimated length of the contagiousness

period for COVID-19 [12–21]), we obtain:

X5 = R1 5 + 4R1 3 R2 + 3R1 2 R3 + 3R1 R2 2 + 2R2 R3 + 2R1 R4 + R5 (5)

The length r of the contagiousness period can be estimated from the ARIMA series of
the stationary random variables Yj ’s, equal to the Xj ’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 R1 = a, R2 = b and R3 = c,
we obtain:
X0 = 1, X1 = a, X2 = a2 + b + , Xc,3X=3 a=
3 +a2ab
3 ++ c, X4 = a4 + 3a2 b + b2 + 2ac,
2ab,
X5 = a + 4a b + 3ab + 3a c + 2bc, X6 = a + 5a b + 4a3 c + 6a2 b2 + 6abc + b3 + c2 ,
5 3 2 2 6 4 (6)
7 5 4 3 2 2 3 2 2
X7 = a + 6a b + 5a c + 10a b + 12a bc + 4ab + 3b c + 3ac
Computation 2021, 9, 109 5 of 31

If R1 and R2 are equal, respectively, to a and b, and if a = b = R/2, c = 0, then, X5

behaves like:
X5 = R5 /32 + R4 /4 + 3R3 /8 (7)
If R = 2, {Xj }i=1,∞ is the Fibonacci sequence, and more generally, for R > 0, the gen-
eralized Fibonacci sequence. Let us suppose now that b = c = 0 and a depends on the
day j: aj = > 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 Ij , we have:

Xj = ∆Ij /∆j = (Ij+1 − Ij )/(j + 1 − j) = νC(j)Ij (8)

Let us suppose, as in Section 2.1, that the first infectious individual 0 recruits from the
centre of its sphere of influence secondary infected individuals remaining in this sphere,
and that the susceptible individuals recruited by the Ij 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:

Xj = ∆Ij /∆j = νC(j)Ij − µIj = νIj S(j)/(c + S(j)) − µIj (9)

This discrete version of epidemic modeling is used much less than the classic continu-
ous version, corresponding to the ODE SIR model, with which we will show a natural link.
Indeed, the discrete Equation (9) is close to SIR Equation (10), if the value of c is greater
than that of S:
dI/dt = νIS/(c + S) − µI (10)

2.3.2. Second Method for Obtaining the SIR Equation from a Stochastic
Discrete Mechanism
Another way to derive the SIR equation is the probabilistic approach, which comes
from the microscopic equation of molecular shocks by Delbrück [17] and corresponds to a
classical birth-and-death process: if at least one event (with rates of contact ν, birth f, death
µ or recovering ρ) occurs in the interval (t, t + dt), and by supposing that births compensate
deaths, leaving constant the total size N of the population, we have:

Probability ({S(t + dt) = k, I(t + dt) = N − k}) = P(S(t) = k, I(t) = N − k) [1 − [µk + νk(N − k)−fk − ρ(N − k)]dt]
+ P(S(t) = k − 1, I(t) = N − k + 1) [f(k − 1) + ρ(N − k + 1)]dt (11)
− P(S(t) = k+1, I(t) = N − k − 1) [µ(k + 1) + ν(k + 1) (N − k − 1)]dt

Hence, we have, if Pk (t) denotes Probability({S(t) = k, I(t) = N − k}):

dPk (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)],

and we obtain:

dPk (t)/dt = −[µk + νk(N − k)−fk − ρ(N − k)]Pk (t) + [f(k − 1) + ρ(N − k + 1)]Pk−1 (t) − [µ(k + 1) + ν(k + 1)(N − k1)]Pk+1 (t)
Computation 2021, 9, 109 6 of 31

Then, by multiplying by sk and summing over k, we obtain the characteristic function

of the random variable S. If births do not compensate deaths, we have:

Probability ({S(t + dt) = k, I(t + dt) = j}) = P(S(t) = k, I(t) = j) (1 − [µk + νkj − fk − ρj]dt)
+ P(S(t) = k − 1, I(t) = j + 1) [f(k − 1) + ρ(j + 1)]dt (12)
− P(S(t) = k + 1, I(t) = j − 1) [µ(k + 1) + ν(k + 1)(j − 1)]dt

If S and I are supposed to be independent and if the coefficients ν, f, µ and ρ are

sufficiently small, S and I are Poisson random variables [27], whose expectations E(S) and
E(I) verify:
dE(S)/dt = fE(S) − νE(SI) − µE(S) + ρE(I)
(13)
or, if f = µ, dE(S)/dt ≈ E(I) [−νE(S) + ρ],
leading to the SIR equation for the variables S, I and R considered as deterministic:

dS/dt = −νSI + ρR, dI/dt = νSI − kI − µI, dR/dt = kI − ρR (14)

