1 Introduction

The spread of COVID-19 since 2019 has involved all kinds of regions—towns, cities, prefectures, and countries—due to the movement and contact of people, even without explicit symptoms. Although testing systems at airports have been introduced and improved, further improvements are needed because the virus has spread despite these efforts (e.g., the development of governmental policies [1]). It should be noted that the first infection in a country started with a few individuals who immigrated without a positive PCR test result or any other sign of infection. Thus, borrowing words from the Susceptible-Exposed-Infectious-Recovered (SEIR) model ([2, 3] and several extensions mentioned later), attention should be paid to individuals in infection states such as exposed (E) or susceptible (S), who may change into (E) after immigration. Furthermore, we should be aware that people can travel from/to regions, such as prefectures or cities within the same country, without any tests or passport controls.

Previous studies examined the influence of travel. Statistical analyses showed that long-distance travel significantly accelerated the spread of infection. For example, countries exposed to high flows of international tourism are more prone to cases and deaths owing to the COVID-19 outbreak [4]. International tourism expenditure, international tourism receipts, international tourist arrivals, and international tourism exports were significantly correlated with the total number of cases, daily growth of COVID-19 cases, and number of cases, especially in places with high incomes [5]. Using network- or population-based models of the spread of infection, long-distance movement across regions has been shown to cause a significant increase in the number of infected individuals [6,7,8]. This finding can be related to the lesson Stay with Your Community (SWYC [9]) for suppressing the spread of infection learned from simulations on a social network model. SWYC means that each individual should avoid meeting as many other unintended people as people to meet intentionally, because the excess triggers an explosive spread of infection. The risk of long-distance travel is regarded as the risk of meeting unintended people. However, we obtain a surprising tendency by combining recent data in [10] and [11], where we find that the increase in the number of travelers and new infection cases co-occurred until close to the end of 2020; however, their trends started to correlate negatively, as shown in Fig. 1. That is, people in the USA started to travel frequently from the beginning of 2021, 2 weeks after the start of vaccination, but the number of new cases was suppressed. Statistical analyses of the temporal changes obtained results correspond to such an observation: a statistically significant, although small, relationship between immigrant flows and COVID-19 rates in border counties existed, but the increase in local cases became non-significant with increasing local vaccination rates [12]. Other results indicate that travel restrictions to and from the country only modestly affect the epidemic trajectory unless combined with additional measures, such as the reduction of transmission in the community [13].

Fig. 1
figure 1

An example of downtrend of COVID-19 infection cases in US after introduction of vaccines in spite of the uptrend of travel activities

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In this paper, we present SEIRS circuits Grid to solve the problem of interregional community confusion—movements due to interregional travel cause complex interactions between individuals from multiple communities and regions. Here, individuals in states S, E, I, or R traveling from one region to another are dealt with as different groups, but may come into infectious contact with those in the destination region. The new method is presented in Sect. 2. In Sect. 3, we present the settings for simulations based on the SEIRS circuit grid in Sect. 4. The results in Sect. 4 will be discussed in Sect. 5, where the concept of social stirring is introduced to explain observed phenomena, such as the effects of activated travel on the suppression of infection cases at a higher rate of vaccination.

2 Grid of SEIRS Circuits

2.1 The SEIR Model and Extensions

In the susceptible–exposed–infectious–recovered (SEIR) model [14,15,16], S, E, I, and R refer to the following numbers of people:

S: number of susceptible individuals. When a susceptible individual comes in contact with a risk of infection (e.g., 15 min within a distance of 2m), the susceptible individual may catch the virus and transition to E below.

E: The number of individuals who have been exposed but are in an incubation period during which one may have caught the virus, but is not yet infective.

I: Number of individuals who have been infected and may have infected individuals in S.

R: Number of individuals recovered from State I. Some analysts deal with dead individuals as a part of R (called removed in such a case), but below, we count the dead as a part of I who do not transit to R.

