**27** Publications

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Bull Math Biol 2020 09 13;82(9):121. Epub 2020 Sep 13.

School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China.

Antiviral treatment remains one of the key pharmacological interventions against influenza pandemic. However, widespread use of antiviral drugs brings with it the danger of drug resistance evolution. To assess the risk of the emergence and diffusion of resistance, in this paper, we develop a diffusive influenza model where influenza infection involves both drug-sensitive and drug-resistant strains. We first analyze its corresponding reaction model, whose reproduction numbers and equilibria are derived. The results show that the sensitive strains can be eliminated by treatment. Then, we establish the existence of the three kinds of traveling waves starting from the disease-free equilibrium, i.e., semi-traveling waves, strong traveling waves and persistent traveling waves, from which we can get some useful information (such as whether influenza will spread, asymptotic speed of propagation, the final state of the wavefront). On the other hand, we discuss three situations in which semi-traveling waves do not exist. When the control reproduction number [Formula: see text] is larger than 1, the conditions for the existence and nonexistence of traveling waves are determined completely by the reproduction numbers [Formula: see text], [Formula: see text] and the wave speed c. Meanwhile, we give an interval estimation of minimal wave speed for influenza transmission, which has important guiding significance for the control of influenza in reality. Our findings demonstrate that the control of influenza depends not only on the rates of resistance emergence and transmission during treatment, but also on the diffusion rates of influenza strains, which have been overlooked in previous modeling studies. This suggests that antiviral treatment should be implemented appropriately, and infected individuals (especially with the resistant strain) should be tested and controlled effectively. Finally, we outline some future directions that deserve further investigation.

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http://dx.doi.org/10.1007/s11538-020-00799-8 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7487074 | PMC |

September 2020

Chaos 2019 Mar;29(3):033122

Department of Mathematics, Shanghai University, Shanghai 200444, People's Republic of China.

This paper investigates feedback pinning control of synchronization behaviors aroused by epidemic spread on complex networks. Based on the quenched mean field theory, epidemic control synchronization models with the inhibition of contact behavior are constructed, combined with the epidemic transmission system and the adaptive dynamical network carrying active controllers. By the properties of convex functions and the Gerschgorin theorem, the epidemic threshold of the model is obtained, and the global stability of disease-free equilibrium is analyzed. For individual's infected situation, when an epidemic disease spreads, two types of feedback control strategies depending on the diseases' information are designed: the first one only adds controllers to infected individuals, and the other adds controllers to both infected and susceptible ones. By using the Lyapunov stability theory, under designed controllers, some criteria that guarantee the epidemic controlled synchronization system achieving behavior synchronization are also derived. Several numerical simulations are performed to show the effectiveness of our theoretical results. As far as we know, this is the first work to address the controlled behavioral synchronization induced by epidemic spread under the pinning feedback mechanism. It is hopeful that we may have deeper insights into the essence between the disease's spread and collective behavior under active control in complex dynamical networks.

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http://dx.doi.org/10.1063/1.5047653 | DOI Listing |

March 2019

Acta Math Sci 2019 27;39(6):1713-1732. Epub 2019 Sep 27.

Department of Mathematics, Shanghai University, Shanghai, 200444 China.

Real epidemic spreading networks are often composed of several kinds of complex networks interconnected with each other, such as Lyme disease, and the interrelated networks may have different topologies and epidemic dynamics. Moreover, most human infectious diseases are derived from animals, and zoonotic infections always spread on directed interconnected networks. So, in this article, we consider the epidemic dynamics of zoonotic infections on a unidirectional circular-coupled network. Here, we construct two unidirectional three-layer circular interactive networks, one model has direct contact between interactive networks, the other model describes diseases transmitted through vectors between interactive networks, which are established by introducing the heterogeneous mean-field approach method. Then we obtain the basic reproduction numbers and stability of equilibria of the two models. Through mathematical analysis and numerical simulations, it is found that basic reproduction numbers of the models depend on the infection rates, infection periods, average degrees, and degree ratios. Numerical simulations illustrate and expand these theoretical results very well.

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http://dx.doi.org/10.1007/s10473-019-0618-3 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7111949 | PMC |

September 2019

J Theor Biol 2019 02 10;462:122-133. Epub 2018 Nov 10.