3. Results
3.1. Distribution of the Daily Reproduction Numbers Rj ’s along the Contagiousness Period of an
Individual. A Theoretical Example Where They Are Supposed to Be Constant during the Epidemics
If R0 denotes the basic reproduction number (or average transmission rate) in a given-
population, we can estimate the distribution V (whose coefficients are denoted Vj = Rj /Ro )
of the daily reproduction numbers Rj along the contagious period of an individual, by
remarking that the number Xj of new infectious cases at day j is equal to Xj = Ij − Ij−1 , where
Ij is the cumulated number of infectious at day j, and verifies the convolution equation
(equivalent to Equation (2)):
Z r
Xj = ∑ Rk Xj−k , giving in continuous time : X(t) =
1
R(s)X(t − s)ds, (15)
k =1,r

where r is the duration of the contagion period, estimated by 1/(ρ + µ), ρ being the
recovering rate and µ the death rate in SIR Equation (14). r and S can be considered as
constant during the exponential phases of the pandemic, and we can assume that the
distribution V is also constant; then, V can be estimated by solving the linear system
(equivalent to Equation (3)):
R = M−1 X (16)
where M is given by Equation (4). Equation (16) can be solved numerically, if the pandemic
is observed during a time greater than 1/(ρ + µ). We will first demonstrate an example of
how the matrix M can be repeatedly calculated for consecutive periods of length equal to
that of the contagiousness period (supposed to be constant during the outbreak), giving
matrix series M1 , M2 , . . . Following Equation (4), we put the values of Xi ’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.
   
X4 X3 X2 X6 X5 X4
M1 = X3
 X2 X1 , M2 = X5
  X4 X3 , . . . ,
X2 X1 Xo X4 X3 X2

where, after Equation (6), M1 and M2 can be calculated from the Rj ’s as:

R41 + 3R21 R2 + 2R1 R3 + R22 R31 + 2R1 R2 + R3 R21 + R2

 

M1 =  R31 + 2R1 R2 + R3 R21 + R2 R1 ,

R21 + R2 R1 1
Computation 2021, 9, 109 7 of 31

and M2 is given by:

R61 + 5R41 R2 + 4R31 R3 + 6R1 R2 R3 + 6R21 R22 + R32 + R23 R51 + 4R31 R2 + 3R21 R2 + 2R2 R3 + 3R3 R21 R41 + 3R21 R2 + 2R1 R3 + R22
 
 R51 + 4R31 R2 + 3R21 R2 + 2R2 R3 + 3R3 R21 R41 + 3R21 R2 + 2R1 R3 + R22 3
R1 + 2R1 R2 + R3 
R41 + 3R21 R2 + 2R1 R3 + R22 R31 + 2R1 R2 + R3 R21 + R2

Additionally, from Equation (2), if, for instance, j = 8 and r = 3, then we have the
expression below, which means that the new cases on the 8th day depend on the new cases
detected on the previous days 7, 6 and 5, supposed to be in a period of contagiousness of
3 days:
X8 = ∑ Rk X8−k = R1 X7 + R2 X6 + R3 X5 (17)
k =1,3

Let us suppose now that the initial Rj ’s on a contagiousness period of 3 days, are equal
to:
   
R1 2
 R2  =  1 , then matrix M defined by Mij = X7−(i+j) gives the Rj ’s from Equation (16),
R3 2
hence allows the calculation of Xj = Σk=1,3 Rk Xj−k .
The inverse of M is denoted by M−1 and verifies: R = M−1 X, where X = (X6 , X5 , X4 ),
with X1 = 1, X2 = 2, X3 = 5, X4 = 14, X5 = 37, X6 = 98 and we obtain:
  −1  
37 14 5 −1/4 1 −3/4
M1−1 =  14 5 2  = 1 −3 1 ,
5 2 1 −3/4 1 11/4

and a deconvolution gives the resulting Rj ’s:

      
−1/4 1 −3/4 98 2 R1
 1 −3 1  37  =  1  =  R2 , thanks to the following calculation:
−3/4 1 11/4 14 2 R3

R1 = −49/2 + 37 − 21/2 = 2

R2 = 98 − 111 + 14 = 1
R3 = −147/2 + 37 + 77 = 2
We obtain for the resulting distribution of daily reproduction numbers the exact replica
of the initial distribution. We obtain the same result by replacing M1 by the matrix M2 .

3.2. Distribution of the Daily Reproduction Numbers Rj ’s When They Are Supposed to Be Random
Let us consider a stochastic version of the deterministic toy model corresponding to
Equation (17), by introducing an increasing noise on the Rj ’s, e.g., by randomly choos-
ing 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
Rj ’s, with conservation of the average of their sum R0 , if the random choice of the values
of the Rj ’s at each generation is repeated, following the stochastic version of Equation (2):
Xj = Σk=1,r (Rk + εk ) Xj−k , where r is the contagiousness period duration and εk is a noise
perturbing Rk , 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 Rk ’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.
For each random choice of the values of the daily reproduction numbers Rj ’s, we can
calculate a matrix M1 corresponding to Equation (3). Its inversion into the matrix M1 −1
makes it possible to solve the problem of deconvolution of Equation (2)—that is to say, to
Computation 2021, 9, 109 8 of 31

obtain new Rj ’s as a function of the observed Xk ’s. We can then calculate a new matrix
M2 from these new Rj’s and thus continue during an epidemic the estimation of the daily
reproduction numbers Rj ’s from the successive matrices M1 , M2 , . . . , and observed Xk ’s.
1. For a = 0.1, let us randomly and uniformly choose the initial distribution of the daily
reproduction numbers R1 in the interval [1.9, 2.1], R2 in [0.95, 1.05] and R3 in [1.9, 2.1]
as R1 = 2.1, R2 = 0.95, R3 = 2.1. Then, the transition matrix M1 is equal to:
 
