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Singh AK, Ramakrishnan S, Kumar M. Instabilities and self-organization in spatiotemporal epidemic dynamics driven by nonlinearity and noise. Phys Biol 2024; 21:046001. [PMID: 38949432 DOI: 10.1088/1478-3975/ad5d6a] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/09/2024] [Accepted: 07/01/2024] [Indexed: 07/02/2024]
Abstract
Theoretical analysis of epidemic dynamics has attracted significant attention in the aftermath of the COVID-19 pandemic. In this article, we study dynamic instabilities in a spatiotemporal compartmental epidemic model represented by a stochastic system of coupled partial differential equations (SPDE). Saturation effects in infection spread-anchored in physical considerations-lead to strong nonlinearities in the SPDE. Our goal is to study the onset of dynamic, Turing-type instabilities, and the concomitant emergence of steady-state patterns under the interplay between three critical model parameters-the saturation parameter, the noise intensity, and the transmission rate. Employing a second-order perturbation analysis to investigate stability, we uncover both diffusion-driven and noise-induced instabilities and corresponding self-organized distinct patterns of infection spread in the steady state. We also analyze the effects of the saturation parameter and the transmission rate on the instabilities and the pattern formation. In summary, our results indicate that the nuanced interplay between the three parameters considered has a profound effect on the emergence of dynamical instabilities and therefore on pattern formation in the steady state. Moreover, due to the central role played by the Turing phenomenon in pattern formation in a variety of biological dynamic systems, the results are expected to have broader significance beyond epidemic dynamics.
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Affiliation(s)
- Aman Kumar Singh
- Department of Mechanical and Aerospace Engineering, University of Dayton, Dayton, OH 45469, United States of America
| | - Subramanian Ramakrishnan
- Department of Mechanical and Aerospace Engineering, University of Dayton, Dayton, OH 45469, United States of America
| | - Manish Kumar
- Department of Mechanical and Materials Engineering, University of Cincinnati, Cincinnati, OH, United States of America
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2
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Luo X, Sun G, He R, Jin Z, Asamoah JKK, Xue Y, Chang L. The relationship between clustering and networked Turing patterns. CHAOS (WOODBURY, N.Y.) 2024; 34:073114. [PMID: 38976800 DOI: 10.1063/5.0195450] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/02/2024] [Accepted: 06/19/2024] [Indexed: 07/10/2024]
Abstract
Networked Turing patterns often manifest as groups of nodes distributed on either side of the homogeneous equilibrium, exhibiting high and low density. These pattern formations are significantly influenced by network topological characteristics, such as the average degree. However, the impact of clustering on them remains inadequately understood. Here, we investigate the relationship between clustering and networked Turing patterns using classical prey-predator models. Our findings reveal that when nodes of high and low density are completely distributed on both sides of the homogeneous equilibrium, there is a linear decay in Turing patterns as global clustering coefficients increase, given a fixed node size and average degree; otherwise, this linear decay may not always hold due to the presence of high-density nodes considered as low-density nodes. This discovery provides a qualitative assessment of how clustering coefficients impact the formation of Turing patterns and may contribute to understanding why using refuges in ecosystems could enhance the stability of prey-predator systems. The results link network topological structures with the stability of prey-predator systems, offering new insights into predicting and controlling pattern formations in real-world systems from a network perspective.
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Affiliation(s)
- Xiaofeng Luo
- School of Mathematics, North University of China, Shanxi, Taiyuan 030051, China
| | - Guiquan Sun
- School of Mathematics, North University of China, Shanxi, Taiyuan 030051, China
- Complex Systems Research Center, Shanxi University, Shanxi, Taiyuan 030006, China
- Key Laboratory of Complex Systems and Data Science of Ministry of Education, Taiyuan 030006, China
| | - Runzi He
- School of Mathematics, North University of China, Shanxi, Taiyuan 030051, China
| | - Zhen Jin
- Complex Systems Research Center, Shanxi University, Shanxi, Taiyuan 030006, China
- Key Laboratory of Complex Systems and Data Science of Ministry of Education, Taiyuan 030006, China
| | - Joshua Kiddy K Asamoah
- Department of Mathematics, Saveetha School of Engineering SIMATS, Chennai 602105, India
- Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi 127901, Ghana
| | - Yakui Xue
- School of Mathematics, North University of China, Shanxi, Taiyuan 030051, China
| | - Lili Chang
- Complex Systems Research Center, Shanxi University, Shanxi, Taiyuan 030006, China
- Key Laboratory of Complex Systems and Data Science of Ministry of Education, Taiyuan 030006, China
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3
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Wang J, Zhao H, Wang H. The role of natural recovery category in malaria dynamics under saturated treatment. J Math Biol 2024; 88:33. [PMID: 38411718 DOI: 10.1007/s00285-024-02051-6] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/19/2023] [Revised: 12/29/2023] [Accepted: 01/19/2024] [Indexed: 02/28/2024]
Abstract
In the process of malaria transmission, natural recovery individuals are slightly infectious compared with infected individuals. Our concern is whether the infectivity of natural recovery category can be ignored in areas with limited medical resources, so as to reveal the epidemic pattern of malaria with simpler analysis. To achieve this, we incorporate saturated treatment into two-compartment and three-compartment models, and the infectivity of natural recovery category is only reflected in the latter. The non-spatial two-compartment model can admit backward bifurcation. Its spatial version does not possess rich dynamics. Besides, the non-spatial three-compartment model can undergo backward bifurcation, Hopf bifurcation and Bogdanov-Takens bifurcation. For spatial three-compartment model, due to the complexity of characteristic equation, we apply Shengjin's Distinguishing Means to realize stability analysis. Further, the model exhibits Turing instability, Hopf bifurcation and Turing-Hopf bifurcation. This makes the model may admit bistability or even tristability when its basic reproduction number less than one. Biologically, malaria may present a variety of epidemic trends, such as elimination or inhomogeneous distribution in space and periodic fluctuation in time of infectious populations. Notably, parameter regions are given to illustrate substitution effect of two-compartment model for three-compartment model in both scenarios without or with spatial movement. Finally, spatial three-compartment model is used to present malaria transmission in Burundi. The application of efficiency index enables us to determine the most effective method to control the number of cases in different scenarios.
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Affiliation(s)
- Jing Wang
- School of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, China
- Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA), MIIT, Nanjing, 211106, China
| | - Hongyong Zhao
- School of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, China.
- Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA), MIIT, Nanjing, 211106, China.
| | - Hao Wang
- Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada
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Zheng Q, Shen J, Pandey V, Yuan X, Guan L. Pattern selection mechanism from the equilibrium point and limit cycle. CHAOS (WOODBURY, N.Y.) 2024; 34:023124. [PMID: 38377290 DOI: 10.1063/5.0187810] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/16/2023] [Accepted: 01/20/2024] [Indexed: 02/22/2024]
Abstract
The outbreak of infectious diseases often exhibits periodicity, and this periodic behavior can be mathematically represented as a limit cycle. However, the periodic behavior has rarely been considered in demonstrating the cluster phenomenon of infection induced by diffusion (the instability modes) in the SIR model. We investigate the emergence of Turing instability from a stable equilibrium and a limit cycle to illustrate the dynamical and biological mechanisms of pattern formation. We identify the Hopf bifurcation to demonstrate the existence of a stable limit cycle using First Lyapunov coefficient in our spatiotemporal diffusion-driven SIR model. The competition between different instability modes induces different types of patterns and eventually spot patterns emerge as stable patterns. We investigate the impact of susceptible, infected, and recovered individuals on the type of patterns. Interestingly, these instability modes play a vital role in selecting the pattern formations, which is directly related to the number of observed spot patterns. Subsequently, we explain the dynamical and biological mechanisms of spot patterns to develop an effective epidemic prevention strategy.
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Affiliation(s)
- Qianqian Zheng
- School of Science, Xuchang University, Henan Joint International Research Laboratory of High Performance Computation for Complex Systems, Xuchang 461000, China
| | - Jianwei Shen
- School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450046, China
| | - Vikas Pandey
- NIPS, National Institute of Natural Science, Okazaki 4448585, Japan
| | - Xiaoping Yuan
- Information Engineering School, Hangzhou Dianzi University, Hangzhou 310018, China
| | - Linan Guan
- School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450046, China
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5
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Acharya S, Kumar Upadhyay R, Mondal B. Exploring the complex dynamics of a diffusive epidemic model: Stability and bifurcation analysis. CHAOS (WOODBURY, N.Y.) 2024; 34:023115. [PMID: 38363954 DOI: 10.1063/5.0159015] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/19/2023] [Accepted: 01/11/2024] [Indexed: 02/18/2024]
Abstract
The recent pandemic has highlighted the need to understand how we resist infections and their causes, which may differ from the ways we often think about treating epidemic diseases. The current study presents an improved version of the susceptible-infected-recovered (SIR) epidemic model, to better comprehend the community's overall dynamics of diseases, involving numerous infectious agents. The model deals with a non-monotone incidence rate that exhibits psychological or inhibitory influence and a saturation treatment rate. It has been identified that depending on the measure of medical resources and the effectiveness of their supply, the model exposes both forward and backward bifurcations where two endemic equilibria coexist with infection-free equilibrium. The model also experiences local and global bifurcations of codimension two, including saddle-node, Hopf, and Bogdanov-Takens bifurcations. Additionally, the stability of equilibrium points is investigated. For a spatially extended SIR model system, we have shown that cross-diffusion allows S and I populations to coexist in a habitat. Also, the Turing instability requirements and Turing bifurcation regime are derived. The relationship between distinct role-playing model parameters and various pattern formations like spot and stripe patterns is validated by carrying out in-depth numerical simulations. The findings in the vicinity of the endemic equilibrium solution demonstrate the significance of positive and negative valued cross-diffusion coefficients in regulating the genesis of spatial patterns in susceptible as well as diseased individuals. The discussion of the findings of epidemiological ramifications concludes the manuscript.
