CIE0406

Elisa Bellotti, András Vörös, Tomáš Diviák, Martin Everett

Mitchell Centre for Social Network Analysis

University of Manchester

Introduction, background, and aim of the study

Following the recent wave of the COVID-19 epidemic, reopening schools in the UK need to implement social contact reduction strategies to protect the health of students, teachers, and families. The aim of social contact reduction is to slow the spread of diseases in a population by limiting physical contact between individuals. These strategies can be effective because:

- they limit the number of people who regularly interact, hence reducing the overall number of students who may get infected.
- by limiting the number of interacting people, they also slow down the peace of the spread, hence increasing the time that it takes for students to get infected.

Potential physical contact between students is reduced by breaking up classrooms and introducing “social bubbles”, groups of students who have contact only with one another during school activities and breaks. These measures will influence the school experience of students over the next academic year and possibly beyond. Despite their importance for health and learning, three key aspects of contact reduction strategies are unexplored:

- How efficient these strategies are in tackling the spread of the virus?
- How do these strategies affect, and how they are affected by, the existing patterns of relationships between students?
- How sustainable are contact reduction strategies over time?

To address these issues, we propose to examine the efficiency of a social network-based approach to reduce contacts in schools. Prior research on disease spread in social networks has a key limitation, as studies typically rely on hypothetical patterns of interactions and not real-world networks. Due to the complexity of these models, the lack of empirical data and realistic assumptions may lead to erroneous conclusions and ineffective strategies.

To understand the efficiency and long-term consequences of contact reduction in schools, we need to consider how physical contacts are actually structured in real life, or in other words students’ social networks. For instance, the spread of the virus may be faster and more extensive in densely connected or highly centralized networks. Combining network simulation models with existing social network data from schools in England, we analyse the potential spread of COVID-19 in real-world settings under contact reduction strategies and we reflect upon the likelihood of such strategies to be maintained over time.

In this document we demonstrate the efficacy of our model in one exemplar classroom. Our finding show that contact reduction strategies are always more effective in reducing the number of students infected with Covid-19 compared to lack of contact reduction, but that they are likely to break natural groupings of students, who over time may rekindle their relationships and diminish the effectiveness of reducing contacts in classrooms.

Data

To illustrate our model, and for the purpose of this document, we conduct our analysis on a single classroom of year 10 students in England (14/15 years old). The analysis can be extended to a representative sample of real-world social network data among students in 214 secondary schools in England (4315 students). The data come from the large-scale Children of Immigrants in Four European Countries (CILS4EU) study, conducted in 2010-2012. The sample includes public and private schools, central and remote geographic regions, plus various school sizes. In each school, two year 10 classrooms were sampled at random. Students in the sample were asked to answer questions about their social networks at school (e.g. who are their friends, whom they spend their free-time with, whom they study with) and outside of school (who are their friends outside schools and what activities they do with them), their social background (e.g. gender, ethnicity, religiosity), their household composition (e.g. number of people living together), and their academic outcomes (aspirations, subjective performance, past achievement). CILS4EU provides the most recent national sample of students’ characteristics and social networks in school settings in England, and despite being several years old, the social network patterns found in this sample, e.g. the mutuality of friendships or presence of groups, are in line with results from recent international studies.

Focusing the attention on teenage students is important, because compared to younger kids 14/15 years old boys and girls develop complex and multifaced relationships with their peers and are more likely to interact in larger groups. Expecting teenagers to distance themselves from their friends and reduce the contact with their reference peer groups is unlikely to be successful in the long term, so understanding how to calibrate contact reduction strategies keeping into account the underlying patterns of social relationships is likely to improve their effectiveness and durability.

Method

Our approach is based on social network analysis (SNA). SNA is concerned with the analysis of relations and interactions among groups of people in various settings such as schools or workplaces. These contacts form the basis of our day-to-day lives, yet they also constitute channels for viral contagion, as viruses like Covid-19 spreads when people get in contact with each other. Therefore, it is important to analyse the structure of real-world contact networks, to be able to subsequently modify their structure in such a way that mitigates the viral spread while preserving as much human interactions as possible.

