#  Systematic description of COVID-19 pandemic using exact SIR solutions and
Gumbel distributions

J.E. Amaro<EMAIL_ADDRESS>Departamento de Física Atómica, Molecular y Nuclear
and Instituto Carlos I de Física Teórica y Computacional, Universidad de
Granada, E-18071 Granada, Spain

###### Abstract

An epidemiological study of deaths is carried out in a dozen countries by
analyzing the first wave of the COVID-19 pandemic. These countries are among
those most affected by the first wave, i.e. where daily-death data series may
closely resemble a solution of the basic SIR equations. The SIR equations are
solved parametrically using the proper time as parameter. Some general
properties of the SIR solutions are studied such as time-scaling and
asymmetry. Additionally, we use approximations to the SIR solutions through
Gumbel functions, which present a very similar behavior. The parameters of the
SIR model and the Gumbel function are extracted from the data and compared for
the different countries. It is found that ten of the selected countries are
very well described by the solutions of the SIR model, with a basic
reproduction number between 3 and 8.

###### keywords:

COVID-19 coronavirus , SIR model , Differential Equations , Gumbel
distribution

##  1 Introduction

Since the declaration of the COVID-19 pandemic by the World Health
Organization in March 2020, studies by the mathematical epidemiologist
community have intensified and models of various kinds have been developed in
order to provide insights and make predictions about the spread of the disease
[1, 2, 3, 4]. Epidemiological models have been used as basic tools in
epidemiology for over a century [5, 6, 7, 8, 9, 10, 11, 12, 13] and have been
extensively used prior to the COVID-19 pandemic [14, 15, 16, 17].

A wide range of models have been proposed and tested in an attempt to describe
the COVID-19 data and forecast the future evolution of the pandemic in
different regions of the planet. From very simple models [18, 19] to numerous
variants of compartmental models based on the SIR model (susceptible,
infected, removed) have been proposed [20, 21, 22, 23]. Additional models have
been employed such as the SEIR model [24], which adds the exposed compartment
of individuals, the uncertain SIR model [25], and others that include
different parameters that statistically describe the many factors that may
influence the pandemic dynamics. More sophisticated approaches include
deterministic or stochastic space-time models [26, 27, 28, 29]. Nonetheless,
it has been argued that complex models with numerous parameters may not
necessarily be advantageous without having enough data for a meaningful
validation [30].

After going through the sixth wave in many countries, the world data [31] show
that a comprehensive description of the entire time series appears to be an
impossible task, since each country presents its own characteristics (e.g.
diverse lock-down and socio-economic measures). Therefore, in this work we
proceed by studying the first wave for those countries that present data with
a similar structure and that can unambiguously be described mathematically
using an epidemiological model of the SIR type.

By inspection, from the recorded worldometer data [31] we found that there are
only nine or ten such countries (we leave aside the case of China that has
been exhaustively studied and where the pandemic apparently died out without
the need for vaccines). Among those countries there are eight Europeans —
Spain, France, Italy, UK, Germany, Belgium, Switzerland and Sweden — together
with Canada and USA. Moreover, we have also added the cases of India and
Brazil. In this work we will carry out a systematic study of the pandemic in
each of them by studying the series of cumulative deaths and daily deaths. Our
hypothesis is that deaths, $D(t)$, can be considered a fraction of removals,
$R(t)$, both cumulative and daily, and therefore both functions follow an
epidemiological curve that will differ essentially in a normalization constant
and a shift in time.

Our purpose is to investigate if data can be described with simple
epidemiological curves using the SIR model and the even simpler Gumbel
function [32]. A fundamental question about the first wave of COVID-19 is
whether the lock-down limitations had an effect in reducing the number of
deaths. Non-pharmaceutical interventions (NPI) are still under debate. A
recent meta-analysis review [33] fails to confirm that lock-downs have had a
large, significant effect on mortality rates. If the daily mortality curves
fit well with a basic SIR model, it would be interesting to conclude
affirmatively or negatively regarding the effect of NPI on them.

The structure of the paper is as follows. In Sect, II we review the solutions
of the SIR model that will be considered here. We describe in detail how to
obtain numerical solutions as a function of the proper time, depending on the
parameters $\beta$ and the basic reproduction number
$\rho=R_{0}=\lambda/\beta$. In Sect. III We examine how well the Gumbel
function fits exact SIR solutions with only one parameter, barring
normalization and a temporal shift. In Sect IV We will discuss the time
scaling of SIR solutions and define an asymmetry parameter that depends
linearly on $\rho$, and therefore can be used to characterize the value of the
basic reproduction number from a set of data. In Sect. V we present our
results of fits of death data with the exact SIR model and Gumbel functions.
In Sect VI we draw our conclusions.

##  2 Solutions of the SIR model and the proper time

In this section we briefly describe the SIR model and discuss its analytical
solution in terms of, what we will call here, proper time, $\tau$, which is a
natural variable to measure time through the proportion of removals, where the
SIR equations have a trivial and easily interpreted solution. The real time is
then obtained by integrating the exact solution.

In the SIR model the individuals of a closed population $N$ affected by a
contagious disease are divided into three types: susceptible, $S$, infected,
$I$, and removed (recovered or dead), $R$. As functions of time, the number of
individuals in each compartment is assumed to verify the following equations

$\displaystyle\frac{dS}{dt}$ $\displaystyle=$
$\displaystyle-\frac{\lambda}{N}IS,$ (1) $\displaystyle\frac{dR}{dt}$
$\displaystyle=$ $\displaystyle\beta I,$ (2) $\displaystyle I$
$\displaystyle=$ $\displaystyle N-R-S.$ (3)

In this transmission-dynamics system the first equation means that the
variation of susceptible individuals decreases by infection and is
proportional to the number of susceptible and the number of infected
individuals. The constant $\lambda$ measures the rate of infection. The second
equation describes the removal variation as proportional to the number of
infected individuals, the removal rate being $\beta$. By the third equation,
the difference between the total number minus the susceptible individuals
minus the removed ones must be the number of infected at each instant $t$.

We will consider the initial values $S(0)=S_{{}_{0}}<N$ and $R(0)=0$.
Therefore $I(0)=N-S_{{}_{0}}>0$. There must be a number, albeit small, of
infected in the system initially for the epidemic to begin. For convenience
below we will work with the percentages of susceptible, infected and recovered
individuals, over the total number of the population, which are obtained by
dividing by $N$:

$s(t)=\frac{S}{N},\kern 28.45274pti(t)=\frac{I}{N},\kern
28.45274ptr(t)=\frac{R}{N},$ (4)

with $s(0)=s_{{}_{0}}$, $i(0)=1-s_{{}_{0}}$, and $r(0)=0$.

The proper time, $\tau$, is defined as the temporal variable that describes
naturally the evolution of the epidemic, by counting the evolution of the
recovered individuals, $r(t)$, which is always an increasing function with
time. From the second SIR equation in differential form, it is defined by

$dR=\beta Idt\equiv Nd\tau,$ (5)

therefore, if we demand that $\tau=0$ for $t=0$, we trivially have

$\tau=r(t).$ (6)

The idea of proper time is based on other equivalent approaches described e.g.
in [34, 35], where the susceptible function $s$ is used as variable instead of
$r$. The proper time is nothing more than a change of the time variable into a
more convenient one. In our case, time is measured by counting the number of
recovered (in percent), since $r(t)$ is an increasing function, although it
does not depend linearly on time. By definition the proper time has a range
limited by

$0\leq\tau\leq 1.$ (7)

From Eq. 5 we have

$I=\frac{N}{\beta}\frac{d\tau}{dt},\kern
28.45274pti=\frac{1}{\beta}\frac{d\tau}{dt}.$ (8)

Thus the change of the physical time is given by

$dt=\frac{1}{\beta}\frac{d\tau}{i(\tau)},$ (9)

where $i(\tau)$ are the infected percent expressed as a function of proper
time.

To obtain the susceptible function note that we can write, inserting Eq. (5)
into Eq. (1)

$dS=-\frac{\lambda}{N}ISdt=-\frac{\lambda}{\beta}Sd\tau=-\rho Sd\tau,$ (10)

where $\rho$ is the so-called basic reproduction number

$\rho\equiv{\cal R}_{0}\equiv\frac{\lambda}{\beta}.$ (11)

The parameter $\rho$ has here the meaning of being the decay constant of the
susceptible population in units of proper time. Eq. (10) is readily integrated
giving

$S=S_{{}_{0}}{\rm e}^{-\rho\tau},$ (12)

which is similar to the exponential decay law of radioactive samples as a
function of proper time. Thus

$\tau_{1/2}=\frac{\ln 2}{\rho}=\frac{\beta\ln 2}{\lambda}$ (13)

represents the half-life in proper time units, i.e., the length of proper time
after which the susceptible population is reduced to half.

