## 1. Introduction

The first case of the novel coronavirus, severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), was reported by Chinese health officials in Wuhan City in December 2019. Coronaviruses are a family of viruses that can cause illnesses, such as severe acute respiratory syndrome and the Middle East respiratory syndrome [

1]. The clinical spectrum of COVID-19 infection is broad, ranging from no symptoms to severe pneumonia. Approximately half of the COVID-19 patients (40–50%) in one study did not show any symptoms [

2]. Other patients developed fever, body aches, nausea, or diarrhea [

3] typically 2 to 14 days after exposure to the virus. During the initial phase of COVID-19 in China (10–23 January 2020), only

$14\%$ of total infections were confirmed. The remaining 86% were not identified or quarantined, contributing to a community spread in China. Later, on 23 January, the Chinese government implemented a total lockdown of Wuhan City, which prevented further community spread [

4]. Owing to this strict intervention, the number of new COVID-19 cases in China dropped to the single-digits in early March. Other countries, however, started to report an increasing number of confirmed cases. COVID-19 spread from Wuhan to the rest of China and neighboring Asian countries such as Thailand, Japan, Korea, Singapore, and Hong Kong. As of 31 March, the number of confirmed cases worldwide had exceeded 820,000, and the number of deaths had reached more than 40,000 in more than 110 countries. The World Health Organization declared COVID-19 a pandemic on 11 March 2020 ([

5,

6]).

In South Korea, the first confirmed case of COVID-19 was reported on 20 January, and the number of cases steadily increased to 30 by 17 February. Beyond the initial sporadic outbreaks from overseas inflows, including China, there was a large-scale community spread around specific regions. Beginning with the 31st confirmed case on 18 February, however, an unprecedented rapid increase of confirmed cases was recorded in the area of Daegu and North Gyeongsang Province (Gyeongbuk). It is still unclear how or from whom the 31st case became infected. Daegu is Korea’s 4th largest city and is surrounded by Gyeongbuk (see

Figure 1). In the process of tracing the contacts of the 31st case on 18 February, it was revealed that she attended worship services at the Sincheonji Daegu Church on 9 February and 16 February, raising the possibility of community spread through the church service. Indeed, an inspection of more than 1000 Sincheonji Daegu members on 20 February resulted in the identification of more than 111 cases. Epidemiological surveillance of these 111 newly identified cases revealed that most cases presented to Cheongdo Daenam Hospital in Gyeongbuk. Health authorities later imposed cohort isolation on the hospital after a large cluster of COVID-19 cases was reported. Although most of the confirmed cases were concentrated in Daegu or Gyeongbuk, COVID-19 spread to nearly every part of the country [

7]. Across the nation, 79.7% of the cases have been proven to be linked to cluster transmission, and 62.5% have been linked to Shincheonji [

8]. As of 31 March 2020, 9786 patients were confirmed in Korea, 7984 of whom (81%) were confirmed in Daegu and Gyeongbuk [

8]. The spread of COVID-19 in Korea displayed a spatial heterogeneity, a clustered rapid spread in Daegu and Gyeongbuk, and a steady slow spread in the rest of the country.

Mathematical modeling can provide useful insights for mitigating emerging infectious diseases such as recent SARS-CoV, MERS-CoV, COVID-19 outbreaks. Particularly, the mathematical modeling of the spread of diseases becomes more important as the movement of people and other living organisms increases due to globalization [

9,

10]. Multi-patch models have been developed to understand the global spread of transmission dynamics in various emerging and reemerging infectious diseases [

11,

12,

13,

14]. Since the COVID-19 outbreak began, many studies have been carried out to investigate the basic or control reproductive numbers, whereas others predicted the outbreak peak time and evolution of the epidemic to guide intervention policies ([

15,

16,

17,

18]). The national and global spread of COVID-19 was explored, accounting for the effect of the metropolitan-wide quarantine of Wuhan and surrounding cities [

19]. A stochastic transmission model was combined with COVID-19 data from Wuhan City to estimate the temporal evolution of transmission [

20]. The study also predicted the probability of how newly introduced cases would generate outbreaks in other areas. Time-varying basic reproduction number has been investigated in Italy and forecasted the COVID-19 spread in the region [

21]. The critical role of a latency period is highlighted under three different modeling of latency periods in SEIR-based models [

22]. Some studies investigated the effectiveness of COVID-19 mitigation strategies, such as travel restrictions ([

23,

24]) and social distancing ([

25,

26,

27,

28]). These nonpharmaceutical measures reduced transmissibility by a maximum of

$34\%$ without resorting to a strict lockdown strategy [

29].

