Since November 2019, the pandemic COVID-19 is still expanding in the world. The infection curves of some countries like Australia, Austria, Japan, Luxemburg and Serbia [1] show clearly a wavy structure. Since the period of the wave is much shorter than that of the wave 
In the SIQR model [2, 3] , population are separated into four compartments; susceptible individuals, infected individuals at large (will be called infecteds for simplicity), quarantined patients in hospitals or at home who are no longer infectious in the community and recovered (and died) patients. The population in each compartment are denoted by S, I, Q and R, respectively, and the total population is given by N(= S + I + Q+ R). The basic equations for the time evolution of the populations are given by a set of ordinary differential equations.
where t is the time. The term βS I N denotes the net rate at which infections spread, where β is a transmission coefficient determined by lockdown measure including social-distancing and self-isolation of people and by the trait of virus. Infected individuals at large, regardless of whether they are symptomatic or asymptomatic, are quarantined at a per capita rate q and become non-infectious to the population. The quarantine rate is determined by the government policy on PCR (Polymerase Chain Reaction) test. Quarantined patients recover at a per capita rate γ ′ (where 1/γ ′ is the average time it takes for recovery) and infected individuals at large become non-infectious at a per capita rate γ (where 1/γ is the average time that an infected patient at large is capable of infecting others). It is apparent that Eqs. (1) ∼ (4) guarantee the conservation of population N = S + I + Q + R.
At the end of September 2020, the total number of infected, quarantined and recovered people is much smaller than the entire population in any countries, and thus the pandemic can be regarded as in its early stage far from the stage of herd immunization. Therefore, I can assume that I + Q + R << N is satisfied and S = N − (I + Q + R) ≃ N. Then the basic equation governing the time evolution of the number of infecteds is written as
where the net rate of change of the number of infecteds denoted as
determines the short-term behavior of the number of infecteds. The number of infecteds increases when λ > 0 and decreases when λ < 0.
It is straightforward to obtain the solution to Eq. (5):
where I(0) is the initial number of infecteds.
The observed data for the outbreak of COVID-19 is the daily confirmed new cases ∆Q(t) ≡ qI, which is given by a convolution of the incubation period distribution function ψ(t) and the number of infecteds I(t). Therefore ∆Q(t) can be expressed as
Since the incubation period distribution is a well behaved function with a single peak [12, 13] , the convolution can be evaluated by the saddle-point method of integration and it is given by [3] 
where τ is a characteristic time representing the peak position of ψ(t) and ψ ′′ (t) = d 2 ψ(t) dt 2 . Therefore, the short-term behavior of the number of daily confirmed new cases can be described by a function similar to Eq. (7).
The first wave of the outbreak of COVID-19 in various countries has been analyzed on the basis of Eq. (7), where ∆Q(t) is approximated by a piece-wise exponential function [3, 4, 5, 6, 7, 8, 9] .
As can be seen in Eq. (7), the time dependence of ∆Q(t) is determined by λ(t) alone and it is a difficult task to convert it to β, q and γ [10, 11] . Here, I first assume that γ is a constant since no treatment could be given to infecteds and set γ = 0.04 [12, 13, 14] .
Next, I assume that β(t) and q(t) change in time continuously between two values represented by a hyperbolic tangent function Table 1 . The quarantine rate is increased from q(0) = 0.02 to q(200)= 0.029 as explained in the text. Fig. 1 . + B is set to satisfy β(t 2i ) = β 2i and β(t 2i+2 ) = β 2i+2 for t i 's listed in Table 1. for t 2i ≤ t ≤ t 2i+2 with β(t 2i ) = β 2i and β(t 2i+2 ) = β 2i+2 and t 2i+1 as the transition point and dt 2i+1 as the width of the transition (i = 0, 1, 2, . . .). Table 1 summarizes parameters determining the time dependence of β(t) used for fitting in Fig. 1 . Figure 2 shows the time dependence of β(t), q(t) and λ(t). It should be emphasized that the assignment of β and q from λ is not unique since adding any amount to β and q at a given time does not change λ. In the present study, I assumed that the time dependence of q is weak since the procedure for the PCR test did show no drastic change in the period for identifying infected individuals at large.
