5.1 Difference in the number of infected individuals simulated between the SIR and the model proposed here.

When the latent period is set equal to the recovery period and when the coefficient of *AL*(*n*) is set to 0, indicating that the activity of the recovered individuals is 0, equivalent to the state that the recovered individuals do not come back to the community, the model proposed here is practically the same as the SIR model. For the SIR model, when the latent period and the recovery period are both 14 days, the population of the community is 1,000,000, the initial number of infected individuals is 1 and the value of the potential infectious capacity of coronavirus (*pfc*(*n*)) is 1.0, the number of infected individuals reaches 2,308 at the peak from the 974th to the 977th, and then decreases down 0 on the 2121st with the total number of infected individuals of 66,771. On the other hand, for the model proposed here, the number of infected individuals reaches 1,589 at the peak from the 928th to the 934th, and then decreases down 0 on the 2012nd with the total number of infected individuals of 45,504, indicating the marked decrease in the number of infected individuals with the shorter duration of infection (Table 1, Fig. 2）.

The differences in the number of infected individuals at the peak, in the total number of infected ones, in the duration of infection between the SIR model and the model proposed here are shown in Table 1 and Figs. 2 and 3 by different values of *pfc*. For any values of *pcf*s, the results of the model proposed here show that the dates of the peak have been brought forward and the durations of infection become short and that the numbers of infected individuals at the peak become smaller and the total number of infected individuals are also smaller. Namely, when the recovered individuals who have immunity come back to the community, the dates of the peak have been markedly brought forward, the durations become considerably shorter and the numbers of infected individuals become significantly smaller.

Since the model proposed here has the latent period different from the recovery period, when the coefficient of *AL*(*n*) is set to 0, meaning that the recovered individuals do not come back to the community, the model proposed here could be called the ‘modified SIR model’. The dates of the peak and the numbers of infected individuals are also different as shown in Table 2 and Fig. 4. It can be also said that, when the recovered individuals who have immunity come back to the community, the dates of the peak have been brought forward, the durations become shorter and the numbers of infected individuals become smaller.

Consequently, the results simulated by the model proposed here, where the recovered individuals who have immunity come back to the community, shows that the dates of the peak are brought forward, the duration becomes shorter and the numbers of infected individuals are smaller in comparison to the results calculated by the SIR model.

5.2 The threshold of the potential infectious capacity, pfc, for limiting the spread of infection.

For the model proposed here, on the first day of the simulation, the number of infected individuals increases from *P*(1) to *P*(1)+*P*(1)** icf*(1) ** pfc*(1)/*lp*(1)). Thus, the number of infected individuals at night,* P*(1(night)), is expressed as:

*P*(1(night)) =*P*(1)+*P*(1)** icf*(1) *(*pfc*(1)/*lp*(1)) =*P*(1)*(1+* icf*(1) *(*pfc*(1)/*lp*(1)) (18)

where *icf*(1) is the infection reduction rate. On the second day, in the morning, the number of infected individuals, *P*(2), is equal to *P*(1(night)), that is, *P*(2)=*P*(1(night)). The number of infected individuals at night on the second day, *P*(2(night)), is given by,

*P*(2(night))= *P*(2)*(1+* icf*(2) *(*pfc*(2)/*lp*(2))

= (*P*(1)*(1+*icf*(1)*(*pfc*(1)/*lp*(1)))*(1+* icf*(2) *(*pfc*(2)/*lp*(2)) (19)

Since *icf*(2) is equal to* icf*(1) and *pfc*(2)/*lp*(2) is also equal to *pfc*(1)/*lp*(1) during the latent period,

*P*(2(night))= (*P*(1)*(1+*icf*(1)*(*pfc*(1)/*lp*(1)))2 (20)

During the latent period, the number of infected individuals increases in the same manner of calculation of the ‘compound interest’, and at night on the last day of the latent period, the number of infected individuals, *P*(*lp*(night)), is given by

*P*(*lp*(night))= *P*(1)*(1+* icf*(1)*(*pfc*(1)/*lp*(1)))lp(1) (21)

On the next day after the latent period is ended, the infected individuals of *P*(1)* *syr*(*lp*) became symptomatic and should be isolated from the community.

