COVID-19 model

The COVID-19 spreading model is represented by a Susceptible-Exposed-Infectious-Recovered (SEIR) process [16] and considers the epidemiological profile depicted in Fig 1A. An epidemiological profile is characterized by the incubation time (time from exposure, E, to onset of symptoms), latency (from exposure to onset of infectiousness, I), infectious period (the period over which the patient is contagious), and the extend of the disease (from the onset of symptoms until recovery or death, R). The values of the corresponding times of SARS-CoV-2 depicted in Fig 1A are obtained from the literature [25–27]. A crucial feature to notice from the epidemiological profile is that the onset of the infectiousness period occurs before the onset of symptoms. In other words, the latency is shorter than the incubation period, and the patient becomes contagious before she/he starts to feel the symptoms of the disease. Furthermore, the peak of infectiousness, that is, when the patient is at its most contagious stage, occurs, in average, about a day or two before the onset of symptoms, according to the study done in [25] at the beginning of the pandemic. These numbers are crucial to understand the rapid spread of COVID-19. They imply that when the patient starts with the symptoms of the disease, she/he has already transmitted the virus to the majority of its infected people during the previous two days. Therefore, even if the patient isolates him/herself after feeling sick, the main transmissions have already occurred.

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Fig 1. Infectiousness profiles of SARS-CoV-1 and SARS-CoV-2.

(A) Infectiousness profile of coronavirus SARS-CoV-2 responsible for COVID-19. The COVID-19 pandemic is modeled by a SEIR model. From exposure (E) the virus is incubated in average for 5.2 days (12.5 days 95^th percentile), starting the symptoms 2 days after infectiousness (I) and lasting the disease up to 17 days to recover (R). We use a window -14/+7 days from the first symptoms to detect infectious and exposure. (B) Infectiousness profile of coronavirus SARS-CoV-1 responsible for SARS-2003. Data obtained from [25]. As opposed to COVID-19, we note that in this case the latency is longer than the incubation period, and the peak of infectiousness then appears after the onset of symptoms. Thus, when the patients present its first symptoms, upon isolation, the transmission of disease is interrupted. In this case, isolating the patients after the symptoms is an effective way to control the pandemic. On the contrary, COVID-19 in (A) is characterized by a latency shorter than incubation, and, even more troublesome, with a pre-symptomatic peak of transmission appearing before the onset of symptoms. Thus, in this case, even if the patient isolates after the symptoms appear, most of its infections have occurred already. This indicates that the only way to stop the chain of transmission of COVID-19 is by going into the past, before symptoms, and performing contact tracing to capture and isolate the contacts of the infected person before the symptoms have appeared. This crucial difference in the epidemiological profiles of these two coronaviruses might explain why SARS was contained successfully in 2003 producing around 8,000 infections and 800 deaths, while COVID-19 kept spreading reaching a much larger worldwide population of 250 million infections and 5 million deaths as of November 2021.

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This peculiar feature of the new coronavirus implies that the only way to stop the chain of transmission (in the absence of vaccines) is to perform contact tracing to track the past contagious contacts of the patient and isolate them. That is, we need to go back in time to identify the contacts that have already occurred before the patient reports the symptoms to the health authorities. Without contact tracing, the chain of transmission cannot be broken, even if the patient enters into isolation after the onset of symptoms. This situation is exacerbated due to the existence of asymptomatic cases, i.e., infected people who do not feel symptoms and can potentially transmit the disease without knowing it.

The relation between latency and incubation of SARS-CoV-2 is inverted in coronavirus SARS-CoV-1 responsible for the SARS pandemic in 2003. As we see in the epidemiological profile of SARS-CoV-1 in Fig 1B, in this case, the latency is longer than the incubation period, according to studies reported in [25]. Patients of SARS 2003 become contagious a few days after the appearance of symptoms. In this case, upon isolation of the patient after reporting symptoms, the chain of transmission can be successfully broken. Thus there is no need to perform contact tracing back in time before the onset of symptoms since all contagious contacts happen during the manifestation of the disease. This situation could explain why SARS in 2003 was contained successfully without spreading worldwide, as this coronavirus infected “only” about 8,000 people with around 800 deaths worldwide. On the other hand, the new coronavirus spread to all continents infecting 250 million people and causing 5 million deaths as of November 2021 across the world. Furthermore, countries that implemented effective earlier contact tracing protocols, like South Korea and China [5–12], were able to contain the pandemic more successfully than counties who did not implement contact tracing protocols. Inspired by this evidence, our study is an attempt to scientifically show how contact tracing works in a real setting.

