Demand response impacts on off-grid hybrid photovoltaic-diesel generator microgrids

1.   Introduction

The basic reproduction number, denoted by $ R_0 $, is an index that represents the expected number of secondary cases from an infected individual during the entire period of infectiousness in a fully susceptible population ^[1]. This number plays a central role in measuring the transmissibility of infectious diseases and thus acts as the fundamental index for planning control strategies. During the COVID-19 epidemic, a new variant of the severe acute respiratory syndrome coronavirus-2 (SARS-CoV-2), the so-called Delta variant (B.1.617), was detected in Japan in March 2021. Estimating the relative reproduction number of the variant compared with the wild type of the virus then became an urgent task. In the early phase of the Delta variant epidemic, common methodologies to estimate $ R_0 $, which includes models based on exponential growth rate and generation time distribution, were difficult to implement due to a limited number of observed cases and the progress of variant replacements from Alpha (B.1.1.7) to Delta. Moreover, the spread of the Delta variant occurred in the middle of the very first vaccination campaign, from spring to summer of 2021, making it even more challenging to calculate the variant's basic reproduction number and relative transmissibility. Common methods for estimating $ R_0 $, as forementioned, rely on major outbreak datasets with an assumption that the population is fully susceptible.

To overcome such problems, the present study focuses on the meticulous observation of secondary transmissions in Wakayama Prefecture of Japan to estimate $ R_0 $. By applying the branching process model, which has been widely used to model minor outbreaks with non-negligible stochasticity, the study presents a method to efficiently estimate $ R_0 $ from very limited data on minor outbreaks. The branching process is particularly useful in quantifying the probability of extinction, a property that is supported by quantifying the offspring distribution and final size, i.e., the cumulative incidence of an outbreak. For a realistic representation of the offspring distribution of directly transmitted infectious diseases, the distribution is frequently assumed to follow a negative binomial distribution with two parameters, $ R_0 $ and the dispersion parameter $ k $ ^[2,3,4,5,6]. The distribution is highly dispersed for $ k < 1 $, and a very small value of $ k $ implies the presence of a super-spreading event ^[2]. Geometric and Poisson distributions are special cases of the negative binomial distribution when $ k \rightarrow \infty $ and $ k = 1 $, respectively.

The branching process model has already been employed to model COVID-19, especially in the early stages of the pandemic ^[6,7,8,9,10]. Published studies clearly indicate the presence of "over-dispersed" secondary transmission, where $ k $ is estimated to range from 0.1–0.3 for the wild type of the virus ^[9,11,12,13]. Preliminary research on the Delta variant has estimated the dispersion parameter to be $ k = 0.23 $ (95% confidence interval [CI]: 0.18–0.30) ^[14], which is broadly consistent with that of the wild type, but published evidence remains scarce.

Wakayama Prefecture of Japan is located 50 km south of Osaka, which is the third-largest prefecture in Japan by population. This prefecture continuously experienced epidemic "waves" of COVID-19, just as other prefectures in Japan did; a "wave" is conventionally recognized as a period of the epidemic with upward and downward trends with substantial and sustaining changes ^[15,16]. The present study focuses on the first and fifth waves caused, respectively, by the wild type and the Delta variant. Each wave is generally recognized as an epidemic around February to May 2020 ^[17,18] and July to October 2021, respectively. The COVID-19 epidemic was controlled early in Japan, involving 16,741 confirmed cases and 898 deaths until the end of May 2020; the indices are far fewer than in other industrialized countries that experienced population-level epidemics. Because of the low-impact setting, a thorough contact tracing practice, including retrospective (backward) tracing of contacts involving earlier generations, was possible and was continued over time. The practice yielded an empirically observed offspring distribution and the final size of minor outbreaks.

The objective of the present study is to interpret the empirically collected epidemic data in a low-impact setting by jointly estimating (ⅰ) the relative transmissibility of the Delta variant compared with the wild type and (ⅱ) the relative transmissibility among vaccinated individuals compared with unvaccinated individuals. Contrary to common methods for estimating $ R_0 $ that rely on large datasets from major outbreaks in a fully susceptible population, our study presents a method to efficiently estimate $ R_0 $ from very limited data on minor outbreaks. This exercise estimates the basic reproduction number of the Delta variant.