Thus, when the value of the symptomatic rate (*syr*(*lp*)) is 1, meaning that all the number of *P*(1) become symptomatic and are isolated, the number of infected individuals having increased during the latent period, *Δ**P*(during *lp*), is given by

*Δ**P*(during *lp*)=* P*(*lp*(night)) – *P*(1)= *P*(1)*(1 +* icf*(1)*(*pfc*(1)/*lp*(1)))lp(1) - *P*(1)

=* P*(1)((1+* icf*(1)*(*pfc*(1)/*lp*(1)))lp(1) -1) (22)

When *Δ**P*(during *lp*) is equal to or less than *P*(1), that is, *Δ**P*(during *lp*) ≤ *P*(1), continuous spread of infection does not occur, though the number of infected individuals increases during the latent period.

Thus, the condition for limiting the spread of infection is as follows:

*P*(1)((1+* icf*(1)*(*pfc*(1)/*lp*(1)))lp(1) -1) ≤ *P*(1) (23)

Eq. (23) is rewritten as

(1+* icf*(1)*(*pfc*(1)/*lp*(1)))lp(1) -1 ≤ 1 (24)

Namely, when the following inequality is satisfied, the infection does not continuously spread. *pfc*(1) ≤ (*lp*(1)*(2(1/*lp*(1)) - l)/*icf*(1) (25)

For example, when the value of the infection reduction rate (*icf*(1)) is 1, meaning that the reduction effect of infectious control measures is not required to be considered, the latent period (*lp*(1)) is 5 days, the condition in which infection does not spread is given by:

*pfc*(1)≤0.7435 (*pfc*(1)/*lp*(1)*≤*0.1487) (26)

It is notable that the threshold of *pfc* is not ‘1 or less’ but ‘0.7435 or less’, indicating that even though the value of *pfc* is less than 1.0, the spread of infection could occur.

The results of simulation satisfying the condition for the case where *pfc*(1)=0.7435 (*pfc*(1)/*lp*(1)*= *0.1487), the initial number of infected individuals (*P*(1)) is 50, the population of the community is 1,000,000, the value of *icf*(*n*) is 1 and *lp*(*n*) is 5, is shown in Table 3 and Fig. 5. The number of infected individuals increases from 50 to 115 at the peak of the 7th and then decreases down to 0 on the 167th. The number of isolated individuals increases to 134 at the peak of the 15th, and then decreases down to 0 on the 182nd. The total number of infected individuals is 462.

5.3 Herd immunity threshold

When any infected individuals are not isolated without any intervention, they continue infecting until the recovery period is ended and then they become the ‘recovered’ individuals who have immunity in the community. For such a case, the number of infected individuals increases to a peak and then decreases. This phenomenon is sometimes explained by the ‘herd immunity’ which is an indirect protection against the spread of infection caused by the immunity of a large proportion of the population. The contact of the infected individuals with the recovered ones must accelerate reduction of the contact rate between the infected individuals and the susceptible ones as the number of recovered individuals increases, inducing decrease in the number of infected individuals. This is a scientific proof of the idea of the herd immunity. The cumulative number of infected individuals at the peak which is the turning point from increase to decrease is one of the ‘herd immunity threshold’, being a target value for vaccination.

For the model propose here, when the symptomatic rate, *syr*(*n*), is set to 0, any infected individuals in the community are not isolated. They stay in the community and continue infecting until the recovery period is ended. And they become the recovered individuals after the recovery period. The results of simulation by different values of *pfc* are shown in Table 4 and Fig. 6. When the value of *pfc* is 1.0, the number of individuals newly infected a day reaches 28,012 at the peak on the 77th with the cumulative number of 383,974, and then decreases to 0 on the 155th. The number of infected individuals reaches 517,318 at the peak on the 89th with the cumulative number of 629,455, and then decreases to 0 on the 189th with the total number of infected individuals of 728,431. On the other hand, when the value of *pfc* is 2.0, the number of individuals newly infected a day reaches 59,029 at the peak on the 42nd with the cumulative number of 464,859, and then decreases to 0 on the 98th. The number of infected individuals reaches 775,424 at the peak on the 56th with the cumulative number of 829,003. After the peak, the number of infected individuals decreases to 0 on the 126th with the total number of infected individuals of 873,160.

