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Wang S, Batool A, Sun X, Pan X. Non-intrusive reduced-order model for time-dependent stochastic partial differential equations utilizing dynamic mode decomposition and polynomial chaos expansion. CHAOS (WOODBURY, N.Y.) 2024; 34:073102. [PMID: 38949526 DOI: 10.1063/5.0200406] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/26/2024] [Accepted: 06/11/2024] [Indexed: 07/02/2024]
Abstract
In this study, we present a novel non-intrusive reduced-order model (ROM) for solving time-dependent stochastic partial differential equations (SPDEs). Utilizing proper orthogonal decomposition (POD), we extract spatial modes from high-fidelity solutions. A dynamic mode decomposition (DMD) method is then applied to vertically stacked matrices of projection coefficients for future prediction of coefficient fields. Polynomial chaos expansion (PCE) is employed to construct a mapping from random parameter inputs to the DMD-predicted coefficient field. These lead to the POD-DMD-PCE method. The innovation lies in vertically stacking projection coefficients, ensuring time-dimensional consistency in the coefficient matrix for DMD and facilitating parameter integration for PCE analysis. This method combines the model reduction of POD with the time extrapolation strengths of DMD, effectively recovering field solutions both within and beyond the training time interval. The efficiency and time extrapolation capabilities of the proposed method are validated through various nonlinear SPDEs. These include a reaction-diffusion equation with 19 parameters, a two-dimensional heat equation with two parameters, and a one-dimensional Burgers equation with three parameters.
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Affiliation(s)
- Shuman Wang
- Department of Mathematics, Shanghai University, Shanghai 200444, China
| | - Afshan Batool
- Department of Mathematical Sciences, Fatima Jinnah Women University, Rawalpindi 46000, Pakistan
| | - Xiang Sun
- School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China
| | - Xiaomin Pan
- Department of Mathematics, Shanghai University, Shanghai 200444, China
- Newtouch Center for Mathematics of Shanghai University, Shanghai University, Shanghai 200444, China
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2
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Persoons R, Sensi M, Prasse B, Van Mieghem P. Transition from time-variant to static networks: Timescale separation in N-intertwined mean-field approximation of susceptible-infectious-susceptible epidemics. Phys Rev E 2024; 109:034308. [PMID: 38632755 DOI: 10.1103/physreve.109.034308] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/03/2023] [Accepted: 02/15/2024] [Indexed: 04/19/2024]
Abstract
We extend the N-intertwined mean-field approximation (NIMFA) for the susceptible-infectious-susceptible (SIS) epidemiological process to time-varying networks. Processes on time-varying networks are often analyzed under the assumption that the process and network evolution happen on different timescales. This approximation is called timescale separation. We investigate timescale separation between disease spreading and topology updates of the network. We introduce the transition times [under T]̲(r) and T[over ¯](r) as the boundaries between the intermediate regime and the annealed (fast changing network) and quenched (static network) regimes, respectively, for a fixed accuracy tolerance r. By analyzing the convergence of static NIMFA processes, we analytically derive upper and lower bounds for T[over ¯](r). Our results provide insights and bounds on the time of convergence to the steady state of the static NIMFA SIS process. We show that, under our assumptions, the upper-transition time T[over ¯](r) is almost entirely determined by the basic reproduction number R_{0} of the network. The value of the upper-transition time T[over ¯](r) around the epidemic threshold is large, which agrees with the current understanding that some real-world epidemics cannot be approximated with the aforementioned timescale separation.
