Self-organization of oscillation in an epidemic model for COVID-19
aa r X i v : . [ q - b i o . P E ] F e b Self-organization of oscillation in an epidemic modelfor COVID-19
Takashi Odagaki ∗ Kyushu UniversityNishiku, Fukuoka 819-0395, JapanandResearch Institute for Science Education, Inc.Kitaku, Kyoto 603-8346, Japan ∗ Corresondence to: Research Institute for Science Education, Inc.,Kitaku, Kyoto 603-8346, Japan
Email address : [email protected]
February 17, 2021
Abstract
On the basis of a compartment model, the epidemic curve is investigated when the netrate λ of change of the number of infected individuals I is given by an ellipse in the λ - I plane which is supported in [ I ℓ , I h ] . With a ≡ ( I h − I ℓ ) / ( I h + I ℓ ) , it is shown that (1) when a < or I ℓ > , oscillation of the infection curve is self-organized and the period of theoscillation is in proportion to the ratio of the difference ( I h − I ℓ ) and the geometric mean √ I h I ℓ of I h and I ℓ , (2) when a = 1 , the infection curve shows a critical behavior where itdecays obeying a power law function with exponent − in the long time limit after a peak,and (3) when a > , the infection curve decays exponentially in the long time limit after apeak. The present result indicates that the pandemic can be controlled by a measure whichmakes I ℓ < . Since the first outbreak in China in November 2019, COVID-19 has been spreading in all con-tinents including Antarctica. According to a recent analysis of infection status of 186 countries11, 2], the time dependence of the daily confirmed new cases in more than 80 countries showoscillations whose periods range from one to five months depending on the country. The periodof the oscillation is much shorter than that of Spanish flu in 1918 ∼ λ of change of thenumber of infected individuals I is a function of I and the function is given by an ellipsein the λ - I plane which is supported in [ I ℓ , I h ] . Here, I h is the upper limit of the number ofinfected individuals above which the government does not allow, and I ℓ is the lowest valuebelow which the government will lift measures. I show that an oscillatory infection curve canbe self-organized when I ℓ > and that the period is determined by the ratio of the difference I h − I ℓ and the geometric mean √ I h I ℓ of I h and I ℓ . I also show that when I ℓ = 0 the infectioncurve in the long time limit after a single peak decays following a power law function withexponent -2 and when I ℓ < it decays exponentially in the long time limit.2 Model country
In most of compartmental models for epidemics, the number of infected individuals I ( t ) isassumed to obey dI ( t ) dt = λI ( t ) . (1)The net rate of change λ of the number of infected individuals is written generally as λ = β SN − γ − α. (2)Here, β and γ are the transmission rate of virus from an infected individual to a susceptibleindividual and a per capta rate for becoming a recovered non-infectious (including dead) indi-vidual (R), respectively, and S and N are the number of susceptible individuals and the totalpopulation. In Eq. (2), α is a model-dependent parameter representing different effect of epi-demics. In the SIR model [8], it is assumed that no effects other than transmission and recoveryare considered and thus α = 0 is assumed. The SEIR model [9] introduces a compartment ofexposed individuals (E), and if one sets α = ( dE/dt ) /I , the basic equation of the SEIR modelreduces to Eq. (1).The SIQR model [10, 11] separates quarantined patients (Q) as a compartment in the pop-ulation and α in Eq. (1) is given by the quarantine rate q ≡ ∆ Q ( t ) /I ( t ) where ∆ Q ( t ) is thedaily confirmed new cases [7]. In the application of the SIQR model to COVID-19, it has beenshown that ∆ Q ( t ) ∝ I ( t − τ ) , (3)where τ is a typical value of the waiting time between the infection and quarantine of an infectedindividual. Therefore, the number of the daily confirmed new cases can be assumed to obeyEq. (1) with the redefined time t − τ . Since ∆ Q ( t ) is treated as an explicit variable instead of I ( t ) , the SIQR model is relevant to COVID-19.3n this paper, I focus on the time evolution of I ( t ) governed by Eq. (1) for COVID-19. Thetransmission coefficient is determined by characteristics of the virus and by government policiesfor lockdown measure and vaccination and by people’s attitude for social distancing. Medicaltreatment of infected individuals affects γ and the government policy on PCR test changes thequarantine rate. The government policies are determined according to the infection status andtherefore the rate of change is considered to be a function of I ( t ) in Eq. (1).Here, I consider a model country in which λ depends on I through λλ ! + (cid:18) I − I ∆ (cid:19) = 1 . (4)This implies that when I becomes large, some policies are employed to reduce λ to the negativearea so that I ( t ) begins to decline and when I becomes small enough, then some measures arelifted and λ becomes positive again. In fact, the plots of λ ( t ) against I ( t ) in many countriesshow similar loops [2]. Note that λ = 0 corresponds to either a maximum or a minimum of thenumber of infected individuals.Figure 1 shows this dependence, namely I h ≡ I + ∆ and I ℓ ≡ I − ∆ are the maximum andminimum of the number of infected individuals set by the policy in the country. When I ℓ < , λ in the region I < is not relevant since no infected individuals exist in this region.Figure 1: The dependence of the net rate λ on the number of infected individuals I in a modelcountry. 4 Infection curve and self-organization of oscillation
In order to solve Eq. (1) with Eq. (4), I introduce a variable x through λ = λ cos x, (5) I − I = ∆ sin x (6)and rewrite Eq. (1) as a dxd ˜ t = 1 + a sin x, (7)where ˜ t ≡ λ t is the time scaled by λ − and a ≡ ∆ /I ≥ is a parameter of the model.Equation (7) can be solved readily under the initial condition I ( t = 0) = I : ˜ t = a √ − a (cid:20) arctan tan( x/