Figure 1:  Solution of the SIR equations as a function of proper time $\tau$
for initial susceptible $s_{{}_{0}}=0.99$, and for several values of the basic
reproduction number $\rho={\cal R}_{0}$.

Finally, the third SIR equation (3) gives directly the infected population as
a function of proper time

$\displaystyle I$ $\displaystyle=$ $\displaystyle N-N\tau-S_{{}_{0}}{\rm
e}^{-\rho\tau}$ (14) $\displaystyle i(\tau)$ $\displaystyle=$ $\displaystyle
1-\tau-s_{{}_{0}}{\rm e}^{-\rho\tau},$ (15)

with the condition $0\leq i\leq 1$. For $\tau=0$ we obtain the initial number
of infected population $i_{{}_{0}}=1-s_{{}_{0}}$. The end of the epidemic is
reached when $i=0$, that is, when the proper time verifies the transcendental
equation

$i(\tau_{f})=0\Longrightarrow\tau_{f}+s_{{}_{0}}{\rm e}^{-\rho\tau_{f}}=1$
(16)

In Fig. 1 we show some numerical examples. The analytical solutions of the SIR
equations are plotted as a function of proper time for various values of the
basic reproduction number $\rho=1.5,$ 2, 3 and 5. In all cases we assume that
$s_{{}_{0}}=0.99$; i.e., that one percent of the population is initially
infected.

Figure 2:  The peak values of $i$, $\tau$ as a function of the basic
reproduction number $\rho$. For $s_{{}_{0}}=0.99$ the peak values are almost
independent of $s_{{}_{0}}$.

In Fig. 1 we see that the number of infected individuals as a function of
$\tau$ first grows to a maximum and then decreases to zero. The maximum of
$i(\tau)$ is the peak of the epidemic. It is reached for $di/d\tau=0$. Then

$-1+\rho s_{{}_{0}}{\rm e}^{-\rho\tau}=0,$ (17)

and therefore

$s_{{}_{0}}{\rm e}^{-\rho\tau}=\frac{1}{\rho}.$ (18)

From here the peak of the epidemic verifies

$\displaystyle\tau_{peak}$ $\displaystyle=$ $\displaystyle\frac{\ln
s_{{}_{0}}\rho}{\rho}$ (19) $\displaystyle s_{peak}=s(\tau_{peak})$
$\displaystyle=$ $\displaystyle\frac{1}{\rho}$ (20) $\displaystyle
i_{peak}=i(\tau_{peak})$ $\displaystyle=$ $\displaystyle 1-\frac{1+\ln
s_{{}_{0}}\rho}{\rho}.$ (21)

A condition for the existence of this maximum for $\tau_{peak}>0$, from Eq.
(19), is that $s_{{}_{0}}\rho>1$. If we assume that, at the very beginning of
the epidemic, $s_{{}_{0}}$ is very close to 1, then it is enough that $\rho>1$
for the epidemic to begin to grow. In that case, $i(\tau)$ starts growing up
to a maximum reached at $\tau_{peak}$, where it starts to decrease to zero.

In Fig. 2 we show the peak values of the infected rate $i_{\rm peak}(\rho)$ in
the epidemic as a function of the basic reproduction number $\rho$, for
$s_{{}_{0}}=0.99$. Since $s_{{}_{0}}$ is very close to one and the dependence
on $s_{{}_{0}}$ is logarithmic, these peak values are almost independent on
the precise value of $s_{{}_{0}}\simeq 1$. The peak value of infected
individuals grows with the basic reproduction number $\rho$. When $\rho$ is
very large, above 10, the logarithmic dependence on the numerator makes the
peak to grow more slowly. For $\rho=10$ we have $i_{\rm peak}=0.67$, that is,
at the peak of the epidemic two thirds of the population will be infected
simultaneously. For $\rho=50$, more than 90% of the population will be
infected simultaneously at the peak.

In Fig. 2 we also show the value of the proper time at the epidemic peak,
$\tau_{\rm peak}(\rho)$. It presents a maximum for

$\displaystyle\frac{d\tau_{\rm peak}}{d\rho}=\frac{1-\ln
s_{{}_{0}}\rho}{\rho^{2}}=0$ (22) $\displaystyle\ln
s_{{}_{0}}\rho=1\Longrightarrow\rho=\frac{e}{s_{{}_{0}}}\simeq 2.72,$ (23)

for $s_{{}_{0}}\simeq 1$. The maximum value of $\tau_{\rm peak}$ is then

$(\tau_{\rm peak})_{\rm max}=\frac{s_{{}_{0}}}{e}\simeq 0.367$ (24)

for $\rho=e=2.72$.

Figure 3:  Solution of the SIR equations as a function of physical time, $t$,
for initial susceptible $s_{{}_{0}}=0.99$, for $\beta=0.3$ d-1, and for
several values of the reproduction number $\rho={\cal R}_{0}$.

Finally, starting from the analytical solution of the SIR equations as a
function of the proper time, $i(\tau)$, we will proceed to obtain the solution
as a function of the physical time, $t$. It is obtained from equation (9) by
integrating between 0 and $\tau$. Assuming that $t=0$ for $\tau=0$ we obtain

$t(\tau)=\frac{1}{\beta}\int_{0}^{\tau}\frac{d\tau}{i(\tau)}=\frac{1}{\beta}\int_{0}^{\tau}\frac{d\tau}{1-\tau-
s_{{}_{0}}{\rm e}^{-\rho\tau}}$ (25)

This integral is not analytical, but it can be calculated numerically with
precision using any numerical algorithm, such as Simpson’s rule or Gaussian
integration, since the integrand is quite smooth.

The final SIR solution is then obtained in parametric form by tabulating
$t(\tau)$, $i(\tau)$, $\tau$ and $s(\tau)$. By plotting $i(\tau)$, $\tau$ and
$s(\tau)$ as a function of $t(\tau)$ we obtain the results of Fig. 3,
corresponding to the exact (numerical) solution of the SIR equations as a
function of physical time, for the same parameters of Fig. 1 and for
$\beta=0.3$ d-1.

Note in Fig. 3, firstly, that the height of the maximum of $i(t)$ coincides
with that of the maximum of the analytical solution $i(\tau)$ of Fig. 1, as it
should be, since we have only made a change of variable —from the proper time
to the physical time. Of course, the dependence on physical time has
drastically changed. For a constant recovery rate $\beta=0.3$ d-1, by
increasing the value of the basic reproduction number, $\rho$, the epidemic
passes more quickly and lasts less time, almost explosively for $\rho=5$ —for
which it lasts only twenty days— compared to $\rho=1.5$ —where it lasts almost
two months.

Therefore, assuming that the recovery rate, $\beta$, in an epidemic is
somewhat constant, the basic reproduction number $\rho$ largely determines the
evolution of the pandemic. The analytical solution allows estimating the
maximum number of simultaneous infections at the peak, $i_{\rm peak}$, from
this number. This does not depend much on $s_{{}_{0}}$, as long as this number
is close to one, due to its dependence on the logarithm of $s_{o}\rho$.

Figure 4:  Gumbel distribution compared to the exact Solution of the SIR
equations for $s_{{}_{0}}=0.99$, $\beta=0.3$ d-1, and for several values of
the reproduction number $\rho$.

Finally, note that the exact solution $i(\tau)$ of the SIR model is
asymmetric. It rises very fast at first — exponentially actually — and then
falls more slowly. The most explosive, severe outbreaks occur for $\rho\geq
5$, where the end of the epidemic is reached for $\tau_{f}\simeq 1$ (see Eq.
(16)), thus most of the initial susceptible individuals have been infected.
For lower reproduction number the epidemic curve becomes more symmetrical and
it is less explosive. Note that In the first wave of the COVID-19 pandemic
daily deaths rose very rapidly and fell more slowly, indicating a high
reproduction number, so that the data could not be fitted with logistic-type
functions, which are symmetric, and a linear combination of two logistic
functions was required to fit the data [18, 19, 26]. In the next section we
will see that some more appropriate functions to describe this situation are
the Gumbel functions, because they have the adequate asymmetry to fit almost
correctly the solutions of SIR equations.