41.7391 15.351 5.36
M1 =  15.351 5.36 2.1  and we have:
5.36 2.1 1
 
−0.2154195 0.92857143 −0.7953515
M1−1 =  0.92857143 −2.95 1.2178571 
−0.7953515 1.2178571 2.705584

From X6 = 113.491, X5 = 41.7391, X4 = 15.351, resulting Rj ’s are: R1 = 2.1, R2 = 0.95,

R3 = 2.1.
The next initial Rj ’s are chosen as: R1 = 2, R2 = 0.95, R3 = 1.9 and we have:

X7 = 2X6 + 0.95X5 + 1.9X4 = 226.982 + 39.652 + 29.17 = 295.8

X8 = 2X7 + 0.95X6 + 1.9X5 = 591.6 + 107.816 + 79.304 = 778.72

Then, we obtain the matrices M2 and M2 −1 :
 
295.8 113.491 41.7391
M2 =  113.491 41.7391 15.351 
41.7391 15.351 5.36
 
−0.07779371 0.20964295 0.00524305
M2−1 =  0.20964295 −1.0123552 1.26721348 
0.00524305 1.26721348 −3.48354228
Then, the resulting Rj ’s equal: R1 = 2.0279, R2 = 7.6158, R3 = −16.426.
The next initial Rj ’s are: R1 = 2, R2 = 1.05, R3 = 1.9 and we have:

X9 = 2X8 + 1.05X7 + 1.9X6 = 1557.44 + 310.59 + 215.63 = 2083.66

X10 = 2X9 + 1.05X8 + 1.9X7 = 4167.32 + 817.656 + 562.02 = 5546.996

From these values of X9 and X10 , we obtain the matrices M3 and M3 −1 :
 
2083.66 778.72 295.8
M3 =  778.72 295.8 113.491 
295.8 113.491 41.7391
 
0.02596375 −0.05192766 −0.04280771
M3−1 =  −0.05192766 0.0256605 0.29823273 
−0.04280771 0.29823273 −0.48358035
Then, the resulting Rj ’s equal: R1 = 2.486, R2 = −2.33, R3 = 7.38769.
2. For a = 1, let us choose the initial R1 in [1, 3], R2 in [0.5, 1.5] and R3 in [1, 3], e.g., R1 = 1,
R2 = 1.355 and R3 = 1.1. Then, the transition matrix M1 is equal to:
Computation 2021, 9, 109 9 of 31

 
9.101 4.81 2.355
M1 =  4.81 2.355 1  and its inverse is given by:
2.355 1 1
 
−1.11983471 2.02892562 0.60828512
M1−1 =  2.02892562 −2.93801653 −1.84010331 
0.60828512 −1.84010331 1.40759184

New cases are: X6 = 18.209, X5 = 9.101, X4 = 4.81, X3 = 2.355, X2 = 1, X1 = 1, and by

deconvoluting, we obtain the resulting Rj ’s equal to: R1 = 1, R2 = 1.355, R3 = 1.1, i.e., the
exact initial distribution.
Let us now consider new initial Rj ’s: R1 = 1, R2 = 1, R3 = 1. That gives a new matrix
M2 , with new X7 and X8 calculated from the new initial Rj ’s, by using the former values of
X6 , . . . , X2 :
X7 = X6 + X5 + X4 = 18.209 + 9.101 + 4.81 = 32.12
X8 = X7 + X6 + X5 = 32.12 + 18.209 + 9.101 = 59.43
Hence, we obtain:
 
32.12 18.209 9.101
M2 = 18.209 9.101 4.81
  and
9.101 4.81 2.36
 
−0.35061537 0.1839519 0.97925345
M2−1 =  0.1839519 −1.47916605 2.31025157 
0.97925345 2.31025157 −8.0783421

and the resulting Rj ’s equal: R1 = 2.90, R2 = 5.4888, R3 = −14.696.

We calculate X9 and X10 using new initial Rj ’s: R1 = 3.0, R2 = 0.5, R3 = 2.9:

X9 = 3X8 + 0.5X7 + 2.9X6 = 178.29 + 16.06 + 52.81 = 247.16

X10 = 3X9 + 0.5X8 + 2.9X7 = 741.48 + 29.715 + 93.148 = 864.343

Hence, we obtain:
 
247.16 59.43 32.12
M3 =  59.43 32.12 18.209  and
32.12 18.209 9.101
 
0.00718287 −0.00805357 −0.00923703
M3−1 =  −0.00805357 −0.22288084 0.47435642 
−0.00923703 0.47435642 −0.80659958

and the resulting Rj ’s equal: R1 = 3.66898, R2 = −33.857, R3 = 61.32.

