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 95th 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.
Fig 2.
(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.
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.
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].
Fig 5.
(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.