A nucleation and growth model for COVID-19 epidemic in Japan
AA nucleation and growth model for COVID-19 epidemic in Japan
Yoshihiko Takase ∗ August 5, 2020
Abstract
COVID-19 epidemics in Japan and Tokyo were an-alyzed by a fundamental equation of the dynamicphase transition. As a result, the epidemic was foundto be in good agreement with the random nucleationand linear growth model suggesting that the epidemicbetween March 13, 2020 and May 22, 2020 was simplyrate-limited by the three constant-parameters: theinitial susceptible, domain growth rate, and nucle-ation decay constant. This model provides a goodpredictor of the epidemic because it consists of oneequation and the initial specific plot is linear.
In Japan, the first case of COVID-19 was reportedon January 16, 2020. The Japanese governmentadmitted the cruise ship ”Diamond Princess” withthe patients to the port on February 3, and beganquarantine[1]. This news was widely reported andprovided an opportunity for most Japanese to knowthe danger of COVID-19. On February 24, 2020,the National Expert Meeting announced its views[2]and on February 27, Prime Minister Abe requestedschools all over the country to close[3]. The govern-ment announced a state of emergency to seven pre-fectures on April 7, 2020, and expanded it nationwideon April 16[4]. The daily number of new infectionsin Japan reached a peak around April 12th, and thenbegan to decline, reaching around 30 in late May.We previously analyzed the ferroelectric polariza-tion reversal phenomenon of polymers[5] by a nu-cleation and domain growth model of the dynamic ∗ Chiba JICA Senior Volunteers Association phase transition, one of the general theory of physics.This time, we have noted that the polymers, whichare complex systems of crystalline and amorphousphases, resemble human society and that the time de-pendence of COVID-19 new infections resembles thepolarization reversal characteristics. The purpose ofthis study is to see if the fundamental equation of themodel is directly applicable to the spread of COVID-19 infection, although the epidemic is commonly an-alyzed by the SIR and its expanded models[6, 7, 8].
We first consider the fraction of domain X (cid:48) that hasbeen transformed by formation and growth of ficti-tious nuclei without mutual impingement. This frac-tion X (cid:48) at time t is expressed in the following form[9], X (cid:48) = (cid:90) t v ( t, τ ) ˙ N ( τ )d τ, (1)where v ( t, τ ) is the volume of a nucleus born at time τ and grown without any restriction until time t ( τ ≤ t ) and ˙ N ( τ ) is the nucleation probability per unitvolume of untransformed region. The term in theintegrand involves the usual random nucleation at arate of ˙ N ( τ ) = N ν e − ντ followed by a steady-statedomain growth, where N is the number of activepoints for nucleus and ν the decay constant.The actual volume fraction X that has undergonetransformation at time t is related to X (cid:48) by[10]d X d X (cid:48) = 1 − X. (2)1 a r X i v : . [ phy s i c s . s o c - ph ] A ug ntegrating Eq.(2), we obtain X = 1 − exp( − X (cid:48) ) . (3)Substituting X (cid:48) in Eq.(1) into Eq.(3), we obtainthe following fundamental equation of the dynamicphase transition. X = 1 − exp (cid:20) − (cid:90) t v ( t, τ ) ˙ N ( τ )d τ (cid:21) . (4)If we assume the total number of daily new in-fections D ( t ) to be proportional to the transformedvolume fraction X , then D ( t ) = D s (cid:20) − exp (cid:18) − (cid:90) t v ( t, τ ) ˙ N ( τ )d τ (cid:19)(cid:21) , (5)where D s is the initial susceptible D ( ∞ ). The volume that is born at time τ and grows one-dimensionally until time t ( τ ≤ t ) is v ( t, τ ) = S c G ( t − τ ) , (6)where G is the growth speed and S c is the growthcross section.Integrating Eq.(4) after substitution of Eq.(6),ln 11 − X = − S c GN ν f , (7)where f = 1 − e − νt − νt .When νt is small enough, Eq.(7) becomesln 11 − X (cid:39) S c GN ν t . (8)The total number of daily new infections is D ( t ) = D s (cid:20) − exp (cid:18) S c GN ν f (cid:19)(cid:21) . (9)The number of daily new infections is driven bydifferentiation of Eq.