The SEIR model has been used with the daily number of passengers using public transportation to determine the effects of human mobility restrictions [17, 18]. According to analyses using the SEIR model, mobility restrictions for individuals with symptomatic infections and high-risk regions had substantial effects on reducing the spread of COVID-19; for example, a 4-week delay of spread if two high-risk regions were locked down [18]. The SEIR model was extended to decompose the transmission of COVID-19 into cases induced by residences and facilities, using mobility data [19]. SEIR has been further extended to reflect the factors influencing the pattern of infection spread in each country, such as interregional travelers [20] and the age of the population [21]. A modified SEIR model has also been proposed to assess the effectiveness of social distancing, banning gatherings, and vaccination strategies [22]. However, the problem addressed below, called interregional community confusion, has not been explicitly highlighted.

To include the influence of travelers from other regions, the number of infective people was previously represented by Iin, which indicates the risk in a region due to receiving travelers [20]. Thus, Eqs. (1)–(4) were used as the analysis models to simulate the spread of infection. See Fig. 2a for an illustration of the model. If we reflect only the infected influx, as in Iin, it disables the consideration of exposed but not yet infected individuals, who should be represented by Ein. Table 1 presents the variables used in this study

Fig. 2
figure 2

Three SEIR-based models. a SEIR considering the number of infective influx travelers (Iin) in the region receiving travelers, corresponding to Eq. (1)–(4). The italicized letters show the variables and parameters in these equations. b The movement of S, E, I, and R with travel, having a circuit where people return from R to S due to the loss of acquired immunity corresponding to Eqs. (5)–(8) including parameter r4. The other parameters are succeeded from a The bottom figure c SEIRS circuit grid where vertical alignment of the SEIR circuit shows the movement of people from the same region and horizontal the interaction of people in a region corresponding to Eqs. (11)–(14). hijk is equal to α Tij /Nki i.e., the percentage of travelers from i to j among those who came from k to i. The other parameters are succeeded from a and b

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Table 1 Variables referred to from multiple sections in this paper
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$$\frac{{\text{d}}S}{{\text{d}}t}= {m}_{SER} \left(N-S\right)-{r}_{3} S (I+{I}_{in}) /N$$
(1)
$$\frac{{\text{d}}E}{{\text{d}}t}= {-({r}_{1}{+ m}_{SER}) E+ r}_{3} S (I+{I}_{in}) /N$$
(2)
$$\frac{{\text{d}}I}{{\text{d}}t}= {-({r}_{2}{+ m}_{I}) I+ {r}_{1}}E$$
(3)
$$\frac{{\text{d}}R}{{\text{d}}t}= {-{ m}_{SER} R+ r}_{2} I.$$
(4)

2.2 SEIRS Circuit Grid to Solve the Interregional Community Confusion

To reflect the exposed influx and represent the return of people in R to the S state, as in the SEIRS model owing to the loss of once-acquired immunity [23, 24] by r4, we consider the model given by Eq. (5)–(8) as shown in Fig. 2b. Sin, Ein, Iin, and Rin refer to the influx of travelers to be merged in the target region with others in S, E, I, or R

$$\frac{{\text{d}}S}{{\text{d}}t}= {S}_{in}{+{r}_{4} R+ m}_{SER} \left(N-S\right)-{r}_{3} S I /N$$
(5)
$$\frac{{\text{d}}E}{{\text{d}}t}= {E}_{in}{-({r}_{1}{+ m}_{SER}) E+ r}_{3} S I /N$$
(6)
$$\frac{{\text{d}}I}{{\text{d}}t}= {{I}_{in}-({r}_{2}{+ m}_{I}) I+ {r}_{1}}E$$
(7)
$$\frac{{\text{d}}R}{{\text{d}}t}= {R}_{in}{-{ ({r}_{4}+m}_{SER}) R+ r}_{2} I.$$
(8)

However, we consider a problem that we call interregional community confusion: movements due to interregional travel cause more complex mutual interactions between individuals from multiple regions than a one-way transition. For example, suppose that only two regions, A and B, exist for simplicity, where a certain number of individuals from region A travel to region B and stay there for a few days. As these travelers are expected to return to region A, the change in the number of individuals in all states in B depends on the difference between the densities of S, E, I, and R in travelers from A to B and those returning from B to A after being assimilated with others in B to A, as represented in Eq. (9). Here, it is assumed that the same number of people, TAB, who travel to region B return to region A. If the returning people are supposed to acquire the states of the others in B, XBin in Eq. (9) indicates the added number of members in State B of State X (S, E, I, or R). Here, NA represents the number of people in A and XA represents the number in state X in region A

$${{X}_{B}}_{in}=\left(\frac{{X}_{A}}{{N}_{A}}-\frac{{X}_{B}}{{N}_{B}}\right) {{T}_{AB}.}$$
(9)