##  3 Approximation to the SIR solutions with Gumbel functions

In reference [32], a Gumbel distribution was used to forecast the series of
daily deaths with COVID-19 positives. The Gumbel distribution describes the
probability of maximum (or minimum) values from the data of many observations
[37, 38]. it is not theoretically clear why the distribution of deaths or
infections roughly recreates the distribution of maxima. In this section we
will compare the Gumbel distribution to the exact solutions of the
epidemiological SIR model. Both functions of time presents similar asymmetry
and depend on three parameters, that can be related numerically.

$s_{{}_{0}}$ | $\rho$ | $\beta$ | $a$ | $t_{0}$ | $b$
---|---|---|---|---|---
0.99 | 1.5 | 0.3 | 2.04 | 20.5 | 10.9
| 2 |  | 2.67 | 14.9 | 6.29
| 3 |  | 3.06 | 8.81 | 3.75
| 5 |  | 3.12 | 5.30 | 2.44
Table 1: Parameters of the Gumbel distribution fitted to the exact SIR model,
for four different values of the basic reproduction number $\rho$.

Although a Gumbel distribution does not exactly verify the SIR equation, the
exact solution can be approximated quite well by the Gumbel distribution by
choosing the parameters appropriately. In this section we will provide
formulas that relate the parameters of SIR model with the parameters of an
appropriate Gumbel function. For this end we will use the proper time method
introduced in the previous section.

The Gumbel function is defined here as a time-dependent function with three
parameters

$G(t)=a{\rm e}^{-{\rm e}^{-(t-t_{0})/b}}$ (26)

and the derivative gives the Gumbel distribution

$g(t)=\frac{dG}{dt}=\frac{1}{b}{\rm e}^{-(t-t_{0})/b}G(t).$ (27)

First we will see how well the Gumbel distribution approximates the rate of
infected of the SIR model, $g(t)\simeq i(t)$.

In Fig. 4 we compare the Gumbel distribution $g(t)$ with the exact SIR
solution $i(t)$. The parameters of the Gumbel distribution have been fitted
with least-squares method to obtain the optimal distribution that best
describes the exact solution. The parameters of the SIR model are
$s_{{}_{0}}=0.99$, $\beta=0.3$, and we use four values of $\rho=1.5,2,3$ and 5
—the same as in Fig. 3. The fitted parameters of the Gumbel distribution are
given in Table 1

The fits of Fig. 4 provide fairly good approximations to the exact SIR
solutions with the Gumbel distribution. Although the fit is not perfect, the
Gumbel distribution clearly shows similar asymmetry as the SIR model. The
similarity is greater when the epidemic is explosive, with a high reproduction
number $\rho\sim 3--5$. Where Gumbel fails most in these fits is in the
initial and final stages of the Pandemic.

To investigate the connection between the Gumbel distribution and the solution
of the SIR equations, it is convenient to express them as functions of proper
time. First we substitute $i(t)$ by $g(t)$ in the definition of proper time

$d\tau=\beta idt=\beta gdt=\beta\frac{dG}{dt}dt$ (28)

Thus integrating between 0 and $t$ we have

$\tau=\int_{0}^{t}\beta\frac{dG}{dt}dt=\beta[G(t)-G_{0}]$ (29)

where $G_{0}=G(0)$. Hence

$G(t)=G_{0}+\frac{\tau}{\beta}=a{\rm e}^{-{\rm e}^{-(t-t_{0})/b}}$ (30)

Taking the logarithm on both sides we obtain

${\rm e}^{-(t-t_{0})/b}=\ln\frac{a}{G_{0}+\tau/\beta}$ (31)

from where we obtain the following approximate relation between $t$ and $\tau$

$t=t_{0}-b\ln\left(\ln\frac{a}{G_{0}+\tau/\beta}\right),$ (32)

and we can write the Gumbel distribution as a function of $\tau$

$g=\frac{1}{b}{\rm
e}^{-(t-t_{0})/b}G=\frac{1}{b}\left(G_{0}+\frac{\tau}{\beta}\right)\ln\frac{a}{G_{0}+\tau/\beta}.$
(33)

Starting with this expression as a function of proper time, we can consider
several alternative ways of adjusting the Gumbel parameters from the
parameters of the SIR model, assuming that the proper time is the same in both
models.

###  3.1 Proper time fit 1

The idea of the proper time fit consists in imposing that the maximum of
$g(\tau)$ coincides with the maximum of $i(\tau)$. In fit 1, we will also
assume that $G_{0}$ is very small can be neglected, $G_{0}\simeq 0$, i.e., we
will not try to impose any additional condition on the initial value. This
allow us to estimate the parameters $a$ and $b$ easily, but we will not be
able to obtain the value of $t_{0}$, which will later be adjusted to fit the
temporal peak of the SIR solution.

We start by writing Eq. (33) for $G_{0}=0$

$g(\tau)=\frac{\tau}{b\beta}\ln\frac{a\beta}{\tau}.$ (34)

To find the maximum of this function we compute the derivative

$g^{\prime}(\tau)=\frac{1}{b\beta}\ln\frac{a\beta}{\tau}-\frac{1}{b\beta}$
(35)

from where we find the maximum condition

$\ln\frac{a\beta}{\tau}=1.$ (36)

Thus the value of the peak position, $\tau_{\rm peak}$, is

$\tau_{\rm peak}=\frac{a\beta}{\rm e}$ (37)

and the height of the peak (maximum of $g$) is

$g(\tau_{\rm peak})=\frac{1}{\rm e}\frac{a}{b}$ (38)

Comparing to the peak values of the SIR solution, Eqs. (19,21), and equating
the values we obtain

$\displaystyle\frac{a\beta}{\rm e}$ $\displaystyle=$ $\displaystyle\frac{\ln
s_{{}_{0}}\rho}{\rho}$ (39) $\displaystyle\frac{1}{\rm e}\frac{a}{b}$
$\displaystyle=$ $\displaystyle 1-\frac{1+\ln s_{{}_{0}}\rho}{\rho}.$ (40)

From here we obtain the values of the parameters $a$ and $b$ of the Gumbel
distribution in terms of the parameters of the SIR model

$\displaystyle a$ $\displaystyle=$ $\displaystyle\frac{{\rm e}\ln
s_{{}_{0}}\rho}{\lambda}$ (41) $\displaystyle\frac{a}{b}$ $\displaystyle=$
$\displaystyle{\rm e}\left(1-\frac{1+\ln s_{{}_{0}}\rho}{\rho}\right)$ (42)

We see that the values of $a$ and $b$ can be calculated directly from the
parameters and initial conditions of the SIR model. The value of $t_{0}$ would
be fitted later to the position of the peak as a function of time.

Note that if the initial number of infected individual is very small, then
$s_{{}_{0}}$ can be approximated by one in the logarithm $\ln
s_{{}_{0}}\rho\simeq\ln\rho$. Then the parameters $a$ and $b$ does not depend
appreciably on the precise value of $s_{{}_{0}}$, but only on $rho$ and
$\lambda$ by the simple relations

$\displaystyle a$ $\displaystyle=$ $\displaystyle\frac{{\rm
e}\ln\rho}{\lambda}$ (43) $\displaystyle\frac{a}{b}$ $\displaystyle=$
$\displaystyle{\rm e}\left(1-\frac{1+\ln\rho}{\rho}\right)$ (44)

Note also that Eq. (44) can be solved numerically to obtain the reproduction
number $\rho$ as a function of $a/b$, and then Eq. (43) gives $\lambda$. So
the constants $\lambda$ and $\rho$ of the SIR system can also be computed from
the values of the constants $a$ and $b$ of the Gumbel distribution.

Figure 5:  Three Gumbel distributions as a function of the proper time,
compared to the exact Solution of the SIR equations for $s_{{}_{0}}=0.99$,
$\beta=0.3$ d-1, and for several values of the reproduction number $\rho$. The
parameters of the Gumbel distributions correspond to the mean-square fit,
proper-time fit 1 and proper-time fit 2. In proper-time fit 1, the value of
$t_{0}$ is taken from Tab. 1. Figure 6:  Three Gumbel distributions compared
to the exact Solution of the SIR equations for $s_{{}_{0}}=0.99$, $\beta=0.3$
d-1, and for several values of the basic reproduction number $\rho$. The
parameters of the Gumbel distributions correspond to the mean-square fit,
proper-time fit 1 and proper-time fit 2. In proper-time fit 1, the value of
$t_{0}$ is taken from Tab. 1.