2.   Materials and methods

2.1.   Data on the final size distribution and offspring distribution

We used two datasets: (a) the epidemiological tracing results of secondary transmission during the first wave in 2020, caused by the wild type, and (b) the number of cases that produced at least one secondary transmission during the fifth wave in 2021, caused by the Delta variant. Other waves (the second, third, and fourth waves) were excluded from our scope since the focus of the present study was to estimate the basic reproduction number of the Delta variant, for which evidence is yet limited compared to other variants.

The observations were made in Wakayama Prefecture, which has a population of $ \sim $917,000. Despite the proximity to the metropolis of Osaka, the first wave within the prefecture was contained at a relatively low level, i.e., only 64 cumulative cases and 3 cumulative deaths were observed from the emergence of COVID-19 until June 30, 2020, compared with 1833 cases and 86 deaths in Osaka in the same period. In Wakayama, reverse transcription-polymerase chain reaction (RT-PCR) testing was conducted intensively, and cluster interventions were handled through contact tracing (i.e., containment effort). The vaccination program in Japan started in the early part of 2021, after the end of the third wave in February 2021. Initially, health care workers were prioritized (starting on February 17), and then the elderly population (over 65 years old) began to be vaccinated starting from April 12. At the peak of the fifth wave, caused by the Delta variant in Japan (August 26, 2021), 41.63% of the entire population had received two doses of vaccine ^[19]. More than 99% of vaccines were mRNA vaccines, either BNT162b2 (Pfizer-BioNTech) or mRNA-1273 (Moderna). Even during the fifth wave, cluster interventions were continued in parallel to vaccination. All cases were confirmed either via RT-PCR testing or antigen testing (including both a qualitative test based on immunochromatogaphy and a quantitative test based on the chemiluminescence enzyme immunoassay (CLEIA) method), and if positive, they were consistently notified to the government health authority (Ministry of Health, Labour and Welfare) in accordance with the Infectious Disease Law.

The first dataset from the first wave covered 25 clusters of COVID-19 infection, and each record included the total number of cases that stemmed from a single primary case. Since each cluster was from a minor outbreak contained at a low level, of which most were reported from households, we could assume that the data was not influenced by public health and social measures (PHSMs) and psychological factors. Minor and major outbreaks are types of epidemics in stochastic SIR models that Whittle characterized in 1955 ^[20]. Minor outbreaks are generally known to have a shorter duration and substantially fewer cases than major outbreaks ^[21], and hence we temporarily define the threshold of the final size between the minor and major to be 500 cases for convenience of the discussion. We emphasize that the dataset was from small settings where transmission was contained within each cluster and did not lead to major outbreaks where transmission is prevalent in wide community settings. Considering that the overall infections at the population level also contained at a low level and that the vaccination programs did not exist during the first wave, we can interpret our estimates as the approximation the basic reproduction number, capturing the intrinsic transmission dynamics in a completely susceptible population.

The second dataset from the fifth wave contained the respective numbers of cases that did and did not produce secondary transmission. We assumed that we could ignore the effects of PHSMs since no such kinds of policies were implemented in the region during the period. Furthermore, since the contact tracing practices were continued in the same manner as in the first wave, our estimates can be interpreted as the basic reproduction number for the Delta variant, just as for the wild type. The data were categorized into two groups by vaccination status of the primary case: the "all" group included all cases, regardless of vaccination (i.e., either 0, 1, or 2 doses), whereas the "2 doses" group included cases of individuals who had received two vaccine doses. We obtained values for the "0 or 1 dose" group by subtracting the "2 doses" group from the "all" group. The proportions of those who produced secondary transmissions for each group are shown in Figure 1.

Figure 1.  Proportion of cases with secondary transmission by vaccine status of primary case, during the fifth wave in Wakayama. Dataset originally included data for "all" and "2 dose" groups. Value for "0 or 1" dose group was calculated by subtracting "2 dose" group from "all" group.

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2.2.   Stochastic model

The goal of this study was to estimate $ R_0 $ and $ k $ from limited observations of transmission in the first and fifth waves, where the wild type and Delta variant were respectively dominant. We estimated four parameters, i.e., $ R_0 $ and $ k $ for the wild type, the relative transmissibility of the Delta variant to the wild type ($ \gamma_\text{delta} $), and the relative transmissibility among fully vaccinated individuals (post 2 doses of vaccination) compared with unvaccinated individuals ($ \gamma_\text{vac} $). We used a negative binomial branching process model and estimated the parameters using maximum-likelihood estimation (MLE). The 95% CI for the estimated parameters was obtained from a parametric bootstrapping procedure.