Thus, XBin reflects the difference between the densities of the population in state X in regions A and B, where XB /NB (or XA /NA) is regarded as the density of state X among all who return to region A from B (or go out to region B from A), assuming that travelers succeed in the state of people in the region from which they move. However, it is mostly just a few days (2.3 days on average in Japan, estimated on the assumption that a traveler goes to one place at once per day [25], which coincides with the computed average in [26]) between the time travelers go from region A to B and the time they return to A. During such a short period, the ratio of travelers in state X changes by an incomparably smaller value than XA/NA. This can be expressed as follows:

The number of travelers from region A to region B: ρ NA.

Number of travelers in state X from region A to region B: ρ XA.

Travelers returning to region A from region B: (1δ1−) ρ NA.

Travelers returning to region A from region B in state X: (1+ δ2) ρ XA.

Here, ρ denotes the ratio of travelers from A to B among all in A, δ1 the ratio of travelers who stay longer in B than those who return in a few days to A, and δ2 the ratio of those who newly enter state X (newly infected minus those who recover if X is I) within a few days. Equation (9) is replaced by Eq. (10), which shows XBin is significantly smaller than XATAB/NA because \(\left| {\delta_{1} } \right| \ll 1\;{\text{and}}\;\left| {\delta_{2} } \right| \ll 1.\)

$${{X}_{B}}_{in}=-\left(\frac{{\updelta }_{1}+{\updelta }_{2}}{1-{\updelta }_{1}}\right) \frac{{X}_{A}}{{N}_{A}}{{T}_{AB}}$$
(10)

However, if regions A and B are significantly different, XBin in Eq. (9) is comparable to that of −XBTAB/NB. Owing to the gap between Eq. (9) and the corrected Eq. (10), equation set [Eqs. (5)–(8) or Fig. 2b)] cannot fit real travel activities. To cope with this problem, we propose a model in which individuals in states S, E, I, or R, traveling from region A to B, are treated as different groups from the others in B but may come into infectious contact. When they return from regions B to A, their probability of being in state X should be estimated to be close to XA /NA instead of XB /NB. Thus, we considered these two effects by extending the SEIRS circuit in Fig. 2b to the grid structure in Fig. 2c.

  1. i)

    Infected individuals in B may infect others staying in B, including those traveling from other regions.

  2. ii)

    An individual traveling from A to region B is added to the group of other individuals in the same state as oneself, staying in B, and coming from A.

Consequently, the model developed in this study is represented by Eqs. (11)–(14).

$$\frac{{\text{d}}{S}_{ij}}{{\text{d}}t}=\alpha {\sum }_{k}({s}_{ik}{T}_{kj}-{s}_{ij}{{T}_{jk})+{r}_{4i}{R}_{ij}-}{r}_{3j}\frac{\sum {{}_{k}{S}_{ij}I}_{kj}}{{N}_{j}} -{{v p}_{vj} N}_{ij}$$
(11)
$$\frac{{\text{d}}{E}_{ij}}{{\text{d}}t}=\alpha {\sum }_{k}({e}_{ik}{T}_{kj}-{e}_{ij}{T}_{jk}){ -r}_{1i }{E}_{ij }+ {r}_{3j} \frac{\sum {{}_{k}{S}_{ij}I}_{kj}}{{N}_{j}}$$
(12)
$$\frac{{\text{d}}{I}_{ij}}{{\text{d}}t}= \alpha {\sum }_{k}({\psi }_{ik}{T}_{kj}-{\psi }_{ij}{T}_{jk})+{r}_{1i}{E}_{ij} {-\left({r}_{2j}{+ m}_{{I}_{i}}\right)I}_{ij}$$
(13)
$$\frac{{\text{d}}{R}_{ij}}{{\text{d}}t}= \alpha {\sum }_{k}({r}_{ik}{T}_{kj}-{r}_{ij}{T}_{jk})+{r}_{2j}{I}_{ij}{-r}_{4i }{R}_{ij.}$$
(14)

The first term xikTkj on the RHS of each equation, with dXij/dt on the LHS, corresponds to those who come from region i to j via k. Here, the number of people who travel from k to j is multiplied by the proportion of those from i to k in state X. The second term xijTjk considers those who leave j for k in state X among those in j who come from i.