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Affiliation(s)
- Sattwika Acharya
- Department of Mathematics and Computing, Indian Institute of Technology (Indian School of Mines), Dhanbad 826004, India
| | - Ranjit Kumar Upadhyay
- Department of Mathematics and Computing, Indian Institute of Technology (Indian School of Mines), Dhanbad 826004, India
| | - Bapin Mondal
- Department of Applied Mathematics, University of Calcutta, Kolkata 700009, India
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Chang L, Wang X, Sun G, Wang Z, Jin Z. A time independent least squares algorithm for parameter identification of Turing patterns in reaction-diffusion systems. J Math Biol 2023; 88:5. [PMID: 38017080 DOI: 10.1007/s00285-023-02026-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/12/2023] [Revised: 09/28/2023] [Accepted: 10/29/2023] [Indexed: 11/30/2023]
Abstract
Turing patterns arising from reaction-diffusion systems such as epidemic, ecology or chemical reaction models are an important dynamic property. Parameter identification of Turing patterns in spatial continuous and networked reaction-diffusion systems is an interesting and challenging inverse problem. The existing algorithms require huge account operations and resources. These drawbacks are amplified when apply them to reaction-diffusion systems on large-scale complex networks. To overcome these shortcomings, we present a new least squares algorithm which is rooted in the fact that Turing patterns are the stationary solutions of reaction-diffusion systems. The new algorithm is time independent, it translates the parameter identification problem into a low dimensional optimization problem even a low order linear algebra equations. The numerical simulations demonstrate that our algorithm has good effectiveness, robustness as well as performance.
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Affiliation(s)
- Lili Chang
- Complex Systems Research Center, Shanxi University, Taiyuan, 030006, China.
- Key Laboratory of Complex Systems and Data Science of Ministry of Education, Taiyuan, 030006, China.
| | - Xinyu 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
| | - Guiquan Sun
- Key Laboratory of Complex Systems and Data Science of Ministry of Education, Taiyuan, 030006, China.
- Department of Mathematics, North University of China, Taiyuan, 030051, China.
| | - 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
| | - Zhen Jin
- Complex Systems Research Center, Shanxi University, Taiyuan, 030006, China
- Key Laboratory of Complex Systems and Data Science of Ministry of Education, Taiyuan, 030006, China
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7
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Wilmott ZM, Goriely A, Raff JW. A simple Turing reaction-diffusion model explains how PLK4 breaks symmetry during centriole duplication and assembly. PLoS Biol 2023; 21:e3002391. [PMID: 37983248 PMCID: PMC10659181 DOI: 10.1371/journal.pbio.3002391] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/19/2023] [Accepted: 10/18/2023] [Indexed: 11/22/2023] Open
Abstract
Centrioles duplicate when a mother centriole gives birth to a daughter that grows from its side. Polo-like-kinase 4 (PLK4), the master regulator of centriole duplication, is recruited symmetrically around the mother centriole, but it then concentrates at a single focus that defines the daughter centriole assembly site. How PLK4 breaks symmetry is unclear. Here, we propose that phosphorylated and unphosphorylated species of PLK4 form the 2 components of a classical Turing reaction-diffusion system. These 2 components bind to/unbind from the surface of the mother centriole at different rates, allowing a slow-diffusing activator species of PLK4 to accumulate at a single site on the mother, while a fast-diffusing inhibitor species of PLK4 suppresses activator accumulation around the rest of the centriole. This "short-range activation/long-range inhibition," inherent to Turing systems, can drive PLK4 symmetry breaking on a either a continuous or compartmentalised Plk4-binding surface, with PLK4 overexpression producing multiple PLK4 foci and PLK4 kinase inhibition leading to a lack of symmetry-breaking and PLK4 accumulation-as observed experimentally.
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Affiliation(s)
- Zachary M. Wilmott
- Sir William Dunn School of Pathology, University of Oxford, Oxford, United Kingdom
- Mathematical Institute, University of Oxford, Oxford, United Kingdom
| | - Alain Goriely
- Mathematical Institute, University of Oxford, Oxford, United Kingdom
| | - Jordan W. Raff
- Sir William Dunn School of Pathology, University of Oxford, Oxford, United Kingdom
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Chang L, Guo L, Liu C, Wang Z, Sun G. The qualitative and quantitative relationships between pattern formation and average degree in networked reaction-diffusion systems. CHAOS (WOODBURY, N.Y.) 2022; 32:093129. [PMID: 36182400 DOI: 10.1063/5.0107504] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/05/2022] [Accepted: 08/26/2022] [Indexed: 06/16/2023]
Abstract
The Turing pattern is an important dynamic behavior characteristic of activator-inhibitor systems. Differentiating from traditional assumption of activator-inhibitor interactions in a spatially continuous domain, a Turing pattern in networked reaction-diffusion systems has received much attention during the past few decades. In spite of its great progress, it still fails to evaluate the precise influences of network topology on pattern formation. To this end, we try to promote the research on this important and interesting issue from the point of view of average degree-a critical topological feature of networks. We first qualitatively analyze the influence of average degree on pattern formation. Then, a quantitative relationship between pattern formation and average degree, the exponential decay of pattern formation, is proposed via nonlinear regression. The finding holds true for several activator-inhibitor systems including biology model, ecology model, and chemistry model. The significance of this study lies that the exponential decay not only quantitatively depicts the influence of average degree on pattern formation, but also provides the possibility for predicting and controlling pattern formation.
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Affiliation(s)
- Lili Chang
- Complex Systems Research Center, Shanxi University, Taiyuan 030006, Shanxi, China
| | - Luyao Guo
- School of Mathematics, Southeast University, Nanjing 210096, China
| | - Chen Liu
- School of Ecology and Environment Science, Northwestern Polytechnical University, Xi'an 710072, Shaanxi, China
| | - Zhen Wang
- Center for Optical Imagery Analysis and Learning, Northwestern Polytechnical University, Xi'an 710072, Shaanxi, China
| | - Guiquan Sun
- Department of Mathematics, North University of China, Taiyuan 030051, Shanxi, China
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9
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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] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/18/2021] [Accepted: 02/08/2022] [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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Chang L, Gao S, Wang Z. Optimal control of pattern formations for an SIR reaction-diffusion epidemic model. J Theor Biol 2022; 536:111003. [PMID: 35026213 DOI: 10.1016/j.jtbi.2022.111003] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/19/2021] [Revised: 12/29/2021] [Accepted: 12/31/2021] [Indexed: 11/19/2022]
Abstract
Patterns arising from the reaction-diffusion epidemic model provide insightful aspects into the transmission of infectious diseases. For a classic SIR reaction-diffusion epidemic model, we review its Turing pattern formations with different transmission rates. A quantitative indicator, "normal serious prevalent area (NSPA)", is introduced to characterize the relationship between patterns and the extent of the epidemic. The extent of epidemic is positively correlated to NSPA. To effectively reduce NSPA of patterns under the large transmission rates, taken removed (recovery or isolation) rate as a control parameter, we consider the mathematical formulation and numerical solution of an optimal control problem for the SIR reaction-diffusion model. Numerical experiments demonstrate the effectiveness of our method in terms of control effect, control precision and control cost.
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Affiliation(s)
- Lili Chang
- Complex Systems Research Center, Shanxi University, Taiyuan 030006, Shanxi, China.
| | - Shupeng Gao
- School of Mechanical Engineering Northwestern Polytechnical University, Xi'an 710072, Shaanxi, China; School of Artificial Intelligence, Optics, and Electronics (iOPEN), Northwestern Polytechnical University, Xi'an 710072, Shaanxi, China
| | - Zhen Wang
- School of Mechanical Engineering Northwestern Polytechnical University, Xi'an 710072, Shaanxi, China; School of Artificial Intelligence, Optics, and Electronics (iOPEN), Northwestern Polytechnical University, Xi'an 710072, Shaanxi, China.
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11
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Li Y, Wei Z. Dynamics and optimal control of a stochastic coronavirus (COVID-19) epidemic model with diffusion. NONLINEAR DYNAMICS 2022; 109:91-120. [PMID: 34803222 PMCID: PMC8595080 DOI: 10.1007/s11071-021-06998-9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/01/2021] [Accepted: 10/14/2021] [Indexed: 05/05/2023]
Abstract
In view of the facts in the infection and propagation of COVID-19, a stochastic reaction-diffusion epidemic model is presented to analyse and control this infectious diseases. Stationary distribution and Turing instability of this model are discussed for deriving the sufficient criteria for the persistence and extinction of disease. Furthermore, the amplitude equations are derived by using Taylor series expansion and weakly nonlinear analysis, and selection of Turing patterns for this model can be determined. In addition, the optimal quarantine control problem for reducing control cost is studied, and the differences between the two models are compared. By applying the optimal control theory, the existence and uniqueness of the optimal control and the optimal solution are obtained. Finally, these results are verified and illustrated by numerical simulation.
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Affiliation(s)
- Yuxi Li
- School of Mathematics and Physics, China University of Geosciences, Wuhan, 430074 China
| | - Zhouchao Wei
- School of Mathematics and Physics, China University of Geosciences, Wuhan, 430074 China
- School of Mathematics and Physics, China University of Geosciences, Wuhan, 430074 China
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12
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Tiomela SA, Macías-Díaz JE, Mvogo A. Computer simulation of the dynamics of a spatial susceptible-infected-recovered epidemic model with time delays in transmission and treatment. COMPUTER METHODS AND PROGRAMS IN BIOMEDICINE 2021; 212:106469. [PMID: 34715516 DOI: 10.1016/j.cmpb.2021.106469] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/04/2021] [Accepted: 10/08/2021] [Indexed: 06/13/2023]
Abstract
BACKGROUND AND OBJECTIVE In this work, we analyze the spatial-temporal dynamics of a susceptible-infected-recovered (SIR) epidemic model with time delays. To better describe the dynamical behavior of the model, we take into account the cumulative effects of diffusion in the population dynamics, and the time delays in both the Holling type II treatment and the disease transmission process, respectively. METHODS We perform linear stability analyses on the disease-free and endemic equilibria. We provide the expression of the basic reproduction number and set conditions on the backward bifurcation using Castillo's theorem. The values of the critical time transmission, the treatment delays and the relationship between them are established. RESULTS We show that the treatment rate decreases the basic reproduction number while the transmission rate significantly affects the bifurcation process in the system. The transmission and treatment time-delays are found to be inversely proportional to the susceptible and infected diffusion rates. The analytical results are numerically tested. The results show that the treatment rate significantly reduces the density of infected population and ensures the transition from the unstable to the stable domain. Moreover, the system is more sensible to the treatment in the stable domain. CONCLUSIONS The density of infected population increases with respect to the infected and susceptible diffusion rates. Both effects of treatment and transmission delays significantly affect the behavior of the system. The transmission time-delay at the critical point ensures the transition from the stable (low density) to the unstable (high density) domain.