In this study, we develop a two-patch model of COVID-19 transmission dynamics, representing a hot spot (patch 1: Daegu and Gyeongbuk) and a slow-spreading area (patch 2: the rest of Korea). In each patch, we employed a mathematical model with five epidemiological compartments (susceptible, exposed, infected, quarantined, and recovered), which is based on the previous work [

14]. The two patches are connected by a mobility matrix that captures the number of people traveling in South Korea (provided by the Korea Transport Institute (KOTI) in 2016) [

30]. It includes origin and destination data on various modes of transportation, including airplanes, cars, buses, ships, and trains. The incubation and recovery rates were assumed to be the same in the two areas and other parameters such as transmission rate were assumed to be different in the two patches and were estimated using cumulative incidence data from 18 February to 24 March 2020. Using the estimated model parameters, we carried out a patch-specific sensitivity analysis. Furthermore, we investigated the impacts of various intervention strategies on the patch-specific transmission dynamics, including limiting traveling between the two regions, implementing social distancing, and early diagnosis.

This paper is organized as follows. In

Section 2, we present a data description followed by a single-patch SEIQR model and a two-patch SEIQR model with a mobility matrix and discuss the basic reproduction numbers (the local reproduction number for a single-patch model and the global reproduction number for a two-patch model). We have carried out parameter estimation and sensitivity analysis in

Section 3. Numerical simulations under various mitigation scenarios have been investigated in

Section 4. The paper concludes with a discussion of results and conclusions in

Section 5 and

Section 6. Following

Section 6, we include the

Appendix A containing the mathematical work, showing the derivation of the basic reproduction number for a two-patch SEIQR model.

## 3. Parameter Estimation

#### 3.1. Estimated Parameter

As shown in

Figure 2, the daily confirmed cases of COVID-19 in the Daegu and Gyeongbuk area show special features; the number of confirmed cases demonstrated a slow initial increase at the beginning, shortly after the 31st confirmed case, a rapid increase of cases. After the peak of confirmed cases, the number of cumulative incidences remained steady (almost no increase). To capture this special feature, the entire time interval ([0, 40] days) was divided into the following 3 subintervals: a small increase interval

$[{t}_{0},{t}_{1})$, a rapid increase interval

$[{t}_{1},{t}_{2})$, and a slower increase interval

$[{t}_{2},{t}_{end})$. Therefore, we estimated three different transmission rates, a small

${\beta}_{1,1}$ during

$[{t}_{0},{t}_{1})$, a rapid increase with a transmission rate

${\beta}_{1,2}$ during time

$[{t}_{1},{t}_{2})$, and a slower increase of incidence after peak with a transmission rate

${\beta}_{1,3}$ during

$[{t}_{1},{t}_{end})$ (

${\beta}_{1,3}<{\beta}_{1,1}<{\beta}_{1,2}$).

We set the initial susceptible population to be 5.12 million and 46.52 million in the Daegu and Gyeongbuk and the rest of Korea, respectively (ie, ${S}_{1}\left({t}_{0}\right)=5.12\times {10}^{6}\phantom{\rule{0.166667em}{0ex}}\&\phantom{\rule{4pt}{0ex}}{S}_{2}\left({t}_{0}\right)=46.52\times {10}^{6}$) as the population size of the two patches. The initial quarantine numbers of ${Q}_{1,2}\left({t}_{0}\right)$ were taken from the cumulative incidence data on 18 February 2020. We assumed the number of exposed ${E}_{1,2}\left({t}_{0}\right)$ and recovered ${R}_{1,1}\left({t}_{0}\right)$ to be zero. The number of initially infected individuals (${I}_{1,2}\left({t}_{0}\right)$) was not measurable and thus was estimated using the method below.