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Affiliation(s)
- Robin Persoons
- Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands
| | - Mattia Sensi
- MathNeuro Team, Inria at Université Côte d'Azur, 2004 Rte des Lucioles, 06410 Biot, France
- Department of Mathematical Sciences "G. L. Lagrange", Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
| | - Bastian Prasse
- European Centre for Disease Prevention and Control (ECDC), Gustav III's Boulevard 40, 169 73 Solna, Sweden
| | - Piet Van Mieghem
- Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands
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3
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Han Z, Liu L, Wang X, Hao Y, Zheng H, Tang S, Zheng Z. Probabilistic activity driven model of temporal simplicial networks and its application on higher-order dynamics. CHAOS (WOODBURY, N.Y.) 2024; 34:023137. [PMID: 38407398 DOI: 10.1063/5.0167123] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/10/2023] [Accepted: 01/27/2024] [Indexed: 02/27/2024]
Abstract
Network modeling characterizes the underlying principles of structural properties and is of vital significance for simulating dynamical processes in real world. However, bridging structure and dynamics is always challenging due to the multiple complexities in real systems. Here, through introducing the individual's activity rate and the possibility of group interaction, we propose a probabilistic activity-driven (PAD) model that could generate temporal higher-order networks with both power-law and high-clustering characteristics, which successfully links the two most critical structural features and a basic dynamical pattern in extensive complex systems. Surprisingly, the power-law exponents and the clustering coefficients of the aggregated PAD network could be tuned in a wide range by altering a set of model parameters. We further provide an approximation algorithm to select the proper parameters that can generate networks with given structural properties, the effectiveness of which is verified by fitting various real-world networks. Finally, we construct the co-evolution framework of the PAD model and higher-order contagion dynamics and derive the critical conditions for phase transition and bistable phenomenon using theoretical and numerical methods. Results show that tendency of participating in higher-order interactions can promote the emergence of bistability but delay the outbreak under heterogeneous activity rates. Our model provides a basic tool to reproduce complex structural properties and to study the widespread higher-order dynamics, which has great potential for applications across fields.
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Affiliation(s)
- Zhihao Han
- Institute of Artificial Intelligence, Beihang University, Beijing 100191, China
- Key laboratory of Mathematics, Informatics and Behavioral Semantics (LMIB), Beihang University, Beijing 100191, China
| | - Longzhao Liu
- Institute of Artificial Intelligence, Beihang University, Beijing 100191, China
- Key laboratory of Mathematics, Informatics and Behavioral Semantics (LMIB), Beihang University, Beijing 100191, China
- State Key Lab of Software Development Environment (NLSDE), Beihang University, Beijing 100191, China
- Zhongguancun Laboratory, Beijing 100094, People's Republic of China
- Beijing Advanced Innovation Center for Future Blockchain and Privacy Computing, Beihang University, Beijing 100191, China
- PengCheng Laboratory, Shenzhen 518055, China
| | - Xin Wang
- Institute of Artificial Intelligence, Beihang University, Beijing 100191, China
- Key laboratory of Mathematics, Informatics and Behavioral Semantics (LMIB), Beihang University, Beijing 100191, China
- State Key Lab of Software Development Environment (NLSDE), Beihang University, Beijing 100191, China
- Zhongguancun Laboratory, Beijing 100094, People's Republic of China
- Beijing Advanced Innovation Center for Future Blockchain and Privacy Computing, Beihang University, Beijing 100191, China
- PengCheng Laboratory, Shenzhen 518055, China
| | - Yajing Hao
- Key laboratory of Mathematics, Informatics and Behavioral Semantics (LMIB), Beihang University, Beijing 100191, China
- School of Mathematical Sciences, Beihang University, Beijing 100191, China
| | - Hongwei Zheng
- Beijing Advanced Innovation Center for Future Blockchain and Privacy Computing, Beihang University, Beijing 100191, China
- Beijing Academy of Blockchain and Edge Computing (BABEC), Beijing 100085, China
| | - Shaoting Tang
- Institute of Artificial Intelligence, Beihang University, Beijing 100191, China
- Key laboratory of Mathematics, Informatics and Behavioral Semantics (LMIB), Beihang University, Beijing 100191, China
- State Key Lab of Software Development Environment (NLSDE), Beihang University, Beijing 100191, China
- Zhongguancun Laboratory, Beijing 100094, People's Republic of China
- Beijing Advanced Innovation Center for Future Blockchain and Privacy Computing, Beihang University, Beijing 100191, China
- PengCheng Laboratory, Shenzhen 518055, China
- Institute of Medical Artificial Intelligence, Binzhou Medical University, Yantai 264003, China
- School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
| | - Zhiming Zheng
- Institute of Artificial Intelligence, Beihang University, Beijing 100191, China
- Key laboratory of Mathematics, Informatics and Behavioral Semantics (LMIB), Beihang University, Beijing 100191, China