More precise simulation results are given in Table 1, which summarizes computations
made for random choices of Rj ’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.
Computation 2021, 9, 109 10 of 31

Table 1. Simulation results obtained for extreme noises a = 0.1 and a = 1, showing great variations of deconvoluted distribution of
daily reproduction numbers Xj ’s and a qualitative conservation of their U-shaped distribution along contagiousness period.

a Initial Rj ’s t Xt Xt+1 Xt+2 Resulting R’j s R0 Distribution Shape, Sign R0

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

3.3. Distribution of the Daily Reproduction Numbers Rj ’s. The Real Example of France
Figure 3 gives the effective transmission rates Re 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.
We have:
  −1  
45, 422 42, 032 41, 622 −0.0000163989812 −0.0000292188776 0.00007142863
M−1 =  42, 032 41, 622 26, 676  =  −0.0000292188776 0.0000938161392 −0.0000628537817 
41, 622 26, 676 20, 468 0.00007142863 −0.0000628537817 −0.00001447698

Hence, we can deduce the daily Rj ’s, i.e., the vector (R1 , R2 , R3 ):
  
−0.0000163989812 −0.0000292188776 0.00007142863 52, 010
 −0.0000292188776 0.0000938161392 −0.0000628537817  45, 422  =
0.00007142863

−0.0000628537817 −0.00001447698 42, 032

−0.852911911949567 −1.32717986039119 3.00228812555347
 −1.51967382631645 4.26131667592337 −2.64187015405365 
3.71500298367996

−2.85494447414886
 
−0.60849658654673

0.82219725466 R1
=  0.0997726955533  =  R2 
0.2515619229844 R3

The effective reproduction number is equal to R0 ≈ 1.174, a value close to that calcu-
lated 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:

H = −Σk=1,r Vk Log(Vk ) = 0.25 + 0.21 + 0.33 = 0.79.

Computation 2021, 9, 109 11 of 31

Re

1.

France
Daily Rj’s

0
1
2
3 Day j

Figure 3. Top: estimation of the effective reproduction number Re ’s for 20 October and the 25 October 2020 (in green, with
their 95%Figure 3
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 Rj ’s along the 3-day contagiousness period of an individual.

3.4. Calculation of the Rj’s for Different Countries

3.4.1. Chile
By using the daily new infected cases given in [30], we can calculate M−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.
1.2 We have:
1  −1
1394 1387 1384 1385 1389 1405

 1387 1384 1385 1389 1405 1362 
 
−1
 1384 1385 1389 1405 1362 1359 
M =   =
 1385 1389 1405 1362 1359 1382  
 1389 1405 1362 1359 1382 1370 
1405 1362 1359 1382 1370 1400
−0.05714222 0.01016059 Chile
−0.00901664 0.01474588 0.00640175 0.03539322
 

 0.01016059 −0.01827291 0.0106261 −0.00763363 0.02139586 −0.01613675  

 −0.00901664 0.0106261 −0.00544051 0.02150289 −0.01468484 −0.00286391  

 0.01474588 −0.00763363 0.02150289 −0.01796266 −0.00553414 −0.00509801  
 0.00640175 0.02139586 −0.01468484 −0.00553414 −0.00305831 −0.00452917 
0.03539322 −0.01613675 −0.00286391 −0.00509801 −0.00452917 −0.00686198

Day j

Figure 4
 2
3 Day j

Figure 3
Computation 2021, 9, 109 12 of 31

1.2
1

Chile

Day j

Figure 4. Top: estimation of the effective reproduction number Re ’s for the 1 November and the 12 November 2020 (in
Figure 4
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 Rj ’s along the infectious 6-day period of an individual.

Hence, after deconvolution, we obtain:

−0.36256122
 

 0.22645436 

 0.01488726 
R= 

 0.33918287 

 0.28557502 
0.50696243

The effective reproduction number is equal to R0 ≈ 1.011, a value close to that calcu-
lated directly, with a maximal daily reproduction number the last day of the contagiousness
period. Due to the negativity of R1 , 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.
The quasi-endemic situation in Chile since the end of August, which corresponds to
the increase of temperature and drought at this period of the year [4], gives a cyclicity of
the new cases occurrence whose period equals the length of the contagiousness period of
about 6 days, analogue to the cyclic phenomenon observed in simulated stochastic data of
Section 3.2. with a similar U-shaped distribution of the Rj ’s.

3.4.2. Russia
By using the daily new infected cases given in [30], we can calculate M−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.
Computation 2021, 9, 109 13 of 31

Re

1.2

Russia
Daily Rj’s

Day j

FigureFigure
5. Top:5 estimation of the effective reproduction number Re ’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 Rj ’s along the 3-day contagiousness period.