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Affiliation(s)
- Sedrique A Tiomela
- Laboratory of Biophysics, Department of Physics, Faculty of Science, University of Yaounde I, Yaoundé P.O. Box 812, Cameroon.
| | - J E Macías-Díaz
- Department of Mathematics, School of Digital Technologies, Tallinn University, Narva Rd. 25, Tallinn 10120, Estonia; Departamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Avenida Universidad 940, Ciudad Universitaria, Aguascalientes 20131, Mexico.
| | - Alain Mvogo
- Laboratory of Biophysics, Department of Physics, Faculty of Science, University of Yaounde I, Yaoundé P.O. Box 812, Cameroon.
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Chu A, Huber G, McGeever A, Veytsman B, Yllanes D. A random-walk-based epidemiological model. Sci Rep 2021; 11:19308. [PMID: 34588487 PMCID: PMC8481482 DOI: 10.1038/s41598-021-98211-5] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/03/2020] [Accepted: 08/30/2021] [Indexed: 12/23/2022] Open
Abstract
Random walkers on a two-dimensional square lattice are used to explore the spatio-temporal growth of an epidemic. We have found that a simple random-walk system generates non-trivial dynamics compared with traditional well-mixed models. Phase diagrams characterizing the long-term behaviors of the epidemics are calculated numerically. The functional dependence of the basic reproductive number [Formula: see text] on the model's defining parameters reveals the role of spatial fluctuations and leads to a novel expression for [Formula: see text]. Special attention is given to simulations of inter-regional transmission of the contagion. The scaling of the epidemic with respect to space and time scales is studied in detail in the critical region, which is shown to be compatible with the directed-percolation universality class.
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Affiliation(s)
- Andrew Chu
- Chan Zuckerberg Biohub, 499 Illinois Street, San Francisco, CA, 94158, USA
| | - Greg Huber
- Chan Zuckerberg Biohub, 499 Illinois Street, San Francisco, CA, 94158, USA
| | - Aaron McGeever
- Chan Zuckerberg Biohub, 499 Illinois Street, San Francisco, CA, 94158, USA
| | - Boris Veytsman
- Chan Zuckerberg Initiative, 601 Marshall Street, Redwood City, CA, 94063, USA
- School of Systems Biology, George Mason University, Fairfax, VA, USA
| | - David Yllanes
- Chan Zuckerberg Biohub, 499 Illinois Street, San Francisco, CA, 94158, USA.
- Instituto de Biocomputación y Física de Sistemas Complejos (BIFI), 50018, Zaragoza, Spain.
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14
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Bertaglia G, Boscheri W, Dimarco G, Pareschi L. Spatial spread of COVID-19 outbreak in Italy using multiscale kinetic transport equations with uncertainty. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2021; 18:7028-7059. [PMID: 34517570 DOI: 10.3934/mbe.2021350] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/13/2023]
Abstract
In this paper we introduce a space-dependent multiscale model to describe the spatial spread of an infectious disease under uncertain data with particular interest in simulating the onset of the COVID-19 epidemic in Italy. While virus transmission is ruled by a SEIAR type compartmental model, within our approach the population is given by a sum of commuters moving on a extra-urban scale and non commuters interacting only on the smaller urban scale. A transport dynamics of the commuter population at large spatial scales, based on kinetic equations, is coupled with a diffusion model for non commuters at the urban scale. Thanks to a suitable scaling limit, the kinetic transport model used to describe the dynamics of commuters, within a given urban area coincides with the diffusion equations that characterize the movement of non-commuting individuals. Because of the high uncertainty in the data reported in the early phase of the epidemic, the presence of random inputs in both the initial data and the epidemic parameters is included in the model. A robust numerical method is designed to deal with the presence of multiple scales and the uncertainty quantification process. In our simulations, we considered a realistic geographical domain, describing the Lombardy region, in which the size of the cities, the number of infected individuals, the average number of daily commuters moving from one city to another, and the epidemic aspects are taken into account through a calibration of the model parameters based on the actual available data. The results show that the model is able to describe correctly the main features of the spatial expansion of the first wave of COVID-19 in northern Italy.
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Affiliation(s)
- Giulia Bertaglia
- Department of Mathematics and Computer Science, University of Ferrara, Via Machiavelli 30, Ferrara 44121, Italy
- Center for Modeling, Computing and Statistic (CMCS), University of Ferrara, Via Muratori 9, Ferrara 44121, Italy
| | - Walter Boscheri
- Department of Mathematics and Computer Science, University of Ferrara, Via Machiavelli 30, Ferrara 44121, Italy
- Center for Modeling, Computing and Statistic (CMCS), University of Ferrara, Via Muratori 9, Ferrara 44121, Italy
| | - Giacomo Dimarco
- Department of Mathematics and Computer Science, University of Ferrara, Via Machiavelli 30, Ferrara 44121, Italy
- Center for Modeling, Computing and Statistic (CMCS), University of Ferrara, Via Muratori 9, Ferrara 44121, Italy
| | - Lorenzo Pareschi
- Department of Mathematics and Computer Science, University of Ferrara, Via Machiavelli 30, Ferrara 44121, Italy
- Center for Modeling, Computing and Statistic (CMCS), University of Ferrara, Via Muratori 9, Ferrara 44121, Italy
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15
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Siebert BA, Hall CL, Gleeson JP, Asllani M. Role of modularity in self-organization dynamics in biological networks. Phys Rev E 2020; 102:052306. [PMID: 33327105 DOI: 10.1103/physreve.102.052306] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/26/2020] [Accepted: 10/13/2020] [Indexed: 12/11/2022]
Abstract
Interconnected ensembles of biological entities are perhaps some of the most complex systems that modern science has encountered so far. In particular, scientists have concentrated on understanding how the complexity of the interacting structure between different neurons, proteins, or species influences the functioning of their respective systems. It is well established that many biological networks are constructed in a highly hierarchical way with two main properties: short average paths that join two apparently distant nodes (neuronal, species, or protein patches) and a high proportion of nodes in modular aggregations. Although several hypotheses have been proposed so far, still little is known about the relation of the modules with the dynamical activity in such biological systems. Here we show that network modularity is a key ingredient for the formation of self-organizing patterns of functional activity, independently of the topological peculiarities of the structure of the modules. In particular, we propose a self-organizing mechanism which explains the formation of macroscopic spatial patterns, which are homogeneous within modules. This may explain how spontaneous order in biological networks follows their modular structural organization. We test our results on real-world networks to confirm the important role of modularity in creating macroscale patterns.
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Affiliation(s)
- Bram A Siebert
- MACSI, Department of Mathematics and Statistics, University of Limerick, Limerick V94 T9PX, Ireland
| | - Cameron L Hall
- MACSI, Department of Mathematics and Statistics, University of Limerick, Limerick V94 T9PX, Ireland
- Department of Engineering Mathematics, University of Bristol, Woodland Road, Clifton BS8 1UB, United Kingdom
| | - James P Gleeson
- MACSI, Department of Mathematics and Statistics, University of Limerick, Limerick V94 T9PX, Ireland
| | - Malbor Asllani
- MACSI, Department of Mathematics and Statistics, University of Limerick, Limerick V94 T9PX, Ireland
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16
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Gai C, Iron D, Kolokolnikov T. Localized outbreaks in an S-I-R model with diffusion. J Math Biol 2020; 80:1389-1411. [PMID: 31950259 DOI: 10.1007/s00285-020-01466-1] [Citation(s) in RCA: 12] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/15/2019] [Revised: 12/31/2019] [Indexed: 11/25/2022]
Abstract
We investigate an SIRS epidemic model with spatial diffusion and nonlinear incidence rates. We show that for small diffusion rate of the infected class [Formula: see text], the infected population tends to be highly localized at certain points inside the domain, forming K spikes. We then study three distinct destabilization mechanisms, as well as a transition from localized spikes to plateau solutions. Two of the instabilities are due to coarsening (spike death) and self-replication (spike birth), and have well-known analogues in other reaction-diffusion systems such as the Schnakenberg model. The third transition is when a single spike becomes unstable and moves to the boundary. This happens when the diffusion of the recovered class, [Formula: see text] becomes sufficiently small. In all cases, the stability thresholds are computed asymptotically and are verified by numerical experiments. We also show that the spike solution can transit into an plateau-type solution when the diffusion rates of recovered and susceptible class are sufficiently small. Implications for disease spread and control through quarantine are discussed.