The infectiousness period of an infected person starts 2 days before and lasts up to 5 days after the onset of symptoms [25]. In this paper, we added two extra days to be conservative in capturing the contacts since the number of days comes from statistical estimations of the different periods characterizing the epidemiological profile of the disease, see Ref. [25]. Thus, in principle, to trace those people potentially infected by COVID-19 patients, we track contacts 4 days before and 7 days after the reported date of first symptoms (see Fig 1A). In addition, we extend the tracing period further back in time to also consider exposures that could come from asymptomatic cases (S1 Appendix, section 7). Exposures start the incubation period of the infected person which can occur up to 12.5 days before onset of symptoms (5.2 days on average, 95% percentile 12.5 days [26, 27], Fig 1A). To conservatively trace these exposure events, we add ∼2 days to this incubation period and obtain the widely used 14 days period. Hence, to trace transmission and exposure cases, we perform contact tracing over -14/+7 days from onset of symptoms (Fig 1A). As noted above, the peak of infectiousness as well as 44% (95% confidence interval, 25–69%) of infected cases occur during the pre-symptomatic stage [25]. Thus, performing contact tracing is essential to stop the spreading of the disease.

Contact model

The GPS geolocation of the trajectories of both infected and susceptible people is used to trace several layers of contacts in the transmission network using the following model (S1 Appendix, section 2). A contact at time stamp n is initiated with an infected user (source) at time t₀ (see Fig 2A). The timestamp n enumerates each GPS datapoint, while t_n refers to the actual time attached with that point. At t₀ we draw a contact area as a circle centered in the source position with a radius r. We then gather all the GPS datapoints from susceptible users (targets) that enter the contact area from t₀ to t₀ + T, where T is the total exposure time. We follow the trajectories of source and target within the time-space area and compute the probability of infection at time stamp n as p_i[n] = p_d[n] ⋅ p_t[n], where p_d[n] is the spatial component, and p_t[n] is the temporal component. When the average overlap between source and target is zero, then p_d[n] = 1, and when the overlap is 2r, then p_d[n] = 0. On the other hand, when the exposure time ≥ T, then p_t[n] = 1, and decreases to p_t[n] = 0 as the exposure time decreases (S1 Appendix, section 2). The probability p_d[n] quantifies the contact probability for two users in the same area defined by r. A contact requires non only a space overlapping but also a time overlap, p_t[n], which quantifies the probability that two users met based on the time commonly spent in the same area. We then combine these two probabilities for each timestamp n into their product.

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Fig 2. COVID-19 contact model.

(A) Contact area used in the contact tracing model. The grey person is at the first datapoint of the source at t₀. We collect all datapoints for every user in a T = 30 min forward window (t₁, t₂, t₃, …, t₀ + T) within an 8 m circle from the initial position. For each target (green and red) we compute the average position and the time spent inside the contact area (red part of the trajectory line). (B) Partial transmission tree of outbreak of confirmed SARS-CoV-2 infection identified by contact tracing during calibration in the month of March 2020. Links goes from the source of infection to the target. The colors represent the day of first symptoms for each node and size is the out-degree.

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Contacts with low probability of infection p_i[n], but repeated throughout time, can also infect the target. To incorporate this effect in the model, we define the probability of infection for a series of repeated contacts P_i[n] as a recursive formula from time 1 to n with P_i[0] = 0: (1) The iteration of contacts between source and target, P_i[n], generates higher probability of infection than a single contact p_i[n]. This means that there is a difference between a short single contact between two people and short repeated contacts between the same people. The latter scenario should have a larger probability than the former to become infected. While the distribution of p_i[n] is homogeneous without a clear threshold for an infectious contact, P_i[n] presents a very polarized distribution where the values are accumulated in the extremes: P_i = 0 or P_i = 1 (see S1 Fig). Thus, P_i[n] is better indicator than p_i[n] to separate infectious from non-infectious contacts. A contact is then considered infectious when this probability exceeds a certain threshold, P_i[n] > p_c. The hyperparameters of the contact model (T, r, p_c) are obtained by calibrating the model using only the contacts between infected people to reproduce the basic reproduction number R₀ = 2.78 in Ceará in the month of March, 2020 (S1 Appendix, section 3). We obtain T = 30 min, r = 8 m and p_c = 0.9. Thus, a contact is defined with probability one when exposure is at least 30 minutes within a distance ≪ 8m. This calibration procedure provides the partial transmission tree of the outbreak from patient zero to the end of the calibration period shown in Fig 2B.