Department of Mathematics, Shanghai University, Shanghai 200444, China. Electronic address:

Many real-world networks exhibit community structure: the connections within each community are dense, while connections between communities are sparser. Moreover, there is a common but non-negligible phenomenon, collective behaviors, during the outbreak of epidemics, are induced by the emergence of epidemics and in turn influence the process of epidemic spread. In this paper, we explore the interaction between epidemic spread and collective behavior in scale-free networks with community structure, by constructing a mathematical model that embeds community structure, behavioral evolution and epidemic transmission. In view of the differences among individuals' responses in different communities to epidemics, we use nonidentical functions to describe the inherent dynamics of individuals. In practice, with the progress of epidemics, individual behaviors in different communities may tend to cluster synchronization, which is indicated by the analysis of our model. By using comparison principle and Gers˘gorin theorem, we investigate the epidemic threshold of the model. By constructing an appropriate Lyapunov function, we present the stability analysis of behavioral evolution and epidemic dynamics. Some numerical simulations are performed to illustrate and complement our theoretical results. It is expected that our work can deepen the understanding of interaction between cluster synchronization and epidemic dynamics in scale-free community networks.

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http://dx.doi.org/10.1016/j.jtbi.2018.11.003 | DOI Listing |

February 2019

Bull Math Biol 2018 08 11;80(8):2049-2087. Epub 2018 Jun 11.

Department of Mathematics, Shanghai University, Shanghai, 200444, China.

Infection age is often an important factor in epidemic dynamics. In order to realistically analyze the spreading mechanism and dynamical behavior of epidemic diseases, in this paper, a generalized disease transmission model of SIS type with age-dependent infection and birth and death on a heterogeneous network is discussed. The model allows the infection and recovery rates to vary and depend on the age of infection, the time since an individual becomes infected. We address uniform persistence and find that the model has the sharp threshold property, that is, for the basic reproduction number less than one, the disease-free equilibrium is globally asymptotically stable, while for the basic reproduction number is above one, a Lyapunov functional is used to show that the endemic equilibrium is globally stable. Finally, some numerical simulations are carried out to illustrate and complement the main results. The disease dynamics rely not only on the network structure, but also on an age-dependent factor (for some key functions concerned in the model).

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http://dx.doi.org/10.1007/s11538-018-0445-z | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7088888 | PMC |

August 2018

Sci Rep 2018 03 19;8(1):4814. Epub 2018 Mar 19.

Department of Physics, Shanghai University, Shanghai, 200444, China.

The threshold model has been widely adopted for modelling contagion processes on social networks, where individuals are assumed to be in one of two states: inactive or active. This paper studies the model on directed networks where nodal inand out-degrees may be correlated. To understand how directionality and correlation affect the breakdown of the system, a theoretical framework based on generating function technology is developed. First, the effects of degree and threshold heterogeneities are identified. It is found that both heterogeneities always decrease systematic robustness. Then, the impact of the correlation between nodal in- and out-degrees is investigated. It turns out that the positive correlation increases the systematic robustness in a wide range of the average in-degree, while the negative correlation has an opposite effect. Finally, a comparison between undirected and directed networks shows that the presence of directionality and correlation always make the system more vulnerable.

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http://dx.doi.org/10.1038/s41598-018-22508-1 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC5859107 | PMC |

March 2018

Math Biosci Eng 2017 Oct/Dec 1;14(5-6):1215-1232

Department of Mathematics, Shanghai University, Shanghai 200444, China email:

When a network reaches a certain size, its node degree can be considered as a continuous variable, which we will call continuous degree. Using continuous degree method (CDM), we analytically calculate certain structure of the network and study the spread of epidemics on a growing network. Firstly, using CDM we calculate the degree distributions of three different growing models, which are the BA growing model, the preferential attachment accelerating growing model and the random attachment growing model. We obtain the evolution equation for the cumulative distribution function F(k,t), and then obtain analytical results about F(k,t) and the degree distribution p(k,t). Secondly, we calculate the joint degree distribution p(k1,k2,t) of the BA model by using the same method, thereby obtain the conditional degree distribution p(k1|k2). We find that the BA model has no degree correlations. Finally, we consider the different states, susceptible and infected, according to the node health status. We establish the continuous degree SIS model on a static network and a growing network, respectively. We find that, in the case of growth, the new added health nodes can slightly reduce the ratio of infected nodes, but the final infected ratio will gradually tend to the final infected ratio of SIS model on static networks.

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http://dx.doi.org/10.3934/mbe.2017062 | DOI Listing |

July 2018

Chaos 2017 Jun;27(6):063101

Department of Mathematics, Shanghai University, Shanghai 200444, China.