(9) as J ( t ) = D s S c GN (1 − e − νt ) exp (cid:18) S c GN ν f (cid:19) . (10) The volume for the two-dimensional growth is v ( t, τ ) = πG ( t − τ ) l c , (11)where l c is the domain thickness.Integrating Eq.(4) after substitution of Eq.(11),ln 11 − X = 2 πG l c N ν f , (12)where f = 1 − e − νt − νt + ( νt ) / νt is small enough, Eq.(12) becomesln 11 − X (cid:39) πG l c N ν t . (13)The total number of daily new infections is D ( t ) = D s (cid:20) − exp (cid:18) − πG l c N ν f (cid:19)(cid:21) . (14)The number of daily new infections is J ( t ) = D s (cid:18) − πG l c N ν f (cid:19) exp (cid:18) − πG l c N ν f (cid:19) . (15) The COVID-19 data of Japan is provided in someweb sites. Using the data[11], and assuming theone-dimensional growth, Fig. 1 shows the ln(1 / (1 − X )) T ime characteristics of Japan. The red markeris the detected infections, the black line is the theo-retical curve of Eq.(7), and the orange line is Eq.(8).The 95% confidence interval (95%CI) was obtainedby the moving standard deviation calculated over asliding window of 7 days across neighboring days.The gray broken lines neighboring the theoretical linerepresent the 95%CI. Appling the same 95%CI datato Eq.(8), the slope of the orange line was estimatedto be 0 . ± . S c GN ν/ D ( t ) − t curveby the least-square scheme using the slopes shownin Fig. 1. The J ( t ) − t curve has larger relative95%CI range, e.g. 509 ± J ( t ), than the D ( t ) − t curve, e.g. 5657 ± . J ( t ) simply because J(t) is the num-ber of daily announce of the PCR-tested-positive per-sons and D ( t ) is the sum of them. The obtainedvalues of the parameter were D s = 15150 [person],0 . − . ≤ S c GN ≤ . . ν = 0 . D ( t ) was the value ob-tained by subtracting the sample value on March 26,2020 as the baseline value and ν was assumed to beconstant.Using the same data, and assuming the two-dimensional growth, Fig. 3 shows the ln(1 / (1 − X )) T ime characteristics of Japan similar to Fig.1. The 95%CI was calculated as same as that inFig. 1. The slope of the orange line was esti-mated to be 1 . × − − . × − ≤ slope ≤ . × − + 1 . × − as shown in the figure,which gives the value of πG l c N ν/ D s = 15720 [person], 0 . − . ≤ πG l c N ≤ . . / day ] , ν =0 . D ( t ) was the value obtained bysubtracting the sample value on March 13, 2020 asthe baseline value and ν was assumed to be constant.The one-dimensional model was superior to thetwo-dimensional model because the standard devia-tions of ln(1 / (1 − X )) data were 0.043 and 0.252, andthose of D ( t ) were 135 and 217, respectively. y = 0.00168 xy = 0.00170 xy = 0.00172 x0.01.02.03.04.05.06.00.0E+00 5.0E+02 1.0E+03 1.5E+03 2.0E+03 2.5E+03 3.0E+03 3.5E+03 4.0E+03 l n ( / ( - X )) Time² [day²] ln(1/(1-X)) - Time² Characteristics of Japan
Theory 95%CI SGN₀νt²/2 Detected Linear (95%CI)
Figure 1: ln(1 / (1 − X )) T ime characteristics ofJapan. The red marker represents the detected in-fections, the black line the theoretical curve (1-dim.),the gray broken lines the 95%CI, the orange line thetheoretical line (1-dim.) for the small νt region, andthe green lines linear approximations of the 95%CIcurves. T o t a l i n f e c t i o n s DD a il y i n f e c t i o n s J [ / d a y ] COVID-19 Epidemic versus Time in Japan
Theoretical J(t) 95%CI_J Daily infectionsTheoretical D(t) 95%CI_D Total infections
Figure 2: COVID-19 epidemic versus time in Japan.The red marker represents the detected total infec-tions, the black line the theoretical curve (1-dim.)of D ( t ), the gray broken lines the 95%CI, the greenmarker the detected daily infections, the blue curvethe theoretical curve (1-dim.) of J ( t ), and the bluebroken lines the 95%CI curves.3 = 1.98E-05xy = 2.12E-05xy = 1.85E-05x0.01.02.03.04.05.06.07.08.00.0E+00 5.0E+04 1.0E+05 1.5E+05 2.0E+05 2.5E+05 3.0E+05 3.5E+05 4.0E+05 l n ( / ( - X )) Time³ [day³] ln(1/(1-X)) - Time³ Characteristics of Japan