Transmission network model

Next, we create the contact network of coronavirus transmission by first tracing the trajectories of confirmed COVID-19 patients to search for contacts -14/+7 days from the onset of symptoms using the above model. From the first contact layer, we add four layers of contacts to constitute the contact network of transmission that is used to monitor the progression of the pandemic. The time-varying network is aggregated to a snapshot defined over a time window of a week [16] (S1 Appendix, section 7). We find that other aggregation windows give similar results as presented.

Next, we analyze the spatio-temporal properties of the contact network. The government of the State of Ceará imposed a mass quarantine on March 19, 2020 which led to a decrease in people’s mobility by 56.5% as shown in Fig 3A. During the lockdown, only the displacements of essential workers were allowed. A large decrease in mobility is also observed across all Latin America, see [24].

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Fig 3. Structural components of transmission networks across the lockdown.

(A) Evolution for different metrics in Ceará, Brazil, previous to the mass quarantine (grey area), right after the imposed quarantine (yellow area) and later. The plot shows the root mean square displacement (MSD) normalized by the maximum value over the total period (blue), the cumulative number of cases (green) and the size of the GCC normalized by the maximum value over the total period (black). The uncertainty corresponds to the standard error (SE). The mobility data is showcased in the Grandata-United Nations Development Programme map shown in https://covid.grandata.com. The initial rise in GCC is due to the lack of data before March 1. (B) The plot shows the 0.5-core size (red), the 0.5-shell size (cyan) all normalized by their respective maximum value pre-lockdown. While the size of the 0.5-shell is reduced drastically during the lockdown, the 0.5-core was not reduced as much and keeps increasing, contributing to sustain the pandemic. The 0.5-core seems to follow the trend in the MSD, which we plot again to show this trend.

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Giant connected component (GCC)

To understand the effect of the lockdown on the contact network, we think by analogy with a “bond percolation” process [16, 17, 28]. In bond percolation, the network connectivity is reduced by removing a small fraction of links (bonds) between nodes, and the global disruption in network connectivity is monitored by studying the normalized size of the giant connected component (S1 Appendix, section 4). Following this analogy, the lockdown acts as a percolation process, and therefore we monitor the GCC of the transmission network before and after the lockdown. We find a large decrease in the size of the GCC [16, 28] within 6 days of the implementation of the lockdown on March 19, when the GCC is almost fully dismantled decreasing by 89.6% of its pre-lockdown size (Fig 3A).

Despite the disintegration of the GCC, the cumulative number of cases kept growing albeit at a lower rate (Fig 3A). We find that the mass quarantine was able to reduce the basic reproduction number from R₀ = 2.78 before lockdown to an effective reproduction number of R_e = 1.2 after the lockdown (Fig 3A). Despite this disruption in the network connectivity, R_e has not decreased below one, as it would have been needed to curb the spread of the disease.

The drastic reduction in the GCC is visually apparent in the contact networks in Fig 4. Before lockdown on March 19 (Fig 4A), the network is a strongly-connected unstructured “hairball”. Eight days into the lockdown on March 27 (Fig 4B), the network has been untangled into a set of strongly-connected modules integrated by tenuous paths of contacts. This structure is even more pronounced a few weeks later on April 28 (Fig 4C).

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Fig 4. Evolution of GCC and k-cores over the quarantine.

Disease transmission networks in the state of Ceará over time before and after the lockdown on March 19, 2020. (A) Transmission network on March 19 (pre-lockdown). A hairball highly-connected network is observed. The disconnected components of the 7-core ( in this network) are colored. These components are well connected into the hairball network as expected since mobility and connectivity is high. (B) The pre-quarantine hairball in (A) has been untangled and the k-cores have emerged 8 days into the lockdown on March 27. Here, we color the nodes according to layers of the transmission network starting at COVID-19 patient (black nodes). Size of nodes is according degree. (C) Network on April 28 including the components of the 5-core in different colors ( for this network). Visible is the high betweenness centrality node representing the weak-link of this k-core. (D) We plot the location of the contacts in the map of Fortaleza constituting the components of the 5-core of the April 28 in (C). The size of the circles in the map corresponds to the number of contacts inside each location. The colors correspond to the clusters of the 5-core in (C). The 5-core sustaining transmission is composed of clusters of contacts localized in hospitals, large warehouses and business buildings. Hospital 3, one of the largest in Fortaleza, constitutes the maximal of the pandemic. The underlying map comes from the Folium library of Python: https://github.com/python-visualization/folium which relies on the OpenStreetMap project [29].