2.2.1.   Estimation of $ R_0 $ and dispersion parameter $ k $ for wild type

With the assumption of $ R_0 > 1 $, because the virus caused a global pandemic, we modelled the final size distribution $ Z $ using the Galton–Watson (GW) process that involves a negative binomially distributed offspring distribution. Using the probability generating function ($ p.g.f. $) for $ Z $,

the probability of the final size being $ z $ is expressed as follows:

Specifically for $ z = 1 $ and $ z \geq 2 $, the probabilities of the final size being $ z $ are the following ^[5,22].

Because the dataset consists of a series of minor outbreaks, we conditioned the final size distribution on extinction. Thus, under the assumption that the offspring distribution follows a negative binomial distribution with dispersion parameter $ k $, the probability of the final size being $ z $ was normalized by the probability of extinction $ \pi $ ^[5].

The likelihood function for estimating the parameters $ R_0 $ and $ k $ for the wild type, given $ j $ complete observations each with a final size of $ z_i $, is expressed as follows:

We excluded all observations of $ z = 1 $, which are sporadic and terminal cases, to adjust for the under-ascertainment of such cases, i.e., compared with the presence of secondary transmission, sporadic primary cases may be far harder to ascertain, especially during the first wave ^[5,6]. We estimated $ k $ indirectly via its reciprocal $ \alpha = \frac{1}{k} $, as numerous studies on the estimation of negative binomial parameters have shown that this is better than using $ k $. The sampling distribution for $ \alpha $ tends to be more symmetric than that for $ k $, which ensures a faster approach to asymptotic normality ^[2,3].

Given 14 data points of final size $ z $ from the first dataset, from which all observations of $ z = 1 $ were excluded, we estimated parameters $ R_0 $ and $ k $. Using the Nelder-Mead method with intial values $ R_0 = 1.5 $ and $ k = 0.15 $, we obtained parameter estimates that maximizes the likelihood function (Eq (2.6)). The optimization was conducted using the R function "optim".

We conducted parametric bootstrapping to obtain the 95% CI of $ R_0 $ and $ k $. Firstly, we derived the covariance matrix from the Hessian matrix that was returned as part of the optimization result in "optim" function. Using the covariance matrix, we generated 3000 pairs of multivariate normal distributions for $ R_0 $ and $ k $ and calculated the 95% CI with the bootstrap percentile method.

2.2.2.   Estimation of relative transmissibility of Delta and relative transmissibility among vaccinated population

We let the reproduction numbers of the Delta variant in the "0 or 1 dose" group and the "2 doses" group to be:

Here, $ \gamma_{\text{delta}} $ expresses the relative transmissibility of the Delta variant compared with that of the wild type, and $ \gamma_{\text{vac}} $ represents the reduced transmissibility among fully vaccinated individuals compared with unvaccinated individuals. $ R_0 $ is the basic reproduction number of the wild type. The vaccine effectiveness (VE), in terms of preventing secondary transmission, can be calculated as $ 100(1 - \gamma_{\rm{vac}}) \% $. We note that $ \gamma_{\rm{delta}} $ and $ \gamma_{\rm{vac}} $ are coefficients that represent relative changes in the number of secondary transmissions produced but not the risk of being infected.

Under the assumption that the offspring distribution $ Y $ follows a negative binomial distribution, the $ p.g.f. $ for having $ y $ offspring is as follows.:

Using such $ p.g.f. $, the probability of having no offspring $ (y = 0) $ ^[22] is derived as:

where $ R_{0_i} $ represents $ R_{0_{\rm{delta}}} $ or $ R_{0_{{\rm{delta}}\_{\rm{vac}}}} $ for $ i = 1 $ and $ 2 $, respectively.

The likelihood function for estimating $ \gamma_{\rm{delta}} $ and $ \gamma_{\rm{vac}} $ is as follows.:

Here, $ i $ represents different groups based on vaccine status, where $ i = 1 $ represents the group with 0 or 1 vaccine doses, and $ i = 2 $ represents fully vaccinated individuals. $ n_i $ refers to the total number of people in the category, and $ m_i $ refers to number of those who had no offspring ($ y = 0 $). For $ i = 1 $ in the 0 or 1 dose vaccination group, $ n_1 $ and $ m_1 $ had values of 177 and 138, respectively. For $ i = 2 $ in the fully vaccinated group, $ n_2 $ and $ m_2 $ were 726 and 344, respectively.