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Affiliation(s)
- Chunyi Gai
- Department of Mathematics and Statistics, Dalhousie University, Halifax, Canada
| | - David Iron
- Department of Mathematics and Statistics, Dalhousie University, Halifax, Canada
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17
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Chacón-Acosta G, Núñez-López M, Pineda I. Turing instability conditions in confined systems with an effective position-dependent diffusion coefficient. J Chem Phys 2020; 152:024101. [DOI: 10.1063/1.5128510] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/16/2023] Open
Affiliation(s)
- G. Chacón-Acosta
- Applied Mathematics and Systems Department, Universidad Autónoma Metropolitana-Cuajimalpa, Vasco de Quiroga 4871, Ciudad de México 05348, Mexico
| | - M. Núñez-López
- Department of Mathematics, ITAM, Río Hondo 1, Ciudad de México 01080, Mexico
| | - I. Pineda
- Basic Sciences Department, Rotational Dynamics Research Lab, Universidad Autónoma Metropolitana-Azcapotzalco, San Pablo 180, Ciudad de México 02200, Mexico
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18
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Chang L, Duan M, Sun G, Jin Z. Cross-diffusion-induced patterns in an SIR epidemic model on complex networks. CHAOS (WOODBURY, N.Y.) 2020; 30:013147. [PMID: 32013486 DOI: 10.1063/1.5135069] [Citation(s) in RCA: 13] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/06/2019] [Accepted: 01/14/2020] [Indexed: 06/10/2023]
Abstract
Infectious diseases are a major threat to global health. Spatial patterns revealed by epidemic models governed by reaction-diffusion systems can serve as a potential trend indicator of disease spread; thus, they have received wide attention. To characterize important features of disease spread, there are two important factors that cannot be ignored in the reaction-diffusion systems. One is that a susceptible individual has an ability to recognize the infected ones and keep away from them. The other is that populations are usually organized as networks instead of being continuously distributed in space. Consequently, it is essential to study patterns generated by epidemic models with self- and cross-diffusion on complex networks. Here, with the help of a linear analysis method, we study Turing instability induced by cross-diffusion for a network organized SIR epidemic model and explore Turing patterns on several different networks. Furthermore, the influences of cross-diffusion and network structure on patterns are also investigated.
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Affiliation(s)
- Lili Chang
- Complex Systems Research Center, Shanxi University, Taiyuan 030006, Shanxi, China
| | - Moran Duan
- Shanxi Key Laboratory of Mathematical Technique and Big Data Analysis on Disease Control and Prevention, Taiyuan 030006, Shanxi, China
| | - Guiquan Sun
- Complex Systems Research Center, Shanxi University, Taiyuan 030006, Shanxi, China
| | - Zhen Jin
- Complex Systems Research Center, Shanxi University, Taiyuan 030006, Shanxi, China
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19
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Putra PS, Susanto H, Nuraini N. Turing Patterns of Non-linear S-I Model on Random and Real-Structure Networks with Diarrhea Data. Sci Rep 2019; 9:8892. [PMID: 31221999 PMCID: PMC6586790 DOI: 10.1038/s41598-019-45069-3] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/07/2019] [Accepted: 05/28/2019] [Indexed: 11/26/2022] Open
Abstract
Most developed models for solving problems in epidemiology use deterministic approach. To cover the lack of spatial sense in the method, one uses statistical modeling, reaction-diffusion in continuous medium, or multi-patch model to depict epidemic activities in several connected locations. Here, we show that an epidemic model that is set as an organized system on networks can yield Turing patterns and other interesting behaviors that are sensitive to the initial conditions. The formed patterns can be used to determine the epidemic arrival time, its first peak occurrence and the peak duration. These epidemic quantities are beneficial to identify contribution of a disease source node to the others. Using a real structure network, the system also exhibits a comparable disease spread pattern of Diarrhea in Jakarta.
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Affiliation(s)
- Prama Setia Putra
- Industrial and Financial Mathematics Research Group, Department of Mathematics, Institut Teknologi Bandung, Ganesha 10, Bandung, 40132, Indonesia.
| | - Hadi Susanto
- Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester, CO4 3SQ, United Kingdom
| | - Nuning Nuraini
- Industrial and Financial Mathematics Research Group, Department of Mathematics, Institut Teknologi Bandung, Ganesha 10, Bandung, 40132, Indonesia
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20
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Huang CY, Chin WCB, Wen TH, Fu YH, Tsai YS. EpiRank: Modeling Bidirectional Disease Spread in Asymmetric Commuting Networks. Sci Rep 2019; 9:5415. [PMID: 30931968 PMCID: PMC6443646 DOI: 10.1038/s41598-019-41719-8] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/19/2018] [Accepted: 03/13/2019] [Indexed: 01/03/2023] Open
Abstract
Commuting network flows are generally asymmetrical, with commuting behaviors bi-directionally balanced between home and work locations, and with weekday commutes providing many opportunities for the spread of infectious diseases via direct and indirect physical contact. The authors use a Markov chain model and PageRank-like algorithm to construct a novel algorithm called EpiRank to measure infection risk in a spatially confined commuting network on Taiwan island. Data from the country's 2000 census were used to map epidemic risk distribution as a commuting network function. A daytime parameter was used to integrate forward and backward movement in order to analyze daily commuting patterns. EpiRank algorithm results were tested by comparing calculations with actual disease distributions for the 2009 H1N1 influenza outbreak and enterovirus cases between 2000 and 2008. Results suggest that the bidirectional movement model outperformed models that considered forward or backward direction only in terms of capturing spatial epidemic risk distribution. EpiRank also outperformed models based on network indexes such as PageRank and HITS. According to a sensitivity analysis of the daytime parameter, the backward movement effect is more important than the forward movement effect for understanding a commuting network's disease diffusion structure. Our evidence supports the use of EpiRank as an alternative network measure for analyzing disease diffusion in a commuting network.
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Affiliation(s)
- Chung-Yuan Huang
- Department of Computer Science and Information Engineering, School of Electrical and Computer Engineering, College of Engineering, Chang Gung University, Taoyuan City, 33302, Taiwan
| | - Wei-Chien-Benny Chin
- Department of Geography, National Taiwan University, Taipei City, 10617, Taiwan.
| | - Tzai-Hung Wen
- Department of Geography, National Taiwan University, Taipei City, 10617, Taiwan
| | - Yu-Hsiang Fu
- Department of Computer Science, National Chiao Tung University, Hsinchu, 30010, Taiwan
| | - Yu-Shiuan Tsai
- Department of Computer Science and Engineering, National Taiwan Ocean University, Keelung City, 20224, Taiwan
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21
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Abstract
With the advent of substantial intercontinental air travel, it is possible for diseases to move from one location to a completely separate location very rapidly. This was an essential aspect of modeling SARS during the epidemic of 2002–2003, and has become a very important part of the study of the spread of epidemics. Mathematically, it has led to the study of metapopulation models or models with patchy environments and movement between patches.
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22
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Wang C, Lv M, Alsaedi A, Ma J. Synchronization stability and pattern selection in a memristive neuronal network. CHAOS (WOODBURY, N.Y.) 2017; 27:113108. [PMID: 29195333 DOI: 10.1063/1.5004234] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
Spatial pattern formation and selection depend on the intrinsic self-organization and cooperation between nodes in spatiotemporal systems. Based on a memory neuron model, a regular network with electromagnetic induction is proposed to investigate the synchronization and pattern selection. In our model, the memristor is used to bridge the coupling between the magnetic flux and the membrane potential, and the induction current results from the time-varying electromagnetic field contributed by the exchange of ion currents and the distribution of charged ions. The statistical factor of synchronization predicts the transition of synchronization and pattern stability. The bifurcation analysis of the sampled time series for the membrane potential reveals the mode transition in electrical activity and pattern selection. A formation mechanism is outlined to account for the emergence of target waves. Although an external stimulus is imposed on each neuron uniformly, the diversity in the magnetic flux and the induction current leads to emergence of target waves in the studied network.
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Affiliation(s)
- Chunni Wang
- Department of Physics, Lanzhou University of Technology, Lanzhou 730050, China
| | - Mi Lv
- Department of Physics, Lanzhou University of Technology, Lanzhou 730050, China
| | - Ahmed Alsaedi
- NAAM-Research Group, Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia
| | - Jun Ma
- Department of Physics, Lanzhou University of Technology, Lanzhou 730050, China
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23
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Chin WCB, Wen TH, Sabel CE, Wang IH. A geo-computational algorithm for exploring the structure of diffusion progression in time and space. Sci Rep 2017; 7:12565. [PMID: 28974752 PMCID: PMC5626785 DOI: 10.1038/s41598-017-12852-z] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/22/2017] [Accepted: 09/14/2017] [Indexed: 01/03/2023] Open
Abstract
A diffusion process can be considered as the movement of linked events through space and time. Therefore, space-time locations of events are key to identify any diffusion process. However, previous clustering analysis methods have focused only on space-time proximity characteristics, neglecting the temporal lag of the movement of events. We argue that the temporal lag between events is a key to understand the process of diffusion movement. Using the temporal lag could help to clarify the types of close relationships. This study aims to develop a data exploration algorithm, namely the TrAcking Progression In Time And Space (TaPiTaS) algorithm, for understanding diffusion processes. Based on the spatial distance and temporal interval between cases, TaPiTaS detects sub-clusters, a group of events that have high probability of having common sources, identifies progression links, the relationships between sub-clusters, and tracks progression chains, the connected components of sub-clusters. Dengue Fever cases data was used as an illustrative case study. The location and temporal range of sub-clusters are presented, along with the progression links. TaPiTaS algorithm contributes a more detailed and in-depth understanding of the development of progression chains, namely the geographic diffusion process.
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Affiliation(s)
| | - Tzai-Hung Wen
- Department of Geography, National Taiwan University, Taipei City, 10617, Taiwan.
| | - Clive E Sabel
- Department of Environmental Science, Aarhus University, 4000, Roskilde, Denmark
| | - I-Hsiang Wang
- Department of Geography, National Taiwan University, Taipei City, 10617, Taiwan
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24
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Khan MA, Ali K, Bonyah E, Okosun KO, Islam S, Khan A. Mathematical modeling and stability analysis of Pine Wilt Disease with optimal control. Sci Rep 2017; 7:3115. [PMID: 28596525 PMCID: PMC5465188 DOI: 10.1038/s41598-017-03179-w] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/11/2016] [Accepted: 04/26/2017] [Indexed: 11/09/2022] Open
Abstract
This paper presents and examine a mathematical system of equations which describes the dynamics of pine wilt disease (PWD). Firstly, we examine the model with constant controls. Here, we investigate the disease equilibria and calculate the basic reproduction number of the disease. Secondly, we incorporate time dependent controls into the model and then analyze the conditions that are necessary for the disease to be controlled optimally. Finally, the numerical results for the model are presented.
