Adaptive Fourier decomposition analysis of different pandemic stages in South Korean cities: policies and trends

Xi Zhang; Weikai Shi; Zige Liu; Aaron Finley; Kerui Cen; Zhiyao Xie; Pengjie Yang; Huitao Li; Unian Leong; Xiaoping Zhang; Honchon Kwong; Matthew Tsunlung Li; Guibin Lu

doi:10.21037/jtd-2024-2141

Original Article

Adaptive Fourier decomposition analysis of different pandemic stages in South Korean cities: policies and trends

Xi Zhang^1#, Weikai Shi^2#, Zige Liu², Aaron Finley³, Kerui Cen², Zhiyao Xie¹, Pengjie Yang⁴, Huitao Li⁵, Unian Leong⁶, Xiaoping Zhang⁷, Honchon Kwong⁶, Matthew Tsunlung Li⁷, Guibin Lu²

¹Faculty of Chinese Medicine, Macau University of Science and Technology, Macau, China; ²Faculty of Innovation Engineering, Macau University of Science and Technology, Macau, China; ³School of Business, Macau University of Science and Technology, Avenida Wai Long, Macau, China; ⁴Department of Psychology and Human Development, IOE Faculty of Education and Society, University College London, London, UK; ⁵Department of Psychology and Neuroscience, University of St Andrews, St Andrews, UK; ⁶Pui Ching Middle School, Macau, China; ⁷Premier School Affiliated to Hou Kong Middle School, Macau, China

Contributions: (I) Conception and design: Z Liu, G Lu; (II) Administrative support: Z Liu; (III) Provision of study materials or patients: Xi Zhang, W Shi, Z Liu, G Lu; (IV) Collection and assembly of data: Xi Zhang, W Shi, A Finley, K Cen, Z Xie, P Yang, U Leong, H Kwong, Z Liu, G Lu; (V) Data analysis and interpretation: Xi Zhang, W Shi, A Finley, K Cen, Z Xie, P Yang, H Li, Xiaoping Zhang, MT Li, Z Liu, G Lu; (VI) Manuscript writing: All authors; (VII) Final approval of manuscript: All authors.

^#These authors contributed equally to this work.

Correspondence to: Zige Liu, ME; Guibin Lu, MBA. Faculty of Innovation Engineering, Macau University of Science and Technology, Avenida Wai Long, Taipa, Macau 999078, China. Email: zgliu@must.edu.mo; gblu@must.edu.mo.

Background: There is a lack of a method supported by mathematical theory to compare infection rate fluctuations with the timing of policy implementation and determine the degree of correlation in South Korean cities. We evaluated the impact of the adaptive Fourier decomposition (AFD) method on the time trends and policies of coronavirus disease 2019 (COVID-19) data in South Korea, aiming to study whether AFD can contribute to future infectious disease prevention and control strategies.

Methods: This study utilized AFD to analyze temporal trends in COVID-19 data, demonstrating an effective method to illustrate the dynamics of the outbreak and comprehend the real-time impact of Korean policies on COVID-19 transmission. AFD is a novel signal decomposition algorithm that characterizes analytic signals through a linear combination of adaptive basis functions. In each AFD decomposition step, basic functions are selected from an overcomplete dictionary, and the process continues until the energy difference between the original and reconstructed signals fall below a predefined tolerance.

Results: We found the reconstructed components of AFD are integrated with policy data spanning various periods, enabling a comprehensive analysis. The decomposition results indicated that the reduced business hours policy was closely associated with the high-frequency components observed in the first two waves. Vaccination policies were initially associated with the fourth wave. In addition, our statistical analyses showed that the third-order components of AFD were significantly and positively correlated with the original infections. In contrast, the correlation coefficients of most components in empirical modal decomposition (EMD) and variational modal decomposition (VMD) did not reach significance.

Conclusions: AFD is a novel algorithm. We found that AFD was able to assess the real-time impact of South Korean policies on the spread of COVID-19 by analyzing data from different stages of COVID-19 in South Korea. We infer that this method can guide future epidemic prevention and control strategies.

Keywords: Adaptive Fourier decomposition (AFD); coronavirus disease 2019 (COVID-19); South Korea; daily fluctuation; epidemic prevention policy

Submitted Dec 12, 2024. Accepted for publication May 14, 2025. Published online Jun 09, 2025.

doi: 10.21037/jtd-2024-2141

Highlight box

Key findings

• Analyses of daily coronavirus disease 2019 (COVID-19) data from South Korea using adaptive Fourier decomposition (AFD) revealed valuable insights into how various government policies influenced infection rates during four different wave periods.

What is known and what is new?

• Analyses of daily COVID-19 data from South Korea using AFD and compare the result with empirical modal decomposition and variational modal decomposition.

• Utilizing AFD to analyze South Korean’s daily COVID-19 data, our study uses data analysis results to find the correlation between the influenced infection rates and policies in different waves of the epidemic in different cities in South Korea.

What is the implication, and what should change now?

• When facing a new wave of epidemics, this approach allows us to compare fluctuations in infection rates with the timing of policy implementation and determine the extent of their correlation.

Introduction

Global epidemics have shown fluctuations, presenting challenges in the fight against novel coronaviruses in recent years. Data from the World Health Organization (WHO) indicate that the coronavirus disease 2019 (COVID-19) infection rate in South Korea has fluctuated daily from January 16, 2020, to September 7, 2023.