Given the secondary transmission dataset from the fifth wave and 3000 bootstrap pairs of pre-estimated $ R_0 $ and $ k $, we estimated parameters $ \gamma_\text{delta} $ and $ \gamma_\text{vac} $ that maximizes the likelihood function. Using the Nelder-Mead method with intial parameter values $ \gamma_\text{delta} = 3 $ and $ \gamma_\text{vac} = 0.2 $, we obtained 3000 pairs of parameter estimates that maximizes the likelihood function. The optimization was conducted using the R function "optim". By the repeating parametric bootstrap for each pairs in the same manner as $ R_0 $ and $ k $, we obtained 95% CIs for $ \gamma_\text{delta} $ and $ \gamma_\text{vac} $.

2.3.   Sensitivity analysis of Delta parameters for different values of dispersion parameter $ k $

We assumed that the dispersion parameter $ k $ is consistent across SARS-CoV-2. Thus, we conducted a sensitivity analysis of the estimated reproduction number for the Delta variant with different values of the dispersion parameter $ k $. We repeated the estimation for $ \gamma_\text{delta} $ and $ \gamma_\text{vac} $ using different values of $ k $, ranging between 0.05 and 1, which were minimum and maximum values found either as estimates or the bounds of the CI in prior research ^{[6,7,10,11,12,23,24,25,26,27]}. Although there is not yet much evidence for the dispersion parameter of the Delta variant, preliminary research estimates $ k = 0.23 $ (95% CI: 0.18–0.30) ^[14], which is within the range of the dispersion parameter for the wild type.

Reproduction numbers of the Delta variant in populations with different vaccination statues were calculated by substituting estimated $ \gamma_\text{delta} $ and $ \gamma_\text{vac} $ into Eq (2.7). 95% CIs for reproduction numbers were also calculated with the bootstrap percentile method.

2.4.   Statistical analysis

All of the analysis in the present study was conducted using R, version 3.6.3 ^[28].

2.5.   Ethics

The present study examined publicly available data that do not contain any personally identifiable information. As such, ethical approval was not required for the present study.

3.   Results

3.1.   Estimates for the wild type

For the wild type of COVID-19, $ R_0 $ was estimated to be 3.78 (95% CI: 3.72–3.83). This value is higher than suggested by previous research, with meta-analysis and a systematic review concluding that $ R_0 $ lies in the range from 2–3 ^[29,30]. The value of $ k $ for the wild type was estimated to be 0.236 (95% CI: 0.233–0.240), which is broadly consistent with prior findings ^{[6,8,9,10,12,13,26]}. The final size distribution given by the estimated $ R_0 $ and $ k $ is shown in Figure 2, which compares observed data (line) and the fitted model (bars). While the model captures the overall trend of frequency, the frequencies for $ z = 2 $ and $ z = 3 $ are underestimated. The fit may be biased by a cluster observed at $ z = 15 $.

Figure 2.  Final size distribution of wild type. Observed data points are shown as dots on a line, and the estimates from the model are shown as bars. The value of $ z = 15 $ in the model shows the total sum for $ z \geq 15 $. Due to the computational limitations, the upper limit contained in the total sum is $ z = 120 $. The sample size was 25 reports. Data include reports before June 4, 2020. Note that the horizontal axis shows the number of secondary (and later) cases involved; primary cases and sporadic cases without any secondary transmission were discarded from the analysis.

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3.2.   Estimates for Delta variant

Using the estimated parameters of the wild type, we then estimated the relative transmissibility of the Delta variant and the relative transmissibility among fully vaccinated individuals. The relative transmissibility of the Delta variant to the wild type, $ \gamma_{\rm{delta}} $, was estimated to be 1.42 (95% CI: 0.94–1.90). The relative effect on transmissibility from two-dose vaccination, $ \gamma_{\rm{vac}} $, was estimated to be 0.09 (95% CI: 0.03–0.14). This indicates that VE for preventing secondary transmission among a fully vaccinated population, compared with unvaccinated individuals, was 91% (95% CI: 85%–97%).

By substituting the estimated parameters into Eq (2.7), the basic reproduction number of the Delta variant, $ R_{0_{\rm{delta}}} $, was calculated as 5.37 (95% CI: 3.55–7.21). This value is consistent with prior research findings, which indicate a mean value of 5.08 ^[31]. The reproduction number of the Delta variant among fully vaccinated individuals, $ R_{0_{\rm{delta\_vac}}} $, was calculated to be 0.44 (95% CI: 0.20–0.69). This result highlights that the reproduction number in the fully vaccinated population was well below 1.

