Rare and extreme events: the case of COVID-19 pandemic
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ORIGINAL PAPER
Rare and extreme events: the case of COVID-19 pandemic J. A. Tenreiro Machado · António M. Lopes
Received: 31 March 2020 / Accepted: 30 April 2020 © Springer Nature B.V. 2020
Abstract Complex systems have characteristics that give rise to the emergence of rare and extreme events. This paper addresses an example of such type of crisis, namely the spread of the new Coronavirus disease 2019 (COVID-19). The study deals with the statistical comparison and visualization of country-based realdata for the period December 31, 2019, up to April 12, 2020, and does not intend to address the medical treatment of the disease. Two distinct approaches are considered, the description of the number of infected people across time by means of heuristic models fitting the real-world data, and the comparison of countries based on hierarchical clustering and multidimensional scaling. The computational and mathematical modeling lead to the emergence of patterns, highlighting similarities and differences between the countries, pointing toward the main characteristics of the complex dynamics. Keywords Coronavirus disease 2019 · Extreme events · Regression · Multidimensional scaling · Hierarchical clustering · Complex systems J. A. Tenreiro Machado Polytechnic of Porto, Dept. of Electrical Engineering, Institute of Engineering, Rua Dr. António Bernardino de Almeida, 431, 4249 – 015 Porto, Portugal e-mail: [email protected] A. M. Lopes (B) UISPA–LAETA/INEGI, Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, 4200 – 465 Porto, Portugal e-mail: [email protected]
1 Introduction Many complex systems generate outputs that are characterized by a frequency-size power law behavior over several orders of magnitude [1,2]. The power laws have been associated with scale-invariance, self-similarity, and fractality and are consistent with self-organized criticallity, that is a process in which a system, by itself, converges to a state characterized by a coherent global pattern, created by local interactions between low-level elements [3,4]. The power laws are characterized by heavy-tails, giving non-negligible probability to large events. However, some extreme events, labeled ‘dragon kings’, while predictable, cannot be foreseen by the extrapolation of power law distributions [5,6]. ‘Dragon kings’ may be associated with positive feedback, bifurcations, and regime changes in out-of-equilibrium complex systems. ‘Dragon kings’ are often discussed in contrast with ‘black swans’, which denote unpredictable catastrophic rare events [7]. These outliers are pervasive in many areas, namely economy, finance, earth sciences, and biology. The recent Coronavirus disease 2019 (COVID-19) outbreak is an example of an extreme event. We must highlight that the occurrence of a rare event and the actual description of its evolution are, however, distinct matters. This paper attempts to understand the dynamics of the spreading across different countries
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J. A. Tenreiro Machado, A. M. Lopes
of COVID-19, but not the prediction of its outbreak
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