key: cord-0882976-trfjv1zj authors: Ahmad, T.; Ashaari, A.; Awang, S. R.; Mamat, S. S.; Wan Mohamad, W. M.; Ahmad Fuad, A. A.; Hassan, N. title: Fuzzy Autocatalytic analysis of Covid-19 outbreak in Malaysia date: 2020-05-21 journal: nan DOI: 10.1101/2020.05.17.20104968 sha: f5848e2426776dc991076da531c545f2ad4ec989 doc_id: 882976 cord_uid: trfjv1zj The objective of this research is to demonstrate a mathematical technique to analyze the Covid-19 outbreak, particularly with respect to Malaysia. The technique is able to accommodate scarcity, quantity, and availability of the data set. The obtained results can offer descriptive insight for reflecting and strategizing actions in combating the pandemic. claimed 105 952 lives worldwide as of 12 April 2020, 08:00 GMT, according to the World 20 Health Organization (WHO) (https://www.who.int/emergencies/diseases/novel-coronavirus-21 2019). Since the first case of pneumonia of unknown cause detected in Wuhan reported to the 22 WHO Country Office in China on 31 December 2019 and followed by its declaration as a 23 Public Health Emergency by the international body on 30 January 2020, researchers, scientists, 24 and mathematicians have been racing in their efforts to stop the potential devastating assault 25 by the coronavirus. 26 27 These efforts include Zhou et al. [1] alerted the world the menace of the virus through their 28 publication in Nature. However, the researchers did not employ any specific mathematical tools 29 in their work. Hamzah et al. [2] utilized a system of ordinary differential equations for 30 Susceptible-Exposed-Infected-Removed (SEIR) in their predictive modeling of the Covid-19 31 outbreak. Similarly, Lin et al. [3] adopted a system of ordinary differential equations that 32 previously used to model the pandemic 1918 Spanish Flu for describing the current Covid-19 33 outbreak. Recently, Forster et al. [4] analyzed the coronavirus genomes using the phylogenetic 34 network, a special type of graph that has been primarily used in archaeological studies. 35 36 There are three main problems with respect to the Covid-19 outbreak, namely, the scarcity, 37 quantity, and availability of data that are essential to produce a good reliable mathematical 38 model. This is due to the fact that the outbreak is about six months old since the first case was 39 reported. Therefore, a mathematical technique must be flexible and robust enough to deal with 40 such identified shortcomings is needed to model the outbreak . In this paper, a suitable 41 mathematical method is proposed, namely a fuzzy autocatalytic set, which is able to 42 accommodate such constraints to analyze the current pandemic. 43 Generally, a graph represents a relationship between objects. Objects are represented as 45 vertices and the relations by edges. Formally, the definition of a graph is as follows 46 47 Definition 1 (see [5] ). A graph is a pair of sets ( , ) where is the set of vertices, and is 48 the set of edges. 49 50 Furthermore, another way to represent a graph is by its adjacency matrix. The definition of an 51 adjacency matrix for a graph is given in Definition 2 below. 52 53 Definition 2 (see [5] ). An adjacency matrix of graph ( , ) with vertices is an An adjacency matrix of a fuzzy graph is defined as follows: 69 70 Definition 4 (see [6] ). An adjacency matrix, of a fuzzy graph = ( , , ) is an × matrix 71 defined as = ( ) such that = ( , ). 72 73 74 75 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 21, 2020. given as follows. 82 83 Definition 5 (see [9] ). An autocatalytic set is a subgraph, each of whose vertices has at least 84 one incoming link from vertices belonging to the same subgraph. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 21, 2020. . https://doi.org/10.1101/2020.05.17.20104968 doi: medRxiv preprint The adjacency matrix in FIGURE 1(b) and FIGURE 2(b) are then processed by the procedure 106 outlined in [10], [11] and improved by [12], respectively. The outcomes of the process are 107 determined via the following steps. 108 109 Step 1: Keeping ( × ) matrix fixed, evolved according to the following equation. 110 , 111 for time , which is large enough for to get reasonably close to its attractor 112 (Perron Frobenius Eigenvector). We denoted ≡ ( ). 