We have:   −1
9081 8704 8371
M−1 =  8704 8371 8056  and
8371 8056 7721
    
0.031553440566948 −0.027594779248393 −0.005417732076268 9473 R1
 −0.027594779248393 −0.00482333528665 0.034950483895551  9081  =  R2 ,
−0.005417732076268 0.034950483895551 −0.030463575061795 8704 R3
where:
R1 = 298.905742490698404 - 250.588190354656833 − 47.155939991836672 = 1.161612144205
R2 = −261.405343820026889−43.80070773806865 + 304.209011826875904 = −0.997039731220
R3 = −51.322175958486764 + 317.385344255498631 - 265.15495733786368 = 0.90821095914
The effective reproduction number is equal to R0 ≈ 1.073, a value close to that calcu-
lated directly, with a maximal daily reproduction number the first day of the contagiousness
period. Due to the negativity of R2 , we cannot derive the distribution V and therefore cal-
culate 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
By using the daily new infected cases given in [30], we can calculate M−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.
Computation 2021, 9, 109 14 of 31

Re

1.2

Nigeria

Day j

Figure
Figure 6 estimation of the effective reproduction number Re ’s for 5 November and 10 November 2020 (in green, with
6. Top:
their 95% confidence interval) [28,29]. Bottom left: Daily new cases in Nigeria between 15 February and 21 November [30].
(a)
Bottom right: increasing evolution of the daily Rj ’s along the 3-day contagiousness period of an individual.
patients
We have:
  −1  
164 161 131 0.01796807 0.01502897 −0.03283028
M−1 = 161
 131 149  =  0.01502897 −0.02832263 0.01575332 
131 149 141 −0.03283028 0.01575332 0.02141264

After deconvolution, we obtain:

Each color corresponds
to one among six patients (c)  (d) 
0.16177513
(b) R =  0.38618314 
0.58115333

The effective reproduction number is equal to R0 ≈ 1.129, value close to that calculated
directly, with a maximal daily reproduction number the last day of the contagiousness
No treatment Antiviral therapy
period. The distribution V equals (0.143, 0.342, 0.515) and its entropy H is equal to:

Figure A1 H = −Σk=1,r Vk Log(Vk ) = 0.29 + 0.37 + 0.34 = 1.

In Appendix C, Table A1 gives the shape of the Rj ’s distribution for 194 countries.

3.5. Weekly Patterns in Daily Infected Cases

Daily new infected cases are highly affected by weekdays, such that case numbers
are lowest at the start of the week and increase afterwards. This pattern is observed at the
world level, as well as at the level of almost every single country or USA state. Hence, in
order to estimate biologically meaningful reproduction numbers, clean of weekly patterns
due to administrative constraints, analyses have to be restricted to specific periods shorter
than a week, or at rare occasions when patterns escape the administrative constraints.
This weekly phenomenon occurs during exponential increase as well as decrease phases
of the pandemic and during endemic periods in numbers of daily cases (Figure 6). In
Computation 2021, 9, 109 15 of 31

addition, the daily new infected case record is discontinuous for many countries/regions,
which frequently publish, on Monday or Tuesday, a cumulative count for that day and the
weekend days. For example, Sweden typically publishes only four numbers over one week,
the one on Tuesday cumulating cases for Saturday, Sunday and the two first weekdays.
Discontinuity in records further limits the availability of data enabling detailed analyses
of daily reproduction numbers and can be considered as extreme weekday effects on new
case records due to various administrative constraints.
We calculated Pearson correlation coefficients r between a running window of daily
new case numbers of 20 consecutive days and a running window of identical duration
with different intervals between the two running windows. These Pearson correlation
coefficients r typically peak with a lag of seven days between the two running windows.
The mean of these correlations are for each day of the week from Tuesday (data making
up for the weekend underestimation) to Monday: 0.571, 0.514 (0.081), 0.383 (0.00008),
0.347 (0.000003), 0.381 (0.000006), 0.468 (0.000444) and 0.558 (0.03916), with, in parentheses,
the p-value of the one-tailed paired t-test showing that the correlation observed with
running windows starting Tuesday are more than the others (see also supplementary
material). This could reflect a biological phenomenon of seven infection days. However,
examination of the frequency distributions of lags for r maxima reveals, besides the median
lag at 7 days, local maxima for multiples of 7 (14, 21, 28, 35, etc.). About 50 percent of all
local maxima in r involve lags that are multiples of seven (seven included).
This excludes a biological causation, except if data periodicity comes from an entrain-
ment by the weekly “Zeitgeber” of census, near the duration of the contagiousness interval.
We tried to control for weekdays using two methods, and combinations thereof. For the
first method, we calculated z-scores for each weekday, considering the mean number of
cases for each weekday, and subtracted that mean from the observed number for a day
(Figure 7). This delta was then divided by the standard deviation of the number of cases
for that weekday. The mean and standard variation are calculated across the whole period
of study for each weekday.
The second method implies data smoothing using a running window of 5 consecutive
days, where the mean number of new cases calculated across the five days is subtracted
from the number of new cases observed for the third day. Hence, data for a given day are
compared to a mean including two previous, and two later days (Figure 8).
We constructed two further datasets, where z-scores are applied in the first to data after
smoothing from the second method and are applied in the second data after smoothing
from the first method (not shown) (Figures 9 and 10).
These four datasets from daily new cases database [30] transformed according to
different methods and combinations thereof designed to control for weekday were analysed
using the running window method. Despite attempts at controlling for weekday effects,
the median lag was always seven days across all four transformed datasets, and local
maxima in lag distributions were multiples of seven. After data transformations, about
50 percent of all local maxima were lags that are multiples of seven, seven included.
Computation 2021, 9, 109 16 of 31

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 8. Z-transformed scores of confirmed world daily new cases [30], from Figure 6, 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.

