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Wang W, Chen G, Wong EWM. Delay-driven phase transitions in an epidemic model on time-varying networks. CHAOS (WOODBURY, N.Y.) 2024; 34:043146. [PMID: 38639346 DOI: 10.1063/5.0179068] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/29/2023] [Accepted: 03/29/2024] [Indexed: 04/20/2024]
Abstract
A complex networked system typically has a time-varying nature in interactions among its components, which is intrinsically complicated and therefore technically challenging for analysis and control. This paper investigates an epidemic process on a time-varying network with a time delay. First, an averaging theorem is established to approximate the delayed time-varying system using autonomous differential equations for the analysis of system evolution. On this basis, the critical time delay is determined, across which the endemic equilibrium becomes unstable and a phase transition to oscillation in time via Hopf bifurcation will appear. Then, numerical examples are examined, including a periodically time-varying network, a blinking network, and a quasi-periodically time-varying network, which are simulated to verify the theoretical results. Further, it is demonstrated that the existence of time delay can extend the network frequency range to generate Turing patterns, showing a facilitating effect on phase transitions.
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Affiliation(s)
- Wen Wang
- School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China
| | - Guanrong Chen
- Department of Electrical Engineering, City University of Hong Kong, Hong Kong SAR, China
| | - Eric W M Wong
- Department of Electrical Engineering, City University of Hong Kong, Hong Kong SAR, China
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2
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Gao S, Chang L, Perc M, Wang Z. Turing patterns in simplicial complexes. Phys Rev E 2023; 107:014216. [PMID: 36797896 DOI: 10.1103/physreve.107.014216] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/18/2022] [Accepted: 12/06/2022] [Indexed: 02/18/2023]
Abstract
The spontaneous emergence of patterns in nature, such as stripes and spots, can be mathematically explained by reaction-diffusion systems. These patterns are often referred as Turing patterns to honor the seminal work of Alan Turing in the early 1950s. With the coming of age of network science, and with its related departure from diffusive nearest-neighbor interactions to long-range links between nodes, additional layers of complexity behind pattern formation have been discovered, including irregular spatiotemporal patterns. Here we investigate the formation of Turing patterns in simplicial complexes, where links no longer connect just pairs of nodes but can connect three or more nodes. Such higher-order interactions are emerging as a new frontier in network science, in particular describing group interaction in various sociological and biological systems, so understanding pattern formation under these conditions is of the utmost importance. We show that a canonical reaction-diffusion system defined over a simplicial complex yields Turing patterns that fundamentally differ from patterns observed in traditional networks. For example, we observe a stable distribution of Turing patterns where the fraction of nodes with reactant concentrations above the equilibrium point is exponentially related to the average degree of 2-simplexes, and we uncover parameter regions where Turing patterns will emerge only under higher-order interactions, but not under pairwise interactions.
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Affiliation(s)
- Shupeng Gao
- School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an 710072, China.,School of Artificial Intelligence, Optics, and Electronics (iOPEN), Northwestern Polytechnical University, Xi'an 710072, China
| | - Lili Chang
- Complex Systems Research Center, Shanxi University, Taiyuan 030006, China.,Shanxi Key Laboratory of Mathematical Techniques and Big Data Analysis for Disease Control and Prevention, Taiyuan 030006, China
| | - Matjaž Perc
- Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, 2000 Maribor, Slovenia.,Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 404332, Taiwan.,Alma Mater Europaea, Slovenska ulica 17, 2000 Maribor, Slovenia.,Complexity Science Hub Vienna, Josefstädterstraße 39, 1080 Vienna, Austria.,Department of Physics, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul, Republic of Korea
| | - Zhen Wang
- School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an 710072, China.,School of Artificial Intelligence, Optics, and Electronics (iOPEN), Northwestern Polytechnical University, Xi'an 710072, China
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3
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Falcó C. From random walks on networks to nonlinear diffusion. Phys Rev E 2022; 106:054103. [PMID: 36559369 DOI: 10.1103/physreve.106.054103] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/20/2022] [Accepted: 10/12/2022] [Indexed: 06/17/2023]
Abstract
Mathematical models of motility are often based on random-walk descriptions of discrete individuals that can move according to certain rules. It is usually the case that large masses concentrated in small regions of space have a great impact on the collective movement of the group. For this reason, many models in mathematical biology have incorporated crowding effects and managed to understand their implications. Here, we build on a previously developed framework for random walks on networks to show that in the continuum limit, the underlying stochastic process can be identified with a diffusion partial differential equation. The diffusion coefficient of the emerging equation is, in general, density dependent, and can be directly related to the transition probabilities of the random walk. Moreover, the relaxation time of the stochastic process is directly linked to the diffusion coefficient and also to the network structure, as it usually happens in the case of linear diffusion. As a specific example, we study the equivalent of a porous-medium-type equation on networks, which shows similar properties to its continuum equivalent. For this equation, self-similar solutions on a lattice and on homogeneous trees can be found, showing finite speed of propagation in contrast to commonly used linear diffusion equations. These findings also provide insights into reaction-diffusion systems with general diffusion operators, which have appeared recently in some applications.
