Currently, the estimated value of the effective reproduction number (ERN), which is an index for grasping the COVID-19 infection status, is used for important planning and evaluation of infection prevention measures. Since ERN in the Sequential SIR model fluctuates in multiple dimensions due to changes in the surrounding environment, it is difficult to set the appropriate accuracy of the uncertainty region of the estimated data. The challenge in this study is to build a mathematical model of infectious disease according to the characteristics and data characteristics of the infectious disease and select an appropriate estimation method. Highly accurate quantitative research that analyzes the validity of “how infectious diseases prevail” from an academic point of view is the key to prediction and estimation in appropriate infection situation analysis. In this study, we adopted a statistical multivariate analysis method (T method) that enables evaluation and prediction of important factors related to ERN estimation and analysis of phenomena that change in real time (time series analysis). It was clarified that it is possible to estimate with higher accuracy by applying the T method to the estimated value of ERN by the current SIR mathematical model.
Currently, the infectious disease sequential SIR model, which expresses various phenomena by ordinary differential equations, plays an important role in the ongoing fight against COVID-19 and various other viral infections around the world. The purpose of the sequential SIR model is to eradicate the infection if it occurs in a population that was previously uninfected, whether it causes a sustained increase in the number of infected people. It is an infection situation analysis such as what kind of intervention is effective. An important index for developing infection status analysis is the effective reproductions number. The effective reproductions number is an index showing “how many people are infected by one infected person on average”. The higher this number is, the more rapidly the infection is spreading, and if the period of less than 1 continues, it can be said that the infection is converging. This estimated effective reproductions number is used for important planning and evaluation of infection prevention measures implemented.
The estimated effective reproductions number depends largely on virus infectivity. The coefficient of determination fluctuates in multiple dimensions due to changes in the environment (human intervention such as behavioral restriction, differences in the surrounding environment such as climate change, and an increase in the number of immune carriers as the infection progresses). Therefore, there is an academic problem that it is difficult to set the appropriate accuracy of the uncertainty region of the estimated data. In this study, we propose a new infection situation analysis method by highly accurate quantitative analysis applying multivariate analysis method regarding the validity of how the infectious disease spreads.
In pandemic under emerging infectious diseases such as COVID-19, pandemic data analysis and scenario analysis with infection mathematical model are important evidence that forms the nucleus of the policy decision. The infectious disease mathematical model describes the spread of infectious diseases in a population with mathematical formulas. Mathematical epidemiology of infectious diseases is a research field with a long history dating back to the 18th century, but in recent years, with the development of computer processing power and computational statistics, research methods for its social implementation have dramatically advanced [
In this section, in addition to the epidemiological findings of COVID-19, the basic concept of the mathematical model of infectious diseases is described.
The basic mathematical model that captures the pandemic dynamics of infectious diseases that propagate directly from person to person is called “Kermack and McKendrick (1927)”, and is also called the “SIR model” from the variables S, I, and R. Each letter of SIR is an acronym for English, dividing the population into three compartments, susceptibility, infectivity, isolation and recovery, according to the stage of infection, and the temporal changes in the status of infection. It is a bottom-up model [
d S ( t ) d t = − β S ( t ) I ( t ) d I ( t ) d t = β S ( t ) I ( t ) − γ I ( t ) d R ( t ) d t = γ I ( t ) (1)
Here, S(t), I(t), and R(t) represent the proportion of susceptible, infected, and recovered/quarantined individuals at a given time in the population. β is a coefficient representing the infection rate per unit time. βI(t) gives the force of infection at time t. In other words, when the population is constant, the infectivity that is a hazard is proportional to the number of infected people in the population. γ is the rate of removal by recovery or quarantine per unit time, and the reciprocal γ−1 gives the average infectivity period from infection to recovery or quarantine.
Here, by transforming the second Equation (1), the following Equation (2) is obtained.
d I ( t ) d t = ( β S ( t ) − γ ) I ( t ) (2)
when the number of newly infected people is increasing, βS(t) − γ > 0, which is a condition for infectious disease pandemics. Furthermore, this Equation (2) can be transformed into βS(t)/γ > 1. At time 0, since all members of the population are susceptible populations, S(0) = 1 can be set, and β/γ is the threshold for infectious disease pandemics. This value is called the basic reproduction number (R0), which is cited as an index of the strength of virus infectivity and is given as a reference value at the initial stage of infection.