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Affiliation(s)
- M A Khan
- Department of Mathematics, Abdul Wali Khan University, Mardan, Khyber Pakhtunkhwa, 23200, Pakistan
| | - K Ali
- Department of Mathematics, Abdul Wali Khan, University Mardan, Khyber Pakhtunkhwa, Pakistan
| | - E Bonyah
- Department of Mathematics and Statistics, Kumasi Technical University, Kumasi, Ghana.
- Department of Mathematics, Vaal University of Technology, X021, Vanderbijlpark, 1900, South Africa.
| | - K O Okosun
- Department of Mathematics, Vaal University of Technology, X021, Vanderbijlpark, 1900, South Africa
| | - S Islam
- Department of Mathematics, Abdul Wali Khan, University Mardan, Khyber Pakhtunkhwa, Pakistan
| | - A Khan
- Department of Physics, Abdul Wali Khan University Mardan, 23200, Mardan, Pakistan
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25
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Xu H, Sun M, Zhao X. Turing mechanism underlying a branching model for lung morphogenesis. PLoS One 2017; 12:e0174946. [PMID: 28376090 PMCID: PMC5380321 DOI: 10.1371/journal.pone.0174946] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/10/2017] [Accepted: 03/19/2017] [Indexed: 11/19/2022] Open
Abstract
The mammalian lung develops through branching morphogenesis. Two primary forms of branching, which occur in order, in the lung have been identified: tip bifurcation and side branching. However, the mechanisms of lung branching morphogenesis remain to be explored. In our previous study, a biological mechanism was presented for lung branching pattern formation through a branching model. Here, we provide a mathematical mechanism underlying the branching patterns. By decoupling the branching model, we demonstrated the existence of Turing instability. We performed Turing instability analysis to reveal the mathematical mechanism of the branching patterns. Our simulation results show that the Turing patterns underlying the branching patterns are spot patterns that exhibit high local morphogen concentration. The high local morphogen concentration induces the growth of branching. Furthermore, we found that the sparse spot patterns underlie the tip bifurcation patterns, while the dense spot patterns underlies the side branching patterns. The dispersion relation analysis shows that the Turing wavelength affects the branching structure. As the wavelength decreases, the spot patterns change from sparse to dense, the rate of tip bifurcation decreases and side branching eventually occurs instead. In the process of transformation, there may exists hybrid branching that mixes tip bifurcation and side branching. Since experimental studies have reported that branching mode switching from side branching to tip bifurcation in the lung is under genetic control, our simulation results suggest that genes control the switch of the branching mode by regulating the Turing wavelength. Our results provide a novel insight into and understanding of the formation of branching patterns in the lung and other biological systems.
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Affiliation(s)
- Hui Xu
- Institute of Robotics and Automatic Information System, Nankai University, Tianjin, China
- Tianjin Key Laboratory of Intelligent Robotics, Nankai University, Tianjin, China
| | - Mingzhu Sun
- Institute of Robotics and Automatic Information System, Nankai University, Tianjin, China
- Tianjin Key Laboratory of Intelligent Robotics, Nankai University, Tianjin, China
| | - Xin Zhao
- Institute of Robotics and Automatic Information System, Nankai University, Tianjin, China
- Tianjin Key Laboratory of Intelligent Robotics, Nankai University, Tianjin, China
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26
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Time delay induces pattern transition in spatial epidemics: Comment on "Pattern transitions in spatial epidemics: Mechanisms and emergent properties" by Gui-Quan Sun et al. Phys Life Rev 2017; 19:85-86. [PMID: 28340931 DOI: 10.1016/j.plrev.2016.10.013] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/01/2016] [Accepted: 10/24/2016] [Indexed: 01/06/2023]
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27
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LUO XIAOFENG, CHANG LILI, JIN ZHEN. DEMOGRAPHICS INDUCE EXTINCTION OF DISEASE IN AN SIS MODEL BASED ON CONDITIONAL MARKOV CHAIN. J BIOL SYST 2017. [DOI: 10.1142/s0218339017500085] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
Demographics have significant effects on disease spread in populations and the topological evolution of the underlying networks that represent the populations. In the context of network-based epidemic modeling, Markov chain-based approach and pairwise approximation are two powerful tools — the former can capture stochastic effects of disease transmission dynamics and the latter can characterize the dynamical correlations in each pair of connected individuals. However, to our best knowledge, the study on epidemic spreading in networks relying on these two techniques is still lacking. To fill this gap, in this paper, a deterministic pairwise susceptible–infected–susceptible (SIS) epidemic model with demographics on complex networks with arbitrary degree distributions is studied based on a continuous time conditional Markov chain. This deterministic model is rigorously derived — using the moment generating function — from the Kolmogorov differential equations for the evolution of individuals and pairs. It is found that demographics will induce the extinction of the disease by reducing the basic reproduction number or lowering the epidemic prevalence after the disease prevails. Moreover, due to the demographical effects, the resulting network tends to a homogeneous network with a degree distribution similar to Poisson distribution, irrespective of the initial network structure. Additionally, we find excellent agreement between numerical solutions and individual-based stochastic simulations using both Erdös–Renyi (ER) random and Barabási–Albert (BA) scale-free initial networks. Our results may provide new insights on the understanding of the influence of demographics on epidemic dynamics and network evolution.
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Affiliation(s)
- XIAOFENG LUO
- School of Computer and Information Technology, Shanxi University, Taiyuan, Shanxi 030006, P. R. China
| | - LILI CHANG
- Complex System Research Center, Shanxi University, Taiyuan, Shanxi 030006, P. R. China
| | - ZHEN JIN
- Complex System Research Center, Shanxi University, Taiyuan, Shanxi 030006, P. R. China
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28
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Sun M, Xu H, Zeng X, Zhao X. Automated numerical simulation of biological pattern formation based on visual feedback simulation framework. PLoS One 2017; 12:e0172643. [PMID: 28225811 PMCID: PMC5321435 DOI: 10.1371/journal.pone.0172643] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/15/2016] [Accepted: 02/07/2017] [Indexed: 12/26/2022] Open
Abstract
There are various fantastic biological phenomena in biological pattern formation. Mathematical modeling using reaction-diffusion partial differential equation systems is employed to study the mechanism of pattern formation. However, model parameter selection is both difficult and time consuming. In this paper, a visual feedback simulation framework is proposed to calculate the parameters of a mathematical model automatically based on the basic principle of feedback control. In the simulation framework, the simulation results are visualized, and the image features are extracted as the system feedback. Then, the unknown model parameters are obtained by comparing the image features of the simulation image and the target biological pattern. Considering two typical applications, the visual feedback simulation framework is applied to fulfill pattern formation simulations for vascular mesenchymal cells and lung development. In the simulation framework, the spot, stripe, labyrinthine patterns of vascular mesenchymal cells, the normal branching pattern and the branching pattern lacking side branching for lung branching are obtained in a finite number of iterations. The simulation results indicate that it is easy to achieve the simulation targets, especially when the simulation patterns are sensitive to the model parameters. Moreover, this simulation framework can expand to other types of biological pattern formation.
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Affiliation(s)
- Mingzhu Sun
- Institute of Robotics and Automatic Information Systems, Nankai University, Tianjin, China
- Tianjin Key Laboratory of Intelligent Robotics, Tianjin, China
| | - Hui Xu
- Institute of Robotics and Automatic Information Systems, Nankai University, Tianjin, China
- Tianjin Key Laboratory of Intelligent Robotics, Tianjin, China
| | - Xingjuan Zeng
- Institute of Robotics and Automatic Information Systems, Nankai University, Tianjin, China
- Tianjin Key Laboratory of Intelligent Robotics, Tianjin, China
| | - Xin Zhao
- Institute of Robotics and Automatic Information Systems, Nankai University, Tianjin, China
- Tianjin Key Laboratory of Intelligent Robotics, Tianjin, China
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29
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Liu W, Zhong S. Web malware spread modelling and optimal control strategies. Sci Rep 2017; 7:42308. [PMID: 28186203 PMCID: PMC5301259 DOI: 10.1038/srep42308] [Citation(s) in RCA: 33] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/11/2016] [Accepted: 01/08/2017] [Indexed: 02/07/2023] Open
Abstract
The popularity of the Web improves the growth of web threats. Formulating mathematical models for accurate prediction of malicious propagation over networks is of great importance. The aim of this paper is to understand the propagation mechanisms of web malware and the impact of human intervention on the spread of malicious hyperlinks. Considering the characteristics of web malware, a new differential epidemic model which extends the traditional SIR model by adding another delitescent compartment is proposed to address the spreading behavior of malicious links over networks. The spreading threshold of the model system is calculated, and the dynamics of the model is theoretically analyzed. Moreover, the optimal control theory is employed to study malware immunization strategies, aiming to keep the total economic loss of security investment and infection loss as low as possible. The existence and uniqueness of the results concerning the optimality system are confirmed. Finally, numerical simulations show that the spread of malware links can be controlled effectively with proper control strategy of specific parameter choice.
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Affiliation(s)
- Wanping Liu
- School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China
- College of Computer Science and Engineering, Chongqing University of Technology, Chongqing 400054, China
| | - Shouming Zhong
- School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China
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30
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Abstract
Hemorrhagic fever with renal syndrome (HFRS) spreading from rodent to human beings is a major public health problem in China, which causes high mortality rate. Data obtained from the China Ministry of Health shows that cases of HFRS in China exhibited monthly periodic outbreak. To well reveal the mechanisms about the outbreak of HFRS, we established a dynamical model to explain the periodic behaviors of HFRS in China. We obtained the basic reproduction number [Formula: see text], analyzed the dynamical behavior of the model, and used the model to fit the monthly data of HFRS cases. Our results demonstrated that periodic transmission rates and rodent periodic birth rate of HFRS in China can give rise to the periodic outbreak of HFRS, hence providing insights into taking measures to control HFRS in China.