Most existing articles use transmission dynamics models, such as susceptible-infective (SI), SI-susceptible (SIS), SI-removed (SIR), susceptible-infected-recovered-susceptible (SIRS), susceptible-exposed-infective-removed (SEIR), and other mathematical models, to analyze the transmission of large-scale epidemics (1-5). Since the 18^th century, mathematical models have been an essential tool in the epidemiological study of infectious diseases, becoming the primary analytical method for understanding epidemic transmission trends. This study analyzed the temporal trends of COVID-19 data using adaptive Fourier decomposition (AFD), demonstrating an effective approach for capturing the dynamics of the epidemic and assessing the real-time impact of Korean policies on COVID-19 transmission. AFD is a novel signal decomposition algorithm that characterizes the analyzed signals by a linear combination of adaptive basis functions. During each AFD decomposition step, basis functions are selected from an overcomplete dictionary, and the process continues until the energy difference between the original and reconstructed signals falls below a predefined tolerance. These reconstructed components, combined with policy data spanning different periods, enable comprehensive analyses of the impact of various policies at different epidemic stages to inform future infectious disease prevention and control strategies.

The existing literature predominantly uses econometric methods and epidemiological models to examine daily fluctuations in COVID-19 cases (6). Existing research has mainly focused on the linear modeling of time-domain properties and daily infection statistics. Daily COVID-19 case counts contain complex nonlinear data with frequency domain properties. Ahn et al. draw on the Deleuzo-Guattarian framework to detail how a surveillance ensemble can be created and manipulated on a rhizomatic basis to monitor a portion of the population (7). In the spatial domain, Kang et al. conducted a spatio-temporal analysis of weekly COVID-19 cases across South Korea, first covering the entire country and then focusing on the metropolitan areas of Seoul, Gyeonggi, and Incheon from February 1, 2020, to May 30, 2021, using Moran’s I and spatial scanning statistics (8). Moreover, Jang et al. conducted a retrospective cohort study of 9,030 COVID-19 patients enrolled between February and November 2021 (9). Relative risks (RRs) and 95% confidence intervals (CIs) were used to assess the effect of vaccination on the incidence of symptoms and viral load as indicated by RdRp and E gene cycle thresholds. By analyzing the content of a press release from the Korean Disease Control and Prevention Agency (KDCA), Chun et al. defined the challenges of the COVID-19 outbreak. They identified the causes of the health and social impacts, developing a problem tree and a logistic model that outlines inputs, activities, outputs, outcomes, and effects (10).

In addition to mathematical models, several studies have analyzed data collected through questionnaires, integrating these findings with policy implications. Hwang et al. conducted a population-based national cohort study using data from the National Health Insurance Service (NHIS) of South Korea (11). Heo et al. demonstrated that adequate medical equipment, technology, human resources, and flexible policy options can motivate healthcare providers and encourage public cooperation during national crises (12). Jeon et al. conducted an extensive retrospective analysis of the aggregation of COVID-19 infections occurring in South Korea from January 2020 to September 2021, with a large-scale epidemiological survey using data provided by KDCA. This study revealed significant clinical and social factors associated with age and regional patterns of infections (13). While these analyses are well-targeted, the challenges of disseminating the number of questionnaires during the epidemic make it more difficult to analyze the data extensively and accurately.

Through preliminary literature research, we found that there is a lack of a method with mathematical theoretical support to compare fluctuations in infection rates with the timing of policy implementation in South Korea and determine the extent of their correlation. Therefore, this study aims to comprehensively analyze the evolution of COVID-19 prevention and control policies in South Korea by AFD analyzing fluctuations in infection rates in South Korean cities and relevant policies to guide future infectious disease prevention and control strategies. We present this article in accordance with the STROBE reporting checklist (available at https://jtd.amegroups.com/article/view/10.21037/jtd-2024-2141/rc).

Methods

Datasets

With a population of approximately 51 million, including about 26 million in metropolitan areas, South Korea’s high population density and mobility have contributed to the concentration of COVID-19 cases in urban regions. Research on South Korea’s successful response to COVID-19 has emphasized the crucial roles of government leadership, the KDCA, and lessons learned from the response to Middle East respiratory syndrome (MERS). The first case of COVID-19 in South Korea was diagnosed on January 20, 2020. As of June 2022, there have been four waves of the outbreak. After the fourth wave, which was the largest, occurring from late April to June 2022, the cumulative number of confirmed cases exceeded 3,835,400 cases.

Study methods

AFD

The decomposition of a signal into a set of oscillatory modes and single components is an essential topic in signal processing (14-16). These signal components provide useful information for classification, filtering, denoising, and feature extraction of signals. Recent advancements have introduced new methods and approaches, with AFD emerging as a method of analyzing nonlinear and non-smooth signals in recent years (17-20).

$\tilde{G} (t) = \sum_{n = 1}^{N} ⟨ G_{n}, e_{{a_{n}}} ⟩ B_{n} (e^{j t})$ [1]

In AFD, we use modified Blaschke products. Eq. [1] as its base (axes):

$B_{k} (z) = \frac{1}{\sqrt{2 π}} \cdot \frac{\sqrt{1 - {| a_{k} |}^{2}}}{1 - a_{k}^{*} z} \cdot \prod_{l = 1}^{k - 1} \frac{z - a_{l}}{1 - a_{l}^{*} z} s . t . a_{l} \in D, \forall l \in Z^{+}$ [2]

Where n represents the number of decomposition layers, z = e^jt = cos(t) + j × sin(t), a* represents the complex conjugation operation, i.e., a* = real{a} − j × imag{a} (a = real{a} − j × imag{a}) and a₁, a₂, ..., a_l are the variables of B_k, and all a_l (l = 1, 2, ..., n) need to satisfy |a_k| <1.

This algorithm shows a clear difference between the AFD method and the conventional decomposition method.