Theory 95%CI πG²lN₀ν/395%CI Detected Linear (95%CI)
Figure 3: ln(1 / (1 − X )) T ime characteristics ofJapan. The red marker represents the detected in-fections, the black line the theoretical curve (2-dim.),the gray broken lines the 95%CI, the orange line thetheoretical line (2-dim.) for the small νt region, theorange dotted lines the 95%CI, and the green lineslinear approximations of the 95%CI curves. The COVID-19 data of Tokyo is provided in someweb sites. Using the data[12], and assuming theone-dimensional growth, Fig. 5 shows the ln(1 / (1 − X )) T ime characteristics of Tokyo similar to Fig. 1.The slope of the orange line was estimated to be0 . ± . D s = 5000 [person], 0 . − . ≤ S c GN ≤ . . ν = 0 . D ( t ) was the value obtained by subtracting the sam-ple value on March 25, 2020 as the baseline value and ν was assumed to be constant.Using the same data, and assuming the two-dimensional growth, Fig. 7 shows the ln(1 / (1 − X )) T ime characteristics of Tokyo similar to Fig.3. The slope of the orange line was estimated to be1 . × − ± . × − as shown in the figure. T o t a l i n f e c t i o n s DD a il y i n f e c t i o n s J [ / d a y ] COVID-19 Epidemic versus Time in Japan
Theoretical J(t) 95%CI_J Daily infectionsTheoretical D(t) 95%CI_D Total infections
Figure 4: COVID-19 epidemic versus time in Japan.The red marker represents the detected total infec-tions, the black line the theoretical curve (2-dim.)of D ( t ), the gray broken lines the 95%CI, the greenmarker the detected daily infections, the blue curvethe theoretical curve (2-dim.) of J ( t ), and the bluebroken lines the 95%CI curves.Fig. 8 shows the number of daily new infectionsand the total number of them versus time in Tokyoin comparison with the two-dimensional theoreticalcurves of Eqs.(15) and (14). The parameters of thetheoretical curves were obtained as mentioned above, D s = 5080 [person], 0 . − . ≤ πG l c N ≤ . . / day ], ν = 0 . D ( t ) was the value obtained by subtracting the sam-ple value on March 13, 2020 as the baseline value and ν was assumed to be constant.The one-dimensional model was a little inferior tothe two-dimensional one because the standard devi-ations of ln(1 / (1 − X ) were 0.119 and 0.081, respec-tively, although those of D ( t ) were 55 and 55, respec-tively. In order for one new phase to form in the basic phase,a new phase must be born and grow. In the caseof COVID-19, the nucleus is the first person whocauses successive infection. The location of nucle-4 = 0.00156 xy = 0.00151 xy = 0.00146 x0.01.02.03.04.05.06.00.0E+00 5.0E+02 1.0E+03 1.5E+03 2.0E+03 2.5E+03 3.0E+03 3.5E+03 4.0E+03 l n ( / ( - X )) Time² [day²] ln(1/(1-X)) - Time² Characteristics of Tokyo
Theory 95%CI SGN₀νt²/295%CI Detected Linear (95%CI)
Figure 5: ln(1 / (1 − X )) T ime characteristics ofTokyo. The red marker represents the detected in-fections, the black line the theoretical curve (1-dim.),the gray broken lines the 95%CI, the orange line thetheoretical line (1-dim.) for the small νt region, theorange dotted lines the 95%CI, and the green lineslinear approximations of the 95%CI curves. T o t a l i n f e c t i o n s DD a il y i n f e c t i o n s J [ / d a y ] COVID-19 Epidemic versus Time in Tokyo
Theoretical J(t) 95%CI_J Daily infectionsTheoretical D(t) 95%CI_D Total infections
Figure 6: COVID-19 epidemic versus time in Tokyo.The red marker represents the detected total infec-tions, the black line the theoretical curve (1-dim.)of D ( t ), the gray broken lines the 95%CI, the greenmarker the detected daily infections, the blue curvethe theoretical curve (1-dim.) of J ( t ), and the bluebroken lines the 95%CI curves. y = 1.81E-05xy = 1.86E-05xy = 1.91E-05x0.01.02.03.04.05.06.07.00.0E+00 5.0E+04 1.0E+05 1.5E+05 2.0E+05 2.5E+05 3.0E+05 3.5E+05 4.0E+05 l n ( / ( - X )) Time³ [day³] ln(1/(1-X)) - Time³ Characteristics of Tokyo
Theory 95%CI πG²lN₀ν/395%CI Detected Linear (95%CI)