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Affiliation(s)
- LI LI
- School of Computer and Information Technology, Shanxi University, Taiyuan, Shanxi 030006, P. R. China
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31
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Bocharov G, Meyerhans A, Bessonov N, Trofimchuk S, Volpert V. Spatiotemporal Dynamics of Virus Infection Spreading in Tissues. PLoS One 2016; 11:e0168576. [PMID: 27997613 PMCID: PMC5173377 DOI: 10.1371/journal.pone.0168576] [Citation(s) in RCA: 28] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/22/2016] [Accepted: 12/03/2016] [Indexed: 12/21/2022] Open
Abstract
Virus spreading in tissues is determined by virus transport, virus multiplication in host cells and the virus-induced immune response. Cytotoxic T cells remove infected cells with a rate determined by the infection level. The intensity of the immune response has a bell-shaped dependence on the concentration of virus, i.e., it increases at low and decays at high infection levels. A combination of these effects and a time delay in the immune response determine the development of virus infection in tissues like spleen or lymph nodes. The mathematical model described in this work consists of reaction-diffusion equations with a delay. It shows that the different regimes of infection spreading like the establishment of a low level infection, a high level infection or a transition between both are determined by the initial virus load and by the intensity of the immune response. The dynamics of the model solutions include simple and composed waves, and periodic and aperiodic oscillations. The results of analytical and numerical studies of the model provide a systematic basis for a quantitative understanding and interpretation of the determinants of the infection process in target organs and tissues from the image-derived data as well as of the spatiotemporal mechanisms of viral disease pathogenesis, and have direct implications for a biopsy-based medical testing of the chronic infection processes caused by viruses, e.g. HIV, HCV and HBV.
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Affiliation(s)
- Gennady Bocharov
- Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russian Federation
- Gamaleya Center of Epidemiology and Microbiology, Moscow, Russian Federation
- RUDN University, Moscow, Russian Federation
| | - Andreas Meyerhans
- Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russian Federation
- Infection Biology Laboratory, Department of Experimental and Health Sciences, Universitat Pompeu Fabra, Barcelona, Spain
- ICREA, Pg. Lluís Companys 23, Barcelona, Spain
| | - Nickolai Bessonov
- Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, Saint Petersburg, Russian Federation
| | - Sergei Trofimchuk
- Instituto de Matemática y Fisica, Universidad de Talca, Talca, Chile
| | - Vitaly Volpert
- Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russian Federation
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, Villeurbanne, France
- INRIA Team Dracula, INRIA Lyon La Doua, Villeurbanne, France
- Laboratoire Poncelet, UMI 2615 CNRS, Moscow, Russian Federation
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32
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Sun GQ, Jusup M, Jin Z, Wang Y, Wang Z. Pattern transitions in spatial epidemics: Mechanisms and emergent properties. Phys Life Rev 2016; 19:43-73. [PMID: 27567502 PMCID: PMC7105263 DOI: 10.1016/j.plrev.2016.08.002] [Citation(s) in RCA: 81] [Impact Index Per Article: 10.1] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/29/2016] [Accepted: 08/04/2016] [Indexed: 12/19/2022]
Abstract
Infectious diseases are a threat to human health and a hindrance to societal development. Consequently, the spread of diseases in both time and space has been widely studied, revealing the different types of spatial patterns. Transitions between patterns are an emergent property in spatial epidemics that can serve as a potential trend indicator of disease spread. Despite the usefulness of such an indicator, attempts to systematize the topic of pattern transitions have been few and far between. We present a mini-review on pattern transitions in spatial epidemics, describing the types of transitions and their underlying mechanisms. We show that pattern transitions relate to the complexity of spatial epidemics by, for example, being accompanied with phenomena such as coherence resonance and cyclic evolution. The results presented herein provide valuable insights into disease prevention and control, and may even be applicable outside epidemiology, including other branches of medical science, ecology, quantitative finance, and elsewhere.
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Affiliation(s)
- Gui-Quan Sun
- Complex Systems Research Center, Shanxi University, Taiyuan, Shanxi 030006, PR China; School of Mathematical Sciences, Fudan University, Shanghai 200433, PR China.
| | - Marko Jusup
- Department of Vector Ecology and Environment, Nagasaki University Institute of Tropical Medicine (NEKKEN), Nagasaki 852-8523, Japan; Center of Mathematics for Social Creativity, Hokkaido University, Sapporo 060-0812, Japan.
| | - Zhen Jin
- Complex Systems Research Center, Shanxi University, Taiyuan, Shanxi 030006, PR China.
| | - Yi Wang
- Department of Mathematics, Southeast University, Nanjing 210096, PR China; Department of Mathematics and Statistics, University of Victoria, Victoria BC V8W 3R4, Canada
| | - Zhen Wang
- Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Fukuoka, 816-8580, Japan.
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33
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Climate change is a potential factor for the emergence of pattern transitions in spatial epidemics: Comment on "Pattern transitions in spatial epidemics: Mechanisms and emergent properties" by Gui-Quan Sun et al. Phys Life Rev 2016; 19:98-99. [PMID: 27810386 DOI: 10.1016/j.plrev.2016.10.003] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/06/2016] [Accepted: 10/21/2016] [Indexed: 11/22/2022]
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34
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Xu Y, Ren J. Propagation Effect of a Virus Outbreak on a Network with Limited Anti-Virus Ability. PLoS One 2016; 11:e0164415. [PMID: 27788157 PMCID: PMC5082916 DOI: 10.1371/journal.pone.0164415] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/21/2016] [Accepted: 09/23/2016] [Indexed: 11/24/2022] Open
Abstract
This paper describes a new computer virus spreading model which takes into account the possibility of a virus outbreak on a network with limited anti-virus ability. Then, the model is investigated for the existence of equilibria and their stabilities are proved and illustrated. Moreover, it is found that these two factors are not only relative to the threshold value determining whether the virus becomes extinct or not, but that they are also relative to the virus epidemic levels. Theoretical and experimental results indicate that, in some ways, it would be practically possible to eradicate the virus or suppress its prevalence below a suitable level. Consequently, some suggestions are proposed that may help eradicate or suppress virus propagation over a real computer network.
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Affiliation(s)
- Yonghong Xu
- The Key Laboratory of Biotechnology for Medicinal Plants of Jiangsu Province, Jiangsu Normal University, Xuzhou, 221116, China
| | - Jianguo Ren
- College of Computer, Jiangsu Normal University, Xuzhou, 221116, China
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35
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Kahiluoto H, Kaseva J. No Evidence of Trade-Off between Farm Efficiency and Resilience: Dependence of Resource-Use Efficiency on Land-Use Diversity. PLoS One 2016; 11:e0162736. [PMID: 27662475 PMCID: PMC5035069 DOI: 10.1371/journal.pone.0162736] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/26/2016] [Accepted: 08/26/2016] [Indexed: 11/18/2022] Open
Abstract
Efficiency in the use of resources stream-lined for expected conditions could lead to reduced system diversity and consequently endanger resilience. We tested the hypothesis of a trade-off between farm resource-use efficiency and land-use diversity. We applied stochastic frontier production models to assess the dependence of resource-use-efficiency on land-use diversity as illustrated by the Shannon-Weaver index. Total revenue in relation to use of capital, land and labour on the farms in Southern Finland with a size exceeding 30 ha was studied. The data were extracted from the Finnish Profitability Bookkeeping data. Our results indicate that there is either no trade-off or a negligible trade-off of no economic importance. The small dependence of resource-use efficiency on land-use diversity can be positive as well as negative. We conclude that diversification as a strategy to enhance farm resilience does not necessarily constrain resource-use efficiency.
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Affiliation(s)
- Helena Kahiluoto
- Lappeenranta University of Technology, Saimaankatu 11, 15140 Lahti, Finland
| | - Janne Kaseva
- Natural Resources Institute Finland, 31600 Jokioinen, Finland
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36
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Zhao K, Jurdak R. Understanding the spatiotemporal pattern of grazing cattle movement. Sci Rep 2016; 6:31967. [PMID: 27555220 PMCID: PMC4995401 DOI: 10.1038/srep31967] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/11/2016] [Accepted: 07/08/2016] [Indexed: 11/21/2022] Open
Abstract
Understanding the drivers of animal movement is significant for ecology and biology. Yet researchers have so far been unable to fully understand these drivers, largely due to low data resolution. In this study, we analyse a high-frequency movement dataset for a group of grazing cattle and investigate their spatiotemporal patterns using a simple two-state 'stop-and-move' mobility model. We find that the dispersal kernel in the moving state is best described by a mixture exponential distribution, indicating the hierarchical nature of the movement. On the other hand, the waiting time appears to be scale-invariant below a certain cut-off and is best described by a truncated power-law distribution, suggesting that the non-moving state is governed by time-varying dynamics. We explore possible explanations for the observed phenomena, covering factors that can play a role in the generation of mobility patterns, such as the context of grazing environment, the intrinsic decision-making mechanism or the energy status of different activities. In particular, we propose a new hypothesis that the underlying movement pattern can be attributed to the most probable observable energy status under the maximum entropy configuration. These results are not only valuable for modelling cattle movement but also provide new insights for understanding the underlying biological basis of grazing behaviour.