###  3.2 Proper time fit 2

A second fit can be done by fixing also the number of initial infected $i(0)$,
in which case a value for $t_{0}$ can be theoretically obtained. We proceed as
in previous section, by computing the maximum of the Gumbel distribution, as a
function of proper time, but in this case we use the exact expression, Eq.
(33). The maximum of $g(\tau)$ is now obtained for

$\ln\frac{a}{G_{0}+\tau/\beta}=1$ (45)

from where the peak (maximum of $g$) is reached at the proper time

$\tau_{\rm peak}=\frac{a\beta}{\rm e}-G_{0}\beta$ (46)

and the value of the maximum is

$g(\tau_{\rm peak})=\frac{1}{\rm e}\frac{a}{b}$ (47)

this is the same value for the peak obtained in Eq. (38).

Comparing to the peak values of the SIR solution, Eqs. (19,21), and equating
the values we obtain

$\displaystyle\frac{a\beta}{\rm e}-G_{0}\beta$ $\displaystyle=$
$\displaystyle\frac{\ln s_{{}_{0}}\rho}{\rho}$ (48) $\displaystyle\frac{1}{\rm
e}\frac{a}{b}$ $\displaystyle=$ $\displaystyle 1-\frac{1+\ln
s_{{}_{0}}\rho}{\rho}$ (49)

Eq. (49) gives the same value for $a/b$ obtained in fit 1. For
$s_{{}_{0}}\simeq 1$, this value only depends on the basic reproduction number
$\rho$.

A third condition is obtained if we demand $g(0)=i(0)=1-s_{{}_{0}}$. Using Eq.
(33) for the Gumbel distribution as a function of proper time, we have

$g(0)=\frac{1}{b}\ln\frac{a}{G_{0}}G_{0}=1-s_{{}_{0}}$ (50)

where

$G_{0}=a{\rm e}^{-{\rm e}^{t_{0}/b}}=a{\rm e}^{-x}$ (51)

where we have defined the parameter

$x={\rm e}^{t_{0}/b}.$ (52)

In terms of the $x$-parameter, the initial value condition can be written as

$g(0)=\frac{a}{b}x{\rm e}^{-x}.$ (53)

The procedure of fit 2 follows the following steps:

  1. 1.

Compute $a/b$ from Eq. 49

$\frac{a}{b}={\rm e}\left(1-\frac{1+\ln s_{{}_{0}}\rho}{\rho}\right).$ (54)

  2. 2.

Write the initial condition $g(0)=i(0)$ as

$x{\rm e}^{-x}=\frac{b}{a}(1-s_{{}_{0}})\equiv\epsilon_{0}$ (55)

and solve this equation numerically for $x>0$

  3. 3.

Once we know the values of $x$ and $a/b$. compute the value of $a$ using Eq.
(48)

$a=\frac{\ln s_{{}_{0}}\rho}{\lambda({\rm e}^{-1}-{\rm e}^{-x})}$ (56)

  4. 4.

Compute $b=(b/a)a$, and finally

  5. 5.

Compute $t_{0}$ as

$t_{0}=b\ln x.$ (57)

In Figs. 5 and 6 the solution of the SIR model is compared with the Gumbel
distributions corresponding to the three fits considered in this work: the
least square fit, and the proper time fits 1 and 2.

The least square fit, with the parameters of Table 1, is represented with
dotted lines as a function of proper time in Fig. 5 and is very close to
$i(\tau)$. We see in Fig. 5 that the proper time fit 1 and fit 2 are
essentially the same and that their maximum coincides (by construction) with
the maximum of the SIR solution $i(\tau)$. The parameter $t_{0}$ in the case
of fit 1 is taken from Table 1, because it cannot be obtained theoretically.
The parameter $t_{0}$ in fit 2 is computed from Eq. (57) in terms of the
computed values of $b$ and $x$.

For low values of the basic reproduction number, $\rho=1.5--3$, both fit 1 and
fit 2 are wider than $i(\tau)$ in Fig. 5, and they extend at the end of the
epidemic up to a proper time larger than the exact solution. When $\rho$ is
larger the width of the Gumbel functions of fits 1 and 2 begin to decrease in
relation to the SIR result, until their widths become smaller than the width
of the SIR solution, for $\rho=5$.

The results of Fig. 5 are translated into the distributions as a function of
physical time in Fig. 6. The Gumbel fits 1 and 2 differ mainly in the value of
$t_{0}$. The maximum of fit 2 is shifted to the left with respect to the
maximum of fit 1. This happens because in fit 2 we are demand that the initial
value of the Gumbel distribution be equal to $i(0)$. As a result $g(t)$ is
shifted to the left of $i(t)$, because the Gumbel distribution increases
slightly faster than the SIR solution as a function of time. By construction,
The maximum number of infected coincides in fits 1 and 2 with the maximum of
$i(t)$, but it occurs slightly earlier in fit 2.

With appropriate parameters, we have seen that the Gumbel distribution
describes the exact solution of the SIR model quite well. This justifies that
epidemic data can be fitted with a Gumbel distribution. As a direct
application, this would allow estimating the parameters of the SIR model in
first approximation from those of Gumbel, using the formulas in this section.
The fact that the Gumbel distribution is analytical as a function of time
allows its parameters to be easily fitted, unlike the SIR model, which,
although simple, must be solved numerically,

##  4 Asymmetry of the SIR solutions

An essential characteristic of the solution $i(t)$ of the SIR equations is the
evident asymmetry that this function presents with respect to its maximum
value. The asymmetry is more pronounced as the reproduction number $\rho$
increases. Indeed, in this section we will define a parameter that uniquely
characterizes the asymmetry and we will see that said asymmetry increases
linearly with the basic reproduction number $\rho$

We begin by mentioning an important property of the $i(t)$ solution concerning
the scaling property with respect to time. It is clearly seen in equation (25)
that the physical time as a function of proper time, $t(\tau)$, is inversely
proportional to $\beta$. Therefore any change in $\beta$ translates into a
change in the time scale. The natural scale of time is thus $\beta t$, that
is, measuring time in units of $1/\beta$. Thus, from the solutions of the SIR
equations for $\beta=1$, all the others are obtained simply by re-scaling the
time with a factor $1/\beta$.

Therefore in Fig. 7, where we represent the solutions of the SIR equations for
a range of $\rho$ values from 1.5 to 500, and for $\beta=1$, all the solutions
are included if they are represented as a function of $\beta t$. Note that
here we use the initial condition $s_{{}_{0}}=0.99$. But this does not detract
from the generality of our affirmations, because a change in the initial
condition ultimately translates into a shift of the time origin such that
$i(0)=0.01$.

Figure 7 shows that as $\rho$ increases, the epidemic progresses faster, that
is, it ends earlier. As a function of natural time (or for $\beta=1$ day-1),
it lasts from about 20 days for $\rho=1.5$, to less than a week for $\rho>10$,
and only a few days if $\rho>20$. Moreover for very large values of $\rho>10$,
the epidemic grows very quickly in a short time interval and then decreases
exponentially-like regardless of the value of $\rho$, with $i(t)\simeq
e^{-\beta t}$. In Fig. 7 (top panel) it is also evident that the asymmetry of
$i(t)$ increases with $\rho$.

Figure 7:  Solutions of the SIR equations for values of the basic reproduction
number $\rho$ from 1.5 to 500. The functions $i(t)$ and $\tau(t)=r(t)$ are
Plotted as a function of the normalized time $\beta t$. Figure 8:  Solutions
of the SIR equations for several values of the basic reproduction number
$\rho$, and for $s_{{}_{0}}=0.99$. The functions $i(t)$ and $\tau(t)=r(t)$ are
plotted as a function of the normalized time $\beta t$. These are compared to
the approximate solution obtained with a Gumbel function $G(t)$ fitted to
$\tau(t)$, and with the corresponding infected function
$i_{G}(t)=1-G(t)-s_{{}_{0}}\exp(-\rho G(t))$ of the extended SIR model.

The recovered function, or proper time, $\tau(t)=r(t)$, is displayed in the
bottom panel of Fig. 7. For small values of $\rho$, it quite resembles a
Gumbel function, but for very large values of the basic reproduction number,
$\rho>10$, it departs from the family of Gumbel functions and approaches the
function $r(t)=1-\exp(-\beta t)$, as the limit for $\rho\rightarrow\infty$.