During the spread of an epidemic, individuals in realistic networks may exhibit collective behaviors. In order to characterize this kind of phenomenon and explore the correlation between collective behaviors and epidemic spread, in this paper, we construct several mathematical models (including without delay, with a coupling delay, and with double delays) of epidemic synchronization by applying the adaptive feedback motivated by real observations. By using Lyapunov function methods, we obtain the conditions for local and global stability of these epidemic synchronization models. Then, we illustrate that quenched mean-field theory is more accurate than heterogeneous mean-field theory in the prediction of epidemic synchronization. Finally, some numerical simulations are performed to complement our theoretical results, which also reveal some unexpected phenomena, for example, the coupling delay and epidemic delay influence the speed of epidemic synchronization. This work makes further exploration on the relationship between epidemic dynamics and synchronization dynamics, in the hope of being helpful to the study of other dynamical phenomena in the process of epidemic spread.

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http://dx.doi.org/10.1063/1.4984217 | DOI Listing |

June 2017

Philos Trans A Math Phys Eng Sci 2017 Jun;375(2096)

Department of Mathematics, Shanghai University, Shanghai 200444, People's Republic of China

In this paper, we propose a model where two strains compete with each other at the expense of common susceptible individuals on heterogeneous networks by using pair-wise approximation closed by the probability-generating function (PGF). All of the strains obey the susceptible-infected-recovered (SIR) mechanism. From a special perspective, we first study the dynamical behaviour of an SIR model closed by the PGF, and obtain the basic reproduction number via two methods. Then we build a model to study the spreading dynamics of competing viruses and discuss the conditions for the local stability of equilibria, which is different from the condition obtained by using the heterogeneous mean-field approach. Finally, we perform numerical simulations on Barabási-Albert networks to complement our theoretical research, and show some dynamical properties of the model with competing viruses.This article is part of the themed issue 'Mathematical methods in medicine: neuroscience, cardiology and pathology'.

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http://dx.doi.org/10.1098/rsta.2016.0284 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC5434075 | PMC |

June 2017

J Math Biol 2017 04 17;74(5):1263-1298. Epub 2016 Sep 17.

Department of Mathematics, Shanghai University, Shanghai, 200444, China.

We introduce three modified SIS models on scale-free networks that take into account variable population size, nonlinear infectivity, adaptive weights, behavior inertia and time delay, so as to better characterize the actual spread of epidemics. We develop new mathematical methods and techniques to study the dynamics of the models, including the basic reproduction number, and the global asymptotic stability of the disease-free and endemic equilibria. We show the disease-free equilibrium cannot undergo a Hopf bifurcation. We further analyze the effects of local information of diseases and various immunization schemes on epidemic dynamics. We also perform some stochastic network simulations which yield quantitative agreement with the deterministic mean-field approach.

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http://dx.doi.org/10.1007/s00285-016-1057-6 | DOI Listing |

April 2017

J Math Biol 2016 12 15;73(6-7):1561-1594. Epub 2016 Apr 15.

Complex Systems Research Center, Shanxi University, Taiyuan, 030006, Shanxi, China.

In the face of serious infectious diseases, governments endeavour to implement containment measures such as public vaccination at a macroscopic level. Meanwhile, individuals tend to protect themselves by avoiding contacts with infections at a microscopic level. However, a comprehensive understanding of how such combined strategy influences epidemic dynamics is still lacking. We study a susceptible-infected-susceptible epidemic model with imperfect vaccination on dynamic contact networks, where the macroscopic intervention is represented by random vaccination of the population and the microscopic protection is characterised by susceptible individuals rewiring contacts from infective neighbours. In particular, the model is formulated both in populations without and then with demographic effects (births, deaths, and migration). Using the pairwise approximation and the probability generating function approach, we investigate both dynamics of the epidemic and the underlying network. For populations without demography, the emerging degree correlations, bistable states, and oscillations demonstrate the combined effects of the public vaccination program and individual protective behavior. Compared to either strategy in isolation, the combination of public vaccination and individual protection is more effective in preventing and controlling the spread of infectious diseases by increasing both the invasion threshold and the persistence threshold. For populations with additional demographic factors, we investigate temporal evolution of infected individuals and infectious contacts, as well as degree distributions of nodes in each class. It is found that the disease spreads faster but is more restricted in scale-free networks than in the Erdös-Rényi ones. The integration between vaccination intervention and individual rewiring may promote epidemic spreading due to the birth effect. Moreover, the degree distributions of both networks in the steady state is closely related to the degree distribution of newborns, which leads to uncorrelated connectivity. All the results demonstrate the importance of both local protection and global intervention, as well as the demographic effects. Our work thus offers a more comprehensive description of disease containment.

