key: cord-0977756-ys95uivs
authors: Scabini, Leonardo F.S.; Ribas, Lucas C.; Neiva, Mariane B.; Junior, Altamir G.B.; Farfán, Alex J.F.; Bruno, Odemir M.
title: Social interaction layers in complex networks for the dynamical epidemic modeling of COVID-19 in Brazil
date: 2020-11-12
journal: Physica A
DOI: 10.1016/j.physa.2020.125498
sha: 554d848af5cb5882ce5d48599c8eff90d41b4703
doc_id: 977756
cord_uid: ys95uivs

We are currently living in a state of uncertainty due to the pandemic caused by the SARS-CoV-2 virus. There are several factors involved in the epidemic spreading, such as the individual characteristics of each city/country. The true shape of the epidemic dynamics is a large, complex system, considerably hard to predict. In this context, Complex networks are a great candidate for analyzing these systems due to their ability to tackle structural and dynamic properties. Therefore, this study presents a new approach to model the COVID-19 epidemic using a multi-layer complex network, where nodes represent people, edges are social contacts, and layers represent different social activities. The model improves the traditional SIR, and it is applied to study the Brazilian epidemic considering data up to 05/26/2020, and analyzing possible future actions and their consequences. The network is characterized using statistics of infection, death, and hospitalization time. To simulate isolation, social distancing, or precautionary measures, we remove layers and reduce social contact’s intensity. Results show that even taking various optimistic assumptions, the current isolation levels in Brazil still may lead to a critical scenario for the healthcare system and a considerable death toll (average of 149,000). If all activities return to normal, the epidemic growth may suffer a steep increase, and the demand for ICU beds may surpass three times the country’s capacity. This situation would surely lead to a catastrophic scenario, as our estimation reaches an average of 212,000 deaths, even considering that all cases are effectively treated. The increase of isolation (up to a lockdown) shows to be the best option to keep the situation under the healthcare system capacity, aside from ensuring a faster decrease of new case occurrences (months of difference), and a significantly smaller death toll (average of 87,000).

the world was not globalized, the means of transport were not as agile as 7 the current ones, and the population was much smaller. The 21st century is 8 marked by globalization and an intricate and intense social network connecting 9 in one way or another to everyone on the planet. The latter fact increases 10 the danger that a local epidemic disease will rapidly evolve into a pandemic 11 like what happened in Wuhan, China, and now is all over the world. 12 The form of propagation and contagion of the SARS-CoV-2 virus is through 13 direct contact between individuals, via secretions, saliva, and especially 14 droplets expelled during breathing, speech, coughing, or sneezing. The virus 15 also spreads by indirect contact, when such secretions reach surfaces, food, and 16 objects [1] . Besides, infected people take a few days to manifest symptoms, 17 which can be severe or as mild as a simple cold. There is even a large 18 proportion of infected people who remain asymptomatic [2] . Therefore, it is 19 hard to quickly identify the infected and apply effective measures to limit 20 the disease's spread. Also, the discovery of SARS-CoV-2 in December 2019 21 is very recent in the face of the current epidemic. Little is known about 22 the COVID-19 disease, which appears to be highly lethal, with no drugs to 23 prevent or treat. The concern is more significant since direct (individual -24 individual) and indirect (individual -objects -individual) social relations are 25 the means of spreading the disease. Thus, the social interaction structure is 26 the key to create strategies and guide health organizations and governments 27 to take appropriate actions to combat the disease. 28 One of the main concerns is overloading the health system. The first 29 known case in Brazil was confirmed on February 26, a 61-year-old man who 30 traveled to the Lombardy region in northern Italy. In the middle of May, 31 there are more than 200,000 cases and 14,000 deaths in all states of Brazil [3] . 32 The concern is even worse due to the country's social inequality. Over 80% of 33 the population relies solely on the public health system, and this distribution 34 is not uniform. According to [4] , there are only nine hospital beds per 100,000 35 people in the northern region, while southeast accounts for 21 hospital beds. 36 The treatment of severe cases requires the use of respirators/ventilation in The proposed model can be used to study any society given sufficient 60 demographic data, such as medium/big cities, countries, or regions. Here sociologist is the author of the well-known idea that there are always chains 104 of up to six people connecting any two individuals globally, which reinforces 105 the importance of analyzing the pandemic spread from a graph view.