$e_{{a_{n}}} (e^{j t}) = \frac{\sqrt{1 - {| a_{n} |}^{2}}}{1 - {\bar{a}}_{n} e^{j t}}$ [3]

AFD decomposes the signal based on the energy distribution of the signal and is suitable for separating signals with overlapping frequencies.

$G_{n} (e^{j t}) = R_{n - 1} (e^{j t}) \prod_{l - 1}^{n - 1} \frac{1 - {\bar{a}}_{l} e^{j t}}{e^{j t} - a_{l}}$ [4]

New components construction based on AFD results

AFD has different variants, including core AFD, unwinding AFD, cyclic AFD, and random AFD. Each variant has been applied across various fields. In the signal decomposition of large-scale data, achieving sufficient decomposition often requires multiple layers. This inevitably leads to redundant data generated by signal decomposition. By merging partial decomposition results, the risk of data errors can be reduced, and the reliability of the data can be improved. Therefore, we attempted to construct new components based on the decomposition results:

$D_{α} (β) \sum_{γ = α β - β + 1}^{α β} C_{γ} B_{γ} (e^{j t}) s . t . C_{γ} = ⟨ G_{γ}, e_{{a γ}} ⟩$ [5]

$E_{γ} (e^{j t}) = \frac{\sqrt{1 - {| a_{γ} |}^{2}}}{1 - {\bar{a}}_{γ} e^{j t}} \prod_{k = 1}^{γ - 1} \frac{e^{j t} - a_{k}}{1 - {\bar{a}}_{k} e^{j t}}$ [6]

$F_{β} = \sum_{α = 1}^{r} C_{α} (β)$ [7]

Pearson correlation coefficient

The Pearson correlation coefficient (21,22), also referred to as the Pearson product-moment correlation coefficient, represents a linear correlation coefficient and is the most widely utilized type of correlation coefficient. Denoted as ‘r’. This coefficient measures the degree of linear correlation between two variables, X and Y. The ‘r’ value ranges from −1 to 1, with a larger absolute value indicating a stronger correlation.

$r = \frac{\sum_{i = 1}^{n} (X_{i} - \bar{X}) (Y_{i} - \bar{Y})}{\sqrt{\sum_{i = 1}^{n} {(X_{i} - \bar{X})}^{2} \sqrt{\sum_{i = 1}^{n} {(Y_{i} - \bar{Y})}^{2}}}}$ [8]

Where $X_{i}$ and $Y_{i}$ are the i^th data points, X and Y are the means of $\bar{X}$ and $\bar{Y}$ , respectively, and n is the number of data points.

Algorithm (Box 1)

Box 1 Algorithm 1: AFD algorithm (23)

Input: s(n): N-sample zero-mean signal with n=0, 1, 2, …, N − 1; S: population size; Tmax: maximum number of iterations

Output: sr(n): reconstructed version of s(n)

a₁=0; i=1;

z = e^{j t}

with

t = 2 π n N

, where

n = 0, 1, 2, \dots, N - 1

B_{1} (z) \leftarrow \frac{1}{\sqrt{2 π}}

(I) Initialization

f (n) \leftarrow s (n) + j H [s (n)]

[the analytic from of s(n)]

d_{1} (n) \leftarrow (\frac{1}{N}) \sum_{n = 0}^{N - 1} f (n)

(II)

e_{a_{i}} (z) \leftarrow \frac{1}{\sqrt{2 π}} \frac{\sqrt{1 - {| a_{i} |}^{2}}}{1 - a_{i}^{*} z}

(III)

B_{i + 1} (z) \leftarrow \frac{1}{\sqrt{2 π}} \frac{\sqrt{1 - {| a_{i + 1} |}^{2}}}{1 - a_{i + 1}^{*} z} \prod_{l = 1}^{i} \frac{z - a_{l}}{1 - a_{l}^{*} z}

(IV)

c_{i + 1} \leftarrow \frac{2 π}{N} \sum_{n = 0}^{N - 1} f (n) B_{i + 1}^{*} (z)

(V)

B_{i + 1} (z) \leftarrow \frac{1}{\sqrt{2 π}} \frac{\sqrt{1 - {| a_{i + 1} |}^{2}}}{1 - a_{i + 1}^{*} z} \prod_{l = 1}^{i} \frac{z - a_{l}}{1 - a_{l}^{*} z}

(VI)

c_{i + 1} \leftarrow \frac{2 π}{N} \sum_{n = 0}^{N - 1} f (n) B_{i + 1}^{*} (z)

(VII)

d_{i + 1} \leftarrow c_{i + 1} B_{i + 1} (z)

(VIII)

f_{i + 1} (n) \leftarrow \sum_{k = 1}^{i + 1} d_{k} (n)

(IX)

S_{r} (n) \leftarrow Re {f_{i} (n)}

AFD introduces an efficient adaptive decomposition method that presents a novel approach to signal analysis. The main understanding and issues involved in signal decomposition emphasize the need for decomposition components to be orthogonal to each other and can be used to represent time-varying signals and extract information efficiently.

Figure 1 is the AFD algorithm flowchart. A critical aspect of this process is determining which components to decompose or what information to extract. When a signal or system lacks sufficient known information, we can anticipate that the extracted components will exhibit the following characteristics:

The extracted components must be orthogonal to each other, ensuring no information redundancy between them.
The information extraction process is efficient, aiming to represent most of the original signal with a small number of components. If we view signal decomposition as feature extraction or dimensionality reduction, we aim to minimize the number of features (dimensions) while ensuring sufficient information extraction.

Figure 1 AFD algorithm flowchart. AFD, adaptive Fourier decomposition.

