key: cord-0997462-i9y68yzs
authors: Omae, Yuto; Sasaki, Makoto; Toyotani, Jun; Hara, Kazuyuki; Takahashi, Hirotaka
title: Theoretical Analysis of SIRVVD Model to Provide Insight on the Target Rate of COVID-19/SARS-CoV-2 Vaccination in Japan
date: 2022-02-13
journal: IEEE Access
DOI: 10.1109/access.2022.3168985
sha: cd33747b360a75f35967aa2101934fe9d1091af1
doc_id: 997462
cord_uid: i9y68yzs

The effectiveness of the first and second dose vaccinations are different for COVID-19; therefore, a susceptible-infected-recovered-vaccination1-vaccination2-death (SIRVVD) model that can represent the states of the first and second vaccination doses has been proposed. By the previous study, we can carry out simulating the spread of infectious disease considering the effects of the first and second doses of the vaccination based on the SIRVVD model. However, theoretical analysis of the SIRVVD Model is insufficient. Therefore, we obtained an analytical expression of the infectious number, by treating the numbers of susceptible persons and vaccinated persons as parameters. We used the solution to determine the target rate of the vaccination for decreasing the infection numbers of the COVID-19 Delta variant (B.1.617) in Japan. Further, we investigated the target vaccination rates for cases with strong or weak variants by comparison with the COVID-19 Delta variant (B.1.617). This study contributes to the mathematical development of the SIRVVD model and provides insight into the target rate of the vaccination to decrease the number of infections.

As of December 2021, the COVID-19 pandemic continues to be a global concern. Telework [1] , lockdown [2] , airport quarantine [3] , and tracing apps [4] are used as countermeasures for overcoming COVID-19 spread. The most important countermeasure is the COVID-19 vaccine (e.g., BNT162b (Pfizer) [5] , mRNA-1273 (Moderna) [6] , and ChAdOx1 (AstraZeneca) [7] , and others). Mathematical simulation is a desirable approach to gain insight into the effect of vaccination on the infectious spread. COVID-19 country-based simulations (Malaysia [8] , Saudi Arabia [9] , Spain [10] , United States [11] , and others) have already been reported. A fast simulation method of differential equations is a susceptible-infected-recovered-vaccination (SIRV) model [12] [13] [14] [15] [16] . However, these models can represent only the first dose of the vaccination. For COVID-19, the effects of the first and second doses on infectious prevention are different; for the COVID-19 Delta variant (B.1.617), the effects of the first and second doses on infectious prevention are 35.6% and 88.0%, respectively [17] . In this context, the SIRVVD model that represents the first and second doses of vaccination was proposed [18] . However, theoretical analysis of the SIRVVD Model is insufficient. Therefore, we find an analytical expression of the infectious number based on the SIRVVD model, by treating the numbers of susceptible persons and vaccinated persons as pa-Figure 1: SIRVVD model [18] rameters.

In order to stop the spread of infection, it is important to achieve herd immunity through vaccination. Some previous studies have reported various target values for achieving herd immunity through vaccination [19] [20] [21] [22] [23] . Because the target vaccination rate to achieve herd immunity depends on the country's lifestyle (wearing masks and social distancing) [24] , it is important to obtain target values of vaccination for each country. Therefore, we calculated the target rate of the vaccinated persons in the scenario of spreading the COVID-19 Delta variant (B.1.617) in Japan based on the solution of infectious numbers of the SIRVVD model. In addition, we investigated target rates in the case of becoming strong or weak variants compared with the COVID-19 Delta variant (B.1.617).

