2.3. Contact Matrix

As Seoul and Gyeonggi provinces are densely populated with diverse people, to enhance the realism of our model, it is beneficial to consider the heterogeneity in contact networks. Two of the most important heterogeneous aspects of a contact network are location and age since different locations are often visited by certain age groups, which leads to consistent contact with specific age groups. For instance, people tend to have contact with people of a similar age outside their households (i.e., schools and workplaces). Since our age-structured model allows us to adjust the transmission rates among different age groups and since the location is closely linked to an individual’s contact pattern with certain age groups, we applied these location-based contact patterns to the transmission rates.

We divided the contact locations into four categories: school, workplace, household, and other locations. For each location category, we used the specific contact matrix of Korea from [21] to build our model. Each contact was defined by either physical or nonphysical contact; physical contact includes skin-to-skin contact like kissing, handshaking, etc., whereas nonphysical contact includes, e.g., a two-way conversation with three or more words in the physical presence of another person but no skin-to-skin contact [24].

Each location-specific contact matrix is a 16 × 16 square matrix, which represents the mean number of instances of contact between individuals of five-year age groups, such as 0–4, 5–9, 10–14, 15–19, 20–24, 25–29, 30–34, 35–39, 40–44, 45–49, 50–54, 55–59, 60–64, 65–69, 70–74, and 75 and above. Each element is the contact rate of an individual in one of the 16 age groups with people in the other 16 age groups at the specific locations. More precisely, the location-specific contact matrix M is written as [25]

where each element mij denotes the mean number of contacts an individual in age group i makes with individuals in age group j per day. Note that contact matrix M is not necessarily symmetrical, which is a general feature that is also found in [26,27,28].

Since the focus areas are Seoul and Gyeonggi province, and the location-specific matrices of the whole region of Korea are only available in [25], we estimated the location-specific matrices of the focus area by using the proportion of the population of the area compared to that of Korea. We assumed the total population to be constant since the period of interest covers less than a year. We used the census data of Korea from January 2020 throughout the simulations. A summary of the data can be found in Figure S2 and Table S2 in Supplementary Section B, which describe how to calculate the contact matrix of the focus area. The calculated location-specific matrices for Seoul and Gyeonggi province are shown in Figure S4 in Supplementary Section B.

A full contact matrix M is composed of a linear combination of the location-specific contact matrices [25]:

where mW is the workplace contact matrix, mS is the school contact matrix, mH is the household contact matrix, and mO is the contact matrix for all other locations, except for the workplace, school, and household; cW, cS, and cO are constants, and cH is a 16 × 16 diagonal matrix, which are each multiplied by their respective matrices. Based on the real policies of school closure and social distancing levels in Korea, we composed five different contact matrices by adjusting cW, cS, cH, and cO as MO, MC, MwC, MmC, and MsC, which denote the contact matrices of the cases of school openings with no social distancing, school closures with no social distancing, school closures with weak social distancing, school closures with medium social distancing, and school closures with strong social distancing, respectively. When the school is closed, cS=0 since there are no contacts made in the school. On the other hand, when the school is closed, cH=diag(1.5, 1.5, 1.5, 1.5, 1.1, 1.1,…, 1.1)16, where diag( )n denotes the diagonal matrix with n diagonal entries, such that for age groups below the age of 20, contact rates increased by 50.0% and for age groups 20 and above, contact rates increased by 10.0% [29]. For social distancing, when there is no social distancing, weak social distancing, medium social distancing, or strong social distancing, we assumed cO=1, 0.7, 0.5, 0.3, respectively, such that cO decreases under stronger social distancing. Note that different types of cO levels were tested while decreasing the orders of cO for stronger social distancing, as shown in Figures S12 and S13 in Supplementary Section D, but we present only one case due to the lack of a significant difference in the fitting and simulation results. An example of a scenario/policy-specific contact matrix of Seoul and Gyeonggi province—school closure with no social distancing, MC—is shown in Figure 3; a comparison with the equivalent version for Korea is provided in Figure S3 in Supplementary Section B. Table 3 shows a summary of the contact matrices for different policies. The contact matrices for each scenario/policy are shown in Figure S5.

Contact matrix for a policy of school closure and no social distancing in Seoul and Gyeonggi province.

Overview of the constants and contact matrixes used for each policy.

* Values with an asterisk (*) are assumed. ** I₁₆ and diag(·)₁₆ denote the 16 × 16 identity matrix and the diagonal matrix with diagonal entries, respectively.