- State Key Lab of Software Development Environment (NLSDE), Beihang University, Beijing 100191, China
- Zhongguancun Laboratory, Beijing 100094, People's Republic of China
- Beijing Advanced Innovation Center for Future Blockchain and Privacy Computing, Beihang University, Beijing 100191, China
- PengCheng Laboratory, Shenzhen 518055, China
- Institute of Medical Artificial Intelligence, Binzhou Medical University, Yantai 264003, China
- School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
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4
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Yin Q, Wang Z, Xia C. Information-epidemic co-evolution propagation under policy intervention in multiplex networks. NONLINEAR DYNAMICS 2023; 111:1-13. [PMID: 37361006 PMCID: PMC10250073 DOI: 10.1007/s11071-023-08581-w] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 10/19/2021] [Accepted: 05/16/2023] [Indexed: 06/28/2023]
Abstract
The emergence of epidemics has seriously threatened the running of human society, such as COVID-19. During the epidemics, some external factors usually have a non-negligible impact on the epidemic transmission. Therefore, we not only consider the interaction between epidemic-related information and infectious diseases, but also the influence of policy interventions on epidemic propagation in this work. We establish a novel model that includes two dynamic processes to explore the co-evolutionary spread of epidemic-related information and infectious diseases under policy intervention, one of which depicts information diffusion about infectious diseases and the other denotes the epidemic transmission. A weighted network is introduced into the epidemic spreading to characterize the impact of policy interventions on social distance between individuals. The dynamic equations are established to describe the proposed model according to the micro-Markov chain (MMC) method. The derived analytical expressions of the epidemic threshold indicate that the network topology, epidemic-related information diffusion and policy intervention all have a direct impact on the epidemic threshold. We use numerical simulation experiments to verify the dynamic equations and epidemic threshold, and further discuss the co-evolution dynamics of the proposed model. Our results show that strengthening epidemic-related information diffusion and policy intervention can significantly inhibit the outbreak and spread of infectious diseases. The current work can provide some valuable references for public health departments to formulate the epidemic prevention and control measures.
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Affiliation(s)
- Qian Yin
- Tianjin Key Laboratory of Intelligence Computing and Novel Software Technology, Tianjin University of Technology, Tianjin, 300384 China
- Faculty of Intelligence Manufacture, Wuyi University , Jiangmen, 529020 China
| | - Zhishuang Wang
- Tianjin Key Laboratory of Intelligence Computing and Novel Software Technology, Tianjin University of Technology, Tianjin, 300384 China
- Faculty of Intelligence Manufacture, Wuyi University , Jiangmen, 529020 China
| | - Chengyi Xia
- Tianjin Key Laboratory of Intelligence Computing and Novel Software Technology, Tianjin University of Technology, Tianjin, 300384 China
- Faculty of Intelligence Manufacture, Wuyi University , Jiangmen, 529020 China
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Zhang S, Zhao D, Xia C, Tanimoto J. Impact of simplicial complexes on epidemic spreading in partially mapping activity-driven multiplex networks. CHAOS (WOODBURY, N.Y.) 2023; 33:2895981. [PMID: 37307162 DOI: 10.1063/5.0151881] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/25/2023] [Accepted: 05/22/2023] [Indexed: 06/14/2023]
Abstract
Over the past decade, the coupled spread of information and epidemic on multiplex networks has become an active and interesting topic. Recently, it has been shown that stationary and pairwise interactions have limitations in describing inter-individual interactions , and thus, the introduction of higher-order representation is significant. To this end, we present a new two-layer activity-driven network epidemic model, which considers the partial mapping relationship among nodes across two layers and simultaneously introduces simplicial complexes into one layer, to investigate the effect of 2-simplex and inter-layer mapping rate on epidemic transmission. In this model, the top network, called the virtual information layer, characterizes information dissemination in online social networks, where information can be diffused through simplicial complexes and/or pairwise interactions. The bottom network, named as the physical contact layer, denotes the spread of infectious diseases in real-world social networks. It is noteworthy that the correspondence among nodes between two networks is not one-to-one but partial mapping. Then, a theoretical analysis using the microscopic Markov chain (MMC) method is performed to obtain the outbreak threshold of epidemics, and extensive Monte Carlo (MC) simulations are also carried out to validate the theoretical predictions. It is obviously shown that MMC method can be used to estimate the epidemic threshold; meanwhile, the inclusion of simplicial complexes in the virtual layer or introductory partial mapping relationship between layers can inhibit the spread of epidemics. Current results are conducive to understanding the coupling behaviors between epidemics and disease-related information.