Figure 7: ln(1 / (1 − X )) T ime characteristics ofTokyo. The red marker represents the detected in-fections, the black line the theoretical curve (2-dim.),the gray broken lines the 95%CI, the orange line thetheoretical line (2-dim.) for the small νt region, theorange dotted lines the 95%CI, and the green lineslinear approximations of the 95%CI curves. T o t a l i n f e c t i o n s DD a il y i n f e c t i o n s J [ / d a y ] COVID-19 Epidemic versus Time in Tokyo
Theoretical J(t) 95%CI_J Daily infectionsTheoretical D(t) 95%CI_D Total infections
Figure 8: COVID-19 epidemic versus time in Tokyo.The red marker represents the detected total infec-tions, the black line the theoretical curve (2-dim.)of D ( t ), the gray broken lines the 95%CI, the greenmarker the detected daily infections, the blue curvethe theoretical curve (2-dim.) of J ( t ), and the bluebroken lines the 95%CI curves.5tion changes randomly as the infected person moves.If the nucleus is born at each place, it spreads in oneor two dimensions. This process fits the random nu-cleation and growth model. The characteristics havebeen determined by three parameters: D s , domaingrowth rate S c GN or 2 πG l c N , and ν .In Japan, there were approximately 16,000 pre-sumed susceptible between March 13 and May 24,2020. This is only about 0.13% of the total Japanesepopulation of 12,616,000. It corresponds to the lim-ited domain growth area rather than the large num-ber of immunes because the antibody positive ratewas announced on June 16, 2020 to be 0.10% in Tokyoand 0.17% in Osaka[13] or 0.43%[14].The value of the one-dimensional growth parame-ter S c GN which is proportional to the growth speed G , was a small constant of 0.378 [1/day] in Japan and0.420 [1/day] in Tokyo. The inverse of these values,2.7 and 2.4 days for Japan and Tokyo, respectively,may be proportional to their serial intervals. Thevalue of ν was a small constant of 0.0090 [1/day] inJapan and 0.0072 [1/day] in Tokyo. When the half-life t / of the remained nuclei is calculated similar tothe radioisotope, for example, t / = ln 2 / . (cid:39)
77 days, which is considered to relate to a period ofthe epidemic.In Japan, the contact chance with infected per-son and the range of random movement of the in-fected are estimated to be small since February 2020.Many Japanese began to pay attention to COVID-19 in February 2020. Experts have taken measuresagainst epidemic clusters since February 2020. It isconsidered that such behavior became a factor thatrestricted the nucleation and growth. There are viewsthat there may be other immunological factors, whichare expected to be verified scientifically.
An attempt was made to analyze the transition ofthe number of newly infected persons with COVID-19 in Japan from March 13 to May 24, 2020 by thedynamic phase transition theory. As a result, theepidemic was in good agreement with the randomnucleation followed by the one- or two- dimensional linear growth basic model. The epidemic in Japan inthat period was simply rate-limited by three param-eters that could be regarded as constant, the initialsusceptible, domain growth rate, and nucleation de-cay constant. This model provides a good predictorof epidemic because it consists of one equation andthe initial ln(1 / (1 − X )) specific plot is linear. References [1] https://diamond.jp/articles/-/236274 [2] [3] https://medical.nikkeibp.co.jp/leaf/mem/pub/eye/202004/565192.html [4] [5] Y. Takase, A. Odajima and T. T. Wang, J. Appl.Phys. 60, 2920 (1986).[6] I. Cooper, A. Mondal and C. G. Antonopoulos,arXiv:2006.10651.[7] P. Priyanka and V. Verma, arXiv:2006.14373.[8] T. Barnes, arXiv:2007.14804.[9] Crystallization of Polymers, Leo Mandelkern(1966).[10] M. Avrami, J. Chem. Phys. 8, 212 (1940).[11] https://github.com/kaz-ogiwara/covid19/blob/master/data/summary.csv [12] https://stopcovid19.metro.tokyo.lg.jp/cards/number-of-confirmed-cases/ [13] https://medical.nikkeibp.co.jp/leaf/all/report/t344/202006/566064.html [14] https://medical.nikkeibp.co.jp/leaf/all/report/t344/202006/565972.html?ref=RL2https://medical.nikkeibp.co.jp/leaf/all/report/t344/202006/565972.html?ref=RL2