Using the report by Workman [

36], we set the percentage of asymptomatic individuals to be 33.3% (i.e.,

$p=0.333$). To estimate a typical number of traveling in Korea, we used data from the 2016 Korea Transport Data Base, a national organization that produces, manages, and provides various transportation data for Korea (KOTI) [

30]. The remaining 10 parameters and 2 initial numbers of infected

$\left({I}_{1,2}\left({t}_{0}\right)\right)$ were estimated. An inverse least-squares formulation was used to find a parameter set (

$\theta $) that minimized the least-squares error between cumulative incidence of the model outputs (

${C}_{1,2}$) and actual cumulative incidence data (

${D}_{1,2}$) from 18 February to 24 March 2020.

The least-squares error

J is defined as

where

$\theta $ represents a set of parameters,

${t}_{j}$ time points (with

$j=1,\cdots ,N$), and

${D}_{1,2}$ actual cumulative incidence data (a subscript denotes a patch;

${C}_{1},{D}_{1}$ for patch 1 and

${C}_{2},{D}_{2}$ for patch 2). The optimal parameter set is given by

We used the Nelder–Mead simplex method [

38], which aims to minimize the least-squares error through a direct search using MATLAB’s built-in optimization function fminsearch. The optimal parameter values for the two-patch model are given in

Table 2, and the fitted curves generated by the estimated parameters are shown in

Figure 5. Cumulative incidence of the two-patch model (solid blue curve) and COVID-19 data (red circles) in each patch are in good agreement.

#### 3.2. Sensitivity Analysis

To assess the variation of the model outputs in response to small perturbations in the parameter values, we performed a global sensitivity analysis that described how a model output changed in response to varying the parameters over a subspace of the parameter space. A global sensitivity method developed by [

39] is a variance decomposition method. First, input parameters are varied, which causes model output to be varied. Next, the variance of the model output is quantified. Then, a partial variance is calculated by partitioning of model output variance to determine what fraction of the model output variance is caused by the variation of each model parameter input.

The model outputs we considered were

${C}_{1}$, cumulative incidences in the Daegu and Gyeongbuk, and

${C}_{2}$, cumulative incidences in the rest of Korea except for Daegu and Gyeongbuk. The sensitivity indices computed by the method described in [

39] are shown in

Figure 6. The left panel for patch 1 shows various sensitivity analysis; the model output

${C}_{1}$ was highly sensitive to the parameters

${\alpha}_{1},\phantom{\rule{4pt}{0ex}}{t}_{1},\phantom{\rule{4pt}{0ex}}{\beta}_{1,2},\phantom{\rule{4pt}{0ex}}{p}_{1},\phantom{\rule{4pt}{0ex}}{t}_{2}$, where

${\alpha}_{1}$ indicates a detection rate from

${I}_{1}$ to

${Q}_{1}$, and

${\beta}_{1,2}$ represents a transmission rate during

$({t}_{1},{t}_{2})$, while

${p}_{1}$ is a proportion from

${E}_{1}$ to

${I}_{1}$.

Next, the right panel for patch 2 shows various sensitivity analysis; the model output ${C}_{2}$ was highly sensitive to the parameters ${\beta}_{2},\phantom{\rule{4pt}{0ex}}{t}_{1},\phantom{\rule{4pt}{0ex}}{\beta}_{1,2},\phantom{\rule{4pt}{0ex}}{p}_{2},\phantom{\rule{4pt}{0ex}}{\alpha}_{1},{\alpha}_{2}$, where ${\beta}_{2}$ indicates a transmission rate of patch 2, ${t}_{1}$ represents the time when a rapid increase of incidences was observed in the Daegu/Gyeongbuk area, ${\beta}_{1,2}$ is a transmission rate during $({t}_{1},{t}_{2})$ in the Daegu/Gyeongbuk area, ${p}_{2}$ indicates a proportion from ${E}_{2}$ to ${I}_{2}$, and ${\alpha}_{1,2}$ represents a detection rate from ${I}_{1,2}$ to ${Q}_{1,2}$, respectively. Cumulative incidence in patch 1, ${C}_{1}$ is sensitive to ${m}_{21}$ (mobility from patch 2 to patch 1) while cumulative incidence in patch 2, ${C}_{2}$ is sensitive to ${m}_{12}$ (mobility from patch 1 to patch 2).