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Affiliation(s)
- Shuofan Zhang
- Tianjin Key Laboratory of Intelligence Computing and Novel Software Technology, Tianjin University of Technology, Tianjin 300384, China
| | - Dawei Zhao
- Shandong Provincial Key Laboratory of Computer Networks, Shandong Computer Science Center (National Supercomputer Center in Jinan), Qilu University of Technology (Shandong Academy of Sciences), Jinan 250014, China
| | - Chengyi Xia
- School of Artificial Intelligence, Tiangong University, Tianjin 300387, China
| | - Jun Tanimoto
- Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Kasuga-koen, Kasuga-shi, Fukuoka 816-8580, Japan
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Yu P, Wang Z, Sun Y, Wang P. Supply Chain Risk Diffusion in Partially Mapping Double-Layer Hypernetworks. ENTROPY (BASEL, SWITZERLAND) 2023; 25:e25050747. [PMID: 37238502 DOI: 10.3390/e25050747] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/08/2023] [Revised: 04/19/2023] [Accepted: 04/28/2023] [Indexed: 05/28/2023]
Abstract
The impact of COVID-19 is global, and uncertain information will affect product quality and worker efficiency in the complex supply chain network, thus bringing risks. Aiming at individual heterogeneity, a partial mapping double-layer hypernetwork model is constructed to study the supply chain risk diffusion under uncertain information. Here, we explore the risk diffusion dynamics, drawing on epidemiology, and establish an SPIR (Susceptible-Potential-Infected-Recovered) model to simulate the risk diffusion process. The node represents the enterprise, and hyperedge represents the cooperation among enterprises. The microscopic Markov chain approach (MMCA) is used to prove the theory. Network dynamic evolution includes two removal strategies: (i) removing aging nodes; (ii) removing key nodes. Using Matlab to simulate the model, we found that it is more conducive to market stability to eliminate outdated enterprises than to control key enterprises during risk diffusion. The risk diffusion scale is related to interlayer mapping. Increasing the upper layer mapping rate to strengthen the efforts of official media to issue authoritative information will reduce the infected enterprise number. Reducing the lower layer mapping rate will reduce the misled enterprise number, thereby weakening the efficiency of risk infection. The model is helpful for understanding the risk diffusion characteristics and the importance of online information, and it has guiding significance for supply chain management.