The offspring distribution of the Delta variant was calculated using the mean of the estimated reproduction number for both the 0 or 1 dose population and the 2 doses population (Figure 3). The dispersion parameter $ k $ was assumed to be identical to the estimated value for the wild type ($ k = 0.236 $). The left panel shows the result for the population with vaccination status of either 0 or 1 dose, while the right panel shows the result for 2 doses. The dotted horizontal line identifies the observed frequency of those who did not cause a secondary infection ($ y = 0 $). For the 2 doses population, the estimated frequency from the model matches that of the observed data.

Figure 3.  Estimated offspring distribution of the Delta variant according to vaccination status. The left panel shows the result for the population with a vaccination status of either 0 or 1 dose, while the right panel shows the result for the 2 doses population. The dispersion parameter $ k $ was assumed to be identical to the estimated value for the wild type ($ k = 0.236 $). The dotted horizontal line identifies the observed frequency of those who did not cause any secondary transmission ($ y = 0 $).

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3.3.   Sensitivity analysis

The results of sensitivity analysis with various values of the dispersion parameter $ k $ are shown in Figure 4. Dotted vertical lines indicate the value of the dispersion parameter ($ k = 0.23 $) estimated from the data for the wild type. At $ k = 0.05 $, the Delta variant is estimated to have an $ R_0 $ value of 2878.75 (95% CI 1248.96–4458.64), while at $ k = 1.0 $, $ R_0 $ is 1.11 (95% CI 0.95–1.28) (Panel C). As shown in Panel B, the relative transmissibility among fully vaccinated individuals is reduced at smaller $ k $ values, meaning that there is a greater reduction in the number of secondary transmissions. The reproduction number in fully vaccinated individuals was estimated to be less than 1 for all $ k $ values greater than 0.15.

Figure 4.  Sensitivity analysis of estimates to variations in dispersion parameter $ k $. Estimations of $ \gamma_{\rm{delta}} $ and $ \gamma_{\rm{vac}} $ were repeated using different values of $ k $. Panel A shows the relative transmissibility of Delta, $ \gamma_{\rm{delta}} $. Panel B shows the relative transmissibility among the vaccinated population, $ \gamma_{\rm{vac}} $. Panels C and D show the reproduction number of the Delta variant for populations with a vaccination status of 0 or 1 doses and 2 doses, respectively. Reproduction numbers were calculated by substituting estimated $ \gamma_\text{delta} $ and $ \gamma_\text{vac} $ into Eq (2.7). 95% CIs of each estimates were calculated with the bootstrap percentile method. The dotted lines identify the value of the dispersion parameter ($ k = 0.23 $) used in the default setting, which was retrieved from the estimation of the wild type. Vertical axes of Panels A and C are in logarithmic scale.

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4.   Discussion

Using meticulously observed contact tracing data from Wakayama prefecture in early 2020 and mid-2021, we estimated the basic reproduction number $ R_0 $ and the dispersion parameter $ k $ of the wild-type COVID-19, as well as the relative transmissibility of the Delta variant and relative transmissibility among fully vaccinated individuals. For the wild type, $ R_0 $ was estimated to be 3.78, and $ k $ was estimated to be 0.24. For the Delta variant, we estimated that $ R_0 $ is 5.37, which is in line with preliminary findings reporting a mean of 5.08 ^[31]. Our estimates also indicated that the number of secondary transmissions was reduced by 91% among fully vaccinated individuals. To the best of our knowledge, no previous research has quantified the $ R_0 $ value of the COVID-19 Delta variant from minor outbreak data while accounting for the relative transmissibility among fully vaccinated individuals.

The present study highlights that the basic reproduction numbers of variants can efficiently be estimated by applying the branching process model to the distribution of minor outbreak data. We emphasize that our method provides "relative transmissibility" of emerging variants on the basic reproduction number, not the effective reproduction number. This provides the basic reproduction numbers of new variants that work as fundamental indices for controlling pandemics (i.e., in planning strategies of vaccination programs). Our methods can be applied if epidemiological surveys are being conducted at a constant level, and the relative susceptibility within the population remains at a comparable level. Moreover, we emphasize that the data from minor outbreaks, which are a type of epidemic that has a small number of cases, can provide further insightful epidemiological estimates, including the dispersion parameter and VE in terms of preventing transmission. While the estimated $ R_0 $ of the wild type was higher than the commonly agreed value, it was within the upper range of published CIs ^[29,30]. Furthermore, we have proved that VE in terms of preventing secondary transmission can also be estimated from limited observed data. Finally, the 95% CI for the estimated $ R_0 $ of the Delta variant overlaps with values reported in existing studies ^[31,32].