The behavior of the SIR solution and the Gumbel functions can be seen more
clearly in Fig. 8. There we plot the infected and recovered functions as a
function of $\beta t$, for the SIR model and fitting a Gumbel function, for
$\rho=1.5,2,3\ldots,9$. In Fig. 8, the fit is an alternative to the one made
in section 2. There, the Gumbel distribution $g(t)$ was fitted directly to the
infected function $i(t)$. In Fig. 8 we first fit the Gumbel function $G(t)$ to
the recovered function $\tau(t)=r(t)$ by least squares method, and then we
calculate the Gumbel-infected function using the SIR Eq. (15)

$i_{G}(t)=1-G(t)-s_{o}\exp(-G(t))$ (58)

for $\beta=1$. The results of fitting $G(t)$ are not exactly the same as
fitting $g(t)$, because a different function if being fitted in each case, but
they are very similar.

We notice that for $\rho=5$, the resulting function $i_{G}(t)$ does not
converges to zero for large $t$. This happens because the least-squares fit of
the Gumbel function does not exactly approach the maximum value of $\tau(t)$.
To remedy this, starting $\rho=6$, we set the value of the $a$ parameter of
$G(t)$ so that $G(t)=\tau(t)$ for large $t$, and then fit the two parameters
$b$ and $t_{0}$.

Fig. 8 shows that the Gumbel distribution is very similar to the SIR solution,
but starting from $\rho=7$, small differences begin to be seen because the
asymmetry of Gumbel and the SIR solution begins to diverge from each other.
For $\rho=9$ such a difference is already quite appreciable, and it increases
for larger values of $\rho$.

Figure 9:  Parameters of the Gumbel function $G(t)$ fitted to $\tau(t)$ in
Fig. 8. These are compared to the functions $a(\rho)$ and $b(\rho)$ from Eqs.
(41,42).

In Fig. 9 we show the values of the Gumbel parameters, $a,b,t_{0}$, fitted in
Fig. 8, as a function of the reproduction number $\rho$. We also show the
values of $a(\rho)$ and $b(\rho)$ calculated with Eqs. (41) and (42), which
give an analytical approximation between the Gumbel function and the SIR
solution. We see that the values of a and b are very close to their analytic
approximations for small $\rho$, and that they start to differ from $\rho=6$.
This difference is due in part to the fact that in equations (41) and (42)
they were obtained by requiring that the maxima of $i(t)$ and $g(t)$ coincide.
But in Fig. 8 we have adjusted the parameter $a$ so that the maximum of $r(t)$
coincides with the maximum of $G(t)$ and that alters Eqs. (41) and (42).

Next we propose a definition of the asymmetry of the SIR solution in terms of
the half-widths of the distribution $i(t)$ to the left and to the right of the
maximum $i_{peak}=i(\tau_{peak})$. First we define $\tau_{1}$ and $\tau_{2}$
as the values of the proper time such that
$i(\tau_{1})=i(\tau_{2})=i_{peak}/2$ and $\tau_{1}<\tau_{peak}<\tau_{2}$.
Specifically, $\tau_{1}$ and $\tau_{2}$ are the two solutions of the equation

$1-\tau-s_{{}_{0}}{\rm e}^{-\rho\tau}=\frac{i_{peak}}{2}$ (59)

where $\tau_{peak}$ and $i_{peak}$ corresponds to the position and value of
the maximum of $i(\tau)$, given by Eqs. (19, 20). Therefore $\tau_{1}$ and
$\tau_{2}$ are the proper-time values at which $i(\tau)$ reaches half of its
maximum value. Two half-widths are defined in term of the physical time as

$\displaystyle\Gamma_{1}$ $\displaystyle=$
$\displaystyle\beta(t(\tau_{peak})-t(\tau_{1}))$ (60)
$\displaystyle\Gamma_{2}$ $\displaystyle=$
$\displaystyle\beta(t(\tau_{2})-t(\tau_{peak}))$ (61)

In view of the previous numerical results we see that $\Gamma_{2}>\Gamma_{1}$
for $\rho>1$, that is, the SIR solution is wider to the right than to the left
of the maximum. The asymmetry is then defined as the quotient

${\cal A}(\rho)=\frac{\Gamma_{2}}{\Gamma_{1}}\geq 1.$ (62)

This function increases with $\rho$. This asymmetry is defined as a ratio that
does not depend on $\beta$ nor on the global normalization of $i(t)$.
Therefore, it is a suitable parameter to express the asymmetry of the
theoretical distribution, as well as that of the experimental data of deaths
described in the next section.

Figure 10:  Asymmetry of the SIR solutions as a function of $\rho$. The points
are the values of the asymmetry computed numerically for discrete values of
$\rho=1.5,2,3,\ldots,9$. The solid line is a fitted right line $A+B\rho$. The
dashed line is the Jacobian $|d\tau^{\prime}/d\tau|_{i_{{}_{0}}}$ evaluated
for $i_{{}_{0}}=0.84i_{peak}$.

To compute the asymmetry of the SIR solution. for a $\rho$ value, we fist
solve the transcendental equation (59) numerically, and obtain the two
solutions $\tau_{1}<\tau_{2}$. The half widths are then computed as the
integrals

$\displaystyle\Gamma_{1}$ $\displaystyle=$
$\displaystyle\int_{\tau_{1}}^{\tau_{peak}}\frac{d\tau}{1-\tau-s_{{}_{0}}{\rm
e}^{-\rho\tau}}$ (63) $\displaystyle\Gamma_{2}$ $\displaystyle=$
$\displaystyle\int_{\tau_{peak}}^{\tau_{2}}\frac{d\tau}{1-\tau-s_{{}_{0}}{\rm
e}^{-\rho\tau}}$ (64)

These integrals are computed numerically and the corresponding asymmetry is
plotted in Fig. 10 for $\rho=1.5,2,3,\ldots,9$, and for $s_{o}=0.99$. We see
that, as a function of $\rho$, the asymmetry of the SIR solutions is quite
approximately a linear function of $\rho$, which is very well fitted by the
parametrization ${\cal A}=A+B\rho$, with $A=0.868$ and $B=0.176$, as seen in
Fig. 10.

To better understand why this quasi-linear dependence on asymmetry occurs, we
proceed as follows. A rigorous proof is not possible to our understanding
because transcendental equations are involved, and because the linearity is
only approximate, but its origin can be roughly understood.

First, in the interval $[\tau_{1},\tau_{peak}]$ the function $i(\tau)=1-\tau-
s_{{}_{0}}{\rm e}^{-\rho\tau}$ is increasing and the change of variable
$\tau\rightarrow i$ can be made inside the integral $\Gamma_{1}$, with
Jacobian $di/d\tau=\rho s_{{}_{0}}{\rm e}^{-\rho\tau}-1>0$. We obtain

$\Gamma_{1}=\int_{i_{peak}/2}^{i_{peak}}\frac{d\tau}{di}\frac{di}{i}$ (65)

Second, in the interval $[\tau_{peak},\tau_{2}]$ the function
$i(\tau^{\prime})=1-\tau^{\prime}-s_{{}_{0}}{\rm e}^{-\rho\tau^{\prime}}$ is
decreasing and the change of variable $i\rightarrow\tau^{\prime}$ can be made
inside the integral (65), with Jacobian $di/d\tau^{\prime}=\rho s_{{}_{0}}{\rm
e}^{-\rho\tau^{\prime}}-1<0$. We obtain

$\Gamma_{1}=\int_{\tau_{2}}^{\tau_{peak}}\frac{d\tau}{di}\frac{di}{d\tau^{\prime}}\frac{d\tau^{\prime}}{i(\tau^{\prime})}=\int_{\tau_{2}}^{\tau_{peak}}\frac{d\tau}{d\tau^{\prime}}\frac{d\tau^{\prime}}{i(\tau^{\prime})}$
(66)

with

$\frac{d\tau}{d\tau^{\prime}}=\frac{d\tau}{di}\frac{di}{d\tau^{\prime}}=\frac{\rho
s_{{}_{0}}{\rm e}^{-\rho\tau^{\prime}}-1}{\rho s_{{}_{0}}{\rm
e}^{-\rho\tau}-1}<0\,.$ (67)

This Jacobian corresponds to the change of variable
$\tau\rightarrow\tau^{\prime}$, inside $\Gamma_{1}$, where
$\tau<\tau^{\prime}$ are the two solutions of the transcendental equation
$i=1-\tau-s_{{}_{0}}{\rm e}^{-\rho\tau}$, for a fixed value of
$i\in[i_{peak}/2,i_{peak}]$.