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Affiliation(s)
- Ping Yu
- School of Science, Dalian Maritime University, Dalian 116026, China
| | - Zhiping Wang
- School of Science, Dalian Maritime University, Dalian 116026, China
| | - Ya'nan Sun
- School of Science, Dalian Maritime University, Dalian 116026, China
| | - Peiwen Wang
- School of Maritime Economics and Management, Dalian Maritime University, Dalian 116026, China
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7
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Masoomy H, Chou T, Böttcher L. Impact of random and targeted disruptions on information diffusion during outbreaks. CHAOS (WOODBURY, N.Y.) 2023; 33:033145. [PMID: 37003816 DOI: 10.1063/5.0139844] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/23/2022] [Accepted: 03/02/2023] [Indexed: 06/19/2023]
Abstract
Outbreaks are complex multi-scale processes that are impacted not only by cellular dynamics and the ability of pathogens to effectively reproduce and spread, but also by population-level dynamics and the effectiveness of mitigation measures. A timely exchange of information related to the spread of novel pathogens, stay-at-home orders, and other measures can be effective at containing an infectious disease, particularly during the early stages when testing infrastructure, vaccines, and other medical interventions may not be available at scale. Using a multiplex epidemic model that consists of an information layer (modeling information exchange between individuals) and a spatially embedded epidemic layer (representing a human contact network), we study how random and targeted disruptions in the information layer (e.g., errors and intentional attacks on communication infrastructure) impact the total proportion of infections, peak prevalence (i.e., the maximum proportion of infections), and the time to reach peak prevalence. We calibrate our model to the early outbreak stages of the SARS-CoV-2 pandemic in 2020. Mitigation campaigns can still be effective under random disruptions, such as failure of information channels between a few individuals. However, targeted disruptions or sabotage of hub nodes that exchange information with a large number of individuals can abruptly change outbreak characteristics, such as the time to reach the peak of infection. Our results emphasize the importance of the availability of a robust communication infrastructure during an outbreak that can withstand both random and targeted disruptions.
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Affiliation(s)
- Hosein Masoomy
- Department of Physics, Shahid Beheshti University, 1983969411 Tehran, Iran
| | - Tom Chou
- Department of Computational Medicine and Department of Mathematics, UCLA, Los Angeles, California 90095, USA
| | - Lucas Böttcher
- Department of Computational Science and Philosophy, Frankfurt School of Finance and Management, 60322 Frankfurt am Main, Germany
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Xu H, Xie W, Han D. A coupled awareness-epidemic model on a multi-layer time-varying network. CHAOS (WOODBURY, N.Y.) 2023; 33:013110. [PMID: 36725628 DOI: 10.1063/5.0125969] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/14/2022] [Accepted: 12/12/2022] [Indexed: 06/18/2023]
Abstract
Social interactions have become more complicated and changeable under the influence of information technology revolution. We, thereby, propose a multi-layer activity-driven network with attractiveness considering the heterogeneity of activated individual edge numbers, which aims to explore the role of heterogeneous behaviors in the time-varying network. Specifically, three types of individual behaviors are introduced: (i) self-quarantine of infected individuals, (ii) safe social distancing between infected and susceptible individuals, and (iii) information spreading of aware individuals. Epidemic threshold is theoretically derived in terms of the microscopic Markov chain approach and the mean-field approach. The results demonstrate that performing self-quarantine and maintaining safe social distance can effectively raise the epidemic threshold and suppress the spread of diseases. Interestingly, individuals' activity and individuals' attractiveness have an equivalent effect on epidemic threshold under the same condition. In addition, a similar result can be obtained regardless of the activated individual edge numbers. The epidemic outbreak earlier in a situation of the stronger heterogeneity of activated individual edge numbers.