## 5. Discussion

In this work, we developed a single-patch and a two-patch model for COVID-19 transmission dynamics, focusing on the largest outbreak in Daegu and Gyeongbuk and the rest of South Korea. We investigate the impacts of three different intervention strategies on the COVID-19 transmission dynamics. These include travel restrictions, social distancing, and the efficient early detection of confirmed cases followed by quarantine.

First, travel restrictions are particularly useful in the early stage of the COVID-19 outbreak confined to a certain area. However, travel restrictions may be less effective once the outbreak is more widespread [

23]. Our simulations showed that travel restrictions delayed the spread of the disease into the rest of Korea from the epicenter and reduced the number of total incidences (see the leftmost panel of

Figure 11, 12,000 cases at 1% vs 16,000 cases at 9%). An increased travel movement between Daegu and Gyeongbuk and the rest of Korea resulted in a reduced number of confirmed cases in the epicenter but not in the rest of Korea. The travel ban in Wuhan and the international travel restrictions did not show noticeable differences in the epidemic in Wuhan but delayed ∼3 days for other locations in mainland China [

24]. Another study demonstrated that internal mobility restrictions are effective only if prohibitively high (

$50\%$ of travel) restrictions are applied [

26]. In another infection study, Ref. [

41] suggested that short-term mobility between heterogeneous patches does not always contribute to overall increases in the prevalence of tuberculosis. In [

14], the strategy of the migration restriction may result in locally bigger outbreaks rather than extinguishing the pandemic spread, even though the spread is delayed. Our results are consistent with other studies showing that travel restrictions may not be particularly effective to reduce the cumulative incidence.

Second, social distancing measures aim to reduce the frequency of contact, and increase the physical distance between individuals, reducing the risks of person-to-person transmission. A large number of published studies examining social distancing conducted during the 2009 influenza pandemic concluded that social distancing controlled the epidemic [

26]. Data-driven models of COVID-19 spread in China [

27] have shown that strict social distancing for 30 days had significant benefits in limiting community spread Wuhan and Hubei. Choi et al suggested that social distancing is crucial in suppressing the spread of the COVID-19 in Korea [

28]. Furthermore, the transmission of viruses was lower with the physical distancing of 1 meter or more, compared with a distance of less than 1 m [

25]. Our simulations also showed that strict social distancing led to a significant reduction in total incidences. Approximately 2000 fewer cases were expected as transmission rates were decreased by 10% (see the middle panel of

Figure 11). From 12 April to 18 April, after four weeks of strict social distancing was enforced, the number of confirmed cases decreased dramatically [

8]. In the situation of a lack of therapeutics and without vaccines, social distancing is one of the basic protective measures against COVID-19.

Lastly, one of the notable things about COVID-19 is the high rate of asymptomatic but infectious individuals. In Korea, approximately 33.3% of the confirmed cases were asymptomatic [

36]. In Italy,

$42.5\%$ of the confirmed infections detected were asymptomatic [

42]. Thus, identification of asymptomatic virus carriers by aggressive testing and contact tracing could potentially reduce the size of the epidemic. Our simulations varying the detection rate

$\alpha $ suggested that approximately 2000 confirmed cases (approximately 20% of the total incidence) could be reduced by doubling detection rates (see the rightmost panel of

Figure 11). Another study also demonstrated that the identification of undocumented COVID-19 cases suppressed the spread of COVID-19 [

4]. Early detection is also a necessary method for preventing transmission. To mitigate the spread of the disease, it is important to detect the outbreak of the patient early, conduct quarantine and treatment, and identify contacts through effective epidemiological surveillance systems and investigations.