Data analysis

Total data on COVID-19 infections in South Korea

The data source is mainly based on the CSV dataset of the number of daily COVID-19 infections in Korea in the WHO from January 16, 2020, to September 7, 2023 (24). The WHO has developed profiles for 207 countries, enabling an in-depth exploration of coronavirus pandemic statistics across various global regions (25).

Evaluating the situation of individual countries in their efforts to examine the COVID-19 epidemic is complex, given the rapidly evolving nature of the pandemic. The WHO continuously monitors the pandemic’s impact to offer a comprehensive evaluation, generating country profiles for 207 nations. As shown in Figure 2 and Table 1, these profiles present interactive visualizations of pandemic statistics, clarify the indicators used in the analyses and provide information about the data sources.

Figure 2 A number of weekly infections (data from WHO). WHO, World Health Organization.

Table 1

South Korea outbreak wave timeline

Wave	Start time	End time	Events	Policies
Wave 1	2020/2/18	2020/4/19	A believer of Daegu New Heaven and Earth Church has been diagnosed with the New Crown	Korea’s CDC has raised the level of alert for infectious disease disasters, from “concern” to “attention”. Beginning of socialization period
Wave 2	2020/8/1	2020/9/11	SaRang Community Church holds offline worship and Gwangmu day (15^th) gathering	Two-stage response to maintain social distance, at which time will enforce measures such as banning indoor gatherings of 50 people
Wave 3	2020/11/1	2021/1/27	The emergence of church congregate infections, which has led to a significant increase in cases of infection	Implementation of a 4-person limit for private gatherings. High-intensity prevention of the epidemic measures. South Korea began to implement the obligation to wear masks
Wave 4	2021/7/1	2022/5/1	“Delta” mutant strain spreads leading to clusters of infections in daily living spaces	Korea’s metropolitan area rises to the highest alert level 4

CDC, Centers for Disease Control and Prevention.

AFD decomposition results discounted graph

This section discusses the process of obtaining the variation of daily COVID-19 infection counts in Korea across different time-frequency scales using AFD (see Figure 3). The first to 8^th vertical lines in column I represent frequency components from the lowest to the highest illustrating changes in infection rates across long- and short-term scales. Similarly, the vertical lines in columns II and III follow a similar approach as in column I. The horizontal lines I to III in Figure 3 represent the construction methods from the first to third order when q=1, 2, and 3, respectively, reflecting the changes in daily COVID-19 infection counts under different AFD construction stages.

Figure 3 AFD comparison chart by order. AFD, adaptive Fourier decomposition.

Specifically, in the first-order decomposition, the raw data were decomposed into eight original components, which were then constructed into eight new components based on the q=1 scenario in Eq. [5]. As shown in column I of Figure 3, Comp1 represents the component with p=1 and q=1, Comp2 represents the component with p=2 and q=1, and so on until Comp8 represents the component with p=8 and q=1. The original signal is obtained when these eight newly constructed components are superimposed by Eq. [6] and Eq. [7].

In the second-order decomposition, the original data is first decomposed into 16 original components and then constructed into eight new components according to q=2 in Eq. [5]. As shown in column II of Figure 3, Comp1 represents the component with p=1 and q=2, Comp2 represents the component with p=2 and q=2, and so on, and when these eight newly constructed components are superimposed by Eq. [6] and Eq. [7], the original signal can be obtained.

For the third-order decomposition, the original data is decomposed into 24 original components, which are then constructed into eight new components according to q=3 in Eq. [5]. As shown in column III of Figure 3, Comp1 represents the component with p=1 and q=3, Comp2 represents the component with p=2 and q=3, and so on. When these eight newly constructed components are superimposed by Eq. [6] and Eq. [7], the original signal is again obtained.

Next, we compare the results obtained from AFD with two commonly used time-frequency methods: empirical modal decomposition (EMD) and variational modal decomposition (VMD). The decomposition results of these two methods are shown in Figure 4, while the correlation between the components and the number of COVID-19 infections per day under the vaccination policy is presented in Figure 4. As shown in Table 2, it can be observed that the first component obtained by VMD is similar to the first component obtained by AFD and their coefficients are much higher than the other components.

Figure 4 Number of the Seoul weekly infections during the first wave. The blue curve is the number of weekly infections in Seoul. The red curve is AFD-decomposed daily infection number decomposition component. AFD, adaptive Fourier decomposition.

Table 2

Correlation of AFD decomposed components

Component	AFD-based		AFD-second		AFD-third		AFD-based & AFD-second		AFD-based & AFD-third		AFD-second & AFD-third
Component	r	P value	r	P value	r	P value	r	P value	r	P value	r	P value
Comp1	0.897	<0.001***	0.937	<0.001***	0.944	<0.001***	0.957	<0.001***	0.951	<0.001***	0.993	<0.001***
Comp2	0.271	<0.001***	0.293	<0.001***	0.312	<0.001***	0.925	<0.001***	0.869	<0.001***	0.939	<0.001***
Comp3	0.111	<0.001***	0.155	<0.001***	0.187	<0.001***	0.719	<0.001***	0.594	<0.001***	0.826	<0.001***
Comp4	0.107	<0.001***	0.151	<0.001***	0.167	<0.001***	0.714	<0.001***	0.645	<0.001***	0.903	<0.001***
Comp5	0.105	<0.001***	0.127	<0.001***	0.167	<0.001***	0.827	<0.001***	0.632	<0.001***	0.764	<0.001***
Comp6	0.072	0.009**	0.130	<0.001***	0.143	<0.001***	0.554	<0.001***	0.504	<0.001***	0.909	<0.001***
Comp7	0.255	<0.001***	0.232	<0.001***	0.224	<0.001***	0.909	<0.001***	0.877	<0.001***	0.965	<0.001***
Comp8	0.104	<0.001***	0.104	<0.001***	0.104	<0.001***	1.000	<0.001***	0.989	<0.001***	0.989	<0.001***

Left three: correlation between decomposition results and raw data; right three: correlation between components of different methods. The correlation coefficients are all based on Pearson correlation coefficient algorithm. ***, P<0.001; **, P<0.01. AFD, adaptive Fourier decomposition.