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http://dx.doi.org/10.1007/s00285-016-1007-3 | DOI Listing |

December 2016

Sci Rep 2016 Mar 31;6:23766. Epub 2016 Mar 31.

Department of Mathematics, Shanghai University, Shanghai 200444, China.

The threshold model has been widely adopted as a classic model for studying contagion processes on social networks. We consider asymmetric individual interactions in social networks and introduce a persuasion mechanism into the threshold model. Specifically, we study a combination of adoption and persuasion in cascading processes on complex networks. It is found that with the introduction of the persuasion mechanism, the system may become more vulnerable to global cascades, and the effects of persuasion tend to be more significant in heterogeneous networks than those in homogeneous networks: a comparison between heterogeneous and homogeneous networks shows that under weak persuasion, heterogeneous networks tend to be more robust against random shocks than homogeneous networks; whereas under strong persuasion, homogeneous networks are more stable. Finally, we study the effects of adoption and persuasion threshold heterogeneity on systemic stability. Though both heterogeneities give rise to global cascades, the adoption heterogeneity has an overwhelmingly stronger impact than the persuasion heterogeneity when the network connectivity is sufficiently dense.

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http://dx.doi.org/10.1038/srep23766 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4815014 | PMC |

March 2016

Phys Rev E Stat Nonlin Soft Matter Phys 2015 Jul 27;92(1):010903. Epub 2015 Jul 27.

Department of Mathematics, Shanghai University, Shanghai 200444, People's Republic of China.

We consider practical synchronization on complex dynamical networks under linear feedback control designed by optimal control theory. The control goal is to minimize global synchronization error and control strength over a given finite time interval, and synchronization error at terminal time. By utilizing the Pontryagin's minimum principle, and based on a general complex dynamical network, we obtain an optimal system to achieve the control goal. The result is verified by performing some numerical simulations on Star networks, Watts-Strogatz networks, and Barabási-Albert networks. Moreover, by combining optimal control and traditional pinning control, we propose an optimal pinning control strategy which depends on the network's topological structure. Obtained results show that optimal pinning control is very effective for synchronization control in real applications.

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http://dx.doi.org/10.1103/PhysRevE.92.010903 | DOI Listing |

July 2015

Chaos 2014 Dec;24(4):043124

School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541004, People's Republic of China.

For many epidemic networks some connections between nodes are treated as deterministic, while the remainder are random and have different connection probabilities. By applying spectral analysis to several constructed models, we find that one can estimate the epidemic thresholds of these networks by investigating information from only the deterministic connections. Nonetheless, in these models, generic nonuniform stochastic connections and heterogeneous community structure are also considered. The estimation of epidemic thresholds is achieved via inequalities with upper and lower bounds, which are found to be in very good agreement with numerical simulations. Since these deterministic connections are easier to detect than those stochastic connections, this work provides a feasible and effective method to estimate the epidemic thresholds in real epidemic networks.

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http://dx.doi.org/10.1063/1.4901334 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7112486 | PMC |

December 2014

Phys Rev E Stat Nonlin Soft Matter Phys 2013 Feb 20;87(2):022813. Epub 2013 Feb 20.

Department of Mathematics, Shanghai University, Shanghai 200444, China.

Vaccination is an important measure available for preventing or reducing the spread of infectious diseases. In this paper, an epidemic model including susceptible, infected, and imperfectly vaccinated compartments is studied on Watts-Strogatz small-world, Barabási-Albert scale-free, and random scale-free networks. The epidemic threshold and prevalence are analyzed. For small-world networks, the effective vaccination intervention is suggested and its influence on the threshold and prevalence is analyzed. For scale-free networks, the threshold is found to be strongly dependent both on the effective vaccination rate and on the connectivity distribution. Moreover, so long as vaccination is effective, it can linearly decrease the epidemic prevalence in small-world networks, whereas for scale-free networks it acts exponentially. These results can help in adopting pragmatic treatment upon diseases in structured populations.

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http://dx.doi.org/10.1103/PhysRevE.87.022813 | DOI Listing |

February 2013

Chaos 2012 Dec;22(4):043137

College of Mathematics and Information Science, Jiangxi Normal University, Nanchang 330022, China.