Here, we set mSER to zero and ignored the terms that vanished. About the parameter values in Eqs. (11)–(14), we set α to 1 for the most recent normal year, 2019. Assuming a decrease of 60% in 2021 compared with 2019 (on the statistical data [27, 28]), α, which is the traveller’s activity, of 0.4 is regarded as approximating the current status. Tkj was obtained from the approximate frequency of movement within each region (prefecture) and from each region to other regions in an ordinary year before 2020 based on the reference data in [26]. Specifically, 2.3T1j + T2j is obtained by referring to this dataset for T1j meaning the number of travelers who stay in region j (for 2.3 days as mentioned above) on average and T2j of a 1-day trip. This value for each region (i.e., j) is then divided into Tkj, which represents the movement from each region (k) in proportion to the population of the destinations (k).

Borrowing the idea from existing models of infection spread with vaccination [20, 22, 29,30,31,32], we integrated the doses of vaccination into the reduction of S as in Eq. (11), with pvj as the pace of vaccination in region j (the ratio of individuals vaccinated per day in the population of region j; to avoid confusion with the percentage of already-vaccinated individuals, here we call vaccination pace instead of rate). pv without suffix j denotes the ratio of the number of vaccinated individuals per day to the entire national population. v is the efficiency of the vaccine in reducing susceptibility. α is the ratio of the number of travelers to that before 2020.

Thus, challenging the interregional community confusion, we obtained an extended model called the Grid of SEIRS circuits, as shown in Fig. 2c, to reflect the interregional travel without the confusion of permanent habitants and individuals from/to other regions suffered in the model shown in Fig. 2b. Each vertical alignment (i.e., column) of the SEIR circuits is linked by vertical arrows in Fig. 2c, which show the movement of people originating from the same region. In this movement, individuals embrace states S, E, I, or R, which change via interactions between people originating from various regions and meeting within a region, as indicated by the horizontal arrows.

3 The Setting for Simulations

We performed simulations by considering the virus variant VOC-202012/01 (lineage B.1.1.7). Here, r1 was set equal to 0.2 r2 to 0.1, r3 to \(0.1\cdot {R}_{t}\) and mI to 0.012, close to the real death rate of infected cases of COVID-19 in Japan. r4 was set to 0.002 based on a previous study [24]. The value of Rt on the first day 7th April in the simulated period was set equal to the value on the same date in 2020 [33] and magnified linearly by 40% for the 50 days from April 2021 according to the increase in the values of Rt for variants according to the literature (an increase of 32% [34], 43–90% [35], etc., according to the literature). The vaccine is supposed to reduce the infectivity by 30% and 80% by the first and second doses, respectively, to obtain v of 55% in Eq. (11). Note that the aim of this study is to show a general tendency regarding the spread of infection and strategies for its control using a vaccine, rather than quantitatively correct predictions. However, to show the generality of the discovered tendencies, we also present the results for B.1.617 (delta variant [36,37,38]). Omicron variants are beyond the scope of this study because of their extraordinarily rapid mutation, strong reduction in antibody neutralization, and enhanced infectivity [39].

The initial values of Sij, Eij, Iij, and Rij are given by Eqs. (15)–(19), Δt1, Δt2, Δt3, Hi, r4i, and γ are constant values, and all other terms are functions of time t. Δt1, Δt2, and Δt3 are set to 2, 14, and 448 days, respectively. γ, the number of days to stay when one travels is set to 2.3. Function infecting (t) represents the number of newly infected cases on day t (data from NHK [40]).