Each of the numbers of 629,455 (about 63% of the population) and 829,003 (about 83% of the population) is a herd immunity threshold without any intervention. It must be called a ‘potential (natural) herd immunity’ to distinguish from such a herd immunity suggested by the SIR model. Though the potential herd immunity could be achieved sooner than expected, it is surely achieved only at the cost of so many infected individuals with so much death.

5.4 The effect of the vaccination on the spread of infection

The number of infected individuals at night on the date *n*, *P*(*n*(night), is given by

*P*(*n*(night))=*RP*(*n*)+*Δ**P*(*n*)=*RP*(*n*) +* AP*(*n*(night))=* RP*(*n*)

+(*pfc*(*n*)*/lp*(*n*))***(*RM*(*n*)/*N*(*n*))**icf*(*n*)*(1-(*alI*(*n*)*(*CRI*(*n*)+*CRT*(*n*))+*al*(*n*)* *CRAS*(*n*)+*alV*(*n*)**V*(*n*))/*N*(*n*))*(*RP*(*n*)/*N*(*n*))**RM*(*n*) (27) =13)

and the number of susceptible individuals in the community at night, *RM*(*n*), is expressed by

*RM*(*n*)=* TN*(*n*)-(*CI*(*n*)+*CP*(*n*)+*V*(*n*)) (28)(= 17)

where *V*(*n*) is the number of vaccinated individuals who are live and work in the community. Since when the number of vaccinated individuals (*V*(*n*)) increases, the number of susceptible individuals (*RM*(*n*)) is decreased as calculated by Eq. (28), Eq. (27), including term *RM*(*n*), indicates that the increase in the number of vaccinated individuals directly decreases the number of infected individuals.

On the other hand, the contact rate (*cr*(*n*)) between the infected individuals and the susceptible ones is given by,

*cr*(*n*)=(*S*(*n*)/*N*(*n*))(1-*δ*(*R*(*n*)/*N*(*n*))) (29)(=5)

where (1-*δ*(*R*(*n*)/*N*(*n*)) is the reduction rate, and as previously mentioned,

(1-*δ*(*R*(*n*)/*N*(*n*))=(1-(* alI*(*n*)*(*CRI*(*n*)+*CRT*(*n*))+*al*(*n*)**CRAS*(*n*)+*alV*(*n*)**V*(*n*))/*N*(*n*))

(30)

The reduction effect on the contact rate increases with decreasing in the value of (1-*δ*(*R*(*n*)/*N*(*n*)). Namely, Eq. (29) indicates that, when the number of vaccinated individuals increases, the reduction rate decreases. As a result, the contact rate is decreased. Consequently, the increase in vaccinated individuals decreases not only the number of susceptible individuals but also the value of the contact rate.

When the vaccination rate is 0, meaning that the number of vaccinated individuals is 0, and any recovered individuals do not come back to the community, the number of infected individuals reaches 20,059 at the peak on the 201st and then decreases to 0 on the 376th with the total number of infected individuals of 196,638 (Table 5). However, when the recovered individuals come back to the community, the number of infected individuals reaches 14,895 at the peak on the 194th and then decreases to 0 on the 354th with the total number of infected individuals of 141,788, indicating that the coming back of the recovered individuals considerably decreases the number of infected individuals (Table 5 and Fig. 7).

On the other hand, when the vaccination rate is set to 0.1(10% of the population) on the 101st, meaning that the number of vaccinated individuals is 100,000 on and after the 101st, the number of infected individuals reaches 368 at the peak on the 102nd, and then decreases to 0 on the 271st with the total infected individuals of 2,487. Even for the modified SIR model, meaning that the recovered individuals do not come back to the community, the number of infected individuals reaches 386 at the peak on the 105th and then decreases to 0 on the 677th with the total number of infected individuals of 6,471 (Table 5 and Fig. 7). For both cases above, the number of infected individuals markedly smaller than those of the case without any vaccinated individuals.