In this document we present a relational event model that we developed to simulate the potential spread of the virus through contact among students, based on their real-world social networks. We represent contact reduction strategies in these model by making certain contacts impossible in the simulations. For example, two friends from different study groups will not be able to meet and pass on the virus. This approach helps to assess the efficiency of each strategy in the empirically observed networks.

As an example of the potential of our method, we demonstrate the effectiveness of contact reduction strategies in the selected year 10 classroom network (Figure 1). Here, each red node represents a student and the ties that connect them represent their connections to other students who they named as either their best friends, and/or the ones they spend time together outside school. In this specific classroom there are 31 students, and the density of the network is 0.24, which means the students have 24% of existing friendship and socialising relationships out of all the possible relationships they could have with their classmates.

Figure 1: Friendship relationships within one classroom

As we can see from Figure 1, despite the fact that not all students are friends or spend time with every other student, there is a path that connects all of them, which means that if any of them gets infected, there is a high probability that the infection will spread to all the students.

To reduce the extent and the speed of the infection, we randomly split the students into three groups of size 10 (2 groups) and 11 (one group) and we ban contacts between these groups. Figure 2 shows the same classroom network where node colours indicate the randomly formed group (on the left) and the removal of contacts across groups (on the right). The remaining ties represent the original friendship relationships between students within the newly created groups.

As we can see, the resulting network is now disconnected in smaller groups of friends, and some students are separated from their pre-existing friendships groups, as they cannot interact with friends who are assigned to a different “bubble”.

Figure 2: Classroom network divided into three groups, and friendship relationships banned across groups

We then simulate the spread of a viral infection, which like Covid-19 requires social proximity and/or social contact to diffuse, in the original network in Figure 1, and compare the rate of contagion in the contact reduced network in Figure 2 under different initial conditions:

- if there is only one student initially infected.
- if there is an initially infected student in each of the contact-reduced groups.
- if there is more than one initially infected student in each group for each run of our simulations.
- if the infected student is the one with the highest or lowest number of friends, or the one who have the shortest or longest distances from all other students.

We simulate the spread of the virus 100 time for each scenario, each time following the rules for selecting the infected seed, and we observe how many students, in each of the 100 simulations, get infected over time (Y axis in the figures below). Time, in our model, is calculated as number of possible contacts between students (X axis in the figures below): a contact event could be, for example, of two students who meet during a break and have a chat; or two students who hang out together after school. As we do not know on average how many contacts students may have with each other every day, we cannot automatically calculate the speed of the viral infection in terms of days, so results should be interpreted accordingly.

Results

Scenario 1

We compare the number of infected students in the original non-reduced network - black curve with grey confidence intervals, to the number of infected students in network with contact reduction - red curve with pink confidence intervals (Figure 3). In each of the 100 simulation there is only one infected student: by splitting the network in 3 groups and having only one student infected, we contain the infection to only one group, and therefore 30% of all students on average get infected, while in the original network every student will eventually get infected. However, the beneficial effect of the grouping is apparent only after a certain number of social contacts between students (around 300 contacts).

Figure 3: Number of infected students in non-reduced network compared to contact-reduced network - one random student infected

Scenario 2

We compare the number of infected students in the original non-reduced network - black curve with grey confidence intervals, to the number of infected students in network with contact reduction - red curve with pink confidence intervals. However, now we have an infectious seed node (student) in each of the newly created groups in the contact-reduced network (Figure 4). Compared to scenario 1, considerably more students (80%) get infected as the infection can now spread in each of the groups. Yet, it still does not reach all of the students as in the non-reduced baseline case. So even if there are more seeds than in the original network, the grouping helps with reducing the overall number of infected students. Compared to scenario 1, having three seeds speed up the spread of the infection considerably, with nearly half of the classroom infected in less than 100 contacts, but grouping eventually slow down the spread of the disease compared to the baseline network.

Figure 4: Number of infected students in non-reduced network compared to contact-reduced network - one student infected in each group

Scenario 3

Figure 5 captures the same scenario as Figure 4 with one key difference – for each of the hundred simulations, we randomly choose a different seed node. This increases the variability of the results as evidenced by the wider pink confidence intervals, but the overall performance of the contact reduction by grouping is still better than no contact reduction. On average, 71% of students get infected in the grouping scenario whereas in the no reduction scenario, all the students still get eventually infected. Like the previous cases, the effect of contact reduction takes some time to be noticeable.