$$N_{{\text{base }}ij} \left( t \right) = \gamma T_{ij} \left( t \right),$$
(15)
$$E_{ij} \left( t \right) = N_{ij} \left( t \right)/N_{ii} \left( t \right){\text{avr}}_{\tau {\text{in}}[t - \Delta t{1} - {2}:t - {2}]} {\text{infect}}_i (\tau )/r_{{1}t} ,$$
(16)
$$I_{ij} \left( t \right) = N_{ij} \left( t \right)/N_{ii} \left( t \right)\sum_{\tau \;{\text{in }}[t - \Delta t{2} - {2}:t - {2}]} {{\text{infect}}_i (\tau )} ,$$
(17)
$$R_{ij} \left( t \right) = ({1} - r_{{4}i})N_{ij} \left( t \right)/N_{ii} \left( t \right)\sum_{\tau \;{\text{in }}[t - \Delta t{3} - {2}:t - {2}]} {{\text{infect}}_i (\tau )} ,$$
(18)
$$S_{ij} \left( t \right) = N_{ij} \left( t \right) - \left\{ {E_{ij} \left( t \right) + I_{ij} \left( t \right) + R_{ij} \left( t \right)} \right\}.$$
(19)

4 Results

4.1 The Effects of Selecting One Prefecture to Vaccinate

The results with pv equal to 1% are shown for the simulated year from April 2021 to March 2022 in Fig. 3a–c, where one prefecture to be vaccinated was selected for each curve representing a sequence of the number of infections. In Fig. 3d, e, the population of the vaccinated prefectures and the number of accumulated infections are plotted for each of the 47 sequences, vaccinating a selected prefecture for one sequence.

Fig. 3
figure 3

The results of vaccinating no or a selected prefecture. a no vaccination, b vaccinating 0.3% of the national population per day, i.e., pv = 0.3%, selecting Tokyo (population\(1.4\times {10}^{7}\), Rt = 1.45) and c:Hyogo (\(5.4\times {10}^{6}\), 1.42). α was set to 40% which is realistic in 2021 according to the data from May 2021. d (pv = 0.3%) and e (pv = 0.1%): the number of accumulated infections (vertical) versus the population of vaccinated prefectures (horizontal)

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As in the comparison between vaccinating people only in Tokyo and Hyogo in Fig. 3b, c, choosing a region with a larger population for vaccination causes a stronger suppression of the spread of infection. Figure 3d, e clarifies this tendency; the Pearson’s correlation R between the population of the selected prefecture for vaccination and the accumulation of infection cases is − 0.969 and − 0.972 for pv of 0.3% and 1% respectively.

4.2 Conditional Entropy of Regional Distribution of Vaccines and the Effect of Movements from/to Prefectures

To investigate the effects of travellers’ activities, we show cases in which vaccines are distributed across multiple prefectures. The diversity in the distribution of vaccines to prefectures can be represented by the conditional entropy Hc defined in Eq. (20). In Eq. (20), ei (\(i\in \{0, 1\}\)) denotes an event in which an individual is vaccinated for i = 1 but not for i = 0, and Cj (\(j\in \{0, 1, \dots \mathrm{the num}.\mathrm{ of regions}-1\}\)) indicates that the individual was vaccinated in the jth region.\(p\left({e}_{1}|{C}_{j}\right)\), \(p\left({e}_{0}|{C}_{j}\right),\) \(p\left({e}_{1},{C}_{j}\right), p\left({e}_{0},{C}_{j}\right)\) are equal to\({{p}_{vj}, {1- p}_{vj}, p}_{vj }{N}_{jj}/N, (1-{p}_{vj }){N}_{jj}/N\). N represents the entire national population

$${H}_{c}= -{\sum }_{i, j}p\left({e}_{i},{C}_{j}\right)\mathit{log}p\left({e}_{i}|{C}_{j}\right).$$
(20)

Conditional entropy, which is prevalent in the selection of variables in machine learning, refers to the extent to which vaccines are distributed diversely without a specific intention or causality when choosing a region to vaccinate.

Each sub-figure in Fig. 4a–l shows a simulated sequence of the number of infection cases, setting a pair of values (pv, α). a–d: without vaccination, e–h: pv = 0.4% of the national population per day, i–l: pv = 1%. Here, we observed the effect of suppressing the spread by accelerating vaccination (increasing pv). From the comparison of Fig.4a–d, e–h, and i–l, we found that the increase in α tends to enhance infections in the low range of pv. In particular, infections in local regions, such as Kumamoto, Ehime, and Okayama, rather than Tokyo or Osaka, have increased. However, this tendency is reversed for the larger value of pv as 1.0 (i through l here). In Fig. 4m, n, we compare the accumulated infection cases for the sequences of various values of Hc.