In addition, when the vaccination rate is set to 0.1 on the 151st, meaning that the number of vaccinated individuals is 100,000 on and after the 151st, for the model proposed here, the number of infected individuals reaches 4,516 at the peak on the 151st and then decreases to 0 at the 305th with the total number of infected individuals of 26,906. For the modified SIR model, the number of infected individuals reaches 4,823 at the peak on the 154th and then decreases to 0 on the 432nd with the total number of infected individuals of 43,260. Though the number of infected individuals markedly smaller than those of the case without any vaccinated individuals, it markedly larger than those of the case that vaccination is carried out on the 101st. Though the simple cases of simulation are shown here, it can be said that the earlier the vaccination is carried out, the larger the effect decreasing the number of infected individuals. It also can be said that the continuous increase in number of vaccinated individuals should quickly decrease the number of infected individuals.

5.5 The effect of the breakthrough infection on the spread of infection

As examined above, when the vaccination rate is set to 0.1 on the 101st, the number of infected individuals reaches 368 at the peak on the 102nd, and then decreases to 61 on the 150th. The subtotal number of infected individuals up to the 150th is 2,297, the number of susceptible individuals is 897,703 and the number of vaccinated individuals is 100,000 (Table 6). When breakthrough infection of which rate (*b *(151) is 0.5 occurs on the 151st, the number of susceptible individuals rises to 947,696, including 50,000 of the vaccinated individuals who have turned to be capable to be infected and are reset to be the susceptible individuals, the subtotal number of infected individuals becomes 2,304 and the number of vaccinated individuals keeping immunity has decreased to 50,000 (Table 6 and Fig. 8).

The number of infected individuals once decreases down to 56 on the 154th, then increases up to 206 at the second peak on the 494th. After the second peak, the spread of infection slowly subsides, showing a tendency to continue for a long time. The number of infected individuals becomes 0 on the 1,498th with the total number of infected individuals of 17,164, whereas when the breakthrough infection does not occur, the infection duration is 271 days with the total number of infected individuals of 2,487 (Table 6). Thus, it can be said that the occurrence of breakthrough infection induces the long duration of infection with the large number of individuals infected by breakthrough infection.

When some of the vaccinated individuals are initially not able to be immune, they are reset to be the susceptible individuals on the date of vaccination adjusted by changing the vaccination rate *v*(*n*). When the quantity of antibody of vaccinated individuals decreases down below a threshold, the breakthrough infection also occurs. This phenomenon must continuously occur through the infection duration. Though the example examined here the breakthrough infection rate is kept as a constant value, when the breakthrough rate increases, the duration of the infection should last a longer time and the total number of infected individuals seriously increases.

5.6 The importance of the PCR test and/or antibody test for preventing the spread of infection

The* T*(*n*) is the number of individuals having PCR test or antibody test. It should be set on the day when test is performed. The incidence rate for the test would be biased, to be higher than that in the community, *ir*(*n*), because those having PCR test are mainly close contacts. Namely, the incidence rate for the test, *tir*(*n*), is given by

*tir*(*n*)=*bp*(*n*)**ir*(*n*) (31)

where *ir*(*n*) is the incidence rate in the community and *bp*(*n*) is the magnification with respect to* ir*(*n*). Thus, the number of infected individuals confirmed by the test, *CP*(*n*), is calculated by,

*CP*(*n*)=*T*(*n*)** tir*(*n*) =* T*(*n*)**bp*(*n*)**ir*(*n*) (32)

As a result, the value of *tir*(*n*) means the positive rate for the test, because * *

*tir*(*n*) = *CP*(*n*)/*T*(*n*) (33)

The individuals who are confirmed to be infected due to test positive are isolated and come back to the community after the isolation period (the recovery period). However, all the infected individuals confirmed are not always isolated. The number of isolated individuals, *I*(*n*), is given by

*I*(*n*)=* CP*(*n*)**i*(*n*) (34)

where *i*(*n*) is the isolation rate for the individuals who are confirmed to be infected due to being test positive. The value of *i*(*n*) indicates the ratio of the number of isolated individuals to the total number of infected individuals confirmed. When all the infected individuals confirmed are isolated, the value of *i*(*n*) should be set to 1. For calculation, the individuals decided to be isolated are isolated on the next day as shown by Eq. (11).

