On the heterogeneity of social activity patterns and social distancing

By Nicola Perra and Andrea Baronchelli

We explore the role of heterogeneous participation to social distancing measures aimed at hampering the spreading of infectious diseases. What is the role of individuals that either can’t or refuse to adopt such measures? And what happens if the first individuals to abandon quarantine are also the most socially active? We tackle these questions by using a toy model of temporally evolving social interactions. We observe that the isolation of the many might be nullified if such minorities are formed by more socially active individuals. By no means the results below are considered to be a realistic account of the current pandemic.

Disclaimer: What this post is NOT about

Considering the seriousness of the current global health emergency and the incredible efforts of modellers at the forefront of the fight against COVID-19, it is extremely important to be clear. The models and results we report below are very far from being, and are not meant to be, realistic. Thus, let us stress this point one more time, they can not be used to describe or to capture the pandemic.

What this post is about

Words that until a few months ago were relegated to academic and research settings, such as social distancing, have entered the global discussion. Here, we aim to provide an informative, high-level, description of how social distancing, and its eventual end, work from the perspective of Network Science and Computational Epidemiology. In particular, we aim to show the possible effects on the spreading of a disease when minorities of individuals can’t or refuse to implement social distancing.

Background

Social distancing was key to arrest the expansion of the COVID-19 epidemic in the Chinese region of Wuhan, according to the Report of the WHO-China Joint Mission on Coronavirus Disease 2019 published on February 24, 2020 as well as data-driven studies [1, 2].

The finding has induced several countries to adopt measures to reduce the contact people have with each other, ranging from reducing socialising in public places or the use of non-essential public transport to recommending more home working and to complete lockdown.

The main goal of social distancing is to delay the surge of infected individuals, thus relieving the expected peak of stress for national health services. Said in other words, the measure aims to buy time for the health infrastructure.

In this unprecedented scenario, models acquire a central role in informing decision makers. Of course, sophisticated data-driven models take the stage. Yet, simple models, such the one we are discussing below, maintain a double validity. First, they are helpful to communicate results to the public. Second, they allow for quick testing of different scenarios. Results, whenever relevant, may then be investigated properly using more realistic approaches.

In this spirit, here we run epidemic simulations on temporally evolving networks to explore the role of population heterogeneity, in terms of activity, on the effects of social distancing.

Time, social interactions and distancing.

Social interactions evolve and emerge in time. They can be modelled as temporal (time-varying) networks. Nodes are people, their interactions links between them. Some people are more social and, overall, interact with more people than others. In fact, observations in a variety of contexts show that social activity, i.e. the rate of social interactions, is indeed heterogeneously distributed [3].

There is more. Not all interactions are the same. Few of them are repeated and renewed in time, for example with one’s partner, close friends, and family. These strong ties are typically organised in tight groups (clusters) or individuals [4, 5, 6]. Many other social interactions are instead sporadic, less frequent, for example with the people met in public transportation, the cashier at the grocery store or other members of a gym. These are weak ties that act as bridges between different groups (i.e clusters or communities).

In “normal” conditions people interact with strong and weak ties as a result of their commitments and daily routines. Let’s call this scenario, we are all missing at the moment, business as usual and let’s assume it can be described by temporal network G1(t) (see Figure 1 below).

It is unfortunately a familiar fact that the spreading of infectious diseases can disrupt such dynamics and modify the way people connect and meet each other. These changes however are not instantaneous. In the current pandemic, several European countries continued to encourage social activities despite evidence of local clusters of transmission. In Italy, for example, in order to protect the economy by preserving G1(t), campaigns such as #MilanoNonChiude (Milan does not close) were spreading on social media while in Venice, people were invited to free spritz in San Marco. Similarly, the Mayor of London still affirmed “We should carry on doing what we’ve been doing” in a video posted on social media on March 11th.

In all European countries, things changed dramatically as soon the number of cases and consequent hospitalisations started to grow. Governments started suggesting social (physical) distancing before mandating it by law. As a result, the time-varying network describing physical interactions was gradually modified. Contacts outside of the household, the strong ties, decreased dramatically for most of us. But not for everybody. Some people are still at work helping to keep essential services up and running. Others instead, for a range of reasons, simply do not comply. In any case, we moved from G1(t), business as usual, to G2(t), social distancing. The aim of social distancing is drastically reducing the routes of transmissions. Cutting as many bridges between communities as possible as well as some strong ties outside each household.