The SIRVVD model [18] for representing the first and second doses of vaccination is defined aṡ

where the dot notation represents the time derivative, i.e.,ḟ (t) := df (t)/dt. Further, S(t) represents the number of susceptible persons; I j∈{0,1,2} (t) represents the number of infected persons of j times the dose of the vaccination (j = 0 means no vaccination). V j∈{1,2} (t) represents the number of the vaccinated persons of j times the dose of the vaccination. R(t) represents the number of recovered persons, and D(t) represents the number of dead persons. Moreover, the total infection number I(t) is defined as

Then, we describe the coefficient parameters as follows: β(t) represents infectivity, θ −1 j∈{0,1,2} represents the average antibody period by the j times the dose of vaccination (j = 0 implies the natural infection). α j∈{1,2} represents the transition rate to V j (t); σ j∈{1,2} represents the effectiveness of j times the dose of vaccination (decreasing the value of infectious probability). γ −1 denotes the average infection period, and δ j∈{0,1,2} denotes the fatality rate of the persons of j times the dose of vaccination. The states transitions are presented in Figure 1 . The infection probability and fatality rate of the vaccinated persons V j∈{1,2} decrease because of the vaccination effects. Then, the total population N is

where R + 0 represents the set of plus real numbers that include zero.

It is desirable that the function of the infectious number I j∈{0,1,2} (t) is clear to investigate the infectious spread. However, it is difficult to obtain an analytical solution in the absence of any assumptions be-causeİ j (t) is complex (including various functions). Therefore, we assume that S(t), V 1 (t), V 2 (t), and β(t) are not time-dependent functions but the constants of S, V 1 , V 2 , and β, respectively. Then, the total number of persons that have an infectious possibility N is given by

If the infectious number is sufficiently small, N N . I j (t) defined by Equations (4)-(6) can be simplified as

where

Herein, if we define

then, Equation (10) can be simplified intȯ

as the first-order separable simultaneous differential equations. Here, we transform it into 1 I(t) dI(t) = A dt, and then, we can represent

where U represents the diagonal matrix of A, and these relationships can be represented by

where P comprises the eigenvectors of the coefficient matrix A, and λ j∈{0,1,2} represents the eigenvalues. In other words, the solution I(t) is clear by calculating the eigenvectors and eigenvalues of the coefficient matrix A.

To be clear, when λ j∈{0,1,2} , let a 3 × 3 unit matrix be denoted by E; we calculate

In addition, the eigenvalues λ are obtained by solving the eigenequation det(A − λE) = 0 as

From the eigenvalues, P comprises eigenvectors that can be represented as

where

The solution I(t) of the differential equationİ(t) can be represented by substituting Equations (12)- (14) into Equation (11) . Then, solutions I j∈{0,1,2} are represented as

The coefficients are given by

where,

These equations require constant values of S(t), V 1 (t), and V 2 (t); thus, these equations are not suitable for long-term simulations.

γ represents the inverse value of the average infection period, and therefore, γ > 0. Thus, lim t→∞ e −γt = 0.

Further, because the coefficients of Equation (15) are constants, each solution I j∈{0,1,2} (t) can be approximated by

S was deleted by Equation (9). Therefore, we get the following theorems:

where, j ∈ {0, 1, 2}. The infectious spread stops by creating a scenario as indicated in Equation (17). We collected terms on virus parameters to the right side and vaccination parameters on the left side to clarify this scenario. As a result, Equation (17) can be simplified into

Here, N denotes the total number of persons with a probability of infection; σ j∈{1,2} represents the effectivity of the vaccination; and V j∈{1,2} represens the number of vaccinated persons. Thus, the left side represents the total number of people who have the probability of infection considering the vaccination (the left side is small when using highly effective vaccination). The longer the infection period and the higher the infectivity, the smaller is the right term because γ −1 represents the average infection period and β represents the infectivity. In addition, in the case a virus has a long infection period and high infectivity, a highly effective vaccination is required to stop the spread of infection.

Here, let V obj.

2 denote the target number of the second dose of vaccination required to stop the spread of infection. We can obtain V obj. 

Further, the target rate of the second dose of vaccination required to stop the infection spread P obj. 

We consider the condition of not stopping the infection spread even if all persons receive the second dose of vaccination. We obtain the following theorem because this scenario would mean substituting V 2 = N , V 1 = 0 for Eq. (16).

Even if all persons receive the second dose of vaccination, the infection number continues to increase when the effectivity of the second dose of the vaccination σ 2 is insufficient. In contrast, the condition of Equation (20) is not satisfied when the vaccination that has perfect prevention (i.e., σ 2 = 1) because N > 0, β > 0, and γ > 0. In other words, the spread of infection stops when developing a vaccination that has a perfect prevention effect (i.e., σ 2 = 1) and getting a vaccination to all persons (i.e., V 2 = N ). However, it is important to know the condition of Equation (20) because achieving this is difficult.

