Evaluation of the effect of the state of emergency for the first wave of COVID-19 in Japan

In this paper, we evaluate the effect of the state of emergency for the first wave of COVID-19 in Japan, 2020 from the viewpoint of mathematical modelling. In Japan, it was announced during the period of the state of emergency from April 7 to May 25, 2020 that the 80% reduction of the contact rate is needed to control the outbreak. By numerical simulation, we show that the reduction rate seems to have reached up to 86%. Moreover, we estimate the control reproduction number Rc during the period of the state of emergency as Rc=0.36 (95%CI, 0.34–0.39), and show that the effective reproduction number Re after the lifting of the state of emergency could be greater than 1. This result suggests us that the second wave of COVID-19 in Japan could possibly occur if any effective intervention will not be taken again.

isolation following this announcement. In fact, it has been reported that the number of people who visited major tourist spots in Japan during the Japanese Golden week holidays (from late April to early May) in 2020 drastically decreased compared to that in 2019 (The Japan . In areas around Ise Jingu Shrine, Mie Prefecture, it has been reported that more than 95% reduction was achieved (The Japan . The purpose of this study is to evaluate the effect of the state of emergency for the first wave of COVID-19 in Japan from the viewpoint of mathematical modelling. In particular, our attention is on whether the 80% reduction of the contact rate was successfully achieved in Japan during the period of the state of emergency. For some prior studies on the effect of the control strategies for COVID-19 in Japan, see (Chen et al., 2020;Kobayashi et al., 2020;Kurita et al., 2020;Sugishita et al., 2020).
In (Kuniya, 2020), the author estimated the epidemic parameters and predicted the epidemic peak for COVID-19 in Japan, 2020 by using the data in the early stage (from January 15 to February 29, 2020). The basic reproduction number R 0 , which implies the expected number of secondary cases produced by a typical infected individual at the initial stage in a completely susceptible population (Diekmann et al., 1990), was estimated as 2.6 (95%CI, 2.4e2.8). The estimated epidemic curve in (Kuniya, 2020) seems to fit well to the actual data until about 2 weeks passed from the start of the state of emergency on April 7, 2020 (see Fig. 2).
From late April, the estimated epidemic curve has left the actual data. We can conjecture that if the state of emergency had not been declared on April 7, then the daily number of newly reported cases might have increased along with the estimated epidemic curve.
In this paper, we assume that the infection (contact) rate is successfully reduced during the period of the state of emergency (that is, from April 7 to May 25, 2020) by multiplying a constant k ð0 < k < 1Þ to the infection rate. We manipulate the epidemic curve by changing k, and find the best k ¼ k * by which the epidemic curve is fitted well to the actual data. ð1 Àk * Þ Â 100 (%) would then be the desired estimated value of the reduction rate of the contact rate during the period of the state of emergency for the first wave of COVID-19 in Japan, 2020.