Fig. 4
figure 4

a–d Newly infected cases per day for different activities α 's) of travelers without vaccination; e–h the results for the activities of travelers for pv = 0.4% of the national population per day, as (d) of the largest Hc among (a, b, c), and (d) in m. On the other hand, i–l are the results for the activities of travelers for pv = 1% of the national population per day, of the middle-valued Hc in n. Here, m and n show the total accumulated cases of all prefectures in 100 sequences for various Hc, represented by 100 dots for the two values of pv

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The sequences of infected cases corresponding to the arrows in Fig. 4m are shown in Fig. 5 for various values of conditional entropy Hc, for a fixed total vaccination pv and constant travel activity α. As shown in Fig. 5, the distribution of vaccines with larger Hc values tended to result in a more substantial suppression of the spread of infection.

Fig. 5
figure 5

Sequences of daily infected cases for varied values of conditional entropy Hc for a fixed total vaccination pv = 0.4%/day and constant travel activity α  = 0.4. The sequences correspond to the arrows in Fig. 4m. The vertical axis shows the obvious reduction of infected cases for larger Hc

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Figure 6a–l shows the effect of conditional entropy Hc on the number of infection cases of B.1.1.7. These figures were obtained by varying the vaccination pace pv from 0 to 1% and α from 0.13 to 2.0, collecting 100 sample sequences for each condition given by a pair of (pv, α), randomly setting pvj of each (jth) region, which is the percentage per day of vaccinated individuals among the population of the region. Here, we use \({\sum }_{j}{p}_{vj}{N}_{jj}{=p}_{v}{\sum }_{j}{N}_{jj}\), that is, the total number of vaccines used per day in the entire country is given by the percentage of vaccination relative to the national population. As a result, the increase in Hc was negatively correlated with the number of infection cases in each condition. This tendency ranges from moderate to strong negative correlations (Pearson’s coefficients R’s in the subfigures range between close to − 0.5 and over − 0.7) for pv of 0.2% and larger, as shown in Fig. 6. In addition, an increase in α tends to enhance infections in the low range of pv, and this tendency is reversed for a larger range of pv.

Fig. 6
figure 6

The effect of conditional entropy Hc (horizontal) on the number of infection cases (accumulation, vertical axiz) of B.1.1.7, varying vaccination pace pv and travel activity α

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On the other hand, to validate the expectation that the average effective reproduction number of vaccinated regions can be a measure of the effect of vaccines, the dependency of the accumulated number of infection cases on Rvac representing the average of effective reproduction number Rtj for all regions (i.e., j’s), weighted by the number of vaccinated individuals, is shown in Fig. 7a–l, varying the vaccination pace pv and travel activity α. Note that Fig. 7 does not show the correlation of the spread with the Rt of the country, averaging Rtj for all j, but shows a correlation with Rvac, the average of Rtj of regions where the government aimed to strongly suppress the spread. Therefore, a negative correlation is expected. By comparing Figs. 6 and 7, we find that the dependency of the vaccination effect on Hc is more significant than that on Rvac. The Pearson’s correlations in the figures support this observation.

Fig. 7
figure 7

The effect of Rvac (weighted average of Rtj for vaccinated regions) on the number of infection cases (accumulation of B.1.1.7), varying vaccination pace pv and travel activity α

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In Fig. 8a, the effect of pv on the average number of infection cases (accumulation) for α of 0.13, 0.3, 1.0, and 2.0 is shown. Information about errors, such as standard deviations or confidence intervals, are not included in Fig. 8a but are in Fig. 8b where the p value as a result of the t test is shown on the vertical axis as an index of the significance of the effect of α on the number of infection cases for each value of pv. For example, the p value for α= 2 was obtained to evaluate the significance of the difference in the number of infections between two conditions: α = 1 and α = 2. Although a p value is usually used to discretely check the significance by comparing it with a borderline value (e.g., p < 0.05); here, we use a smaller p value as evidence of a more significant difference. Here, we find a noteworthy tendency regarding the traveling activity. That is, more frequent travel across the borders of prefectures, represented by the larger α causes a greater increase in the number of infected cases below the vaccination pace pv of 0.1%. However, the increase is moderate as pv is improved to close to 0.4% and decreases if pv is further increased to close to 1%. As shown in Fig. 8a, pv of approximately 0.4% is the borderline for this reversal. As shown in Fig. 8b, the suppression of infection spread with an increase in α was found for pv > 0.4% but was not as significant as the acceleration with an increase in α for pv < 0.3%.