In the context of epidemic propagation analysis, one of the best known 107 and widely used models is the susceptible-infected-recovered (SIR) model, 

• Infected: the ones that have the disease.

• Recovered: a person usually recovers after some time and maybe will 114 not susceptible to infection again due to the immunity process (in this 115 case, this is an assumption of the process). The recovery rate of infected Also, the model can be described as

where s, i and r represents the ratio of susceptible, infected and recovered 119 people in the population, respectively and N represents their absolute sum.

Usually, the SIR model is solved with differential equations. can be compared to the statistical analysis in [26] , however here we introduce 264 a more detailed model of social contacts with specific layers and connection 265 patterns better to fit the particularities of a given country or city. connected at group j of layer i, its edge weight is then defined by

where t i represents the average weekly contact time on layer i, k i is the . The dotted lines represent the standard deviation, and in the case of the real data, the curve is the average over a 5-day window, and the solid lines the real raw data. The highest average number of deaths produced by the proposed model may be related to underdetection (See Figure 5) . possible to notice that the number of undiagnosed cases is much higher than 536 the diagnosed cases. This reflects the number of asymptomatic cases and the 537 lack of tests for mild cases. In the worst scenario, which means ending the 538 isolation, the total infected number may go above 5 million. The recovered 539 rate is directly proportional to the infected rate, as one needs to be infected 540 to either die or become resistant to the disease. If the infected rate is high, 541 so is the recovered rate, e.g., the scenarios of keeping or ending isolation. A 542 high recovered rate also helps in mitigating the epidemic propagation (natural 543 immunization). However, increasing isolation decreases the spread much 544 faster than natural immunization, with a considerably smaller death toll. It 545 is also possible to observe the differences at the start of effective recovering, 546 i.e., when the recovered rate surpasses the infected rates, this is due to the 547 early increase in isolation levels. beds, which is around three times higher than the entire country's capacity.

The other alternative, which is the increase of isolation levels (lockdown), 641 appears to be the only alternative to stop the healthcare system from entering Although the proposed method includes various demographic information 652 for the network construction, and an improved SIR approach to COVID-653 19, it still does not cover all factors that impact the epidemic propagation.

As future works, more information can be included, such as an increased 

Another important point regarding the obtained results 665 is related to the "keep isolation" scenario, which may be underestimated as 666 we take various optimistic assumptions and also consider a fixed isolation level 667 based on previously observed data

Therefore, during the network evolution, a 669 possible improvement is the use of dynamic isolation levels to represent reality 670 better. It is also possible to consider various scenarios for future actions, such 671 as two or more measures of increasing/reducing isolation. This may allow the 672 discovery of new epidemic waves if social activities return too soon after the 673 isolation period, for instance

This study was financed in part by the Coordenação de Aperfeiçoamento de 676

Pessoal de Nível Superior -Brasil (CAPES) -Finance Code 001

Ribas acknowledge support from São Paulo Research 678

Foundation (FAPESP) (Grant #2019/07811-0 and #2016/23763-8)

Junior acknowledges support from CNPq (Grant # 144323/2019-2)

Neiva acknowledges support from CAPES 681 (Grant number #PROEX-9527567/D and CAPES PROEX-9056169/D re-682 spectively

#307897/2018-4) and FAPESP (grant #2014/08026-1 and

He is also grateful to Henrique Pott Junior and Francisco Fambrini, for the 685 fruitful conversations regarding the pandemic. The authors are also grateful 686 to the NVIDIA GPU Grant Program for the donation of the Titan Xp GPUs

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