Although it is not clear when, hopefully as soon as we will get a better handle on the situation and the number of cases and hospitalisations will go down we will be moving to G1(t). Arguably, we could in fact move to a G3(t), the new normal

Figure 1: Schematic representation of the changes of social ties as function of the progression of the disease. In the first phase, social interactions are described by the temporal network G1. As soon as the prevalence goes above a threshold (horizontal dashed line) social distancing is implemented. Social interactions change and are described by the temporal network G2. Note how connections become more clustered in tight communities. After the prevalence goes below the threshold, we enter phase 3. Social distancing measures are lifted and the system goes back to the business as usual.

Including time in the modelling of the social network is key, as an infected individual can transmit the virus only to contacts she has after becoming infectious. As an example, imagine that Alice gets exposed to a virus at time t and becomes infectious at time t’. All her past interactions are possible routes for her exposure but they are not available for the future propagation of the virus. Now, “follow” Alice. For some days after t’, she goes about her life, interacts with her partner, with her friends, co-workers, as well as random encounters in the bus, at work, and in other locations. These interactions emerge and evolve in time and are the possible routes that the virus can use to propagate further. In other words, the virus “sees” all ties that the “host” (Alice in this case) activates from the moment the viral load is enough to allow for transmission (time t’) until self-isolation. The interactions she had even the day before this are not relevant for the spreading.

Details on the adopted model of temporal network are in the Appendix

Behavioural changes and diseases spreading

One of the main challenges of epidemiology is to understand how behavioural changes induced by diseases, such as social distancing, affect the spreading of diseases [7, 8, 9]. In fact, diseases and behaviours are linked by a feedback loop. The spreading of a virus induces changes in behaviours which in turn affect the unfolding of the disease. While there are several efforts to understand and characterise such changes with data [10, 11 ,12], in this short note we explore — with the toy and prototypical SIR model — how the course of a disease might be affected by dramatic changes in the time-varying network it is spreading upon. Furthermore, we investigate — in the same SIR setting — the impact of individuals who do not adhere to social distancing either out of necessity or other reasons.

Figure 2: The SIR model is one a prototypical epidemic model. Individuals can be in one of three compartments according to their status. Healthy individuals that are susceptible to the diseases are in the compartment S. Infectious individuals are instead in the compartment I. People no longer susceptible nor infected are in the recovered compartment R. The natural history of the disease describes the transition between compartments. Susceptibles in contact with infectious become infectious themselves. These then spontaneously recovered acquiring permanent immunity.

To this end, we do the following simulations. We consider (see the details below) the business as usual network G1(t) and start a disease following the SIR model. The disease unfolds freely until the prevalence (total number of infected nodes at a given time) reaches a threshold. At this point, the network changes and transitions to G2(t). Social distancing is put in place. Connections become much more clustered within small groups of nodes (see Figure 1). However, only a fraction c of nodes implements social distancing so that a fraction 1-c does not. We then consider three scenarios, where these (1-c)% of non-complying individuals are selected A) at random, B) among a pool of more socially active people, or C) exactly among the most socially active individuals. As soon the prevalence goes below the threshold, social distancing is relaxed and people go back to the business as usual.

In Figure 3 we show the prevalence, I(t), as a function of time, for different values of c and considering the three different scenarios

Figure 3: Number of infected nodes as function of time for different values of c and mechanisms to select those that do not implement social distancing. In the plots, we show the average of 20 stochastic simulations. Details about the parameters are reported below.

Few observations are in order:

• The larger the fraction of nodes implementing social distancing the lower the impact of the disease in the population. In fact, the peak is lower and the peak-time is delayed. This scenario is the hope of people invoking “flattening the curve”.
• However, even a small minority of non self-isolating individuals (5%) are enough to nullify the effect of social distancing if this 5% is composed by the most socially active nodes in the population. Why is that? Intuitively, these active individuals are still keeping the connectivity between communities, clusters, thus reducing the effect of isolation of the others.
• Also if the non-complying nodes are selected in a larger pool of active nodes (in the top 24% in case of 1-c=5%), thus not exactly among the top (1-c)%, the positive effects are significantly weakened with respect to case A) but anyway much more beneficial than the case C). To provide a more direct comparison, in Figure 4 we plot the prevalence for c=95% in the three scenarios. The difference between the three is quite striking

Figure 4: Prevalence as function of time for the three mechanism adopted to select the nodes not adopting social distancing. In all three cases we set c=0.95

In order to provide some visual cues on the effect of each scenario, in Figure 4 we show the social networks at the peak time for the different scenarios. In particular, to simplify the representation, each node in the figure describes a community of nodes. Such communities are the tight clusters of people with whom nodes are more likely to interact in the business as usual scenario, strong ties, and become essentially the only people with whom they interact in the social distancing scenario. A link between two nodes captures the connection between two or more member nodes. If a community contains, at that moment, at least an infected individual is colored in red. In the figure the left column describes c=0.95 (95% of the population implements social distancing) and the right c=0.5 (50% of the population implements social distancing). The rows instead capture the different mechanisms to select nodes that do not implement social distancing. The order is the same considered in Figure 1. The figure confirms how social distancing can have a large impact on the disease limiting the number of communities affected by it. However, the effects of high levels of social distancing might be drastically reduced if the top active people do not or can not comply.