Fig. 8
figure 8

The effect of vaccination pace pv on the average number of infection cases (accumulation of B.1.1.7) for α of 0.13, 0.3, 1.0, and 2.0 is shown in a. In b, the p value as a result of the t test is shown on the horizontal axis as an index of the significance of the effect of α on the number of infection cases for each value of pv

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In Figs. 9, 10, 11, we present the results for B.1.617 (the delta variant), setting it to reach an increase in Rt of 27% from the 51st to the 100th day of the simulated period. In Fig. 9, for the cases with B.1.617, similar to Fig. 6, we observe a dependency of the number of infection cases on Hc. The negative correlation of the number of infection cases with Hc for pv > 0.2% is here found to be more significant for B.1.617. In Fig. 10, the positive correlation of the number of infected cases with the average Rvac in the case of pv = 0.1% for B.1.617 was negative, as expected for B.1.1.7. The correlations for other values of pv were even lower in Fig. 10 than in Fig. 7. Finally, as shown in Fig. 11, for cases of B.1.617 corresponding to Fig. 8 of B.1.1.7, the suppression of infection spread for an increase in α was found for pv > 0.6%, although it was not as significant as the acceleration with an increase in α for pv < 0.5%. This tendency was similar to that shown in Fig. 8. These results are referred to in the discussions in Sect.  5.

Fig. 9
figure 9

The effect of conditional entropy Hc on the number of infection cases (accumulation), varying the vaccination pace pv and the travel activity a, in the case of the variant B.1.617 (Delta variant)

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Fig. 10
figure 10

The effect of effective reproduction number on the number of infection cases (accumulation), varying the vaccination pace pv and the travel activity α, in the case of the variant B.1.617

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Fig. 11
figure 11

The effect of vaccination pace pv on the average number of infection cases (accumulation) for α of 0.13, 0.3, 1.0, and 2.0 is shown in a. In b, the p value as a result of the t test is shown on the horizontal axis, as an index of the significance of the effect of α on the number of infection cases for each value of pv, in the case of the variant B.1.617

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5 Discussion

5.1 The Overall Observations of the Results

The tendency is shown in Fig. 3, and the spread of infection is suppressed when a prefecture with a larger population is selected for vaccination, which may be counterintuitive, because the number of vaccinated individuals was equal in all simulated cases. However, it is natural that the spread in the prefectures selected for vaccination is substantially suppressed; therefore, the above tendency is comprehensible.

On the other hand, as shown in Fig. 4, even the tendency of enhancement of the spread in local regions due to the activation of travel (increase in α) was reversed by accelerating the vaccination pace, i.e., for the larger pv, which is an essential finding in this study. In addition, the larger the value of Hc, the more efficient the vaccination results in suppression, as shown in Figs. 4m–n, 5, and 6.

The above observations can be explained by introducing the social network shown in Fig. 12; a conceptual illustration of networks of infectious contacts was manually generated by the author, where the solid lines show the infectious connections (contacts) among individuals represented by nodes. As shown in Fig. 12a, the infective connections without disturbance by vaccinated individuals caused the infection to spread across the regions. If the vaccines are distributed equally to regions corresponding to a large Hc, the infection spread is suppressed and the range of nodes infected owing to the spread from each initially infected individual (nodes with thick rims) becomes narrower, as shown in Fig. 12b. However, if infected (infected and not yet recovered) individuals travel across regions, the social stirring of non-vaccinated individuals by traveling across regions causes a faster reproduction that corresponds to the change from Fig. 12a–c. In contrast, the social stirring caused by vaccinated individuals traveling across regions causes a slower pace of reproduction because of the elimination of infectious interregional connections. If the vaccines are distributed to multiple prefectures unequally, that is, at a low or moderate value of conditional entropy Hc, as shown in Fig. 12a, this type of social stirring is expected to increase Hc with a cutting-off effect over the entire network, as shown in Fig. 12d.