When the test is started on the 101st under the condition that the magnification (*bp*(*n*)) is 5, the isolation rate (*i*(*n*)) is 1, meaning that all the infected individuals confirmed are isolated, the latent period is 5 days, the recovery period is 14 days, the population of the community is 1,000,000 and the initial number of infected individuals is 1, the changes in the number of infected individuals and isolated ones are shown in Table 7 and Fig. 9 and Table 8 and Fig. 10. When the test with 1,000 tested individuals is started on the 101st, the number of individuals confirmed to be infected and isolated due to being test positive a day is 2 (accurately 1.8) on the 101st, and then increases to 49 at the peak from the 198th to the 200th, and then decreases down 0 on the 359th with the total number of test positive individuals of 3,817. The number of individuals infected and isolated a day due to being symptomatic in the community reaches 1,396 at the peak on the 198th, indicating the considerably large number compared to 49 of test positive individuals. The number of infected individuals then decreases to 0 on the 353rd with the total number of infected individuals in the community of 109,467. For both the number of individuals newly infected a day and the number of infected individuals, it should be recognized that the numbers of individuals confirmed to be infected by testing is relatively smaller than the actual numbers in the community. However, the total number of infected individuals, 113,284 (3,817+109,467), including the test positive individuals, is less by 28,500 than that for the case without any test, that is, *T*=0 (Table 7 and Fig. 9).

On the other hand, when the test with 10,000 tested individuals is started on the 101st, the number of infected individuals which is the sum of the test positive individuals and the symptomatic ones in the community is 406 at the peak on the 103rd, then decreases down 0 on the 203rd with the total number of infected individuals of 2,463 (Table 8 and Fig. 10). However, when the test with 10,000 tested individuals is started late on the 151st, the number of infected individuals becomes 4,937 at the peak on the 153rd, then decreases down 0 on the 242nd with the total number of infected individuals of 27,388. It shows more than ten times in the number of infected individuals for the case of the test started on the 101st. Thus, it can be said that the earlier the start of test, the smaller the number of infected individuals.

Comparing those numbers of tested individuals and the dates of start, it can be pointed out that the larger the number of tested individuals and the earlier the start of test, the markedly smaller the number of infected individuals. In other words, it is important that confirming early and large number of infected individuals in the community, isolating them from the community and returning them to the community after the recovery period. For the test, however, the more interesting point is as follows:

Table 9 and Fig. 11 show the difference in the number of infected individuals between the cases of test and vaccination. When the test with the number of tested individuals of 10,000/day is started on the 101st, as mentioned above, the number of infected individuals reaches 406 at the peak on the 103rd, then rapidly decreases down to 0 on the 203rd with the total number of infected individuals of 2,463. On the other hand, when the vaccination with a rate of 0.1 is carried out on the 101st, meaning that the number of vaccinated individuals is 100,000 on and after the 101st, the number of infected individuals reaches 368 at the peak on the 102nd, and then quickly decreases down to 0 on the 271st with the total number of infected individuals of 2,487. The curves showing the changes in the number of infected individuals are similar each other (Fig. 11), and both of the total number of infected individuals are also similar. The test started on the 151st and the vaccination carried out also on the 151st show the similar results, indicating that both have a comparable effect on preventing the spread of infection. The PCR test and/or antibody test which are effective and convenient for any time and for any regions should be highly evaluated.

5.7 Intensified potential infectious capacity of coronavirus

When the potential infectious capacity of coronavirus, *pfc*, is intensified, the re-spread of infection must occur. For example, under the condition that the population of the community is 1,000,000, the initial number of infected individuals is 1, the latent period is 5 and the recovery period is 14, when the *pfc* is 1.0 during the period from the 1st to the 351st, the number of infected individuals reaches 14,895 at the peak on the 194th, and then decreases down to 1 on the 351st with the total number of infected individuals of 141,787, suggesting that the infection is almost ended.

However, when the *pfc*(351) is intensified to be 2.0 on and after the 351st, the number of infected individuals increases again, slowly at first and rapidly reaches 73,262 at the second peak on the 457th, and then decreases down to 0 on the 522nd with the total number of infected individuals of 446,280 (Table 10 and Fig. 12). During the second period of infection after the 351st, the infected individuals increase by 304,493, three times as many as those infected individuals during the first period from the 1st to the 351st, indicating the main period of infection is the second period after the *pfc* is intensified. As previously mentioned, the value of *pfc*(*n*) of the model proposed here is changed not by the interventions controlling contact rate such as self-isolation and lockdown, but by the seasonal variation and others including a new strain, suggesting that the possibility of intensifying of the* pfc* is considerably high.