Visual inspection of plots of these transformed data versus time for daily new infected
cases from the whole world shows systematic local biases in daily new infected cases
(after transformation) on Sundays and Mondays, for all four transformed datasets, with
Sundays and/or Mondays as local minima and/or local maxima, according to which
method or combination thereof was applied to the data. Hence, the methods we used failed
to neutralize the weekly patterns in daily new cases due to administrative constraints. This
issue highly limits the data available for detailed analyses of daily new cases aimed at
estimating biologically relevant estimates of reproduction numbers at the level of short
temporal scales.
Computation 2021, 9, 109 17 of 31

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.
Computation 2021, 9, 109 18 of 31

By smoothing on five consecutive days of raw data (confirmed world daily new
infected cases [24]) and applying the z-transformation, we obtain a better result in Figure 11
than in Figure 10 in order to neutralize the weekly pattern. The reason is that the smoothing
largely eliminates the counting defect during weekends due either to fewer hospital
admissions and/or less systematic PCR tests or to a lack of staff at the end of the week to
perform the counts.

Figure 11. Z-transformed scores of smoothed confirmed world daily new cases [30] smoothed data from Figure 9, 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.

4. Discussion
The duration of the contagiousness period, as well as the daily virulence, are not
constant over time. Three main factors, which are not constant during a pandemic, can
explain this:
- In the virus transmitter, the transition between the mechanisms of innate (the first de-
fense barrier) and adaptive (the second barrier) immunity may explain a transient de-
crease in the emission of the pathogenic agent during the phase of contagiousness [15],
- In the environmental transmission channel, many geophysical factors that vary over time
can influence the transmission of the virus (temperature, humidity, altitude, etc.) [4–8],
- 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].
It is therefore very important to seek to estimate the average duration of the period of
contagiousness of individuals and the variations, during this phase of contagiousness, of
the associated daily reproduction numbers [33–39]. If the duration of the contagiousness
Computation 2021, 9, 109 19 of 31

phase is more than 3–5 days, for example ±7 days, the periodicity of seven days observed
for the new daily cases could result of an entrainment of the dynamics of new cases driven
by the social “Zeitgeber” represented by the counting of new cases, less precise during
the weekend (probably underestimated in many countries not working at this time). That
questions the deconvolution over 3 and 5 days, giving some negative Rj . 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 Rj ’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 Rj ’s,
more often positive and of more realistic magnitude, while keeping a sum approximately
equal to R0 .
Rhodes and Demetrius have pointed out the interest of the distribution of the daily re-
production numbers [24] with respect to the classical unique R0 , 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.
We have neglected in the present study the natural birth and death rates by supposing
them identical, but we could have taken into account the mortality due to the COVID-19.
The discrete dynamics of new cases can be considered as Leslie dynamics governed by the
matrix equation:
Xj = L Xj−1 ,
where Xj is the vector of the new cases living at day j and L is the Leslie matrix given by:
 
R1 R2 R3 ... ... Rr Xj−1
 

 b1 0 0 ... ... 0 


 Xj−2 


 0 b2 0 ... ... 0 


 Xj−3 

L= .. .. .. ..  and Xj−1 =  .. ,


 . . . ... ... . 


 . 
 .. .. .. .. ..   .. 
. . . . ... .
 
   . 
0 0 0 ... br − 1 0 Xj−r

where bj = 1 − µj ≤ 1, ∀ i = 1, . . . , r, is the recovering probability between days j and j + 1.

The dynamical stability for L2 distance to the stationary infection age pyramid
P = limj Xj /Σi=j,j−r+1 Xi is related to |λ − λ0 |, the modulus of the difference between the
dominant and sub-dominant eigenvalues of L, namely λ = eR and λ0 , where R is the Malthu-
sian 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 [26–32], which is defined
if lj = ∏i=1,j−1 bi and pj = lj Rj /λj , as follows:

H = −Σj=1,r pj Log(pj )/Σj=1,r j pj (18)

The mathematical characterization by the population entropy defined in Equation (16)

of the stochastic stability of the dynamics described by Equation (16) has its origin in
the theory of large deviations [40–42]. This notion of stability pertains to the rate at
which the system returns to its steady state after a random exogenous and/or endogenous
Computation 2021, 9, 109 20 of 31

perturbation and it could be useful to quantify further the variations of the distribution of
the daily reproduction numbers observed for many countries [43–53].
In summary, the main limitations of the present study are:
- 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 tempera-
ture 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 [10,54–58].