Now, using the mean value theorem, we can factor the Jacobian out of the
second integral of Eq. (66), evaluated at some intermediate value of
$i=i_{{}_{0}}$ between $i_{peak/2}$ and $i_{peak}$, obtaining

$\Gamma_{1}=\left|\frac{d\tau}{d\tau^{\prime}}\right|_{i_{{}_{0}}}\int_{\tau_{peak}}^{\tau_{2}}\frac{d\tau^{\prime}}{i(\tau^{\prime})}=\left|\frac{d\tau}{d\tau^{\prime}}\right|_{i_{{}_{0}}}\Gamma_{2}$
(68)

Computing numerically both sides of (68) for different values of $i_{{}_{0}}$
we found that the best value is $i_{{}_{0}}=0.84$, for which we plot the
Jacobian as a function of $\rho$ in Fig. 10. We see that quite a straight line
is obtained that coincides with the calculated value of the asymmetry.

The quasi-linear behavior of the asymmetry of the SIR solutions as a function
of $\rho$ allows us to obtain the value of $\rho$ from experimental data,
after estimating the asymmetry empirically.

| $G(t)$ | $g(t)$ | SIR |  |  |  $g(t)$ fit to SIR
---|---|---|---|---|---|---
Country |  $b$ [d] |  $b$ [d] | $\rho$ |  $\beta$ [d1] |  $\beta^{-1}$ [d] |  $b$ [d]
Spain | 13.4 | 12.6 | 7 | 0.05 | 20 | 12
UK | 17.7 | 16.15 | 7 | 0.038 | 26 | 16.2
France | 13.35 | 11.6 | 6 | 0.06 | 16.7 | 10.8
Italy | 18.4 | 17.1 | 5.5 | 0.045 | 22.2 | 15.2
Germany | 15.4 | 14.6 | 5.5 | 0.05 | 20 | 13.6
Canada | 18.01 | 18.1 | 5.1 | 0.041 | 24.4 | 16.9
Belgium | 12.8 | 11.65 | 3 | 0.1 | 10 | 11.27
Switzerland | 11.7 | 11.79 | 4 | 0.085 | 11.8 | 10.16
Sweden | 26.7 | 25.66 | 7 | 0.03 | 33 | 18.8
USA | 21.1 [0:150] | 19.6 | 8 | 0.028 | 35.7 | 17.7
|  | 47.4 (2nd wave) |  |  |  |
Brazil | 60.6 [0:250] | 35.3 |  |  |  |
|  | 37 (2nd wave) |  |  |  |
India | 60 | 66.4 |  |  |  |
|  | 32 (2nd wave) |  |  |  |
Table 2: Parameters of the Gumbel and SIR models fitted for the different
countries considered in this work. Figure 11:  Total deaths $D(t)$ during the
first wave of the COVID-19 pandemic for several countries, compared to the
Gumbel function $G(t)$. The parameter $b$ of the Gumbel function fitted to the
data is given in Table 2. Data are form Ref.[39].  Figure 12:  Total deaths
$D(t)$ during the first wave of the COVID-19 pandemic for several countries,
compared to the Gumbel function $G(t)$. The parameter $b$ of the Gumbel
function fitted to the data is given in Table 2. Data are form Ref.[39].

##  5 Results

In this section we present results comparing with the data of total deaths
$D(t)$ and daily deaths $\Delta D(t)$ as a function of time for the first wave
of the Covid-19 pandemic. These data are supplied by government ministries in
different countries and are available from various sources. The data can vary
in different sources, as it can often be found averaged or smoothed, or in
many cases the data can be later revised and updated differently at different
sites.

To study the daily deaths we are assuming that a certain constant portion of
removals ends in death, that is, the death rate verifies an equation similar
to the removal rate with a different constant

$\frac{dD}{dt}=\alpha I.$ (69)

this means that the daily deaths, $\Delta D(t)\simeq\frac{dD}{dt}$, are
described, except normalization constant, by the solution $i(t)$ of the SIR
equations and we can use the formalism of the previous sections.

One of our purposes of this work is to check if the SIR model is capable of
describing the data, in which case it can be assumed that the hypotheses of
this very simple model are justified; in particular if averaged values can be
adopted for the two basic constants of the model: the reproduction number
$\rho$ and the recovery rate $\beta$. This would indicate the universal
validity of the SIR model. Comparing the parameters of the model between
different countries will tell us the degree of universality of these
parameters when passing from one country to another.

For the present study we have selected the countries where the first wave of
the COVID-19 pandemic is visually similar to the solution of the SIR
equations. The data that we have studied come from Ref. [39]. But they can
also be found in other sources, such as the worldometer website [31].
Surprisingly, after an inspection of the death data for each country, it turns
out that the solutions of the SIR equations can only be applied in a handful
of countries, which can be counted with the fingers of both hands. These
countries are: Spain, France, Italy, UK, Germany, Canada, Belgium,
Switzerland, Sweden and USA (we left aside the curious case of China where the
epidemic ended mysteriously and prematurely). Here we only consider the first
wave of the pandemic, because the subsequent ones are much more irregular and
require a separate study. Note that we only analyze the data on daily deaths
and cumulative deaths. The homogeneity of the reported number of infected
individuals is questioned because it is proportional to the number of tests
performed, and the experimental error of the tests is not given.

To begin with, in Figs. 11 and 12 we show the accumulated deaths as a function
of time in the ten countries mentioned with the addition of Brazil and India,
as they are the top countries in number of infections and perhaps also in
number of deaths. Time is measured in days. Data are from Ref.[39] and day one
is 2020 02 01\. A Gumbel function has been fitted for each country. The only
tabulated parameter in the second column of Table 2 is the value of the $b$
parameter in days. Note that the value of $a$ in the Gumbel function is simply
a normalization constant, and the value of $t_{0}$ is a shift in time. Thus
what really characterizes the dynamics is the value of the $b$, which is
related to the duration of the epidemic. Note that the USA data have been
fitted until day 150, when the second wave starts to appear. In the case of
Brazil and India the fit is performed until day 250. Note that Spain, France
and Belgium and Switzerland have similar $b$-values in the range $b\sim 12$–13
days. Italy, UK and Canada are well fitted with $b\sim 18$ days. Germany in in
between with $b\sim 15$, and Sweden is the European country with the highest
value $b\sim 27$ days.

Cumulative deaths, $D(t)$, are fairly smooth distributions and very similar
among different countries because we are dealing with sums —or integrals on
the continuous limit. More detailed information is obtained by describing the
daily death data $\Delta D(t)$, shown in Figs. 13 and 14. Although these data
show more fluctuations, they can be fitted well with the Gumbel $g(t)$
distribution, although the fit parameters differ somewhat from those obtained
by fitting the Gumbel function $G(t)$, since different functions and data are
being involved. the fitted parameter $b$ is in the third column of Table 2.
Again we only tabulate the parameter $b$, because $a$ and $t_{0}$ give simply
the relative height and the position of the peaks. in the case of the USA,
Brazil and India, we fit two Gumbel distributions, since it is apparent that
there are at least two overlapping waves. In these cases, in Table 2 we
tabulate the two values $b_{1}$ and $b_{2}$ of the two adjusted waves.

Figure 13:  Total deaths $D(t)$ during the first wave of the COVID-19 pandemic
for several countries compared to the Gumbel function $G(t)$. The parameter
$b$ of the Gumbel function fitted to the data is given in Table 2. Data are
form Ref.[39].  Figure 14:  Total deaths $D(t)$ during the first wave of the
COVID-19 pandemic for several countries compared to the Gumbel function
$G(t)$. The parameter $b$ of the Gumbel function fitted to the data is given
in Table 2. Data are form Ref.[39].  Figure 15:  Total deaths $D(t)$ during
the first wave of the COVID-19 pandemic for several countries compared to the
Gumbel function $G(t)$. The parameter $b$ of the Gumbel function fitted to the
data is given in Table 2. Data are form Ref.[39].  Figure 16:  Total deaths
$D(t)$ in the first wave of coronavirus pandemic for several countries,
compared to the Gumbel function $G(t)$. The parameter $b$ of the Gumbel
function fitted to the data is given in Table 2. Data are form Ref.[39].

We see that all countries fit one or two Gumbel distributions quite well, so
this function is an optimal candidate to quantitatively describe an epidemic
of these characteristics with only one parameter, $b$, plus the normalization
and position of the peak. The fact that the nine countries considered with an
isolated first wave (Spain, France, Italy, UK, Germany, Canada, Belgium,
Switzerland and Sweden) only require a time-independent parameter is
remarkable. This does not happen in the following waves or in other countries,
where the data show different behavior with large overlaps and stochastic
fluctuations.