Meanwhile, the 4^th to 6^th components obtained by EMD are similar to the peaks obtained by AFD. In addition, the coefficients of the VMD components decrease with increasing frequency, which is consistent with the results obtained from the AFD decomposition.

Pearson’s coefficient validation

In this study, Pearson’s correlation coefficient was calculated for all decomposition results. The correlation analysis focused on the interval where the P value is less than 5%. The coefficients of the first component were much higher than the rest of the components of the same order across different AFD decompositions, and the distribution of the coefficients decreased from low frequency to high frequency. The reconstruction results for the first-, second-, and third-order components were consistent, reflecting the relationships between each component and the daily COVID-19 diagnosis data. The strong correlation between the components and the raw data showed that this component can better analyze the source data fluctuations. As shown in Table 2, within 5% of the correlation coefficients, all the components except component 8 satisfy the positive correlation trend with the raw data. Table 2 The fourth to sixth columns of the correlation data of each order of AFD were all within 1%, which means that there was a strong correlation between the results of the AFD interpretation, the third order coefficient of the first component is 0.944, which was the highest compared to the other seven components; there was a negative correlation between the coefficients and the frequency, i.e., the coefficients decreased with the increase of the frequency.

As shown in Table 3, we also estimated the Pearson correlation coefficients between the components obtained from these models to further explore the correlation between the AFD, EMD, and VMD decomposition results. The AFD components were positively correlated with EMD and VMD at the 5% significance level for 75%, while the correlation coefficients between the remaining components were insignificant. This indicates that AFD is highly correlated with most of the components of the other two methods. Whereas most of the coefficients for the EMD and VMD components were not significant, indicating a weak correlation between their decomposition results. Therefore, AFD performs best in absorbing useful information obtained from these methods on the effectiveness of anti-epidemic measures.

Table 3

Correlation of different method decomposed

Component	ADF-third		EMD-based		VMD-based		ADF-third & EMD-based		ADF-third & VMD-based		EMD-based & VMD-based
Component	r	P value	r	P value	r	P value	r	P value	r	P value	r	P value
Comp1	0.944	<0.001***	0.330	<0.001***	0.741	<0.001***	0.331	<0.001***	0.763	<0.001***	0.527	<0.001***
Comp2	0.312	<0.001***	0.290	<0.001***	0.736	<0.001***	−0.215	<0.001***	0.295	<0.001***	0.005	0.846
Comp3	0.187	<0.001***	0.339	<0.001***	0.169	<0.001***	0.122	<0.001***	−0.002	0.938	0	0.988
Comp4	0.167	<0.001***	0.437	<0.001***	0.139	<0.001***	0.064**	0.021	0.073**	0.008	0.001	0.970
Comp5	0.167	<0.001***	0.131	<0.001***	0.126	<0.001***	−0.124	<0.001***	−0.003	0.923	0.001	0.976
Comp6	0.143	0.009**	−0.104	<0.001***	0.099	<0.001***	0.142	<0.001***	0.057	0.038**	0.005	0.860
Comp7	0.224	<0.001***	0.160	<0.001***	0.075	0.006**	0.335	<0.001***	0.253	<0.001***	0.009	0.753
Comp8	0.104	<0.001***	0.114	<0.001***	0.090	0.001**	0.250	<0.001***	0.314	<0.001***	0.417	<0.001***

Left three: correlation between decomposition results and raw data; right three: correlation between components of different methods. The correlation coefficients are all based on Pearson correlation coefficient algorithm. ***, P<0.001; **, P<0.01. AFD, adaptive Fourier decomposition; EMD, empirical modal decomposition; VMD, variational modal decomposition.

The AFD method analyzes the daily new infection counts, with line graphs for each epidemic wave decomposed individually. This decomposition allows for a detailed examination of the links between wave-specific components and implemented policies. From the whole to each wave, the analysis evaluates the strength of the relationships between government policies, significant events, and epidemic trends. The study further incorporates Pearson’s correlation coefficient to assess the best performance of the AFD algorithm relative to other methods. The algorithm with the best results ultimately represents the Korean epidemic’s evolution and the policies’ effectiveness.

Connection between AFD decompositions and policy

Combined with policy analysis

South Korea, with a population of approximately 51 million, including around 26 million in the metropolitan area, faces challenges associated with high population density and mobility. These factors contributed to the concentration of COVID-19 cases in urban areas. Studies examining Korea’s successful response to COVID-19 have primarily emphasized government leadership, KDCA, and strategies refined from experiences with MERS (26-32). The first case of COVID-19 in Korea was diagnosed on January 20, 2020. By June 2022, there had been four waves of the epidemic. Following the largest fourth wave, the cumulative number of confirmed cases from the end of April to June 2022 exceeded 3,835,400.

Priority waves and their analyses

As shown in Table 4, the South Korean government has targeted epidemic prevention measures at different stages of the COVID-19 pandemic. The diagnosis of a member of the Xintiandi Church in Daegu triggered the first wave of the epidemic. The government implemented blockade measures in Daegu City and North Qingshang Road, strengthened the tracking and detection of church members, and restricted public places and religious gatherings. The offline worship of the first church in Seoul and the Liberation Day rally exacerbated the second wave of the epidemic. The government further restricted religious activities, promoted online worship, and strengthened social distancing measures and mask-wearing requirements. In the third wave of the epidemic, the government implemented high-intensity prevention measures, limiting the number of private gatherings to four and closing high-risk venues such as nightclubs and bars. The spread of the Delta variant triggered the fourth wave of the epidemic. The government has strengthened travel restrictions and quarantine measures, promoted vaccination, and mandated wearing masks in public places to ensure public health (27).