5. Conclusions and Perspectives

Concerning contagious diseases, public health physicians are constantly faced with
four challenges. The first concerns the estimation of the basic reproduction number R0 . The
systematic use of R0 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 [54–56]. 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 [57–59]), 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 [59–61], the best example being
that of the pandemic linked to obesity, for which many concepts and modelling methods
remain available.
Eventually, our approach using marginal daily reproduction numbers involving a
certain level of noise in the dynamics of new daily infected cases defines a stochastic
framework which describes phenomenologically the exponential phase as our results show
for countries such as France, Russia, Sweden, etc. This stochastic modelling allows a better
understanding of the role of the contagiousness period length and of the heterogeneity
(e.g., the U-shape) of its daily reproduction number distribution in the COVID-19 outbreak
dynamics [62–65]. On the medical level, the important message about the U-shape is
that COVID-19 is similar to other viral diseases, such as influenza, with two successive
reactions from the two immune defense barriers, innate cellular immunity first, which is
not sufficient if symptoms persist, then adaptive immunity (cellular and humoral), which
results in a transient decrease in contagiousness between the two phases. The medical
recommendations are, in this case, never to take a transient improvement for a permanent
disappearance of the symptoms. One could indeed, for a public health use, be satisfied
after estimating the sum of the Rj ’s, that is to say, R0 or the effective Re . For an individual
health use, it is important to know the existence of a minimum of the Rj ’s, which generally
corresponds to a temporary clinical improvement, after the partial success of the innate
Computation 2021, 9, 109 21 of 31

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: The following are available online at https://www.mdpi.com/article/10

.3390/computation9100109/s1. Table S1. Presentation of the Pearson correlation coefficients between
20 numbers of world daily new cases observed between the days 34 to 53 after the 24 January 2020
(date of the start of the Covid-19 outbreak with confirmed cases in Europe) and series of 20 numbers
Re of world daily new cases observed in running windows of length 20 days until day 213.
Author Contributions: Conceptualization, J.D. and J.W.; methodology, J.D., K.O., M.R., H.S. and J.W.;
1.2 K.O. and F.T. have performed the calculations and Figures. All authors have equally participated to
the other steps of the article elaboration. All authors have read and agreed to the published version
of the manuscript.
1
Funding: This research received no external funding.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Data are available on public databases https://renkulab.shinyapps.io/
Nigeria
COVID-19-Epidemic-Forecasting/_w_e213563a/?tab=ecdc_pred&country=France, (accessed on 22
November 2020). and https://www.worldometers.info/coronavirus/, (accessed on 2 November 2020).
Acknowledgments: The authors hereby give their thanks to the framework of the University of
Excellence Concept “Research University in Helmholtz Association I Living the Change”.
Conflicts of Interest: The authors declare no conflict of interest.

Appendix A Day j
Figure A1 shows a U-shaped evolution for the viral load in real [57] and in simu-
lated [58] COVID-19 patients, and in real influenza-infected animals for the viral load and
Figure 6 the body temperature [59].

(a)
patients

Each color corresponds

to one among six patients (c) (d)
(b)

No treatment Antiviral therapy

Figure A1. (a) Viral load in real COVID-19 patients [10], (b) in influenza-simulated patients [57] and (c) in real influenza-
Figure(red
infected animals A1 curve [58]), and (d) body temperature in real influenza-infected animals (red curve [58]).
Computation 2021, 9, 109 22 of 31

Appendix B
1. Beginning of the pandemic in France from 21 February 2020 to 9 March 2020
The numbers of new cases are:
21 February 2, 4, 19, 18, 39, 27, 56, 20, 67, 126, 209, 269, 236, 185 9 March
Then, the matrix M is defined by:

236 269 209 126 67 20 56

 

 269 209 126 67 20 56 27 


 209 126 67 20 56 27 39 

M=
 126 67 20 56 27 39 18 


 67 20 56 27 39 18 19 

 20 56 27 39 18 19 4 
56 27 39 18 19 4 2

and we have:
M−1 =
−5.884 × 10−5 5.399 × 10−5 −1.555 × 10−4 7.241 × 10−3 −5.146 × 10−3 −1.255 × 10−2 −1.277 × 10−2
 

 5.399 × 10−5 −1.714 × 10−4 7.324 × 10−3 −6.862 × 10−3 −1.139 × 10−2 1560 × 10−2 −3.242 × 10−3 


 −1.555 × 10−4 7.324 × 10−3 −6.862 × 10−3 −1.177 × 10−2 −1.592 × 10−2 −2.441 × 10−3 −4.780 × 10−4 

7.241 × 10−3 −6.862 × 10−3 −1.177 × 10−2 −2.164 × 10−2 −6.654 × 10−3 −1.0780 × 10−2 −9.514 × 10−3
 
 
−5.146 × 10−3 −1.139 × 10−2 1.592 × 10−2 −6.654 × 10−3 −3.692 × 10−3 2.797 × 10−2 2.637 × 10−2
 
 
1.255 × 10−2 1.560 × 10−2 −2.441 × 10−3 −1.078 × 10−2 2.797 × 10−2 2.555 × 10−2 −3.125 × 10−2
 
 
1.277 × 10−2 −3.242 × 10−3 −4.780 × 10−4 9.514 × 10−3 2.637 × 10−2 −3.125 × 10−2 −7.828 × 10−4

185 0.239
   
 236   0.052 
   
 269   −0.783 
−1
   
Because, X =   209 , hence R = M X =
 
 −0.295  and we can represent

 126   1.189 
   
 67   3.060 
20 3.122
the evolution of Xj ’s on Figure A2.

