key: cord-0927665-dipntsni
authors: Odagaki, Takashi
title: Self-organization of oscillation in an epidemic model for COVID-19
date: 2021-03-18
journal: Physica A
DOI: 10.1016/j.physa.2021.125925
sha: 6f0bb813ebc69ef47a33d1971f74261b616c1cca
doc_id: 927665
cord_uid: dipntsni

On the basis of a compartment model, the epidemic curve is investigated when the net rate [Formula: see text] of change of the number of infected individuals [Formula: see text] is given by an ellipse in the [Formula: see text]- [Formula: see text] plane which is supported in [Formula: see text]. With [Formula: see text] , it is shown that (1) when [Formula: see text] , oscillation of the infection curve is self-organized and the period of the oscillation is in proportion to the ratio of the difference [Formula: see text] and the geometric mean [Formula: see text] of [Formula: see text] and [Formula: see text] , (2) when [Formula: see text] , the infection curve shows a critical behavior where it decays obeying a power law function with exponent [Formula: see text] in the long time limit after a peak, and (3) when [Formula: see text] , the infection curve decays exponentially in the long time limit after a peak. The present result indicates that the pandemic can be controlled by a measure which makes [Formula: see text].

, the time dependence of the daily confirmed new cases in more than 80 countries show oscillations whose periods range from one to five months depending on the country. The period of the oscillation is much shorter than that of Spanish flu in 1918 1919 which is the result of the mutation of virus, and it is an open question why the infection curve of COVID-19 shows oscillation in some countries.

There have been several compartmental models which explain epidemic oscillations [3, 4, 5] . The simplest idea to explain the oscillation is to introduce a sinusoidal time dependence of parameters of the model. Recently, Greer et al [6] introduced a dynamical model with timevarying births and deaths which shows oscillations of epidemics.

Since the infection curve of COVID-19 shows different features depending on the country, the infection curve must have a strong relation to the government policy, and the conventional approach may not be appropriate to COVID-19. In fact, different measures have been employed in each country by its government and citizens have been restricting the social contact among them, both of which depend on the infection status. Therefore, parameters including transmission coefficient of the virus can be considered to be a function of the infection status, and the non-linear effects due to this dependence must be clarified.

In this paper, I introduce a compartment model in which the net rate of change of the number of infected individuals Á is a function of Á and the function is given by an ellipse in the -Á plane which is supported in Á Á . Here, Á is the upper limit of the number of infected individuals above which the government does not allow, and Á is the lowest value below which the government will lift measures. I show that an oscillatory infection curve can be self-organized when Á ¼ and that the period is determined by the ratio of the difference Á Á and the geometric mean Ô Á Á of Á and Á . I also show that when Á ¼ the infection curve in the long time limit after a single peak decays following a power law function with exponent -2 and when Á ¼ it decays exponentially in the long time limit. 

The net rate of change of the number of infected individuals is written generally as

Here, ¬ and are the transmission rate of virus from an infected individual to a susceptible individual and a per capta rate for becoming a recovered non-infectious (including dead) individual (R), respectively, and Ë and AE are the number of susceptible individuals and the total population. In Eq. (2), « is a model-dependent parameter representing different effect of epidemics. In the SIR model [8] , it is assumed that no effects other than transmission and recovery are considered and thus « ¼ is assumed. The SEIR model [9] introduces a compartment of The SIQR model [10, 11] separates quarantined patients (Q) as a compartment in the population and « in Eq. (1) is given by the quarantine rate Õ ¡É´Øµ Á´Øµ where ¡É´Øµ is the daily confirmed new cases [7] . In the application of the SIQR model to COVID-19, it has been shown that

where is a typical value of the waiting time between the infection and quarantine of an infected individual. Therefore, the number of the daily confirmed new cases can be assumed to obey Here, I consider a model country in which depends on Á through

This implies that when Á becomes large, some policies are employed to reduce to the negative area so that Á´Øµ begins to decline and when Á becomes small enough, then some measures are lifted and becomes positive again. In fact, the plots of ´Øµ against Á´Øµ in many countries show similar loops [2] . Note that ¼ corresponds to either a maximum or a minimum of the number of infected individuals. In order to solve Eq. (1) with Eq. (4), I introduce a variable Ü through

and rewrite Eq. (1) as

where Ø ¼ Ø is the time scaled by ½ ¼ and

Equation (7) can be solved readily under the initial condition Á´Ø ¼ µ Á ¼ :

The infection curve is given in terms of Ø Ò´Ü ¾µ by

The infection curves are shown for Fig. 2 (a) and for ½ ¾ in Fig.   2 (b). Therefore, the infection curve is a periodic function when ½ and a decaying function with a single peak when ½.

Characteristics of the infection curve are in order:

(1) When ½, the infection curve shows a self-organized oscillation which can be characterized as follows:

1. The location of the peak Á Ñ Ü Á ¼ ½ · Á Á ¼ and the bottom Á Ñ Ò Á ¼ ½ Á Á ¼ are given by 

½ the infection curve is a decaying function with a single peak. The infection curve for ½ shown in both panels obeys a power-law decay in the long time limit after a peak.

respectively, where Ò ½ ¾ .

2. Therefore, the period Ì is given by

Namely, the period is given by the ratio of a half of the difference ¡ Á Á ¾ and the geometrical mean Ô Á Á of Á and Á .

(2) When ½, the infection curve shows a peak, after which it decays to zero. It can be characterized as follows:

1. The infection curve reaches its maximum Á Ñ Ü Á ¼ ¾ at Ø ¼ ½ .

2. In the long time limit, it decays as Ø ¾ .

J o u r n a l P r e -p r o o f (3) When ½, the infection curve shows a peak, after which it decays to zero. It can be characterized as follows:

1. The infection curve reaches its maximum Á Ñ Ü Á ¼

3. In the long time limit, the effective relaxation time defined by ÐÒ Á Ø ½ is given (1) In order to control the outbreak, a policy is needed to make

Since is determined by ¬, Ë, and «(or Õ), this can be achieved by the lockdown measure to reduce ¬, by the vaccination to reducing Ë and by the quarantine measure to increase Õ.

(2) The worst policy is Á ¼. In this case, oscillation continues until becomes negative due to the herd immunity by vaccination and/or infection of a significant fraction of the population.

(3) In order to make negative, it has been rigorously shown that increasing the quarantine rate Õ is more efficient than reducing the transmission coefficient ¬ by the lockdown measure [14] .

This result indicates that the pandemic can be controlled only by keeping measures of ¼ till

(4) It should be remarked that the change in the infectivity of the virus due to mutation can be included in ´Á µ in the present model. Namely effects due to new variants of SARS-CoV-2 found in UK, in South Africa or in Brazil can be included by moving the state to a new vs Á relation.

In this study, I assumed that Á ¼ is fixed and the dependence of on Á is symmetric. It is easy to generalize the present formalism to the case of non-symmetric dependence of on Á.

•The infection curve of COVID-19 is investigated in a model country where the correlation between the number of infected individuals and the infection rate is described by an ellipse.

• Oscillation of the infection curve is self-organized when the all measures are lifted before the number of infected individuals becomes zero.

• The period of the oscillation is given by the ratio of the difference and the geometrical mean of the maximum and minimum of the number of infected individuals allowed in the country.

• Convergence of the infection curve is realized only when the minimum number of the infected individuals allowed is zero.

• The present model provides a basic formalism for investigation of effects due to new variants of SARS-CoV-2.

Mathematical epidemiology

Proc. Natl. Acad. Sci