113 Step 2: The set of nodes with the least value of is determined, i.e. 114 This is the set of "least fit" nodes, identifying the relative concentration of a variable 116 in the attractor (or, more specifically, at ) with its "fitness" in the environment 117 defined by the graph. The least fit node is removed from the system along with its 118 links, leading a graph of − 1 variables. 119 Step 3: is now reduced to ( − 1) × ( − 1) matrix. The remaining nodes and links of 120 remained unchanged. All these (0 ≤ ≤ 1) are rescaled to keep 121 Repeat all the steps until the 2 × 2 matrix is attained. 123 124 FIGURE 5 illustrates the initial step (Step 1). Then one of the nodes with the least eigenvector 125 is removed from the graph ( Step 2). The node is removed along with its links, and the graph is 126 left with a reduced number of nodes and links (Step 3). This process is then repeated until a 127 graph with at least two nodes is attained. 128 129 130 FIGURE 5: Schematic portrayal of the graph dynamics. All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 21, 2020. . https://doi.org/10.1101/2020.05.17.20104968 doi: medRxiv preprint The procedure to transform the graph into 2D-Euclidean space is adopted from [13] , which is 133 based on the Laplacian matrix and solving a unique one-dimensional optimization problem in 134 order to determine their coordinates. The general overview of the transformation is depicted in 135 the following FIGURE 6. 136 137 138 139 The technique described in Section 2 is implemented on two sets of data; Malaysia and its 142 neighboring countries and states in Malaysia. 143 The daily new reported cases of Covid-19 for Malaysia, Singapore, Thailand, Indonesia, and 145 Brunei are obtained (publicly available) from European Centre for Disease Prevention and 146 Control's website (see https://www.ecdc.europa.eu/en/publicationsdata) from 1 February 2020 147 until 27 March 2020 (see FIGURE 7) . In order to determine the pandemic signature of Covid-148 19 for these countries, we sampled the data from 12 to 27 March only. This is due to the fact 149 that the plotted lines are clearly erratic for the sampled countries (refer FIGURE 7 and FIGURE 150 8) during that period. 151 152 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 21, 2020. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 21, 2020. Similarly, the fuzzy analysis of the graphs is divided into two identified demographic areas and 173 presented in the following subsections. 174 The graph and its adjacent matrix for data from 12 to 27 March are constructed and exhibited 176 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 21, 2020. The dominance of each country (outcome) is identified for each day of the erratic interval and 185 presented in the following Malaysia (S1) and Singapore (S2) nodes are closer to each other. This characteristic hint that 198 daily reported cases for these two countries are quite similar, followed by Thailand (S3) and 199 Indonesia (S4), whereas Brunei (S5) is isolated from the rest. 200 201 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 21, 2020. Selangor, Negeri Sembilan, Melaka and Johor whereas Cluster 3 is made of Pahang, 219 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. 248 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. A. Ashaari, T. Ahmad, S. Zenian, and N. A. Shukor, "Selection Probe of EEG Using Dynamic Graph of 288 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 21, 2020. 294 295 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 21, 2020. . https://doi.org/10.1101/2020.05.17.20104968 doi: medRxiv preprint A pneumonia outbreak associated with a new coronavirus of probable bat origin COVID-19 Outbreak Data Analysis and Prediction A conceptual model for the coronavirus disease 2019 (COVID-19) outbreak in Wuhan China with individual reaction and governmental action Phylogenetic network analysis of SARS-CoV-2 272 genomes Preference graph of potential method as a fuzzy graph Fuzzy Graphs, Fuzzy Sets and their Applications to Cognitive and Decision Process, M Cellular homeostasis, epigenesis and replication in randomly aggregated 278 macromolecular systems Ein systemtheoretisches Modell zur Biogenese / A System Theoretic Model of 280 Autocatalytic sets and the growth of complexity in an evolutionary model