key: cord-0901753-s7l9rlnr authors: Ahmad, T.; Ashaari, A.; Awang, S. R.; Mamat, S. S.; Wan Mohamad, W. M.; Ahmad Fuad, A. A.; Hassan, N. title: Identification of severity zones for mitigation strategy assessment COVID-19 outbreak in Malaysia date: 2020-05-26 journal: nan DOI: 10.1101/2020.05.19.20107359 sha: eb7c3dfca97b16c9004ae723c707d80d907dfc54 doc_id: 901753 cord_uid: s7l9rlnr The objective of this research is to identify severity zones for the COVID-19 outbreak in Malaysia. The technique employed for the purpose is fuzzy graph that can accommodate scarcity, quantity, and availability of data set. Two published sets of data by the Ministry of Health of Malaysia are used to implement the technique. The obtained results can offer descriptive insight, reflection, assessment, and strategizing actions in combating the pandemic. followed by its declaration as a Public Health Emergency by the international body on 30 th 27 January 2020, researchers, scientists, and mathematicians have been racing in their efforts to 28 stop the potential devastating assault by the coronavirus. 29 30 Zhou et al. [1] first tipped off the world to the menace of the virus through their publication 31 in Nature. However, the researchers did not employ any specific mathematical tools in their 32 work. Most mathematical modelers have employed Ordinary Differential Equation (ODE), 33 such as Liang [2] as a tool in their predictive modeling of COVID-19. Similarly, Qianying et 34 al. [3] adopted the system of ordinary differential equations that previously used to model the 35 pandemic 1918 Spanish Flu for describing the current COVID-19 outbreak. Recently, Krantz 36 and Rao [4] described underreporting cases of COVID-19 for several countries using coupled 37 ODE-wavelets model. There are more than 25 papers and preprints in the literature on ODE 38 or ODE coupled with other methods as 2 nd May 2020 to model COVID-19 and related issues. 39 For instance, Hamzah et al. [5] and Prem et al. [6] utilized a system of ordinary differential 40 equations in their Susceptible-Exposed-Infected-Removed (SEIR) models and Jia et al. [ There are three main downsides for modeling COVID-19, namely, scarcity, quantity, and 49 availability of data that are essential to produce a good reliable mathematical model. This is 50 due to the fact that the outbreak is about six months old since the first case was reported. 51 Therefore, a flexible and robust mathematical technique that can handle such identified 52 shortcomings is necessary to model the outbreak. In this paper, a fuzzy graph analysis method 53 is presented, namely fuzzy autocatalytic set, that is capable of accommodating such constraints 54 to analyze the current pandemic. 55 Generally, a graph represents a relationship between objects. Objects are represented as 57 vertices and the relations by edges. A graph is formally defined as the following. 58 59 Definition 1 (see [9] ). A graph is a pair of sets ( , ) where is the set of vertices and is 60 the set of edges. 61 62 Furthermore, another way to represent a graph is by its adjacency matrix. The definition of an 63 adjacency matrix for a graph is given in Definition 2 below. 64 65 Definition 2 (see [9] ). An adjacency matrix of graph ( , ) with vertices is an 69 The concepts of graph and fuzzy set have given 'birth' to a new mathematical structure, 71 namely, fuzzy graph. Definition 3 indicates that vertices and edges are both fuzzy. In other 72 words, the vertices and edges have values between 0 and 1. Figure 0 .2 illustrates a fuzzy graph. 73 74 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. An adjacency matrix of a fuzzy graph is defined as follows: 78 79 Definition 4 (see [10] ). An adjacency matrix, of a fuzzy graph = ( , , ) is an × 80 matrix defined as = ( ) such that = ( , ). ACS is given as follows. 90 91 Definition 5 (see [13] ). An autocatalytic set is a subgraph, each of whose vertices has at least 92 one incoming link from vertices belonging to the same subgraph. Definition 6 (see [14] ). A fuzzy autocatalytic set is a subgraph each of whose vertices has at 103 least one incoming link with membership value, ( ) ∈ (0,1], ∀ ∈ from any other vertices 104 are belonging to the same subgraph. 105 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 26, 2020. 107 The adjacency matrix in Step 1: Keeping ( × ) matrix fixed, evolved according to the following equation. 113 , 114 for time , which is large enough for to get reasonably close to its attractor 115 (Perron Frobenius Eigenvector). We denoted ≡ ( ). 116 Step 2: The set of nodes with the least value of is determined, i.e. 117 This is the set of "least fit" nodes, identifying the relative concentration of a variable 119 in the attractor (or, more specifically, at ) with its "fitness" in the environment 120 defined by the graph. The least fit node is removed from the system along with its 121 links, leading a graph of − 1 variables. 122 Step 3: is now reduced to ( − 1) × ( − 1) matrix. The remaining nodes and links of 123 remained unchanged. All these (0 ≤ ≤ 1) are rescaled to keep 124 Repeat all the steps until the 2 × 2 matrix is attained. 126 127 Figure 0 .5 illustrates the initial step (Step 1). Then one of the nodes with the least eigenvector 128 is removed from the graph ( Step 2). The node is removed along with its links, and the graph is 129 left with a reduced number of nodes and links (Step 3). This process is then repeated until a 130 graph with at least two nodes is attained. 131 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 26, 2020. 141 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. 28 th March to 5 th April is considered for the first case. The period is selected due to the 147 erraticness of the data, as depicted in Figure 0 .7a. 148 The data 28 th March 2020 -5 th April 2020 149 The data of new cases of COVID-19 is tabulated in The reported new cases in states of Malaysia from 28 th March to 5 th April 2020 are depicted 154 in Figure 3 .1 b, and its adjacency matrix is given in Figure 0 .2b. 155 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. where Cluster 2 and 4 popped up in the zone. Hence, the government has to pay attention to 173 the states in Cluster 3 because these states have the potential to move into Zone 3. On top of 174 that, Zone 2 is clearly closed adjacent to Zone 3. 175 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 26, 2020. In fact, the government has gazetted 23 districts in these states as the red zone, namely, 186 Putrajaya, Jasin, Negeri Sembilan, Hulu Langat, Petaling, Johor Bahru, Kuching, and Tawau. 187 The district of Hulu Selangor in the state of Selangor has announced another red zone on 10 th 188 April (Figure 0.10a) . Our FACS analysis concurred with the list of states in red zones released 189 by Crisis Preparedness and Response Centre (CPRC), Ministry of Health of Malaysia. 190 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. Pahang, Terengganu, Kelantan, Sabah 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 26, 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 26, 2020. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (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 26, 2020. 18 31 14 14 14 46 44 27 69 27 26 19 14 19 4 13 16 29 49 39 16 30 14 8 10 32 A pneumonia outbreak associated with a new coronavirus of probable bat origin Mathematical model of infection kinetics and its analysis for COVID-19, SARS and MERS A conceptual model for the coronavirus disease 2019 (COVID-19) outbreak in Wuhan China with individual reaction and governmental action Level of underreporting including underdiagnosis before the first 265 peak of COVID-19 in various countries: Preliminary retrospective results based on wavelets and 266 deterministic modeling COVID-19 outbreak data analysis and prediction Slow potential waves in the human brain associated with expectancy, attention and 271 decision Modeling the control of COVID-19: impact of policy 273 interventions and meteorological factors Phylogenetic network analysis of SARS-CoV-2 275 genomes Preference Graph of Potential Method as a Fuzzy Graph Fuzzy graphs, Fuzzy Sets and their Applications Cellular Homeostasis, Epigenesis and Replication in Randomly Aggregated V State