2) + a √ − a − arctan a √ − a (cid:21) when a < , x/ x/ when a = 1 ,a √ a − " ln tan( x/
2) + a − √ a − x/
2) + a + √ a − − ln a − √ a − a + √ a − when a > . (8)The infection curve is given in terms of tan( x/ by I ( t ) I = 1 + a tan( x/ ( x/ . (9)The infection curves are shown for a = 0 . , . , . , in Fig. 2(a) and for a = 1 , , , in Fig.2(b). Therefore, the infection curve is a periodic function when a < and a decaying functionwith a single peak when a ≥ .Characteristics of the infection curve are in order:(1) When a < , the infection curve shows a self-organized oscillation which can be character-ized as follows:1. The location of the peak I max /I = 1+ a = I h /I and the bottom I min /I = 1 − a = I ℓ /I are given by ˜ t max ( n ) = 2 a √ − a arctan s − a a + ( n − π , (10)5a) (b)Figure 2: The infection curve for the model country. (a) When a < , a wavy infection curve isself-organized. (b) When a > the infection curve is a decaying function with a single peak.The infection curve for a = 1 shown in both panels obeys a power-law decay in the long timelimit after a peak. ˜ t min ( n ) = 2 a √ − a arctan s a − a + ( n − π , (11)respectively, where n = 1 , , . . . .2. Therefore, the period T is given by T λ = 2 πa √ − a = 4 π ( I h − I ℓ ) √ I h I ℓ . (12)Namely, the period is given by the ratio of a half of the difference ∆ = I h − I ℓ and thegeometrical mean √ I h I ℓ of I h and I ℓ .(2) When a = 1 , the infection curve shows a peak, after which it decays to zero. It can becharacterized as follows:1. The infection curve reaches its maximum I max /I = 2 at tλ = 1 .2. In the long time limit, it decays as t − . 6igure 3: The period when a < and the relaxation time in the long time behavior when a > are shown as functions of a .(3) When a > , the infection curve shows a peak, after which it decays to zero. It can becharacterized as follows:1. The infection curve reaches its maximum I max /I = 1 + a at tλ = a √ a − ln( a + √ a − .2. The infection curve returns to the initial state I ( t ) = I at tλ = a √ a − ln a + √ a − a −√ a − .3. In the long time limit, the effective relaxation time defined by τ ≡ − (cid:16) d ln Idt (cid:17) − is givenby τ λ = a √ a − .Figure 3 shows the period for a < and the relaxation time for a > as functions of a . I have shown that oscillation of the infection curve can be self-organized in the epidemic modeldescribed by an ordinary differential equation which exploits the net rate of change Eq. (4)7a) (b)Figure 4: (a) The time dependence of daily confirmed new cases in Japan from April 5, 2020to February 11, 2021. The dependence is fitted by piece-wise quadratic functions. (b) The netrate is shown as a function of the number of new cases. The spiral nature of this plot indicatesan enhancing wavy behavior of the infection curve. The jumps seen in the plot are due to theprocedure which does not impose the continuity of the curvature.depending on the number of infected individuals. All countries employ their own policy whichdepends on the infection status of the country and the relation Eq. (4) represents general trendof the policy. Namely, when the number of infected individuals approaches the maximumnumber acceptable in a country, a strong measure is introduced to make the net rate of change λ negative, and the measure will be lifted when the number of infected individuals is consideredto be small enough, which makes λ > . Therefore, the policy with I ℓ > itself is consideredto be the origin of the oscillation of the infection curve and the policy with I ℓ < seems tohave succeeded in controlling the pandemic [2]. As an example, I show in Fig. 4(a) the timedependence of daily confirmed new cases in Japan from April 5, 2020 to February 11, 2021which consists of three waves. Using λ determined by fitting the data by piece-wise quadraticfunctions as shown by the solid curve [2], I show the correlation between λ and ∆ Q in Fig.4(b).Some of oscillations in biological systems such as the prey and predator system have been8xplained by the Lotka-Volterra model [12, 13], which is essentially a coupled logistic equationand it is reducible to a second order non-linear differential equation for one variable. Since thepresent model is based on a first order non-linear differential equation, the origin of oscillatorysolution of the present model is different from that of the Lotka-Volterra model.Several important implications of the present results are:(1) In order to control the outbreak, a policy is needed to make a > or I ℓ < and λ < .Since λ is determined by β , S , γ and α (or q ), this can be achieved by the lockdown measure toreduce β , by the vaccination to reducing S and by the quarantine measure to increase q .(2) The worst policy is I ℓ > . In this case, oscillation continues until λ becomes negative dueto the herd immunity by vaccination and/or infection of a significant fraction of the population.(3) In order to make λ negative, it has been rigorously shown that increasing the quarantine rate q is more efficient than reducing the transmission coefficient β by the lockdown measure [14].This result indicates that the pandemic can be controlled only by keeping measures of λ < till I = 0 .(4) It should be remarked that the change in the infectivity of the virus due to mutation canbe included in λ ( I ) in the present model. Namely effects due to new variants of SARS-CoV-2found in UK, in South Africa or in Brazil can be included by moving the state to a new λ vs I relation.In this study, I assumed that I is fixed and the dependence of λ on I is symmetric. It is easyto generalize the present formalism to the case of non-symmetric dependence of λ on I . Acknowledgments
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