5.8 The effect of the ‘restrictions on movement of persons’

The increase and/or decrease in the number of susceptible individuals occurs due to external factors such as travelling/staying home and migration (immigration/emigration). The *NAP*(*n*) of the model proposed here gives the value of increase and/or decrease in the number of susceptible individuals, *RM*(*n*) (Fig. 1). Especially, the emergency measures such as ‘avoiding any unnecessary outings/travel’, ‘staying home (self-isolation)’ and ‘lockdown’ to prevent the spread of the infection by reducing the contact rate practically induces a reduction of the number of susceptible individuals in the real community of Fig. 1. When the emergency measures mentioned above are taken, the value of *NAP*(*n*) should be given by negative value, the susceptible individuals decrease in the community, accompanied by decrease in the population, *TN*(*n*), including *N*(*n*) and *RM*(*n*).

For example, when the initial population of the community is 1,000,000, the initial number of infected individuals is 1, the *pfc* is 1.0, the latent period is 5 days and the recovery period is 14 days, and when the value of *NAP *(101) is set to -500,000 on the 101st, expressed as ‘*NAP *(101)-500,000’, meaning that the 50 % reduction of population of the real community by, for example, the ‘self-isolation’, the number of infected individuals reaches 7,453 at the peak of 181st, then decreases down 0 on the 301st. The total number of infected individuals is 70,910, meaning the 50% decrement in the number of infected individuals compared to that for the case without any measures, ‘*NAP*(*n*)0’, 141,788 (Table 11 and Fig. 13). However, when the value of the *NAP *(201) is set to -500,000 on the 201st, the number of infected individuals reaches 14,895 at the peak of 194th, then decreases down 0 on the 275th. The total number of infected individuals is 111,769, meaning 21% reduction compared to that for the case without any measures, 141,788. Table 11 indicates that the larger the number of self-isolated susceptible individuals and the earlier the carrying out of ‘self-isolation’, the smaller the number of infected individuals.

For another example, when the value of the *NAP*(*n*) is set to -500,000 on the 101st and is reset to 500,000 on the 301st, '(*NAP*)(101) -500,000, (301) 500,000', meaning that 500,000 ‘self-isolated’ individuals starts on the 101st and the 500,000 ‘self-isolated’ individuals come back to the community on the 301st, the number of infected individuals reaches 7,453 at the peak on the 181st, then decreases to 5 on the 301st, and down 0 on the 506th. The total number of infected individuals is 70,958, indicating that the total number of infected individuals is slightly increased compared to that for the case of the ‘*NAP *(101) -500,000’, 70,910.

On the other hand, when the value of the *NAP*(*n*) is set to -500,000 on the 101st and is reset to 500,000 on the 251st, '(*NAP*)(101) -500,000, (251) 500,000', the number of infected individuals reaches 7,453 at the peak on the 181th, then decreases to 230 on the 251st, and down 0 on the 737th. The total number of infected individuals is 73,073, indicating that the total number of infected individuals is about 2,000 more than that for the case that the self-isolation is lifted on the 301st. Moreover, the infection duration becomes considerably longer. Thus, it is pointed that the increase in the number of susceptible individuals induces the increase in the number of infected individuals and if the self-isolation is lifted too early, the number of infected individuals increases and the infection duration becomes significantly longer. The duration long lasted could give the chance of recurrence of the infection by the intensified infectious capacity of coronavirus, as shown by the section** 5-7.**

Incidentally, when the value of the *NAP*(*n*) is set by a positive value, meaning that the susceptible individuals is increased by, for example, immigration of the susceptible individuals, the number of infected individuals increases. When the value of the* NAP*(101) is set to 1,000 on the 101st, meaning that 1,000 susceptible individuals, that is 0.1% of the population, immigrate into (come and live in) the community, the number of infected individuals reaches 14,910 at the peak on the 194th, then decreases down 0 on the 354th. The total number of infected individuals is 141,929, meaning the 0.1% increment compared to that for the case without any immigrant, 141,788 (Table 11).