Figure 5: Networks between communities at the peak-time. Networks on the left column consider c=0.95. On the left c=0.5. Each row describes a different mechanism to select the nodes that do not comply with social distancing. In the first row we consider the random mechanism, in the second nodes do not implement social distancing are selected in a top pool of active nodes, in the such nodes are selected exactly in decreasing order of activity.

Finally, we explore another hypothetical scenario in which as soon as the prevalence decreases for a certain period of time, individuals might be lured to a false sense of security and thus stop their social distancing. For simplicity, let’s call this scenario ii) while we refer to scenario i) to the models described above. Importantly, we consider the case in which the probability of abandoning social distancing is equal to the activity of each node. The rationale behind this choice is that the propensity of going back to the business as usual might be more appealing for people that are more prone to social interactions either due to their personality or for their job. In Figure 6 we report the results of the simulations comparing scenario i) and ii) across the different models described above for c=0.95.

Figure 6. Prevalence as function of time for the two scenarios i) and ii). In the second, nodes, as soon as the prevalence goes down for a certain period of time (10 time steps) the stop implementing social distancing with a probability equal to their activity. The three panels A, B and C refer to the three mechanisms described above to select nodes that do not apply social distancing independently of the prevalence

It is interesting to note the major effects are in panel A) where nodes that do not adopt social distancing, independently of the prevalence, are selected at random. In this scenario in fact, as soon as the trend of the number of infected nodes is decreasing, the most active nodes are the first to go back to business as usual. In doing so, they re-activate the bridges between communities thus providing more routes of transmission to the virus. As result, we observe a resurgence of the diseases with a slightly higher and delayed peak.

Conclusion

We addressed the role of social heterogeneity in the stylised context of a SIR model unfolding on an artificial temporally evolving network. In particular, we explored the scenario in which the most socially active individuals either do not adopt social distancing or are the first to let it go. We saw that, if this is the case, the positive effects of social distancing are severely compromised in both scenarios

The (toy) model

Following Ref 13 we consider N nodes characterised by two properties. The first is the activity: their propensity, at each time-step, to be socially active and willing to establish connections with others. To reproduce observations from real social networks we consider a heterogenous activity distribution. The second is the membership to a particular community. For simplicity we consider communities of equal size.

In these settings, a time-varying network is created as follows. At each time step:

1. The network starts with N disconnected nodes
2. Nodes become active with probability function of their activity
3. Active nodes create m connections with probability mu at random with members of their community, with probability 1-μ at random with nodes in other communities.
4. All edges are deleted and the dynamics start from point 1.

The probability μ regulates the modularity: the higher μ the tighter the clusters will be. It is important to notice that communities form in time. For simplicity we consider m=1.

These dynamics aim to capture the business as usual. In order to introduce social distancing and study its effect on a disease, we introduce the following modifications.

In the first scenario (i), after the incidence surpasses a certain threshold, a fraction c of nodes change the modularity going from μ (business as usual) to μ’ (social distancing). As a result these nodes interact, with higher probability, with others in their community. In contrast, the remaining 1-c fraction keeps the business as usual dynamics. Such groups describe those nodes that either do not comply with the measures of social distancing (defectors) or cannot comply due to the nature of their jobs. As soon as the incidence goes below the threshold all nodes return to the business as usual dynamics.

How do we select nodes not implementing social distancing? We consider three mechanisms. In particular, nodes not implementing social distancing are selected

1. At random, independently of their activity.
2. In decreasing order of activity with some noise. We pick, at random, among the top N[1-c(1-f)] nodes for activity. We set f=0.2. Simply, in this scenario the nodes that do not implement social distancing are selected from a pool of active nodes but they are not exactly the top active ones.
3. Exactly in decreasing order of activity. Thus, we pick the (1-c)N top nodes for activity.

Note that in case f=0, b) and c) become equivalent.

In the second scenario (ii) we introduce a little variation. In particular, we repeat the same dynamics described above but nodes, as soon as the prevalence goes down, on average for ten days, relax the social distancing with a probability proportional to their activity.

In all simulations, we set N=10⁴, m=1, β=0.6 (infection probability), γ=0.01 (recovery probability), number of communities=10³, and the social distancing is implemented when the prevalence is larger and equal than 1%. Furthermore, we extract activities from a power law with exponent -2.1 and the activities are defined from 0.01 to 1. Finally, we fix μ=0.4 and μ’=0.95