Since Gumbel provides a good analytical approximation to the SIR model
solution, it is natural to wonder if the exact SIR solution would give an even
better description of the data. So in Figs. 15 and 16 we compare data with
exact SIR solutions given by the equations of the previous sections. We know
from the last section that there is a linear relationship between the
asymmetry of data and the basic reproduction number, $\rho$. This has allowed
us to obtain an approximate value of $\rho$ and then we have fitted the value
of $\beta$ and a normalization factor to the width and height of the data, we
have added a shift in time to get the position of the peak. The parameters
$\rho$, and $\beta$ are given in Table 2. In Fig. 16 we have fitted the US
data only to where the first peak is clearly seen. For this reason the other
two countries, Brazil and India have not been fitted.

In Figs. 15, 16, we also plot the results of the Gumbel distribution, but this
time the parameters have not been fitted to the data, but to the respective
SIR solution. Note that, from Eqs. (43) and (44)

$\beta b=\frac{\ln\rho}{\rho-1-\ln\rho}$ (70)

From inspections of the numbers given Table 2, columns 4, 5 and 7, we see that
this equation is approximately verified. For instance, for $\rho=7$ (Spain and
UK), the right-hand side of Eq. (70) gives 0.48, and roughly $1/\beta\simeq
2b$. For $\rho=3$ the equation gives $1/\beta\simeq b/1.2$.

We see that in general the fit of the Gumbel distribution to the SIR solutions
in these countries is good, although it begins to fail, for high values of
$\rho$, in the tail part, as we have already seen in the previous section

From Figs. 15 and 16 we conclude that the data of the countries considered are
globally well described with an exact SIR solution, without the need for any
time dependence of the parameters. Again we underline that this only happens
in the first wave of the countries that we have considered here and not in the
other countries or in the remaining waves. The cause of this requires a
detailed study of the epidemiological causes that is beyond the scope of this
paper.

Figure 17:  Locations of the different countries studied in the plane of SIR
parameters $(\rho,1/beta)$.

Finally, in Fig. 17 we plot the values of the parameters of the SIR solution
in the $(\rho,1/\beta)$ plane. If we fit a right line to these data, we see
that the general trend seems to be for $1/\beta$ to increase with $\rho$. If
we exclude the ’outsider’ countries, Belgium and Switzerland (which have small
values of $\rho=3$ or 4) and Sweden and USA (with large values of $\rho=7$,
8), we are left with six countries with similar parameters, in the central
zone, around $\rho=6$ and $1/\beta=20$ d —Spain, France, Italy, UK, Canada and
Germany. In these countries a relationship between $\beta$ and $\rho$ is not
found. On the other hand, the fact that $\beta$ is similar in these six
countries indicates that the probability of recovery is similar in all of
them. The recovery probability $\beta$ for an individual should in principle
be independent of location. But if we consider that there could be important
effects due to medical treatments and hospital capacity to treat severe cases
in different countries, this could explain the differences between the $\beta$
values.

Note that in most countries the value of $\rho$ exceeds five, indicating an
explosive increase in the first stage of the epidemic in each country. An
important consequence of the SIR equations for these high values of $\rho$ is
that the total number of people infected during the epidemic reaches almost
100%. Indeed, this is mathematically given by the value of the proper time
$\tau$ at the end of the epidemic, which is the solution of Eq. (16).
Numerically it is easy to verify that this end value is practically one, for
$\rho>5$. This can also be seen in the lower panel of Fig. 7, where the value
of $\tau(t)=r(t)$ for large $t$ is practically one. As a consequence, our
results indicate that the data from these countries where $\rho>5$ are
compatible with an epidemic where practically all initially susceptible
individuals were infected, according to the SIR model.

At this point we can already link with the question raised in the introduction
of this work, which is whether the lock-downs and other restrictions over the
population had any effect in reducing mortality. According to the meta-data
study of Ref.[33], the NPI had practically no effect on mortality. This seems
to be corroborated in our study for three reasons: (i) that the data are
compatible with SIR solutions with high value ($R_{0}=\rho\simeq 6$) of the
basic reproduction number, which implies that all the susceptible individuals
were infected; (ii) that the SIR parameters do not depend on time, but if
there were some NPI effects the parameters should be time dependent and the
epidemiological curves should differ from the SIR solutions; (iii) there does
not seem to be a relationship between the intensity of the lock-down measures
and the basic reproduction number. For example, Spain, which had very harsh
restrictions, is fitted with the same reproduction number as Sweden, which
practically did not have, and is greater than Italy, where the measures were
introduced a week earlier.

##  6 Conclusions

In this work we have solved the differential equations of the SIR model in a
parametric way using the proper time as a parameter, defined as the relative
number of recovered individuals $\tau=r(t)$. As a function of $\tau$, the SIR
solution is analytical, which allows us to study some of its properties, such
as, for example, calculating the maximum number of infected $i_{\rm peak}$ and
the asymptotic number of recovered at the end of the epidemic $r(\infty)$.

Secondly, we have studied the possibility of approximating the SIR solutions
using Gumbel distributions $g(t)$, because this family of functions presents a
similar asymmetry as the SIR solutions, and only depends on one parameter,
plus the normalization and the position. We have proposed various methods of
fitting Gumbel distributions to exact SIR solutions. In particular, using the
proper time, we have found simple relationships between the Gumbel parameters
and the SIR model parameters.

Third, we have discussed the scaling properties of the exact SIR solutions
when plotted as a function of $\beta t$, where $\beta$ is the probability of
removal per unit of time. Next we have defined an asymmetry parameter, as the
ratio between the right and left half-widths at half-height, of the SIR
solutions. We have shown numerically that the asymmetry ${\cal A}(\rho)$ grows
almost linearly with the reproduction number $\rho$ and that it is independent
of $\beta$. The asymmetry uniquely characterizes the value of $\rho$ and vice-
versa. Therefore, a measure of the asymmetry of some data of $i(t)$ at the
middle of the height, allows to extract the value of $\rho$.

Finally, we have applied the SIR model and the Gumbel distribution to study
the daily death-data in the first wave of the COVID-19 pandemic in a dozen
countries. The countries have been selected because they are the only ones
that present a peak that closely resembles a SIR solution. The data from
Spain, France, Italy, UK, Canada, Germany, Belgium, Switzerland and Sweden can
be fitted quite well with a SIR solution and also with a Gumbel function.
Except for Belgium and Switzerland, data in the rest of the countries are
compatible a reproduction number $\rho>5$. This seem to indicate that in
practically all susceptible individuals were infected and eventually
recovered. This raises questions about the effectiveness of non-pharmaceutical
interventions, such as lock-downs, in many countries.

##  7 Acknowledgments

The author thank Dr. Nico Orce for critical reading of the manuscript. This
work has been supported by the Spanish Agencia Estatal de investigacion
(D.O.I. 10.13039/501100011033, Grant No. PID2020-114767GB-I00), the Junta de
Andalucia (Grant No. FQM-225).

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11institutetext: Northwestern University, IL 60201, USA
††thanks: Those authors contribute equally to this paper. 11email:
<EMAIL_ADDRESS>
# Transformer based Generative Adversarial Network for Liver Segmentation

Ugur Demir  Zheyuan Zhang  Bin Wang  Matthew Antalek  Elif Keles  Debesh Jha
Amir Borhani  Daniela Ladner  Ulas Bagci

###### Abstract

Automated liver segmentation from radiology scans (CT, MRI) can improve
surgery and therapy planning and follow-up assessment in addition to
conventional use for diagnosis and prognosis. Although convolutional neural
networks (CNNs) have became the standard image segmentation tasks, more
recently this has started to change towards Transformers based architectures
because Transformers are taking advantage of capturing long range dependence
modeling capability in signals, so called attention mechanism. In this study,
we propose a new segmentation approach using a hybrid approach combining the
Transformer(s) with the Generative Adversarial Network (GAN) approach. The
premise behind this choice is that the self-attention mechanism of the
Transformers allows the network to aggregate the high dimensional feature and
provide global information modeling. This mechanism provides better
segmentation performance compared with traditional methods. Furthermore, we
encode this generator into the GAN based architecture so that the
discriminator network in the GAN can classify the credibility of the generated
segmentation masks compared with the real masks coming from human (expert)
annotations. This allows us to extract the high dimensional topology
information in the mask for biomedical image segmentation and provide more
reliable segmentation results. Our model achieved a high dice coefficient of
0.9433, recall of 0.9515, and precision of 0.9376 and outperformed other
Transformer based approaches. The implementation details of the proposed
architecture can be found at
https://github.com/UgurDemir/tranformer_liver_segmentation.