The strength of the present study lies in the successful estimation of the basic reproduction number $ R_0 $ of the Delta variant from minimal observed data of minor outbreaks. Understanding $ R_0 $ is essential in planning effective strategies to contain the spread of the virus, particularly when it is known to be growing with the emergence of new variants. Furthermore, estimations regarding the Delta variant were made possible by assuming that the dispersion parameter is equivalent to that of the wild type, which we estimated from another dataset collected in the same geographic region. We could have separately estimated the dispersion parameter if either the final size or offspring distribution had also been observed for the Delta variant.

Our results imply that contact tracing is a critical factor that can enhance future data collection during the pandemic. Our findings prove that contact tracing data contribute to estimations of transmissibility and VE. To further understand the nature of the virus and its mechanisms of transmission, surveillance should be customized to include information as follows: vaccination status, number of secondary transmissions (not only whether they did or did not cause secondary transmission), detailed demographic and epidemiological attributes (i.e., sex and age group for each reported case). This information allows a more explicit and detailed analysis of the transmission dynamics specific to particular population groups and settings.

We must address some limitations of our study. First, the data on secondary transmission were identified through epidemiological investigation, which relied on the local capacity of contact tracing and patient cooperation in the tracing practice. The tracing method means that there is always the possibility of recall bias, which may lead to the presence of unascertained cases. However, because backward tracing was combined with forward tracing in Japan, it is likely that most secondary transmission, including asymptomatic cases, was captured. More active testing strategies, including daily PCR testing, may capture more comprehensive transmission dynamics. Second, the sample size of the final size distribution was limited to only 25 clusters. While it is still very rare for data to be observed, the estimates for $ R_0 $ and $ k $ could have been more consistent with existing research if more reports were available. A serological survey could have been conducted so that the actual final size could be determined without the possibility of any biases. Third, the fifth wave dataset lacked a detailed offspring distribution, but it instead included the respective frequencies of those who did or did not cause secondary transmissions. If the full distribution of secondary transmission per person was observed (e.g., $ y = 1, 2, 3, \dots $), $ R_0 $ and $ k $ for the Delta variant could have been separately estimated with greater precision. While modeling from a limited dataset is still possible under certain assumptions, a detailed offspring distribution is essential in understanding the full nature of the transmission dynamics of infectious diseases. Finally, the waning of vaccine-induced immunity was ignored. Because the fifth wave occurred amid the vaccination program, we believe it was not essential to meticulously measure the time since vaccination at an individual level.

5.   Conclusions

In conclusion, the present study has demonstrated a method to estimate the basic reproduction number of a new variant, denoted by the relative transmissibility from an existing type of virus, through the application of branching process models on very limited data. We emphasize that our method does not rely on exponential growth rate and generation time distribution models that require major outbreak datasets from fully susceptible populations. Using the final size and offspring distribution of minor outbreaks aggregated based on intensive contact tracing, we have estimated that the Delta variant is 1.42 times as transmissive as the wild type of COVID-19, which results in $ R_0 $ of 5.37. The VE in the reduction in the number of secondary transmissions was estimated to be 91% among fully vaccinated individuals compared with unvaccinated individuals.

The study has also presented the importance of collecting pieces of information on minor outbreaks, where transmissions are contained in small settings without spreading to a wide range of communities in the population. We emphasize that offspring and final size distributions accumulated based on intensive contact tracing practices or serological surveys can significantly contribute to determining the fundamental indices for understanding and controlling outbreaks.

Acknowledgments

H. N. received funding from Health and Labour Sciences Research Grants (20CA2024, 20HA2007, 21HB1002 and 21HA2016), the Japan Agency for Medical Research and Development (JP20fk0108140, JP20fk0108535, and JP21fk0108612), the Japan Society for the Promotion of Science (JSPS) KAKENHI (21H03198), the Environment Research and Technology Development Fund (JPMEERF20S11804) of the Environmental Restoration and Conservation Agency of Japan, and the Japan Science and Technology Agency SICORP program (JPMJSC20U3 and JPMJSC2105). We thank local governments, public health centers, and institutes for surveillance, laboratory testing, epidemiological investigations, and data collection. We thank Stuart Jenkinson, PhD, from Edanz (https://jp.edanz.com/ac) for editing a draft of this manuscript.

Conflict of interest

The authors declare there is no conflict of interest.