Figure 5: Number of infected students in non-reduced network compared to contact-reduced network - different infected students in each group

Scenario 4

In the last set of simulations (Figure 6), we investigate the effects of having the most central students in the classroom network as initial infected nodes. Central nodes are defined, in our case, as students who have the highest number of friends (degree centrality) and students who are at the shortest path of distance from all other students (closeness centrality). Such central nodes can act as superspreaders the in cases of epidemic outbreaks because due to their position in the network they will infect a higher number of individuals.

We compare the same non-reduced network (black curve) against the network split into three groups (red curve), where the initial infected actors are respectively the ones with the highest and lowest degree and with the highest and lowest closeness centrality. In the network with no contact reduction, all students will eventually get infected. In the cases where the seed nodes have the lowest degree or closeness centrality, the epidemic outbreak does not even take off because these nodes are isolated (they do not have any friends to interact with in their own group). The most interesting results are the ones in which the initial infected students are the most central one. In both the cases of infected students with highest degree and highest closeness, the infection eventually reaches 81% of the students on average, which is still lower, and therefore more effective, that the contagion in the original non-reduced network. Again the effect of contact reduction takes some time to be noticeable.

Figure 6: Number of infected students in non-reduced network compared to contact-reduced network - most and least central nodes infected

Discussion

The proposed relational event model offers several advantages compared to existing models that rely on hypothetical patterns of interactions and not real-world networks. In our model, we have shown the effectiveness of contact reduction strategies in an exemplary classroom, but also the limits that such strategies may face when implemented over an underlying pattern of existing social relationships.

Effectiveness of contact reduction strategies in classrooms

- Contact reduction strategies do contain the rate of infection, but the effects are visible only after a certain period of time.
- Breaking social contact between students within max 10 people per group significantly decreases the overall number of infected students.
- Grouping still reduces the number of infected students even if each group contains an initial infected student.
- Grouping reduces the number of infected students if the initial infected student changes over time.
- Grouping still reduces the number of infected students even if the initial infected student is highly central.

Impact of contact reduction strategies on students’ relationships and long-term sustainability

- When randomly assigning students to groups, we may split up students who naturally spend time together.
- Students may end up in groups they are not comfortable with or may have underlying negative relationships with.
- Over time isolated students may

- Integrate in the new group (if accepted), which means they will increase contacts and therefore the rate of infection.

- Exacerbate conflicts within their groups, which means that students may suffer from social isolation.

- Interact with friends outside their own groups despite the ban, which will seriously undermine the efficacy of contact reduction strategies and therefore increase the rate of infection.

Advices to schools and future work

Overall, contact reduction strategies should be implemented in schools, as they are effective in containing the rate of infection. However, strategies for grouping students should consider the underlying existing relationships between students, which are not always visible to teachers, as students interact not only in classrooms but also outside schools. Attention to students’ social networks can:

- Improve the sustainability of groups, by aggregating students based on their existing social networks.
- Improve the effectiveness of grouping, for example by calibrating the size of groups depending on the number of active relationships students may have. Highly connected students should be kept within their natural friendship groups, as they may otherwise act as superspreaders across groups.
- Monitor marginalisation in classrooms, by making sure less connected students are not grouped with others they are not familiar with, or they have conflicts with.

To improve our understanding of how contact reduction strategies can be efficiently and successfully implemented in schools, we plan in the coming months:

- To extend the analysis to a representative sample of classrooms. Students’ social networks can vary extensively depending on the size and composition of classrooms. The same contact reduction strategy will succeed differently in different underlying structures.
- To modify the criteria for grouping students and observe the effect on viral contagion and students’ interactions over time. For example, students can be grouped based on similarity (gender, geographical distance, ethnicity); on repetition of contact (students are only allowed to interact with other students they previously interacted with); on conflict avoidance (students are only allowed to interact with other students they are not in conflict with). Different contact reduction strategies can be more or less sustainable over time, and therefore more effective in keeping the viral contagion under control over extended periods of time.
- To understand how different type of grouping may affect the underlying existing pattern of social relationships. Different contact reduction strategies may be more effective in reducing the viral contagion, but may exacerbate already existing inequalities: isolated students may become more peripheral and ethnic and socio-economic boundaries may become stronger in student networks.

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