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Affiliation(s)
- Kun Zhao
- CSIRO Data61, Brisbane, Queensland, Australia
| | - Raja Jurdak
- CSIRO Data61, Brisbane, Queensland, Australia
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37
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Wang T, Duan J, Liu T. Competition promotes the persistence of populations in ecosystems. Sci Rep 2016; 6:30477. [PMID: 27460143 PMCID: PMC4962045 DOI: 10.1038/srep30477] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/24/2016] [Accepted: 07/05/2016] [Indexed: 11/29/2022] Open
Abstract
Competition is one of the most common form in ecological systems, which plays important roles in population dynamics. However, the influences of competition on persistence of populations remain unclear when space effect is included. In this paper, we investigated a predator-prey model with competition and spatial diffusion. Based on pattern formations and time series of populations, we found that competitions induce the persistence of populations, which denies competitive exclusion principle. Moreover, we testify the robustness of these effects. Our results also suggest that space may lead to the emergence of new phenomenon in ecosystems.
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Affiliation(s)
- Tao Wang
- College of Science, Shihezi University, Shihezi, Xinjiang 832003, P. R. China
| | - Jinqiao Duan
- Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA
- Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan, 430074, P. R. China
| | - Tong Liu
- College of Life Science, Shihezi University, Shihezi, Xinjiang 832003, P. R. China
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38
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Song LP, Ding XQ, Feng LP, Shi Q. Hopf Bifurcation of an Epidemic Model with Delay. PLoS One 2016; 11:e0157367. [PMID: 27304674 PMCID: PMC4909215 DOI: 10.1371/journal.pone.0157367] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/25/2016] [Accepted: 05/29/2016] [Indexed: 11/19/2022] Open
Abstract
A spatiotemporal epidemic model with nonlinear incidence rate and Neumann boundary conditions is investigated. On the basis of the analysis of eigenvalues of the eigenpolynomial, we derive the conditions of the existence of Hopf bifurcation in one dimension space. By utilizing the normal form theory and the center manifold theorem of partial functional differential equations (PFDs), the properties of bifurcating periodic solutions are analyzed. Moreover, according to numerical simulations, it is found that the periodic solutions can emerge in delayed epidemic model with spatial diffusion, which is consistent with our theoretical results. The obtained results may provide a new viewpoint for the recurrent outbreak of disease.
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Affiliation(s)
- Li-Peng Song
- Department of Computer Science and Technology, North University of China, Taiyuan, Shan’xi 030051, People’s Republic of China
| | - Xiao-Qiang Ding
- Department of Computer Science and Technology, North University of China, Taiyuan, Shan’xi 030051, People’s Republic of China
| | - Li-Ping Feng
- Department of Computer Science and Technology, North University of China, Taiyuan, Shan’xi 030051, People’s Republic of China
| | - Qiong Shi
- Department of Computer Science and Technology, North University of China, Taiyuan, Shan’xi 030051, People’s Republic of China
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39
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Malik R, Deardon R, Kwong GPS. Parameterizing Spatial Models of Infectious Disease Transmission that Incorporate Infection Time Uncertainty Using Sampling-Based Likelihood Approximations. PLoS One 2016; 11:e0146253. [PMID: 26731666 PMCID: PMC4701410 DOI: 10.1371/journal.pone.0146253] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/09/2015] [Accepted: 12/15/2015] [Indexed: 11/18/2022] Open
Abstract
A class of discrete-time models of infectious disease spread, referred to as individual-level models (ILMs), are typically fitted in a Bayesian Markov chain Monte Carlo (MCMC) framework. These models quantify probabilistic outcomes regarding the risk of infection of susceptible individuals due to various susceptibility and transmissibility factors, including their spatial distance from infectious individuals. The infectious pressure from infected individuals exerted on susceptible individuals is intrinsic to these ILMs. Unfortunately, quantifying this infectious pressure for data sets containing many individuals can be computationally burdensome, leading to a time-consuming likelihood calculation and, thus, computationally prohibitive MCMC-based analysis. This problem worsens when using data augmentation to allow for uncertainty in infection times. In this paper, we develop sampling methods that can be used to calculate a fast, approximate likelihood when fitting such disease models. A simple random sampling approach is initially considered followed by various spatially-stratified schemes. We test and compare the performance of our methods with both simulated data and data from the 2001 foot-and-mouth disease (FMD) epidemic in the U.K. Our results indicate that substantial computation savings can be obtained--albeit, of course, with some information loss--suggesting that such techniques may be of use in the analysis of very large epidemic data sets.
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Affiliation(s)
- Rajat Malik
- Department of Mathematics & Statistics, University of Guelph, Guelph, Ontario, Canada
| | - Rob Deardon
- Department of Mathematics & Statistics, University of Guelph, Guelph, Ontario, Canada
- Faculty of Veterinary Medicine, University of Calgary, Calgary, Alberta, Canada
- Department of Mathematics & Statistics, University of Calgary, Calgary, Alberta, Canada
| | - Grace P. S. Kwong
- Department of Mathematics & Statistics, University of Calgary, Calgary, Alberta, Canada
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40
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Chen P, Han Y, Pan J. High-Dynamic-Range CT Reconstruction Based on Varying Tube-Voltage Imaging. PLoS One 2015; 10:e0141789. [PMID: 26544723 PMCID: PMC4636367 DOI: 10.1371/journal.pone.0141789] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/10/2015] [Accepted: 10/13/2015] [Indexed: 12/01/2022] Open
Abstract
For complicated structural components characterized by wide X-ray attenuation ranges, the conventional computed tomography (CT) imaging using a single tube-voltage at each rotation angle cannot obtain all structural information. This limitation results in a shortage of CT information, because the effective thickness of the components along the direction of X-ray penetration exceeds the limitation of the dynamic range of the X-ray imaging system. To address this problem, high-dynamic-range CT (HDR-CT) reconstruction is proposed. For this new method, the tube's voltage is adjusted several times to match the corresponding effective thickness about the local information from an object. Then, HDR fusion and HDR-CT are applied to obtain the full reconstruction information. An accompanying experiment demonstrates that this new technology can extend the dynamic range of X-ray imaging systems and provide the complete internal structures of complicated structural components.
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Affiliation(s)
- Ping Chen
- National Key Laboratory for Electronic Measurement Technology, North University of China, Taiyuan, 030051, China
- Key Laboratory of Instrumentation Science & Dynamic Measurement, North University of China, Taiyuan, 030051, China
| | - Yan Han
- National Key Laboratory for Electronic Measurement Technology, North University of China, Taiyuan, 030051, China
- Key Laboratory of Instrumentation Science & Dynamic Measurement, North University of China, Taiyuan, 030051, China
| | - Jinxiao Pan
- National Key Laboratory for Electronic Measurement Technology, North University of China, Taiyuan, 030051, China
- Key Laboratory of Instrumentation Science & Dynamic Measurement, North University of China, Taiyuan, 030051, China
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41
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Liccardo A, Fierro A. Multiple Lattice Model for Influenza Spreading. PLoS One 2015; 10:e0141065. [PMID: 26513580 PMCID: PMC4626091 DOI: 10.1371/journal.pone.0141065] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/21/2015] [Accepted: 10/05/2015] [Indexed: 11/19/2022] Open
Abstract
Behavioral differences among age classes, together with the natural tendency of individuals to prefer contacts with individuals of similar age, naturally point to the existence of a community structure in the population network, in which each community can be identified with a different age class. Data on age-dependent contact patterns also reveal how relevant is the role of the population age structure in shaping the spreading of an infectious disease. In the present paper we propose a simple model for epidemic spreading, in which a contact network with an intrinsic community structure is coupled with a simple stochastic SIR model for the epidemic spreading. The population is divided in 4 different age-communities and hosted on a multiple lattice, each community occupying a specific age-lattice. Individuals are allowed to move freely to nearest neighbor empty sites on the age-lattice. Different communities are connected with each other by means of inter-lattices edges, with a different number of external links connecting different age class populations. The parameters of the contact network model are fixed by requiring the simulated data to fully reproduce the contact patterns matrices of the Polymod survey. The paper shows that adopting a topology which better implements the age-class community structure of the population, one gets a better agreement between experimental contact patterns and simulated data, and this also improves the accordance between simulated and experimental data on the epidemic spreading.
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Affiliation(s)
- Antonella Liccardo
- Physics Department, Università degli Studi di Napoli “Federico II”, Napoli, Italy
- Istituto Nazionale Fisica Nucleare (INFN) - Sezione di Napoli, Napoli, Italy
| | - Annalisa Fierro
- Physics Department, Università degli Studi di Napoli “Federico II”, Napoli, Italy
- Consiglio Nazionale delle Ricerche (CNR) - Institute Superconductors, oxides and other innovative materials and devices (SPIN), Napoli, Italy
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42
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Zhang T, Wang K, Zhang X. Modeling and Analyzing the Transmission Dynamics of HBV Epidemic in Xinjiang, China. PLoS One 2015; 10:e0138765. [PMID: 26422614 PMCID: PMC4589324 DOI: 10.1371/journal.pone.0138765] [Citation(s) in RCA: 23] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/20/2015] [Accepted: 09/03/2015] [Indexed: 12/04/2022] Open
Abstract
Hepatitis B is an infectious disease caused by the hepatitis B virus (HBV) which affects livers. In this paper, we formulate a hepatitis B model to study the transmission dynamics of hepatitis B in Xinjiang, China. The epidemic model involves an exponential birth rate and vertical transmission. For a better understanding of HBV transmission dynamics, we analyze the dynamic behavior of the model. The modified reproductive number σ is obtained. When σ < 1, the disease-free equilibrium is locally asymptotically stable, when σ > 1, the disease-free equilibrium is unstable and the disease is uniformly persistent. In the simulation, parameters are chosen to fit public data in Xinjiang. The simulation indicates that the cumulated HBV infection number in Xinjiang will attain about 600,000 cases unless there are stronger or more effective control measures by the end of 2017. Sensitive analysis results show that enhancing the vaccination rate for newborns in Xinjiang is very effective to stop the transmission of HBV. Hence, we recommend that all infants in Xinjiang receive the hepatitis B vaccine as soon as possible after birth.