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Affiliation(s)
- Carles Falcó
- Mathematical Institute, University of Oxford, OX2 6GG Oxford, United Kingdom
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Liu C, Gao S, Song M, Bai Y, Chang L, Wang Z. Optimal control of the reaction-diffusion process on directed networks. CHAOS (WOODBURY, N.Y.) 2022; 32:063115. [PMID: 35778117 DOI: 10.1063/5.0087855] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/10/2022] [Accepted: 05/16/2022] [Indexed: 06/15/2023]
Abstract
Reaction-diffusion processes organized in networks have attracted much interest in recent years due to their applications across a wide range of disciplines. As one type of most studied solutions of reaction-diffusion systems, patterns broadly exist and are observed from nature to human society. So far, the theory of pattern formation has made significant advances, among which a novel class of instability, presented as wave patterns, has been found in directed networks. Such wave patterns have been proved fruitful but significantly affected by the underlying network topology, and even small topological perturbations can destroy the patterns. Therefore, methods that can eliminate the influence of network topology changes on wave patterns are needed but remain uncharted. Here, we propose an optimal control framework to steer the system generating target wave patterns regardless of the topological disturbances. Taking the Brusselator model, a widely investigated reaction-diffusion model, as an example, numerical experiments demonstrate our framework's effectiveness and robustness. Moreover, our framework is generally applicable, with minor adjustments, to other systems that differential equations can depict.
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Affiliation(s)
- Chen Liu
- School of Ecology and Environment, Northwestern Polytechnical University, Xi'an 710072, China
| | - Shupeng Gao
- School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an 710072, China
| | - Mingrui Song
- School of Ecology and Environment, Northwestern Polytechnical University, Xi'an 710072, China
| | - Yue Bai
- School of Ecology and Environment, Northwestern Polytechnical University, Xi'an 710072, China
| | - Lili Chang
- Complex Systems Research Center, Shanxi University, Taiyuan 030006, China
| | - Zhen Wang
- School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an 710072, China
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Gao S, Chang L, Romić I, Wang Z, Jusup M, Holme P. Optimal control of networked reaction-diffusion systems. J R Soc Interface 2022; 19:20210739. [PMID: 35259961 PMCID: PMC8905157 DOI: 10.1098/rsif.2021.0739] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/16/2022] Open
Abstract
Patterns in nature are fascinating both aesthetically and scientifically. Alan Turing's celebrated reaction-diffusion model of pattern formation from the 1950s has been extended to an astounding diversity of applications: from cancer medicine, via nanoparticle fabrication, to computer architecture. Recently, several authors have studied pattern formation in underlying networks, but thus far, controlling a reaction-diffusion system in a network to obtain a particular pattern has remained elusive. We present a solution to this problem in the form of an analytical framework and numerical algorithm for optimal control of Turing patterns in networks. We demonstrate our method's effectiveness and discuss factors that affect its performance. We also pave the way for multidisciplinary applications of our framework beyond reaction-diffusion models.