A colored network model, corresponding to a colored graph in mathematics, is used for describing the complexity of some inter-connected physical systems. A colored network is consisted of colored nodes and edges. Colored nodes may have identical or nonidentical local dynamics. Colored edges between any pair of nodes denote not only the outer coupling topology but also the inner interactions. In this paper, first, synchronization of edge-colored networks is studied from adaptive control and pinning control approaches. Then, synchronization of general colored networks is considered. To achieve synchronization of a colored network to an arbitrarily given orbit, open-loop control, pinning control and adaptive coupling strength methods are proposed and tested, with some synchronization criteria derived. Finally, numerical examples are given to illustrate theoretical results.

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http://dx.doi.org/10.1063/1.4769991 | DOI Listing |

December 2012

Chaos 2012 Dec;22(4):043113

School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541004, People's Republic of China.

There are certain correlations between collective behavior and spreading dynamics on some real complex networks. Based on the dynamical characteristics and traditional physical models, we construct several new bidirectional network models of spreading phenomena. By theoretical and numerical analysis of these models, we find that the collective behavior can inhibit spreading behavior, but, conversely, this spreading behavior can accelerate collective behavior. The spread threshold of spreading network is obtained by using the Lyapunov function method. The results show that an effective spreading control method is to enhance the individual awareness to collective behavior. Many real-world complex networks can be thought of in terms of both collective behavior and spreading dynamics and therefore to better understand and control such complex networks systems, our work may provide a basic framework.

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http://dx.doi.org/10.1063/1.4766677 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7112518 | PMC |

December 2012

J Theor Biol 2013 Jan 9;317:133-9. Epub 2012 Oct 9.

Department of Mathematics, Shanghai University, Shanghai 200444, PR China.

The heterogeneous patterns of interactions within a population are often described by contact networks, but the variety and adaptivity of contact strengths are usually ignored. This paper proposes a modified epidemic SIS model with a birth-death process and nonlinear infectivity on an adaptive and weighted contact network. The links' weights, named as 'adaptive weights', which indicate the intimacy or familiarity between two connected individuals, will reduce as the disease develops. Through mathematical and numerical analyses, conditions are established for population extermination, disease extinction and infection persistence. Particularly, it is found that the fixed weights setting can trigger the epidemic incidence, and that the adaptivity of weights cannot change the epidemic threshold but it can accelerate the disease decay and lower the endemic level. Finally, some corresponding control measures are suggested.

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http://dx.doi.org/10.1016/j.jtbi.2012.09.036 | DOI Listing |

January 2013

Chaos 2012 Jun;22(2):023127

College of Mathematics and Information Science, Jiangxi Normal University, Nanchang 330022, China.

In this paper, synchronization of a network coupled with complex-variable chaotic systems is investigated. Adaptive feedback control and intermittent control schemes are adopted for achieving adaptive synchronization and exponential synchronization, respectively. Several synchronization criteria are established. In these schemes, the outer coupling matrix is not necessarily assumed to be symmetric or irreducible. Further, for a class of networks with an irreducible and balanced outer coupling matrix, a pinning control scheme is adopted for achieving synchronization. Numerical simulations are demonstrated to verify the effectiveness of the theoretical results.

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http://dx.doi.org/10.1063/1.4717525 | DOI Listing |

June 2012

Chaos 2012 Mar;22(1):013101

College of Mathematics and Information Science, Jiangxi Normal University, Nanchang 330022, China.

We explore the impact of awareness on epidemic spreading through a population represented by a scale-free network. Using a network mean-field approach, a mathematical model for epidemic spreading with awareness reactions is proposed and analyzed. We focus on the role of three forms of awareness including local, global, and contact awareness. By theoretical analysis and simulation, we show that the global awareness cannot decrease the likelihood of an epidemic outbreak while both the local awareness and the contact awareness can. Also, the influence degree of the local awareness on disease dynamics is closely related with the contact awareness.

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http://dx.doi.org/10.1063/1.3673573 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7112450 | PMC |

March 2012

Chaos 2011 Sep;21(3):033111

School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541004, People's Republic of China.

Many realistic epidemic networks display statistically synchronous behavior which we will refer to as epidemic synchronization. However, to the best of our knowledge, there has been no theoretical study of epidemic synchronization. In fact, in many cases, synchronization and epidemic behavior can arise simultaneously and interplay adaptively. In this paper, we first construct mathematical models of epidemic synchronization, based on traditional dynamical models on complex networks, by applying the adaptive mechanisms observed in real networks. Then, we study the relationship between the epidemic rate and synchronization stability of these models and, in particular, obtain the conditions of local and global stability for epidemic synchronization. Finally, we perform numerical analysis to verify our theoretical results. This work is the first to draw a theoretical bridge between epidemic transmission and synchronization dynamics and will be beneficial to the study of control and the analysis of the epidemics on complex networks.