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Affiliation(s)
- Tailei Zhang
- School of Science/Chang’an University, Xi’an, China
| | - Kai Wang
- Department of Medical Engineering and Technology/ Xinjiang Medical University, Urumqi, China
| | - Xueliang Zhang
- Department of Medical Engineering and Technology/ Xinjiang Medical University, Urumqi, China
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43
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Wu Z, Haack SE, Lin W, Li B, Wu L, Fang C, Zhang Z. Soil Microbial Community Structure and Metabolic Activity of Pinus elliottii Plantations across Different Stand Ages in a Subtropical Area. PLoS One 2015; 10:e0135354. [PMID: 26267338 PMCID: PMC4533972 DOI: 10.1371/journal.pone.0135354] [Citation(s) in RCA: 21] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/26/2015] [Accepted: 07/21/2015] [Indexed: 11/20/2022] Open
Abstract
Soil microbes play an essential role in the forest ecosystem as an active component. This study examined the hypothesis that soil microbial community structure and metabolic activity would vary with the increasing stand ages in long-term pure plantations of Pinus elliottii. The phospholipid fatty acids (PLFA) combined with community level physiological profiles (CLPP) method was used to assess these characteristics in the rhizospheric soils of P. elliottii. We found that the soil microbial communities were significantly different among different stand ages of P. elliottii plantations. The PLFA analysis indicated that the bacterial biomass was higher than the actinomycic and fungal biomass in all stand ages. However, the bacterial biomass decreased with the increasing stand ages, while the fungal biomass increased. The four maximum biomarker concentrations in rhizospheric soils of P. elliottii for all stand ages were 18:1ω9c, 16:1ω7c, 18:3ω6c (6,9,12) and cy19:0, representing measures of fungal and gram negative bacterial biomass. In addition, CLPP analysis revealed that the utilization rate of amino acids, polymers, phenolic acids, and carbohydrates of soil microbial community gradually decreased with increasing stand ages, though this pattern was not observed for carboxylic acids and amines. Microbial community diversity, as determined by the Simpson index, Shannon-Wiener index, Richness index and McIntosh index, significantly decreased as stand age increased. Overall, both the PLFA and CLPP illustrated that the long-term pure plantation pattern exacerbated the microecological imbalance previously described in the rhizospheric soils of P. elliottii, and markedly decreased the soil microbial community diversity and metabolic activity. Based on the correlation analysis, we concluded that the soil nutrient and C/N ratio most significantly contributed to the variation of soil microbial community structure and metabolic activity in different stand ages of P. elliottii plantations.
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Affiliation(s)
- Zeyan Wu
- Fujian Agriculture and Forestry University, Fujian, China
| | - Stacey Elizabeth Haack
- Department of Plant Pathology and Microbiology, University of California Riverside, Riverside, California, United States of America
| | - Wenxiong Lin
- Fujian Agriculture and Forestry University, Fujian, China
| | - Bailian Li
- Ecological Complexity and Modeling Laboratory, University of California Riverside, Riverside, California, United States of America
| | - Linkun Wu
- Fujian Agriculture and Forestry University, Fujian, China
| | - Changxun Fang
- Fujian Agriculture and Forestry University, Fujian, China
| | - Zhixing Zhang
- Fujian Agriculture and Forestry University, Fujian, China
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44
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Zhang C, Zhou S, Chain BM. Hybrid epidemics--a case study on computer worm conficker. PLoS One 2015; 10:e0127478. [PMID: 25978309 PMCID: PMC4433115 DOI: 10.1371/journal.pone.0127478] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/12/2014] [Accepted: 04/14/2015] [Indexed: 02/07/2023] Open
Abstract
Conficker is a computer worm that erupted on the Internet in 2008. It is unique in combining three different spreading strategies: local probing, neighbourhood probing, and global probing. We propose a mathematical model that combines three modes of spreading: local, neighbourhood, and global, to capture the worm's spreading behaviour. The parameters of the model are inferred directly from network data obtained during the first day of the Conficker epidemic. The model is then used to explore the tradeoff between spreading modes in determining the worm's effectiveness. Our results show that the Conficker epidemic is an example of a critically hybrid epidemic, in which the different modes of spreading in isolation do not lead to successful epidemics. Such hybrid spreading strategies may be used beneficially to provide the most effective strategies for promulgating information across a large population. When used maliciously, however, they can present a dangerous challenge to current internet security protocols.
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Affiliation(s)
- Changwang Zhang
- Department of Computer Science, University College London, London, United Kingdom
- Security Science Doctoral Research Training Centre, University College London, London, United Kingdom
| | - Shi Zhou
- Department of Computer Science, University College London, London, United Kingdom
| | - Benjamin M. Chain
- Division of Infection and Immunity, University College London, London, United Kingdom
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45
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Wang Y, Cao J, Sun GQ, Li J. Effect of time delay on pattern dynamics in a spatial epidemic model. PHYSICA A 2014; 412:137-148. [PMID: 32308253 PMCID: PMC7156238 DOI: 10.1016/j.physa.2014.06.038] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 03/19/2014] [Revised: 06/10/2014] [Indexed: 06/11/2023]
Abstract
Time delay, accounting for constant incubation period or sojourn times in an infective state, widely exists in most biological systems like epidemiological models. However, the effect of time delay on spatial epidemic models is not well understood. In this paper, spatial pattern of an epidemic model with both nonlinear incidence rate and time delay is investigated. In particular, we mainly focus on the effect of time delay on the formation of spatial pattern. Through mathematical analysis, we gain the conditions for Hopf bifurcation and Turing bifurcation, and find exact Turing space in parameter space. Furthermore, numerical results show that time delay has a significant effect on pattern formation. The simulation results may enrich the finding of patterns and may well capture some key features in the epidemic models.
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Affiliation(s)
- Yi Wang
- Department of Mathematics, Southeast University, Nanjing 210096, People's Republic of China
| | - Jinde Cao
- Department of Mathematics, Southeast University, Nanjing 210096, People's Republic of China
- Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
| | - Gui-Quan Sun
- Complex Systems Research Center, Shanxi University, Taiyuan, Shanxi 030006, People's Republic of China
- Department of Mathematics, North University of China, Taiyuan, Shanxi 030051, People's Republic of China
| | - Jing Li
- Department of Mathematics, North University of China, Taiyuan, Shanxi 030051, People's Republic of China
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46
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Nian F, Wang K. A two-step high-risk immunization based on high-risk immunization. NONLINEAR DYNAMICS 2014; 78:1729-1737. [PMID: 32214670 PMCID: PMC7089430 DOI: 10.1007/s11071-014-1555-2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 04/21/2014] [Accepted: 06/20/2014] [Indexed: 06/10/2023]
Abstract
In this paper, the two-step high-risk immunization was investigated based on high-risk immunization for the SIRS model in small-world networks and scale-free networks. First, the effects of various immunization schemes are studied and compared. When the number of immune is same, the research result shows that the immune effect of the two-step high-risk immunization strategy is not the best nor the worst. However, the practicability is better compare with others. Furthermore, by changing the proportional of immunization the optimal immune effect can be achieved in the two-step high-risk immunization. Computation results verify that the two-step high-risk immunization is effective, and it is economic and feasible in practice.
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Affiliation(s)
- Fuzhong Nian
- School of Computer & Communication, Lanzhou University of Technology, Lanzhou, 730050 China
| | - Ke Wang
- School of Computer & Communication, Lanzhou University of Technology, Lanzhou, 730050 China
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47
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Wang T. Pattern dynamics of an epidemic model with nonlinear incidence rate. NONLINEAR DYNAMICS 2014; 77:31-40. [PMID: 32214668 PMCID: PMC7089073 DOI: 10.1007/s11071-014-1270-z] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 12/11/2013] [Accepted: 01/20/2014] [Indexed: 06/10/2023]
Abstract
All species live in space, and the research of spatial diseases can be used to control infectious diseases. As a result, it is more realistic to study the spatial pattern of epidemic models with space and time. In this paper, spatial dynamics of an epidemic model with nonlinear incidence rate is investigated. We find that there are different types of stationary patterns by amplitude equations and numerical simulations. The obtained results may well explain the distribution of disease observed in the real world and provide some insights on disease control.
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Affiliation(s)
- Tao Wang
- Department of Mathematics, Shihezi University, Shihezi, 83200 Xinjiang People’s Republic of China
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Barbera E, Consolo G, Valenti G. Spread of infectious diseases in a hyperbolic reaction-diffusion susceptible-infected-removed model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:052719. [PMID: 24329308 DOI: 10.1103/physreve.88.052719] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/29/2013] [Revised: 08/02/2013] [Indexed: 05/23/2023]
Abstract
A one-dimensional hyperbolic reaction-diffusion model of epidemics is developed to describe the dynamics of diseases spread occurring in an environment where three kinds of individuals mutually interact: the susceptibles, the infectives, and the removed. It is assumed that the disease is transmitted from the infected population to the susceptible one according to a nonlinear convex incidence rate. The model, based upon the framework of extended thermodynamics, removes the unphysical feature of instantaneous diffusive effects, which is typical of parabolic models. Linear stability analyses are performed to study the nature of the equilibrium states against uniform and nonuniform perturbations. Emphasis is given to the occurrence of Hopf and Turing bifurcations, which break the temporal and the spatial symmetry of the system, respectively. The existence of traveling wave solutions connecting two steady states is also discussed. The governing equations are also integrated numerically to validate the analytical results and to characterize the spatiotemporal evolution of diseases.
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Affiliation(s)
- Elvira Barbera
- Department of Mathematics and Computer Science, University of Messina, V. le F. D'Alcontres 31, I-98166 Messina, Italy
| | - Giancarlo Consolo
- Department of Mathematics and Computer Science, University of Messina, V. le F. D'Alcontres 31, I-98166 Messina, Italy
| | - Giovanna Valenti
- Department of Mathematics and Computer Science, University of Messina, V. le F. D'Alcontres 31, I-98166 Messina, Italy
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