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Affiliation(s)
- Shupeng Gao
- School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an 710072, People's Republic of China.,School of Artificial Intelligence, Optics, and Electronics (iOPEN), Northwestern Polytechnical University, Xi'an 710072, People's Republic of China
| | - Lili Chang
- Complex Systems Research Center, Shanxi University, Taiyuan 030006, People's Republic of China.,Shanxi Key Laboratory of Mathematical Techniques and Big Data Analysis for Disease Control and Prevention, Taiyuan 030006, People's Republic of China
| | - Ivan Romić
- School of Artificial Intelligence, Optics, and Electronics (iOPEN), Northwestern Polytechnical University, Xi'an 710072, People's Republic of China.,Statistics and Mathematics College, Yunnan University of Finance and Economics, Kunming 650221, People's Republic of China.,Graduate School of Economics, Osaka City University, Osaka 558-8585, Japan
| | - Zhen Wang
- School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an 710072, People's Republic of China.,School of Artificial Intelligence, Optics, and Electronics (iOPEN), Northwestern Polytechnical University, Xi'an 710072, People's Republic of China
| | - Marko Jusup
- Tokyo Tech World Hub Research Initiative (WRHI), Institute of Innovative Research, Tokyo Institute of Technology, Yokohama 152-8550, Japan
| | - Petter Holme
- Tokyo Tech World Hub Research Initiative (WRHI), Institute of Innovative Research, Tokyo Institute of Technology, Yokohama 152-8550, Japan
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Van Gorder RA. Pattern formation from spatially heterogeneous reaction-diffusion systems. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2021; 379:20210001. [PMID: 34743604 DOI: 10.1098/rsta.2021.0001] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/03/2023]
Abstract
First proposed by Turing in 1952, the eponymous Turing instability and Turing pattern remain key tools for the modern study of diffusion-driven pattern formation. In spatially homogeneous Turing systems, one or a few linear Turing modes dominate, resulting in organized patterns (peaks in one dimension; spots, stripes, labyrinths in two dimensions) which repeats in space. For a variety of reasons, there has been increasing interest in understanding irregular patterns, with spatial heterogeneity in the underlying reaction-diffusion system identified as one route to obtaining irregular patterns. We study pattern formation from reaction-diffusion systems which involve spatial heterogeneity, by way of both analytical and numerical techniques. We first extend the classical Turing instability analysis to track the evolution of linear Turing modes and the nascent pattern, resulting in a more general instability criterion which can be applied to spatially heterogeneous systems. We also calculate nonlinear mode coefficients, employing these to understand how each spatial mode influences the long-time evolution of a pattern. Unlike for the standard spatially homogeneous Turing systems, spatially heterogeneous systems may involve many Turing modes of different wavelengths interacting simultaneously, with resulting patterns exhibiting a high degree of variation over space. We provide a number of examples of spatial heterogeneity in reaction-diffusion systems, both mathematical (space-varying diffusion parameters and reaction kinetics, mixed boundary conditions, space-varying base states) and physical (curved anisotropic domains, apical growth of space domains, chemicalsimmersed within a flow or a thermal gradient), providing a qualitative understanding of how spatial heterogeneity can be used to modify classical Turing patterns. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.
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Affiliation(s)
- Robert A Van Gorder
- Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin 9054, New Zealand
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Krause AL, Gaffney EA, Maini PK, Klika V. Modern perspectives on near-equilibrium analysis of Turing systems. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2021; 379:20200268. [PMID: 34743603 PMCID: PMC8580451 DOI: 10.1098/rsta.2020.0268] [Citation(s) in RCA: 17] [Impact Index Per Article: 5.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 06/18/2021] [Indexed: 05/02/2023]
Abstract
In the nearly seven decades since the publication of Alan Turing's work on morphogenesis, enormous progress has been made in understanding both the mathematical and biological aspects of his proposed reaction-diffusion theory. Some of these developments were nascent in Turing's paper, and others have been due to new insights from modern mathematical techniques, advances in numerical simulations and extensive biological experiments. Despite such progress, there are still important gaps between theory and experiment, with many examples of biological patterning where the underlying mechanisms are still unclear. Here, we review modern developments in the mathematical theory pioneered by Turing, showing how his approach has been generalized to a range of settings beyond the classical two-species reaction-diffusion framework, including evolving and complex manifolds, systems heterogeneous in space and time, and more general reaction-transport equations. While substantial progress has been made in understanding these more complicated models, there are many remaining challenges that we highlight throughout. We focus on the mathematical theory, and in particular linear stability analysis of 'trivial' base states. We emphasize important open questions in developing this theory further, and discuss obstacles in using these techniques to understand biological reality. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.
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Affiliation(s)
- Andrew L. Krause
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
- Department of Mathematical Sciences, Durham University, Upper Mountjoy Campus, Stockton Rd, Durham DH1 3LE, UK
| | - Eamonn A. Gaffney
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
| | - Philip K. Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
| | - Václav Klika
- Department of Mathematics, FNSPE, Czech Technical University in Prague, Trojanova, 13, 12000 Praha, Czech Republic
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