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Superspreading k-core structures

The highly connected modules found in Fig 4B and 4C are k-core structures [30–33] of higher complexity than the GCC (which is a 1-core), that are known to sustain an outbreak even when the GCC has been disintegrated [16, 33]. The k-core of a graph is the maximal subgraph in which all nodes have a degree (number of connections) larger or equal than k [30–33]. The k-shell is the periphery of the k-core and is composed by all the nodes that belong to the k-core but not to the (k+1)-core (S1 Appendix, section 5). The k-core is obtained by iteratively pruning the nodes with degree smaller than k. For instance, the 3-core is obtained by removing the 1-shell and 2-shell in a k-shell decomposition process (S1 Appendix, section 5). Thus, all nodes in a k-core have at least degree k, and are connected to other nodes with degree at least k too. K-cores are nested and can be made of disconnected components (see S4 Fig). High k-cores are those with large k up to a maximal , and constitute the inner most important part of the network. In theory, the high k-cores are known from network science studies to be the reservoir of disease transmission persistence [16, 33]. On the contrary, low peripheral k-shells (see S2 Fig) do not contribute as much to the spread as the high inner k-cores.

Fig 3B shows that despite the disappearance of the GCC, there is a significant maximal k-core that was not dismantled by the mass quarantine. The figure shows that the outer k-shells of the transmission network (i.e., the 0.5-shell defined as the union of the k-shells with ) are disintegrated in the lockdown, decreasing by 91% with respect to their pre-quarantine size, in tandem with the GCC. However, the inner k-core (i.e., the 0.5-core defined as the k-core with ) persists in the lockdown. The figure shows that the decrease of the 0.5-core is only 50% compared to the 91% decrease of the 0.5-shell; the former even increases slightly at the end of April, following the same trend in mobility (see Fig 3B). This process is visually corroborated in the evolution of the networks seen from Fig 4A and 4C where we observe the disappearance of the peripheral k-shells and the persistence of the maximal k-core. Indeed, the unessential contacts in the peripheral k-shells may have been first pruned during social distancing.

Using numerical simulations, we corroborate previous results indicating that the infection can persist in these high k-cores of the network while virus persistence in outer k-shells is less important [16, 33]. We use a SIR model on the transmission network (see S15 Fig) showing that the maximal k-cores of the network sustain the spreading of the disease more efficiently than the outer k-shells. Thus, the maximal k-core components of the contact network are plausible drivers of disease transmission. Apart from this structural explanation (i.e., k-core), epidemiological factors may also play a role in the persistence of the disease, such as a transition of the disease to vulnerable communities with high demographic density, or with large inhabitants per household where isolation is poorly fulfilled.

When we plot the geolocation of the contacts forming the maximal k-core in the map of Ceará, we find that these contacts take place in highly transited areas of the capital Fortaleza, such as hospitals, business buildings, warehouses as well as large condominiums, see Fig 4D. These contacts generate superspreading k-core events that generalize the conventional notion of superspreaders, which refer mainly to individuals with large number of transmission contacts [34–36]. However, connections are not everything [18, 19]. K-core superspreaders not only generate a large number of transmission contacts, but their contacts are also highly connected people, and so forth.

Optimized quarantine

The existence of k-cores in the transmission network suggests that a more structured quarantine could be deployed to either isolate or destroy those cores that help maintain the spread of the virus. We perform an optimal percolation analysis [18–20] to find the minimal number of people necessary to quarantine that will dismantle the transmission network. We compare different strategies to find the best among them to break the network by ranking the nodes based on (1) the number of contacts (hub-removal) [16, 18, 19], (2) the largest k-shells and then by the degree inside the k-shells [16, 33], (3) the collective influence algorithm for optimal percolation [20], (4) the generalized k-core strategy [37], and (5) betweenness centrality [38–41].

