RESUMO
Targeted immunization or attacks of large-scale networks has attracted significant attention by the scientific community. However, in real-world scenarios, knowledge and observations of the network may be limited thereby precluding a full assessment of the optimal nodes to immunize (or remove) in order to avoid epidemic spreading such as that of current COVID-19 epidemic. Here, we study a novel immunization strategy where only n nodes are observed at a time and the most central between these n nodes is immunized (or attacked). This process is continued repeatedly until 1 - p fraction of nodes are immunized (or attacked). We develop an analytical framework for this approach and determine the critical percolation threshold pc and the size of the giant component P{infty}; for networks with arbitrary degree distributions P(k). In the limit of n [->] {infty} we recover prior work on targeted attack, whereas for n = 1 we recover the known case of random failure. Between these two extremes, we observe that as n increases, pc increases quickly towards its optimal value under targeted immunization (attack) with complete information. In particular, we find a new scaling relationship between |pc({infty}) - pc(n) | and n as |pc({infty}) - pc(n)| ~ n-1 exp(-n). For Scale-free (SF) networks, where P(k) ~ k-{gamma}, 2 <{gamma} < 3, we find that pc has a transition from zero to non-zero when n increases from n = 1 to order of logN (N is the size of network). Thus, for SF networks, knowledge of order of logN nodes and immunizing them can reduce dramatically an epidemics.
RESUMO
The new coronavirus known as COVID-19 is spread world-wide since December 2019. Without any vaccination or medicine, the means of controlling it are limited to quarantine and social distancing. Here we study the spatio-temporal propagation of the first wave of the COVID-19 virus in China and compare it to other global locations. We provide a comprehensive picture of the spatial propagation from Hubei to other provinces in China in terms of distance, population size, and human mobility and their scaling relations. Since strict quarantine has been usually applied between cities, more insight about the temporal evolution of the disease can be obtained by analyzing the epidemic within cities, especially the time evolution of the infection, death, and recovery rates which affected by policies. We study and compare the infection rate in different cities in China and provinces in Italy and find that the disease spread is characterized by a two-stages process. At early times, at order of few days, the infection rate is close to a constant probably due to the lack of means to detect infected individuals before infection symptoms are observed. Then at later times it decays approximately exponentially due to quarantines. The time evolution of the death and recovery rates also distinguish between these two stages and reflect the health system situation which could be overloaded.