Figure A2. Values of the daily reproduction numbers Rj along the period of contagiousness of length
7 days.

The evolution of the Xj’s along the period of contagiousness shows at day 4 a sharp
increase and a saturation.
2. Exponential phase in France from 25 October 2020 to 7 November 2020
The numbers of new cases are:
Computation 2021, 9, 109 23 of 31

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
Then, the matrix M is defined by:

58, 581 56, 292 39, 880 35, 912 51, 104 45, 258 33, 447
 

 56, 292 39, 880 35, 912 51, 104 45, 258 33, 447 46, 185 


 39, 880 35, 912 51, 104 45, 258 33, 447 46, 185 44, 705 

M=
 35, 912 51, 104 45, 258 33, 447 46, 185 44, 705 34, 194 


 51, 104 45, 258 33, 447 46, 185 144, 705 34, 194 31, 360 

 45, 258 33, 447 46, 185 44, 705 34, 194 31, 360 25, 123 
33, 447 46, 185 44, 705 34, 194 31, 360 25, 123 48, 808

and we obtain
2.867
 

 −1.231 


 1.351 

R=
 −2.705 


 −0.155 

 0.223 
0.769
The Figure A3 shows an evolution of the Xj’s with a U-shape on the three first days
along the period of contagiousness with a sum of Rj ’s equal to 1.11, close to the effective
reproduction number Re = 1.13 [28].

Figure A3. Values of the daily reproduction numbers Rj 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
The number of new cases are:
21 February 20, 0, 0, 18, 4, 3, 0, 3, 5, 7, 25, 24, 34, 63 5 March
Then, we have:
0.466
 
 0.584 
 
 1.547 
 
R=  −1.044 

 0.174 
 
 0.297 
0.692
Computation 2021, 9, 109 24 of 31

The evolution of the Xj’s shows in Figure A4 a U-shape on day 4 with a sum of Rj ’s
equal to 2.72, less than the effective reproduction number Re = 3.27 [28].

Figure A4. Values of the daily reproduction numbers Rj along the period of contagiousness of length
7 days.

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

The numbers of new cases are:
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
Then, we have:
0.020
 
 −0.439 
 
 0.583 
 
R=  −0.367 

 0.497 
 
 −0.056 
1.113
The evolution of the Xj’s shows in Figure A5 a U-shape on the four last days with a
sum of Rj ’s equal to 1.35, close to the effective reproduction number Re = 1.24 [28].

Figure A5. Values of the daily reproduction numbers Rj along the period of contagiousness of length
7 days.
Computation 2021, 9, 109 25 of 31

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

The number of new cases are:
23 February 4, 0, 0, 0, 3, 4, 3, 12, 3, 11, 33, 26, 43, 41 7 March
Then, we have:
−0.388
 
 −1.189 
 
 1.334 
 
R=  1.960 

 4.862 
 
 −0.170 
3.479
Figure A6 shows an evolution of the Xj’s with a U-shape on the three last days along
the period of contagiousness with a sum of Rj ’s equal to 9.88, higher than the effective
reproduction number Re = 2.95 [28].

Figure A6. Values of the daily reproduction numbers Rj along the period of contagiousness of length
7 days.

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

The numbers of new cases are:
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
Then, we have:
0.020
 
 0.334 
 
 0.462 
 
R=  −0.098 

 −0.134 
 
 −0.043 
0.526
Figure A7 shows an evolution of the Xj’s with a U-shape on the five last days along the
period of contagiousness with a sum of Rj ’s equal to 1.07, close to the effective reproduction
number Re = 1.06 [28].
Computation 2021, 9, 109 26 of 31

Figure A7. Values of the daily reproduction numbers Rj along the period of contagiousness of length
7 days.

Appendix C
Table A1 is built from new COVID-19 cases at the start of the first and second waves
for 194 countries; it shows 42 among these 194 countries having a U-shape evolution of
their daily Rj ’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 Rj ’s,
which confirms the comparison with the behavior of seasonal influenza (see Section 2.2).

Table A1. Calculation of the daily Rj ’s and shape of their distribution for 194 countries and for the two first waves.

All Countries First Wave Second Wave

No Country Name R0 Rj’ s U-Shape R0 Rj ’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
Computation 2021, 9, 109 27 of 31

Table A1. Cont.

All Countries First Wave Second Wave

No Country Name R0 Rj’ s U-Shape R0 Rj ’s U-Shape
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
Computation 2021, 9, 109 28 of 31

Table A1. Cont.

All Countries First Wave Second Wave

No Country Name R0 Rj’ s U-Shape R0 Rj ’s U-Shape
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
Computation 2021, 9, 109 29 of 31

Table A1. Cont.

All Countries First Wave Second Wave

No Country Name R0 Rj’ s U-Shape R0 Rj ’s U-Shape
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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