Fig 5B shows the normalized size of the GCC versus the fraction of removal nodes following different strategies, as well as a random null model of removal in a typical network under lockdown in April 28 (March 19 pre-lockdown results are plotted in S15 Fig). While the disease can persist in the k-cores (Fig 5A), quarantining people directly inside the maximal k-core is not an optimal strategy. The reason is that k-cores are populated by hyper-connected hubs that require many removals to break the GCC [40] (around 7%, see Fig 5B). For the same reason, removing directly the hubs is not the optimal strategy either, since the hubs are within the maximal k-core and not outside. A collective influence strategy [20] improves over hub-removal since it takes into account how hubs are spatially distributed, yet, it is far from optimal. A generalized k-core strategy, which consists in sequentially removing the nodes in the k-leaf (where k = k_max), has been recently reported to be more suitable to study spreading behavior [37]. Fig 5B shows that, in this case, it performs similarly like k-core. The reason for this can be found in the tree structure of the network and its low average degree. Clearly, Fig 5B shows that the best strategy is to quarantine people by their betweenness centrality. By removing just the top 1.6–2% of the high betweenness centrality people, the GCC is disintegrated. This result is consistent with the particular structure of the transmission networks seen in Figs 4B and 4C and 5.

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Fig 5. Weak links and k-cores.

(A) Average size of infected population, M [33], in an outbreak average over all starting nodes in a k-shell as a function of the probability of infection β for a SIR model on the network in Fig 4C during the lockdown. The black is the average value over all the network. The average divides the k-shell contribution to the spreading of the virus in two groups: above and below the average. The 0.5-cores have maximal spreading and the 0.5-shell have minimal spreading. Error bars correspond to a confidence interval of 95%. (B) Optimal percolation analysis performed over the network in Fig 4C during the lockdown in following different attack strategies and their effect on the size of the largest connected component G(q) versus the removal node fraction, q. Nodes are removed (in order of increasing efficiency): randomly (blue); by the highest k-shell followed by high degree inside the k-shell [33]; by highest degree (orange); by collective influence (red) [20]; by the highest generalized k-core (brown) [37]; and by the highest value of betweenness centrality (green) [38, 39]. After each removal we re-compute all metrics. The most optimal strategy among those studied is removing the nodes by the highest value of betweenness centrality. (C)-(D) Effect of removing three high betweenness centrality nodes shown in Fig 5B in the network of Fig 4C. (C) We show the 2-core component of the network after the removal of 12 high betweenness centrality nodes. The red node is the one with the highest betweenness centrality value (next node to remove, 13th) and the blue node is the 14th removal. Different k-cores and k-shell are in different colors. (D) Network k-cores are disintegrated after the removal of the high BC nodes.

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The betweenness centrality of a node is proportional to the number of shortest paths in the network going through that node. Thus, given the particular structure of the networks in Figs 4B and 4C and 5C, the high betweenness centrality nodes are the bottlenecks of the network, i.e., loosely-connected bridges between the largely-connected k-cores components. These connectors are the “weak links”, fundamental concept in sociology proposed by Granovetter [42], according to which, strong ties (i.e., contacts in the k-cores) clump together forming clusters. A strategically located weak tie between these densely “knit clumps”, then becomes the crucial bridge that transmits the disease (or information [42]) between k-cores. These weak links are people traveling among the different k-cores components allowing the disease to escape the cores into the rest of society. These bridges are displayed in the network of Fig 5C as the yellow, blue and red nodes. The removal of these high betweenness centrality people disconnects the k-core components of the network entirely, as shown in Fig 5D, halting the disease transmission from one core to the other [40, 43].

An important finding is that quarantining the large superspreading k-cores is neither optimal (as shown in Fig 5B, green curve) nor practical, since they are mainly comprised by chiefly essential workers who need to remain operational (Fig 4D). Thus, the best strategy, in conjunction with a mass quarantine, is then to disconnect these k-cores from the rest of the social network (Fig 5C and 5D), rather than quarantining the people inside the k-cores. This can be performed by quarantining the high betweenness centrality weak-links that simultaneously preserve the operational k-cores. However, individuals belonging to the maximal k-cores should be tested at a higher frequency to promptly detect their infectiousness before the symptoms start, to help control the spreading inside the k-cores.