Table 4

COVID-19 waves onset and marker events (27)

Waves	Start date	Marker event	Peak date	Peak number of infections per day
Wave 1	February 18^th 2020	A believer of Daegu New Heaven and Earth Church has been diagnosed with the New Crown	February 29^th 2020	909 cases
Wave 2	August 15^th 2020	SaRang Community Church holds offline worship and Gwangmu day (15^th) gathering	August 27^th 2020	441 cases
Wave 3	November 1^st 2020	High-intensity preventive measure, such as limiting the number of members in private gatherings to four, were implemented	December 25^th 2020	1,240 cases
Wave 4	July 1^st 2021	Spread of Delta mutant strains led to cluster infections in daily life spaces	December 15^th 2021	7,848 cases

COVID-19, coronavirus disease 2019.

First wave

The first wave of the epidemic in South Korea began on January 20, 2020, and the government promoted voluntary national participation by building trust in the Ministry of Administration and Self-Government and government policies through open and transparent communication (28). Starting on February 4, 2020, South Korea began restricting entry for foreigners as a precautionary measure. According to the KDCA, there was a sudden jump on February 21, 2020, which was mainly attributed to the “31^st International Convention on the Prevention and Control of Diseases in South Korea”. This was mainly attributed to “Patient 31”, who attended a meeting at the New Heaven and Earth Church in Daegu. To contain the outbreak, public gatherings in affected cities were canceled, and large numbers of soldiers in Daegu were quarantined. From March 19, all South Korean nationals and foreigners entering South Korea were required to follow special entry procedures. These included temperature checks, completing special quarantine declarations, and submitting health status questionnaires (29). Figure 5 is Seoul weekly infections during the first wave, it shows the blue curve of the number of weekly infections in Seoul and the red curve of the decomposition of the number of daily infections by AFD.

Figure 5 Number of the Seoul weekly infections during the second wave. The blue curve is the number of weekly infections in Seoul. The red curve is AFD-decomposed daily infection number decomposition component. AFD, adaptive Fourier decomposition.

Second wave

Table 5 is the Korea Epidemic Control Policy in 2020, and it contains changes in South Korea’s second wave epidemic policy (30). In August 2020, South Korea encountered the second wave of Shinkwon, primarily triggered by the “8-15” rally in the metropolitan area, fostering a sustained trend of collective infections within small and medium-sized groups. In June 2020, Korea’s Ministry of Health and Welfare issued an epidemic prevention strategy termed the “K-Prevention Test-Trace-Treat (3T) International Standardisation Roadmap” to address outbreak (31). This model entails three specific steps: (I) conducting diagnostic examinations and confirmation (Test); (II) performing epidemiological investigations and contact tracing (Trace); and (III) implementing isolation and treatment procedures (Treat), commonly known as the ‘3T strategy’. To enhance the infectious disease response system’s efficacy and bolster its autonomy and specialization, the Disease Management Headquarters, previously under the Ministry of Health and Welfare, was upgraded to an independent agency. This independent body has the authority to conduct various surveys and research on disease management and health promotion. The government established a regional bed-sharing system and expanded capacity for critically ill patients to improve hospital bed allocation and enhance utilization. However, a notable infection surge during this wave was attributed to emerging religious groups in the Seoul metropolitan area, further highlighting the challenge of controlling group-related outbreaks. Figure 6 is Seoul weekly infections during the second wave, it shows the blue curve of the number of weekly infections in Seoul and the red curve of the decomposition of the number of daily infections by AFD.

Table 5

Korea epidemic control policy (30)

Time announcement [2020]	Policy
March 22^nd to April 19^th	Socially strict observation period
August 16^th	Seoul City, Gyeonggi Province: level 2 epidemic response
August 23^rd	Nationwide: level 2 epidemic response
August 30^th	Nationwide: level 2.5 epidemic response
December 1^st	Non-metropolitan area: level 1.5; metropolitan area: level 2 response

Figure 6 Number of the Seoul weekly infections during the third wave. The blue curve is the number of weekly infections in Seoul. The red curve is AFD-decomposed daily infection number decomposition component. AFD, adaptive Fourier decomposition.

Third wave

The third wave of the outbreak, from November 13, 2020, to January 20, 2021, primarily concentrated in group living facilities such as nursing homes and training colleges, resulting in approximately 45,000 confirmed cases. In response to the new coronavirus pandemic, South Korea bolstered the management of infection-prone facilities such as nursing homes, prohibited private gatherings exceeding five people, and imposed business hour restrictions until 21:00. As the number of confirmed cases surged in the metropolitan area, an emergency medical response plan was announced, which included expanding hospital capacity by adding 10,000 hospital beds. Since November 7, 2020, the Central Disaster Safety Countermeasures Headquarters has enhanced the existing policy regarding maintaining social distance, further refining its contents. Considering the reinforced epidemic prevention measures and enhancements in the medical system during this period, the criteria for social distancing have been revised. Additionally, the prevention system has been restructured based on the known characteristics of the new coronavirus and the issues arising from the adjustments in social distancing. The previous three-stage system was expanded into a five-stage framework, introducing intermediate stages for greater flexibility and clarity. These stages are categorized into epidemic prevention (stage 1), regional outbreaks (stages 1.5 and 2), and national outbreaks (stages 2.5 and 3). Figure 7 is Seoul weekly infections during the third wave, it shows the blue curve of the number of weekly infections in Seoul and the red curve of the decomposition of the number of daily infections by AFD.