We consider conditions under which the vaccine is not required; this implies substituting V 2 = 0, V 1 = 0 for Equation (18) . We can obtain a simplified equation as

2.6 Validity of the target number V obj.

Here, we check whether the infection spread stops by satisfying V obj.

. Therefore, we conduct the SIRVVD model-based simulations (i.e., Equations (1) -(8)) assuming a simple scenario. The fixed parameters are • Initial number of vaccinated persons (first dose)

We adopted V 2 ∈ {0×V obj.

, 0.2×V obj.

, · · · , 2.0×V obj.

2 } over the initial number of persons who get the second dose of vaccination V 2 to verify the cases that satisfy V obj.

The simulation results for cases V 1 = 0 and V 1 = N/2 are shown in Figures 2 and 3 , respectively. The horizontal axis represents V 2 ; the vertical axis, I(t end )/I(0). In addition, the black dashed line represents I(t end )/I(0) = 1; the red dashed line, V 2 = V obj.

. I(t end )/I(0) > 1 indicates that the infection number increases, and I(t end )/I(0) < 1 implies that it decreases. In all cases, when V 2 = V obj. 

Let β delta denote the infectivity of the COVID-19 Delta variant (B.1.617) in Japan; we estimate this value. According to previous research [25] , infectivity β can be represented as

where r denotes the effective reproduction number. Further, β calculated by Equation (22) [27] . Therefore, we consider that the infectivity of the Delta variant (B.1.617) is underestimated owing the effect of the vaccination when considering the newer cases of infection during this period. Thus, we estimated the infectivity of the Delta variant (B.1.617) in Japan using data from the period when the vaccination was not started (before February 2021). According to the COVID-19 advisory board for the Japanese government [28] , the Delta variant (period: August 2021) had a 1.9 times higher infection rate compared to the that of the regular COVID-19 virus (period: December 2020) in Japan because no people were vaccinated in December 2020. Thus, we consider the infectivity of the Delta variant (B.1.617) as the 1.9 times the infectivity of this period. We assume

where β delta and β regular represent the infectivity of the Delta variant (B.1.617) and regular COVID-19 virus, respectively. The maximum value of the effective reproduction number in December 2020 in Japan was r = 1.18 [29] . [19] World 82.5% 45 -60 % Gumel [20] US 70% 80 % Gumel [20] US 95% 60 % Kadkhoda [21] -100% 62 -72 % Kadkhoda [21] -95% 63 -76 % Fontanet [22] France -67 % Liu [23] China 90% 83 -92% Ours Japan** 88% 63% *: Prevention effect of vaccine. **: The case of lifestyle on Dec. 2020 in Japan.

Kobayashi et al. estimated 0.17 as the γ parameter of COVID-19 in Japan from 0.13 to 0.17 [30] . We adopted γ = 0.13 because we consider optimism to be undesirable, and this leads to a long infection period. The total population in Japan is approximately S = 1.2×10 8 persons. We obtain β regular 1.28×10 −9 as the infectivity of the COVID-19 regular virus by substituting these parameters in Equation (22) . Moreover, the infectivity of the Delta variant (B.1.617) is β delta 2.43 × 10 −9 by substituting β regular for Equation (23) and by substituting β regular .

In a previous study, Bernal et al. [17] reported the effect of the BNT162b (Pfizer) vaccine on reducing the probability of infection with the COVID-19 Delta variant (B.1.617). The effect of the first dose was 35.6% (22.7∼46.4%), and that of the second dose was 88.0% (85.3∼90.1%). Thus, we adopted σ 1 = 0.356 and σ 2 = 0.880.

2 and target rate P obj.

We consider the target number V obj. 2 and target rate P obj. 2 in Japan for the infectious spread of the Delta variant (B.1.617). We assume that all persons who received the first dose will receive the second dose, i.e., V 1 = 0. Further, the total population in Japan is about 1.2 × 10 8 . V obj. can be represented by substituting these values for Equation (19) as The target rate of 63% is for the case of the lifestyle of December 2020 in Japan. If we obtain the lifestyle values before the appearance of COVID-19 (before December 2019), a target rate of about 63% will appear to be insufficient.