Basic reproduction number (R0) is the most basic index of infectious disease in infectious disease epidemiology. Interpreted as the average number of secondary infections that a typical infected person reproduces during the infectious period
in a population that is all susceptible to an infectious disease. If the basic reproduction number is greater than 1, a large-scale pandemic can occur, but if it is less than 1, the pandemic disappears spontaneously. Since the value of the basic reproduction number is related to the population density, social structure, and contact mode between individuals, the estimated value varies depending on the situation of the area to be analyzed. The basic reproduction number of COVID-19 in China is estimated to be 1.5 to 3.5 from the data of the early stage of the pandemic [
It is known that COVID-19 has a high degree of heterogeneity related to secondary infection, and the distribution of secondary infections reproduced by one infected person varies widely. In other words, most infected people do not produce secondary infections, and some infected people become super spreaders, producing many secondary infections. A study analyzing epidemiological data outside of China estimated that 80% of infected people originated from a small number of infected people (~10%) [
On the other hand, as a situation where many secondary infections occur, the environment where they are densely gathered in a closed space has been clarified from the observation data [
R 0 = c b d (3)
Here, c is the average number of effective contacts (rate) that one person makes effective contact per unit time, b is the probability of infection from infected person to susceptible person per effective contact, and d is the average infectivity period. ERN (RS) is expressed by the following Equation (4), where p (0 < p < 1) is the effect coefficient (decrease rate) of infectivity due to measures against infectious diseases such as wearing a mask, washing hands, and reducing contact.
R S = S β γ ( 1 − p ) = S N ( 1 − p ) R 0 (4)
The estimated value of RS from Equation (4) largely depends on the virus infectivity R0. However, γ, β, and p are unknown coefficients that fluctuate due to changes in the environment (human intervention such as behavioral restriction, differences in the surrounding environment such as climate change, and an increase in immune carriers as infection progresses). In this respect, there is a practical problem that it is difficult to set the appropriate accuracy of the uncertainty region of the estimated data [
Intervention effect Vaccination also can be considered similarly. Let the total population be 1 and the vaccination ratio of the population be x. Assuming that the vaccinated person can be uniformly immunized, ERN (RS) can be expressed by replacing the right-hand side p of Equation (4) with x. Conditions for infection eradication is to ERN (RS) is below 1, ( 1 − x ) R 0 < 1 . The ratio ( 1 − 1 / R 0 ) when this equation is solved for x is called the critical immunity ratio and is often used as the target value for the vaccination rate [
When actually monitoring ERN using observational data in an infectious disease pandemic, various reporting biases need to be considered. A typical example is the delay in the time from infection until the test is actually positive and the person is reported as infected. Since the number of infected persons currently observed is less than that of actual infected persons, the current situation is that the reporting delay is adjusted by statistical estimation [
The purpose of this study is to analyze the validity of “how infectious diseases prevail” from an academic point of view. By building a new infection situation analysis with accurate quantitative analysis, it can create a permanent infection prevention plan. In this study, we propose a new statistical mathematical model of infectious diseases that combines the SIR model in infection situation analysis with the multivariate analysis method that enables appropriate prediction and estimation. Multivariate analysis is a general term for methods that statistically analyze data consisting of multiple variables [
MTS is a mathematical and method system for pattern recognition and prediction in quality engineering (Taguchi method). In addition to the MT method based on statistical mathematics, it includes a unique feature extraction technology, so it is a system that emphasizes practical use rather than simply multivariate data analysis theory.
(a) Mathematical pattern recognition: MT method, RT method.
Both MT/RT methods are methods for pattern recognition. As a common point, the normal group is used as a reference (unit space), and the difference in pattern from that is calculated as the distance. A large distance indicates that it is far from the reference pattern. The difference is that the MT method has the highest pattern recognition accuracy and may be regarded as one of AI (Artificial Intelligence). In the RT method, the scale of unit data is 2 × 2, regardless of
the number of variables. Therefore, it is effective when the number of patterns to be recognized such as character recognition is large [
(b) Mathematical prediction/estimation: T method.
Similar to multiple regression analysis, the T method is a means for predicting and estimating output values (objective variables) from multivariate data (explanatory variables). It has the advantage that there is no instability or impossibility of calculation when the number of samples is small and the multi-collinearity problem, which is a weak point of multiple regression analysis.
(c) Feature value extraction.
In pattern recognition for waveforms and images, the success or failure of feature extraction technique determines. MTS includes feature extraction technique called “variation value” and “abundance value” from these patterns. Although it is a simple method, it often has better anomaly detection sensitivity than frequency analysis when targeting waveforms. It is also used as an extension for image inspection, etc., and by using it in combination with the MT method; it is possible to obtain faster and more sensitive results than conventional technology [
In this study, we adopt the “T method” which is a method for estimating output values from multivariate data in MTS. The T method is a mathematical method that estimates the output value in the same way as multiple regression analysis. Comprehensive estimation is performed from the relationship between each item and the output value by the method of estimating the output value from multiple item variables [
The T Method defines the Unit Space where the output value is in the medium position and homogeneous (densely populated). The computation procedure of the T Method is explained below [
Let’s suppose that, as shown in