Figure 7 Number of the Seoul weekly infections during the fourth wave. The blue curve is the number of weekly infections in Seoul. The red curve is AFD-decomposed daily infection number decomposition component. AFD, adaptive Fourier decomposition.

Forth wave

Figure 8 is the number of Seoul weekly infections during the fourth wave. Following the fourth wave of the epidemic in July 2021, the cumulative number of COVID-19 cases from the end of April to June 2022 was more than 3,835,400 cases. However, the number of reported deaths remained surprisingly low compared to the scale of confirmed cases, leading to skepticism regarding the reliability of data from North Korea (32). According to the development of the epidemic, the Korean government issued a universal free vaccination program and implemented it sequentially. The first quarter of the vaccination program aimed at personnel in nursing homes, medical and welfare facilities for the elderly, and high-risk practitioners in medical institutions; the second quarter of the vaccination program aimed at people over 65 years of age and medical personnel in medical institutions and pharmacies; the third quarter of the program focused on patients with chronic diseases, and adults; and the fourth quarter of the program focused on second-time vaccination recipients and those who had not yet been vaccinated. The Korean government has also prepared a medical response system that manages up to 20,000 newly diagnosed patients daily, developing and promoting enhanced prevention and control measures. On July 7, 2021, the fourth pandemic wave, the Shin-Kwan Epidemic, emerged, leading to widespread infections nationwide. By this time, more than 110,000 patients had been diagnosed. In response to this surge, the government implemented the strictest measures for keeping social distance in the metropolitan area. For example, no more than four people were allowed to gather at a private party before 18:00. No more than two people after 18:00. Additionally, all gatherings in the metropolitan area were prohibited, entertainment activities in public facilities were suspended, and remaining facilities were permitted to operate only until 22:00. Only spectator-less games were allowed in sports, distance learning was mandated in all schools, and religious facilities were restricted to “no-contact” worship practices (33-37).

Figure 8 Flow chart of the research programme. AFD, adaptive Fourier decomposition; EMD, empirical modal decomposition; VMD, variational modal decomposition.

Statistical analysis

We used AFD decomposition with Python v3.7.9, EMD with EMD-signal, and VMD with vmdpy. Statistical analysis and visualization of the data were performed with Python 3.7.9.

Results

Analyses of daily COVID-19 data from South Korea using AFD revealed valuable insights into how various government policies influenced infection rates during four different wave periods. The decomposition results indicated that the reduced business hours policy was closely associated with the high-frequency components observed in the first two waves. Vaccination policies were initially associated with the fourth wave. In addition, our statistical analyses showed that the third-order components of AFD were significantly and positively correlated with the original infections, while the correlation coefficients of most components in EMD and VMD did not reach significance.

Discussion

In this paper, the focus is on examining the fluctuations in the daily count of new COVID-19 infections in South Korea alongside the diverse anti-epidemic measures enacted by the government (38-40). The objective of the study is to assess the efficacy of these policies in influencing the course of the epidemic. Employing AFD, we decomposed the daily COVID-19 infection numbers based on epidemic prevention policies. The resulting multilevel composite components were reconstructed to extract elements across different combination scales. Where feasible, comparative experiments employing EMD and VMD algorithms were conducted to assess the effectiveness of the AFD algorithm. These components at varying combinatorial scales were thoroughly analyzed. To assess the correlation, we utilized the Pearson coefficient. A higher Pearson coefficient signifies a stronger correlation, while a lower one indicates a weaker correlation. This approach allows us to compare fluctuations in infection rates with the timing of policy implementation and determine the extent of their correlation.

This study applied the AFD algorithm to epidemic data analysis for the first time. Its core advantage lies in its ability to adaptively decompose the high- and low-frequency components of the signal and extract features that are highly correlated with the time point of policy implementation. For example, the high-frequency components of the first and second waves of the epidemic are significantly associated with short-term control policies such as “shortening business hours”. In contrast, the fourth wave’s low-frequency trend reflects the vaccination policy’s long-term cumulative effect. Compared with traditional methods (such as EMD and VMD), the strong correlation between the third-order component of AFD and the original infection data (Pearson coefficient of 0.944) shows its superiority in capturing complex nonlinear relationships. This finding provides a more accurate tool for policy effect evaluation: AFD can quantify the immediate impact of policies and distinguish the superposition effects of policies at different time scales, such as the synergy between short-term blockades and long-term immunization strategies. In addition, the multi-layer decomposition capability of AFD provides theoretical support for the “dynamic adaptation” of policies. For example, in the third wave of the epidemic, the government achieved refined control of the outbreak’s spread by adjusting the social distance level in stages (from level three to level five). The matching of AFD’s hierarchical results with the phased implementation of policies indicates that this method can help decision-makers identify the dominant fluctuation components at different stages of the epidemic, thereby optimizing the timing arrangement of policy combinations (such as prioritizing short-term intervention during high-frequency fluctuation periods and focusing on long-term prevention and control during low-frequency trend periods).