###### Keywords:

Liver segmentation Transformer Generative adversarial network

##  1 Introduction

Liver cancer is among the leading causes of cancer-related deaths, accounting
for 8.3% of cancer mortality [14]. The high variability in shape, size,
appearance, and local orientations makes liver (and liver diseases such as
tumors, fibrosis) challenging to analyze from radiology scans for which the
image segmentation is often necessary [3]. An accurate organ and lesion
segmentation could facilitate reliable diagnosis and therapy planning
including prognosis [5].

As a solution to biomedical image segmentation, the literature is vast and
rich. The self-attention mechanism is nowadays widely used in the biomedical
image segmentation field where long-range dependencies and context dependent
features are essential. By capturing such information, transformer based
segmentation architectures (for example, SwinUNet [2]) have achieved promising
performance on many vision tasks including biomedical image segmentation [7,
15].

In parallel to the all advances in Transformers, generative methods have
achieved remarkable progresses in almost all fields of computer vision too
[4]. For example, Generative Adversarial Networks (GAN) [6] is a widely used
tool for generating one target image from one source image. GAN has been
applied to the image segmentation framework to distinguish the credibility of
the generated masks like previous studies [11, 9]. The high dimensional
topology information is an important feature for pixel levell classification,
thus segmentation. For example, the segmented mask should recognize the object
location, orientation, and scale prior to delineation procedure, but most
current segmentation engines are likely to provide false positives outside the
target region or conversely false negatives within the target region due to an
inappropriate recognition of the target regions. By introducing the
discriminator architecture (as a part of GAN) to distinguish whether the
segmentation mask is high quality or not, we could proactively screen poor
predictions from the segmentation model. Furthermore, this strategy can also
allow us to take advantage of many unpaired segmentation masks which can be
easily acquired or even simulated in the segmentation targets. To this end, in
this paper, we propose a Transformer based GAN architecture as well as a
Transformer based CycleGAN architecture for automatic liver segmentation, a
very improtant clinical precursor for liver diseases. By combining two strong
algorithms, we aim to achieve both good recognition (localization) of the
target region and high quality delineations.

##  2 Proposed method

We first investigated the transformer architecture to solve the liver
segmentation problem from radiology scans, CT in particular due to its
widespread use and being the first choice in most liver disease
quantification. The self-attention mechanism of the Transformers has been
demonstrated to be very effective approach when finding long range
dependencies as stated before. This can be quite beneficial for the liver
segmentation problem especially because the object of interest (liver) is
large and pixels constituting the same object are far from each other. We also
utilized an adversarial training approach to boost the segmentation model
performance. For this, we have devised a conditional image generator in a
vanilla-GAN that learns a mapping between the CT slices and the segmentation
maps (i.e., surrogate of the truths or reference standard). The adversarial
training forces the generator model to predict more realistic segmentation
outcomes. In addition to vanilla-GAN, we have also utilized the CycleGAN [17],
[13] approach to investigate the effect of cycle consistency constraint on the
segmentation task. Figure 1 demonstrates the general overview of the proposed
method.

Figure 1: Block diagram of the Transformer GAN. (a) Vanilla GAN and (b) Cycle-
GAN with Transformer generator architectures.

###  2.1 Transformer based GAN

Like other GAN architectures [10], Transformer based GAN architecture is
composed of two related sub-architectures: the generator and the
discriminator. The generator part could generate the segmentation mask from
the raw image (i.e., segmentation task itself), while the discriminator tries
to distinguish predictions from the human annotated ground truth. GAN provides
a better way to distinguish the high-dimensional morphology information. The
discriminator can provide the similarity between the predicted masks and the
ground truth (i.e., surrogate truth) masks. Vanilla GAN considers the whole
segmentation to decide whether it is fake or not.

###  2.2 Transformer based CycleGAN

One alternative extension to the standard GAN approach is to use transformer
based segmentation model within the CycleGAN setup. Unlike a standard GAN,
CycleGAN consists of two generators and two discriminator networks. While the
first generator accepts the raw images as input and predicts the segmentation
masks, the second generator takes the predicted segmentation maps as input and
maps them back to the input image. The first discriminator classifies the
segmentation masks as either real or fake, and the second discriminator
distinguishes the real and the reconstructed image. Figure 1 illustrates this
procedure with liver segmentation from CT scans.

To embed transformers within the CycleGAN, we utilized the encoder-decoder
style convolutional transformer model [13]. The premise behind this idea was
that the encoder module takes the input image and decreases the spatial
dimensions while extracting features with convolution layers. This allowed
processing of large-scale images. The core transformer module consisted of
several stacked linear layers and self-attention blocks. The decoder part
increased the spatial dimension of the intermediate features and makes the
final prediction. For the discriminator network, we tried three convolutional
architectures. The vanilla-GAN discriminator evaluates the input image as a
whole. Alternatively, we have adopted PatchGAN discriminator architecture [8]
to focus on small mask patches to decide the realness of each region. It
splits the input masks into NxN regions and asses their quality individually.
When we set the patch size to a pixel, PatchGAN can be considered as pixel
level discriminator. W have observed that the pixel level discriminator tends
to surpass other architecture for segmentation. Figure 1 demonstrates the
network overview. In all of the experiments, the segmentation model uses the
same convolutional transformer and pixel level discriminator architectures.

##  3 Experimental setup

We have used Liver Tumor Segmentation Challenge (LiTS)[1] dataset. LiTS
consists of 131 CT scans. This dataset is publicly available under
segmentation challenge website and approved IRB by the challenge organizers.
More information about the dataset and challenge can be found here
111https://competitions.codalab.org/competitions/17094#learn_the_details.

All our models were trained on NVIDIA RTX A6000 GPU after implemented using
the PyTorch [12] framework. We have used 95 samples for training and 36
samples for testing. All models are trained on the same hyperparameters
configuration with a learning rate of $2e^{-4}$, and Adam optimizer with beta1
being 0.5 and beta2 being 0.999. All of the discriminators use the pixel level
discriminator in both GAN and CycleGAN experiments. We have used recall,
precision, and dice coefficient for quantitative evaluations of the
segmentation. Further, segmentation results were qualitatively evaluated by
the participating physicians. Our algorithms are available for public use.

Figure 2: Transformer based GAN liver segmentation results. Green: True positive, Red: False Positive, Blue: False Negative.  Table 1: Performance of Transformer based methods on the LITS dataset. [1] Method | Dice coefficient | Precision | Recall
---|---|---|---
Transformer [16, 13] | 0.9432 | 0.9464 | 0.9425
Transformer - CycleGAN (ours) | 0.9359 | 0.9539 | 0.9205
Transformer - GAN (ours) | 0.9433 | 0.9376 | 0.9515

##  4 Results

We presented the evaluation results in Table 1. Our best performing method was
Transformer based GAN architecture, achieved a highest dice coefficient of
0.9433 and recall rate of 0.9515. Similarly, our transformer based CycleGAN
architecture has the highest precision, 0.9539. With Transformer based GAN, we
achieved 0.9% improvement in recall and 0.01% improvement in dice coefficient
with respect to the vanilla Transformers. It is to be noted that we have used
also post-processing technique which boosts the performance for ”all” the
baselines to avoid biases one from each other.

Figure 2 shows our qualitative results for the liver segmentation. We have
examined all the liver segmentation results one-by-one and no failure were
identified by the participating physicians. Hence, visual results agreed with
the quantitative results as described in Table 1.

##  5 Conclusion

In this study, we explored the use of transformer-based GAN architectures for
medical image segmentation. Specifically, we used a self-attention mechanism
and designed a discriminator for classifying the credibility of generated
segmentation masks. Our experimental result showed that the proposed new
segmentation architectures could provide accurate and reliable segmentation
performance as compared to the baseline Transfomers. Although we have shown
our results in an important clinical problem for liver diseases where image-
based quantification is vital, the proposed hybrid architecture (i.e.,
combination of GAN and Transformers) can potentially be applied to various
medical image segmentation tasks beyond liver CTs as the algorithms are
generic, reproducible, and carries similarities with the other segmentation
tasks in biomedical imaging field. We anticipate that our architecture can
also be applied to medical scans within the semi-supervised learning, planned
as a future work.

## Acknowledgement

This study is partially supported by NIH NCI grants R01-CA246704 and
R01-R01-CA240639.

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