The COVID-19 outbreak has motivated a large number of numerical studies using epidemiology models [1, 2] . A commonly used model is the Susceptible-Exposed-Infected-Removed (SEIR) model [3] . This model is formulated as a system of nonlinear ordinary differential equations, for which no exact analytic solution has yet been found. For this reason, most of the recent works focus on the numerical analysis of statistical ensembles of initial data for these equations. However, due to the uncertainty and often unreliability of the clinical data, the prediction about the real evolution of the epidemic is rather difficult, if not impossible [4, 5] . On the other hand, the SEIR epidemic model provides a deterministic evolution for some given initial state. Therefore, the aim of this work is to provide simple expressions of the main characteristics of the population of individuals that have been in contact with the disease, as of instance the peak of the infected population and the time after which it occurs, the final number of individuals who have contracted the disease and the temporal shape of the infectious population's curves. These analytical expressions can become useful through their application to the COVID-19, to obtain fundamental parameters as the reproduction number r and the epidemic starting time.
The paper is organized as follow, In sec. II we recall the SEIR model;
in sec. III we study the linear regime with the exponential growing and decaying evolution, depending on the reproduction number r; in sec. IV we investigate the nonlinear regime in the free spread evolution with r > 1. We approximate the exact model of equations by a reduced model where the decaying mode is adiabatically eliminated. This reduced model allows to obtain analytical results which have been seen to be in good agreement with the exact numerical solution. Sec. V summarizes the results and draws the conclusions.
We used the susceptible-exposed-infected-removed (SEIR) compartment model [3, 6, 7, 8] to characterize the early spreading of COVID-19, where each individual could be in one of the following states: susceptible (S), exposed (E, being infected but without infectiousness), infected (I, with infectiousness), recovered (R) and dead (D). At later times a susceptible individual in the state S would turn to be an individual in the exposed state E with a rate r/τ I , where r is the reproduction number (i.e. the average number of infected people generated by each infected person during the desease) and τ I = 1/γ 2 is the average time in the infected state I. An exposed individual in the state E becomes infected, i.e. in the state I in an average time τ E = 1/γ 1 . Then the infected individual is removed from the total population with the rate γ 2 either by recovering (R) or dying (D) with a mean case fatality proportion p.
The dynamical process of SEIR is described by the following set of equations:
Here S(t), E(t), I(t), R(t) and D(t) respectively represent the number of individuals in the susceptible, exposed, infectious, recovered and death states at time t and N is the total number of individuals in the system such that
Finally, the cumulative population C is
equal to the total population of individuals who have contracted the infection.
If E(t), I(t), R(t) N (t), then the susceptible population S can be approximated by the total population N (i.e. S ∼ N ) and the equations for the exposed and infected population are linear:
Introducing the Laplace transforms
with Reλ > 0, Eqs. (7) and (8) becomes
where E(0) and I(0) are the initial conditions. The eigenvalues λ are solution
giving det(λ) = λ 2 + (γ 1 + γ 2 )λ + γ 1 γ 2 (1 − r) = 0 (11) with solutions
where
Since ∆ > 0 the eigenvalues are real. Depending on r, we distinguish three cases:
(a) If r > 1 then √ ∆ > γ 1 + γ 2 , so that λ + > 0 and λ − < 0. The solution grows exponentially (explosive regime);
(b) If r < 1 then √ ∆ < γ 1 + γ 2 , so that both λ + < 0 and λ − < 0. The solution decays exponentially (relaxation regime);
(c) If r = 1 then √ ∆ = γ 1 + γ 2 , so that λ + = 0 and λ − = −(γ 1 + γ 2 ). The solution remains partially constant (marginally stable regime).
whereas in the case (c) (r = 1) the solution is
The only parameter which can be controlled by confinement measures is the reproduction number r. In the following we assume that for COVID-19 the characteristic times are τ E = 3.69 days and τ I = 3.48 days [9] . We consider the time evolution of the population E and I for r > 1, r = 1 and r < 1, corresponding to the explosive, marginally stable and relaxation regimes, respectively.
For r > 1 and λ + t 1,
with λ + > 0.
When r = 1, in the asymptotic limit t τ E , τ I , E and I are constant,
and the death population grows linearly in time
where D(0), E(0) and I(0) are the values taken at time when r starts to be r = 1.
When r < 1, λ + is negative and E and I tend to zero, whereas D tends to the following constant value,
where D(0), E(0) and I(0) are the values taken at time when r starts to be r < 1. Fig. 1 shows a typical temporal evolution of I(t) and D(t) starting with r > 1, then subsequently changed to r = 1 and later on to a value r < 1. 
In the following, we investigate the nonlinear regime with a constant reproduction number r > 1. This corresponds to a free spread of the infection, with an initial exponential growth of the exposed population E, and so also of I and D. The exponential growth stops when susceptible population S be- 
These equations have a single steady-state solution (i.e.ẋ =ẏ =ż = 0) with x = y = 0 (end of the epidemics) and z = z 0 with 0 < z 0 < 1. This solution is stable if r 0 = r(1 − z 0 ) < 1. We see that the stability condition
In Fig. 2 We observe that C/N tends to a steady-state value of about 0.6, whereas the peak of I/N is about 0.03: it means that for these parameters the 60% of the total population has contracted the infection and the peak the infected population is about 3% of the total population. Note that these results are independent on p and depend only on τ E , τ I and r.
In this section we find an approximated analytic solution of Eqs. (7) and (8) in the form
the normalized eigenvectors associated to the eigenvalues λ ± of Eq.(12) are where
In the new basis Eqs.(27)-(29) take the form:
Notice that as expected in the linear regime the dynamics ofx andȳ are uncoupled. Now we consider the free spread regime with r > 1 such that λ + is positive and λ − is negative. If r − 1 is small, then |λ − | λ + and we can adiabatically eliminate the 'slave' variableȳ. Neglectingẏ in (37) we obtain
which when inserted in Eqs.(36) and (38) yieldṡ
where k = (r − 1)/r and β = λ + /k √ ∆. Finally, the original variables are
Eqs.(42) and (43) may provide some analytical result. Rescaling the time
and defining
Eqs.(42) and (43) take the form:
In the limit β → 0 they have the form of Lotka-Volterra equations [11] . From them, dividing member by member, it results which when integrated yields
where we assumed s → 0 when z → 0. On the other hand, s → 0 when z → z ∞ (see Fig.3 ), where z ∞ is the solution of the transcendental equation
The same transcendental equation (53) for z ∞ has been obtained for the SIR compartmental model [12, 13] . Here we have demonstrated its validity also for the SEIR model.
We see from Fig. 3 that s = 0 for z = 0 and z = z ∞ . The maximum value of s occurs when z = k = 1 − 1/r so that
We have obtained analytical expressions for the asymptotic value of the cumulative population fraction C/N and the peak of the infectious population fraction I/N in the case of free spread evolution of COVID-19. Furthermore, we have obtained approximated expressions of these quantities as a function of time and the times at which the peak and the end of the epidemics is expected. We summarize here below these results: and (54),
The agreement of this expression with the exact result shown in Fig.4(b) is better for values of r closer to the threshold r = 1.
(c) We have obtained an approximated temporal profile of C(t)/N ,
where time:
where β = λ + /k √ ∆ and
The good agreement of Eqs. (62) and (63) (d) The time at which the peak of I/N is reached is 