Snehota et al. [31] reported that about 72.5% of the population in various countries have taken the COVID-19 vaccine. We believe that the target vaccination rate can be achieved in Japan because the target rate P obj. 2 = 0.63 is lower than the reported global value of 72.5%. In Japan, on December 27, 2021, 78% of the total population received the second dose of vaccination [27] , and the infection number decreased (e.g., the average number of new cases of infection per day from November 1, 2021, to December 31, 2021 was lower than 3 persons per million persons [32] .).

The target rates of the vaccinated persons reported by other researchers are listed in Table 1 . These are target rates for achieving herd immunity to stop infection spread. We check the various values in this study.

One reason for the different values is that the target rate is dependent on the lifestyle of each country. For example, Shen et al. [24] reported that the target rate of vaccination is dependent on wearing masks and social distancing.

The target rate of vaccinated persons to stop the infection spread is not the same because the lifestyles of each country are not the same. Thus, it is necessary to calculate the target rate of each country. P obj. 2 = 0.63 represents the target rate for the lifestyle in December 2020 in Japan.

In the near future, there is a possibility of new variants occurring; for example, the COVID-19 Omicron variant (B.1.1.529) [33] . Thus, we consider the cases of strong or weak new variant comparisons with the Delta variant (B.1.617). We consider the following conditions:

• Changing the average infection period γ −1

• Changing infectivity β

• Changing vaccination effectivity σ 2

For the simulation parameters, we considered the average infection periods ranging from 2 -20 days, and the infectivity ranging from 0.2 -3 times. In addition, we consider three patterns as vaccination effectiveness: the case of perfect prevention (σ 2 = 1); the case of BNT162b (Pfizer) effectivity (σ 2 = 0.88 [17] ); and the case of a lower effectivity comparison with BNT162b (Pfizer) (σ 2 = 0.70). Thus, we adopted the combinations of the following parameters γ −1 ∈ {2, 4, · · · , 20}, β ∈ {0.2 × β delta , 0.4 × β delta , · · · , 3.0 × β delta }, σ 2 ∈ {1, 0.88, 0.70}, and calculated the target rate P obj. 2 to stop the infection spread.

The target rates P obj.

2 in the case of perfect prevention (σ 2 = 1) are shown in Figure 4 . The vertical axis represents the average infection period γ −1 [day], and the horizontal axis represents the infectivity β (1.0 × β delta represents the Delta variant infectivity). The percentage values are P obj. However, the perfect prevention vaccine is not realistic. Therefore, we calculate the case of σ 2 = 0.88, as shown in Figure 5 . Painted black markers represent the results satisfying Equation (20) , i.e., even if all persons receive the second dose of vaccination, the infection spread does not stop. This scenario occurs in the cases of long infection periods and high infectivity. For such a scenario, we consider not only vaccinations but also lockdowns, such as stay-at-home orders.

Finally, the case of a new variant that leads to a decrease in the vaccination effect (σ 2 = 0.70) is shown in Figure 6 . There are many cases wherein only the vaccination cannot stop the spread of the infection caused by a new strong variant. The results indicate that it is important to develop more high-effect vaccines to overcome the new strong variant.

We found the analytical expressions of I j∈{0,1,2} (t) of the differential equationsİ j∈{0,1,2} (t) of the SIRVVD model. Furthermore, we proposed a method to determine P obj.

, which is the target rate of the vaccination required to stop the spread of infection. Assuming the COVID-19 Delta variant (B.1.617) in Japan, we estimated P obj. 2 to be approximately 63%. Further, we calculated the target rate of the vaccination by assuming a new strong or weak variant (Figures 4, 5, and 6 ). We consider that these values are important to control the infection spread.

We did not consider the validity term of the vaccination. The vaccination effect decreases based on elapsed time [33] . In future studies, we plan to investigate vaccination strategies that consider the validity of the vaccination. 

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