Methods
In prior studies, various compartmental models have been used to study COVID-19 (for instance, SIQR model (Crokidakis, 2020a, b), SIRX model (Maier & Brockmann, 2020) and SIRD model (Reis et al., 2020)). As the incubation period of COVID-19 is not negligible (Linton et al., 2020), there would be a merit for using an SEIR model, in which the latent class E is taken into consideration. In this paper, as in (Kuniya, 2020), we use the following SEIR model (see Fig. 3) with the detected infective population Y. S 0 ðtÞ ¼ ÀbSðtÞIðtÞ; E 0 ðtÞ ¼ bSðtÞIðtÞ À εEðtÞ; I 0 ðtÞ ¼ εEðtÞ À gIðtÞ; R 0 ðtÞ ¼ gIðtÞ; YðtÞ ¼ pIðtÞN; (1) where S, E, I and R denote the susceptible, exposed, infective and removed populations, respectively. b, ε, g and p denote the infection rate, the onset rate, the removal rate and the detection rate, respectively.
As stated below, each population implies the fraction to the total population. Hence, we can fit the daily data of newly reported cases by Y ¼ pIN, where N denotes the total population in Japan. The baseline values of each parameter are as shown in Table 1.
The initial condition is given as follows.
This implies that one infective individual is confirmed at t ¼ 0 (that is, Yð0Þ ¼ pIð0ÞN ¼ 1) and each population indicates the fraction to the total population as SðtÞ þ EðtÞ þ IðtÞ þ RðtÞ ¼ 1 for all t ! 0. The basic reproduction number R 0 is calculated as R 0 ¼ b=g. Comparison of the actual data of COVID-19 in Japan, 2020 and the predicted epidemic curve for R 0 ¼ 2:6 (95%CI, 2.4e2.8), which was estimated in (Kuniya, 2020) using the early data (from January 15 to February 29, 2020).
Let the unit time be 1 day and regard t ¼ 0 as January 15, 2020. Let T 1 ¼ ½0; 83 be the time period before the state of emergency was declared on April 7 (t ¼ 83), and let T 2 ¼ ð83; 131 be the time period during the state of emergency, which was lifted on May 25 (t ¼ 131). We assume that the epidemic process obeys the model (1) for t2T 1 , whereas it obyes the following alternative model for t2T 2 : S 0 ðtÞ ¼ ÀkbSðtÞIðtÞ; E 0 ðtÞ ¼ kbSðtÞIðtÞ À εEðtÞ; I 0 ðtÞ ¼ εEðtÞ À gIðtÞ; R 0 ðtÞ ¼ gIðtÞ; YðtÞ ¼ pIðtÞN; (2) where 0 < k < 1. That is, the infection rate b is reduced to kb during the period T 2 of the state of emergency. For each k, we define the following weighted least squares function as in (Capaldi et al., 2012, Section 3). Here, to specify the dependence on k, we write YðtÞ ¼ Yðt; kÞ: LðkÞ : ¼ X t2T2∩N ½Yðt; kÞ À ZðtÞ 2 Yðt; kÞ ; where ZðtÞ denotes the actual number of newly reported cases at time t, which is collected from the situation reports in (WHO, 2020). We then find k ¼ k * that minimizes LðkÞ.

Estimation of the effect of the state of emergency
The weighted least square function LðkÞ is numerically calculated as in Fig. 4. From Fig. 4, we see that k ¼ k * ¼ 0:14 minimizes LðkÞ. The fitted epidemic curve for k ¼ k * ¼ 0:14 is shown in Fig. 5. Here, R c denotes the control reproduction number (Inaba, 2017, Section 5.5.3), which is given by R c ¼ k * R 0 z0:36 (95%CI, 0.34e0.39). This result suggests us that the state of emergency in Japan for the first wave of COVID-19 resulted in ð1 À k * Þ Â 100 ¼ 86% reduction of the contact rate.

Discussion
In this paper, we have evaluated the effect of the state of emergency for the first wave of COVID-19 in Japan, 2020 by using the SEIR epidemic model (1)e(2). We have obtained k * ¼ 0:14, which implies that 86% reduction of the contact rate was achieved during the period of the state of emergency in Japan. On the other hand, we have obtained k * 2 ¼ 0:45, which implies that the effective reproduction number R e as of June 30, 2020 after the lifting of the state of emergency on May 25, 2020 is greater than 1, and the second wave of COVID-19 in Japan could possibly occur. To avoid this worse scenario, some strong intervention might be required again.
Our simulation was based on the assumption that R 0 ¼ 2:6 (95%CI, 2.4e2.8), which was estimated in (Kuniya, 2020). This assumption could be reasonable because the epidemic curve in Fig. 2, which was estimated by using the early data until February 29, 2020, seems to fit well to the data before the large intervention started on April 7, 2020. For the readers' convenience, we refer to the estimated values of R 0 for COVID-19 in some prior studies (see Table 2).
From Table 2, we can conjecture that R 0 for COVID-19 in Japan could be lower than the average in the world.

Conclusions
The conclusions in this paper are as follows.
The 80% reduction of the contact rate in Japan seems to have been successfully achieved during the period of the state of emergency from April 7 to May 25, 2020. More precisely, the reduction rate seems to have reached up to 86%. The control reproduction number R c during the period of the state of emergency in Japan was estimated as R c ¼ 0:36 (95%CI, 0.34e0.39).
The effective reproduction number R e as of June 30, 2020 after the lifting of the state of emergency on May 25, 2020 seems to be greater than 1. This implies that the second wave of COVID-19 in Japan could possibly occur if any effective intervention will not be taken again.
The actual future pattern of COVID-19 might be unpredictable because it would be affected by many factors such as the social behavior and the number of PCR tests. However, our result suggests that the state of emergency might have been highly effective on the first wave of COVID-19 in Japan. If the second wave becomes realistic in Japan, then taking a strong intervention again without any hesitation could be important to avoid a catastrophic scenario.

Declaration of competing interest
The author declares no conflict of interest.