The time dependence of the transmission coefficient β(t) must be attributed to the attitude of people to self-isolation under government policy and massive information from news media.
Here I consider the transmission coefficient is a function of I and dI dt , and I introduce a model country in which β is given by
β h when dI dt > 0 and I ≤ I h β ℓ when dI dt < 0 and I ℓ ≤ I < I h , where β ℓ < q + γ < β h is satisfied. Figure 3 (a) shows β(I).
The infection curve for β(I) given by Eq. (13) is shown in Fig. 3 
According to the data available at Coronavirus Resource Center, Johns Hopkins University [1] , the daily confirmed new cases in each country up to the end of September seems to be classified into the following five types:
Type I The infection curve keeps increasing, like India, Iraq and Argentina.
Type II After the first peak with a certain length of tail, the infection curve increases again like type I. This is seen in Russia and many European countries.
Type III The infection curve shows oscillation like in Austria, Australia, Luxemburg, Japan and Serbia.
Type IV Infection curve is characterized by a sharp peak followed by more or less constant infection for a long time. This infection curve is seen in USA and Sweden.
Type V After a small peak, few new cases have been observed like in China, Taiwan, Thailand and Viet Nam.
As discussed in Sec. 3, the relative magnitude of β and q + γ must be responsible for the structure of the infection curve. Here, keeping q + γ constant, I discuss the relative magnitude of these parameters for different types of infection curve. It should be emphasized that the difference β − (q + γ) determines the infection curve. Here, for the sake of simplicity, I fix q + γ and attribute all effects to change in β. It is possible to discuss in the same way by changing q with a fixed β.
For Type I infection curve, β > q + γ is satisfied ( Fig. 4(a) ) and thus the infection curve keeps increasing. The infection curve will reach eventually its maximum and start to decline because of the non-linear term SI/N in Eqs. (1) and (2) . The Spanish flu belongs to this type.
Type II infection curve will be realized when a strong lockdown measure is introduced at the outbreak and it is lifted in fear of economic break down ( Fig. 4(b) ). After a little peak and some length of tail, the infection curve will follow the same trend as Type I.
Wavy infection curve (Type III, Fig. 4 (c) ) has already been discussed in Sec. 3.
Infection curve of Type IV is characterized by a fixed point in the β -I plane which is reached after the first peak ( Fig. 4(d) ), and I ℓ determines the size of the daily confirmed new cases.
Type V infection curve represents the most efficient measure; the transmission coefficient is brought below q + γ (or q is increased) so that β < q + γ is satisfied, and the measure is kept.
The trajectory in the β -I plane has a fixed point near I = 0 as shown in Fig. 4(e) . 
In this paper, I discussed the infection curves of COVID-19 observed in many countries and showed that the infection curve can be classified into five types. In particular, a wavy infection curve can be self-organized due to change in self-isolation and/or quarantine measures making β above or below q + γ. It is shown that these different infection curves are caused by relative strength of lockdown measure and quarantine measure.
It will be possible to formulate the optimum policy specific to the country for controlling the outbreak on the basis of the present theoretical framework, if the cost function and the aim of policy in each country are given [15] .
The pandemic in countries whose infection curve is of Type I or Type II will stamp out when sufficient number of population get immunity. According to percolation theory, the condition for herd immunity is that the fraction of immunized individuals is larger than a critical value [16] p c = 1 − 4.5 nβ ,
where β is the transmission coefficient and n is the average number of people with whom an infected individual meets while it is infectious. The critical value depends on β and n and it could be as large as 50 ∼ 80 %. Therefore, it could take much longer time before the herd immunity is realized in any countries in the world.