Fig. 12
figure 12

A conceptual illustration of networks of infectious contacts, manually generated by the author: the white nodes represent vaccinated individuals, segmenting regions by dotted lines. a the spread due to the vaccination in few regions, b the spread suppressed due to vaccines distributed equally to regions, c fostered spread due to interregional contacts due to infective travelers, d suppression due to interregional contacts caused by the travels of vaccinated individuals in a

Full size image

This explanation is consistent with the results shown in Fig. 4a–h, where the travel of non-vaccinated or weakly vaccinated individuals spread the virus to foster infection. On the other hand, Fig. 4i–l, which may be surprising in that frequent travels are found to suppress the spread of infection, are also explainable, because vaccinated individuals are spread by travels and cause social stirring, corresponding to the increase in Hc, which cuts the paths of infection spread. Thus, the social stirring effects due to traveling were found to depend on pv in such a way as to enhance the spread of infection for the smaller pv and suppress it for the larger pv. As the distribution of vaccines with a large Hc tends to result in suppressing the spread as shown in Figs. 5 and 6, maximizing Hc can be regarded as an effective policy to suppress the spread of infection.

To maximize Hc, we propose (1) and (2) as political vaccination strategies:

  1. (1)

    A region with a larger population is recommended if a single region is selected for vaccination.

  2. (2)

    If more than one region can be considered, the given quantity of vaccine should be distributed without intentional bias in a restricted region.

It is expected that the distribution of vaccines can be improved by increasing Rvac, the average effective reproduction number of vaccinated regions, for a smaller pv. However, it should be noted that Rtj of each region j is not easy to use, because it tends to be unstable as in the target time range (April 2021 through March 2022), and it is not trustworthy to estimate its future value if there are some causes of change in people’s social activities, such as Olympic games involving the studied regions. Furthermore, the less significant correlation, compared to Hc, of the average Rvac with the number of infection cases indicates a lower reliability of using effective reproduction numbers for suppressing the spread. The even lower performance of Rvac in Fig. 10 compared to Fig. 7 is inferred to be due to the extremely fast infection spread as B.1.617 is difficult to conquer, especially by slow vaccination, which may work if the spread is as slow as B.1.1.7. Thus, Hc can be regarded as more useful than effective reproduction numbers for improving the distribution of vaccines, because of its stable correlation with the number of infection cases.

On the other hand, as shown in Figs. 8 and 11, more frequent travel across the borders of prefectures represented by the larger α accelerates infection spread for a low vaccination pace, but the acceleration is moderate as pv is improved, and then decelerates if pv is further increased. However, the deceleration for a large pv was not as significant as the acceleration with an increase in α for a small pv. Thus, we should say “the risk due to travels can be suppressed” rather than “it is encouraged to travel” across prefectures by setting large pv and Hc.

6 Conclusions

Mixing state-transition models, such as SEIR, its various extensions, and network models, is emerging as an established approach for obtaining unified models of interacting microscopic agents and macroscopic events in society [41,42,43]. In comparison, the method proposed in this study can be positioned as a method to model the networks of local societies to consider social stirring.

The findings of this study, that is, one that focused on vaccination in regions of the larger population as well as of the larger Rt tend to be effective for vaccination, and another that traveling causes not only enhancement but also suppression of infection expansion during the period of accelerated vaccination, coincide with the general tendencies shown in facts and studies that have worked on so far. These results partially support the reliability of this method in estimating the risks in local regions and the following discoveries regarding the tendencies of a group of regions, such as a nation, considering the interaction of micro-(among individuals within each region) and macro-(among regions) level interactions in society. Social stirring is a useful concept for explaining the findings of this study. The practical findings here are, in the third place, that a restricted quantity of vaccine can be used efficiently by maximizing conditional entropy. Fourth, travel across the borders of regions accelerates the spread of infection if the vaccine is distributed at a slower pace but may suppress it if the pace of vaccination is accelerated.

So far, the principle of staying with one’s community has been shown to reduce the risk of travel by involving habitats in the target region in the process of embodying the research results into their own wisdom for living [44]. The findings of this study will be translated into political wisdom, including vaccination strategies.