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Affiliation(s)
- Haidong Xu
- School of Mathematical Sciences, Jiangsu University, Zhenjiang, Jiangsu 212013, China
| | - Weijie Xie
- School of Management, Zhenjiang, Jiangsu 212013, China
| | - Dun Han
- School of Mathematical Sciences, Jiangsu University, Zhenjiang, Jiangsu 212013, China
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Fan J, Zhao D, Xia C, Tanimoto J. Coupled spreading between information and epidemics on multiplex networks with simplicial complexes. CHAOS (WOODBURY, N.Y.) 2022; 32:113115. [PMID: 36456318 DOI: 10.1063/5.0125873] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/14/2022] [Accepted: 10/10/2022] [Indexed: 06/17/2023]
Abstract
The way of information diffusion among individuals can be quite complicated, and it is not only limited to one type of communication, but also impacted by multiple channels. Meanwhile, it is easier for an agent to accept an idea once the proportion of their friends who take it goes beyond a specific threshold. Furthermore, in social networks, some higher-order structures, such as simplicial complexes and hypergraph, can describe more abundant and realistic phenomena. Therefore, based on the classical multiplex network model coupling the infectious disease with its relevant information, we propose a novel epidemic model, in which the lower layer represents the physical contact network depicting the epidemic dissemination, while the upper layer stands for the online social network picturing the diffusion of information. In particular, the upper layer is generated by random simplicial complexes, among which the herd-like threshold model is adopted to characterize the information diffusion, and the unaware-aware-unaware model is also considered simultaneously. Using the microscopic Markov chain approach, we analyze the epidemic threshold of the proposed epidemic model and further check the results with numerous Monte Carlo simulations. It is discovered that the threshold model based on the random simplicial complexes network may still cause abrupt transitions on the epidemic threshold. It is also found that simplicial complexes may greatly influence the epidemic size at a steady state.
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Affiliation(s)
- Junfeng Fan
- Tianjin Key Laboratory of Intelligence Computing and Novel Software Technology, Tianjin University of Technology, Tianjin 300384, China
| | - Dawei Zhao
- Shandong Provincial Key Laboratory of Computer Networks, Shandong Computer Science Center (National Supercomputer Center in Jinan), Qilu University of Technology (Shandong Academy of Sciences), Jinan 250014, China
| | - Chengyi Xia
- School of Control Science and Engineering, Tiangong University, Tianjin 300387, China
| | - Jun Tanimoto
- Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Kasuga-koen, Kasuga-shi, Fukuoka 816-8580, Japan
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Co-evolution dynamics of epidemic and information under dynamical multi-source information and behavioral responses. Knowl Based Syst 2022. [DOI: 10.1016/j.knosys.2022.109413] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
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Nakagiri N, Sato K, Sakisaka Y, Tainaka KI. Serious role of non-quarantined COVID-19 patients for random walk simulations. Sci Rep 2022; 12:738. [PMID: 35031645 PMCID: PMC8760292 DOI: 10.1038/s41598-021-04629-2] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/10/2021] [Accepted: 12/21/2021] [Indexed: 02/07/2023] Open
Abstract
The infectious disease (COVID-19) causes serious damages and outbreaks. A large number of infected people have been reported in the world. However, such a number only represents those who have been tested; e.g. PCR test. We focus on the infected individuals who are not checked by inspections. The susceptible-infected-recovered (SIR) model is modified: infected people are divided into quarantined (Q) and non-quarantined (N) agents. Since N-agents behave like uninfected people, they can move around in a stochastic simulation. Both theory of well-mixed population and simulation of random-walk reveal that the total population size of Q-agents decrease in spite of increasing the number of tests. Such a paradox appears, when the ratio of Q exceeds a critical value. Random-walk simulations indicate that the infection hardly spreads, if the movement of all people is prohibited ("lockdown"). In this case the infected people are clustered and locally distributed within narrow spots. The similar result can be obtained, even when only non-infected people move around. However, when both N-agents and uninfected people move around, the infection spreads everywhere. Hence, it may be important to promote the inspections even for asymptomatic people, because most of N-agents are mild or asymptomatic.
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Affiliation(s)
- Nariyuki Nakagiri
- School of Human Science and Environment, University of Hyogo, Himeji, 670-0092, Japan
| | - Kazunori Sato
- Department of Mathematical and Systems Engineering, Shizuoka University, Hamamatsu, 432-8561, Japan
| | - Yukio Sakisaka
- Institute of Preventive and Medicinal Dietetics, Nakamura Gakuen University, Fukuoka, 814-0198, Japan
- Division of Early Childhood Care and Education, Nakamura Gakuen University Junior College, Fukuoka, 814-0198, Japan
| | - Kei-Ichi Tainaka
- Department of Mathematical and Systems Engineering, Shizuoka University, Hamamatsu, 432-8561, Japan.
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