The results show that the effects of South Korea’s epidemic prevention policies show apparent stage heterogeneity. For example, the precise control of specific groups (such as religious gatherings) in the first wave of the epidemic quickly suppressed high-frequency transmission. In contrast, the vaccination policy in the fourth wave of the epidemic significantly reduced the severe disease rate, but its effect lagged behind the peak of infection. This phenomenon suggests that the timeliness of the policy and the coverage of the target population need to be closely combined. The AFD analysis further showed that the low-frequency correlation of vaccination may be due to the delayed effect of establishing the immune barrier. Therefore, future policy design needs to take into account both “rapid response” and “long-term planning”, such as strengthening detection and isolation (high-frequency intervention) in the early stage of the epidemic while deploying vaccine production and distribution in advance (low-frequency layout). In addition, the policy synergy effect was indirectly verified in this study. For example, during the second wave of the epidemic, coordinating the “3T strategy” and restrictions on religious activities curbed clustered transmission through high-frequency intervention. This multi-policy linkage is manifested as the superposition correlation of multiple components in the AFD decomposition. This shows that a single policy may be challenging to cope with complex transmission chains, and the coordinated design of policy combinations (such as “testing + isolation + public communication”) can achieve quantitative evaluation through AFD’s multi-scale analysis, thereby improving the overall prevention and control efficiency.

Based on the AFD analysis framework, this study provides the following directions for future infectious disease prevention and control strategies:

Dynamic policy toolbox: establish a hierarchical policy response mechanism, incorporate high-frequency interventions (temporary blockades and mandatory mask orders) and low-frequency strategies (such as vaccine development and medical resource reserves) into a unified model, and adjust policy priorities through real-time data updates.
Regional differentiated prevention and control: the high population density and mobility of Korean cities have led to the concentration of epidemics in urban areas. The spatial extension application of AFD (such as combining the signal decomposition results of each town) can identify regional transmission characteristics and provide a basis for formulating differentiated policies (such as strengthening mobility control in key cities and maintaining economic openness in low-risk areas).
Public behavior modeling: high-frequency fluctuations are often related to changes in public behavior (such as gathering on holidays). Future research can combine AFD with behavioral survey data to quantify the impact of policies on public compliance, thereby optimizing communication strategies (such as targeted publicity or incentives).

This study still has some limitations:

AFD analysis relies on the integrity of epidemic data and the accurate record of policy implementation time, while the actual data may be delayed or biased; secondly, the impact of external factors (such as virus mutations and international imported cases) on the decomposition results is not deeply explored. Future work can be expanded in the following directions:
Develop a fusion model of AFD and multi-source data (such as traffic flow and medical resources) to enhance the comprehensiveness of policy evaluation.
Combine machine learning algorithms to predict the long-term effects of different policy combinations.
Extend the AFD framework to other countries or regions, verify its universality, and optimize parameter settings.

Conclusions

This study utilized AFD to analyze temporal trends in COVID-19 data, showcasing an effective method to illustrate the dynamics of the outbreak and comprehend the real-time impact of Korean policies on COVID-19 transmission. AFD is a novel signal decomposition algorithm that characterizes analytic signals through a linear combination of adaptive basis functions. In each AFD decomposition step, basic functions are selected from an overcomplete dictionary, and the process continues until the energy difference between the original and reconstructed signals falls below a predefined tolerance. These reconstructed components are integrated with policy data spanning various periods, enabling a comprehensive analysis. The results highlighted the impact of diverse policies during varying epidemic phases.

In conclusion, our study used AFD, a novel algorithm, to analyze the time trends of South Korea’s COVID-19 data and found that AFD is a method capable of assessing the real-time impact of South Korea’s policies on the spread of COVID-19. We found that the AFD method can guide future infectious disease prevention and control strategies by analyzing the impact of various policies at different epidemic stages.

Acknowledgments

None.

Footnote

Reporting Checklist: The authors have completed the STROBE reporting checklist. Available at https://jtd.amegroups.com/article/view/10.21037/jtd-2024-2141/rc

Data Sharing Statement: Available at https://jtd.amegroups.com/article/view/10.21037/jtd-2024-2141/dss

Peer Review File: Available at https://jtd.amegroups.com/article/view/10.21037/jtd-2024-2141/prf

Funding: The study was supported by the Major Project of Guangzhou National Laboratory (No. GZNL2024A01004), the Science and Technology Development Fund of Macau SAR (No. 0002/2024/RDP), and the National Key Research and Development Program of China (No. 2024YFE0214800).

Conflicts of Interest: All authors have completed the ICMJE uniform disclosure form (available at https://jtd.amegroups.com/article/view/10.21037/jtd-2024-2141/coif). The authors have no conflicts of interest to declare.

Ethical Statement: The authors are accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved.

Open Access Statement: This is an Open Access article distributed in accordance with the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International License (CC BY-NC-ND 4.0), which permits the non-commercial replication and distribution of the article with the strict proviso that no changes or edits are made and the original work is properly cited (including links to both the formal publication through the relevant DOI and the license). See: https://creativecommons.org/licenses/by-nc-nd/4.0/.

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Cite this article as: Zhang X, Shi W, Liu Z, Finley A, Cen K, Xie Z, Yang P, Li H, Leong U, Zhang X, Kwong H, Li MT, Lu G. Adaptive Fourier decomposition analysis of different pandemic stages in South Korean cities: policies and trends. J Thorac Dis 2025;17(6):3516-3531. doi: 10.21037/jtd-2024-2141

Adaptive Fourier decomposition analysis of different pandemic stages in South Korean cities: policies and trends

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Introduction

Methods

Datasets

Study methods

AFD

New components construction based on AFD results

Pearson correlation coefficient

Algorithm (Box 1)

Data analysis

Total data on COVID-19 infections in South Korea

Table 1

AFD decomposition results discounted graph

Table 2

Pearson’s coefficient validation

Table 3

Connection between AFD decompositions and policy

Combined with policy analysis

Priority waves and their analyses

Table 4

First wave

Second wave

Table 5

Third wave

Forth wave

Statistical analysis

Results

Discussion

Conclusions

Acknowledgments

Footnote

References

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