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http://dx.doi.org/10.1063/1.3622678 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7112447 | PMC |

September 2011

J Syst Sci Complex 2011 11;24(4):619. Epub 2011 Jun 11.

3Department of Mathematics, Shanghai University, Shanghai, 200444 China.

In this paper, epidemic spread with the staged progression model on homogeneous and heterogeneous networks is studied. First, the epidemic threshold of the simple staged progression model is given. Then the staged progression model with birth and death is also considered. The case where infectivity is a nonlinear function of the nodes' degree is discussed, too. Finally, the analytical results are verified by numerical simulations.

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http://dx.doi.org/10.1007/s11424-011-8252-8 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7089252 | PMC |

June 2011

Phys Rev E Stat Nonlin Soft Matter Phys 2009 Jun 10;79(6 Pt 2):067201. Epub 2009 Jun 10.

Department of Mathematics, Shanghai University, Shanghai 200444, People's Republic of China.

During the process of adding links, we find that the synchronizability of the classical Barabási-Albert (BA) scale-free or Watts-Strogatz (WS) small-world networks can be statistically quantified by three essentially structural quantities of these networks, i.e., the eccentricity, variance of the degree distribution, and clustering coefficients. The results indicate that both the eccentricity and clustering coefficient are positively linearly correlated with synchronizability, while the variance is negatively linearly correlated. Moreover, the efficiency of some particular strategies of adding links to change the synchronizability is also investigated. This information can be used to guide us to design corresponding strategies of structure-evolving processes to manipulate the synchronizability of a given network.

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http://dx.doi.org/10.1103/PhysRevE.79.067201 | DOI Listing |

June 2009

Chaos 2009 Jun;19(2):023106

Department of Mathematics, Shanghai University, Shanghai 200444, People's Republic of China.

In this paper dynamical networks with community structure and nonidentical nodes and with identical local dynamics for all individual nodes in each community are considered. The cluster synchronization of these networks with or without time delay is studied by using some feedback control schemes. Several sufficient conditions for achieving cluster synchronization are obtained analytically and are further verified numerically by some examples with chaotic or nonchaotic nodes. In addition, an essential relation between synchronization dynamics and local dynamics is found by detailed analysis of dynamical networks without delay through the stage detection of cluster synchronization.

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http://dx.doi.org/10.1063/1.3125714 | DOI Listing |

June 2009

Phys Rev E Stat Nonlin Soft Matter Phys 2008 Mar 12;77(3 Pt 2):036113. Epub 2008 Mar 12.

Department of Mathematics, Zhejiang Normal University, Jinhua, China.

We examine epidemic thresholds for disease spread using susceptible-infected-susceptible models on scale-free networks with variable infectivity. Infectivity between nodes is modeled as a piecewise linear function of the node degree (rather than the less realistic linear transformation considered previously). With this nonlinear infectivity, we derive conditions for the epidemic threshold to be positive. The effects of various immunization schemes including ring and targeted vaccination are studied and compared. We find that both targeted and ring immunization strategies compare favorably to a proportional scheme in terms of effectiveness.

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http://dx.doi.org/10.1103/PhysRevE.77.036113 | DOI Listing |

March 2008

Nonlinear Biomed Phys 2008 May 1;2(1). Epub 2008 May 1.

School of Mathematics and Computational Science, Anhui University, Hefei 230039, China.

In the study of epidemic spreading two natural questions are: whether the spreading of epidemics on heterogenous networks have multiple routes, and whether the spreading of an epidemic is a local or global behavior? In this paper, we answer the above two questions by studying the SIS model on heterogenous networks, and give the global conditions for the endemic state when two distinct routes with uniform rate of infection are considered. The analytical results are also verified by numerical simulations.

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http://dx.doi.org/10.1186/1753-4631-2-2 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2409347 | PMC |

May 2008

Phys Rev E Stat Nonlin Soft Matter Phys 2007 Nov 16;76(5 Pt 2):056213. Epub 2007 Nov 16.

Department of Mathematics, Shanghai University, Shanghai 200444, People's Republic of China.

We consider discrete dynamical networks, and analytically demonstrate the relation between transverse stability in the Milnor sense and contraction stability, the stability for synchronous manifolds obtained via the partial contraction principle. By contraction for a system, we mean that initial conditions or temporary disturbances are forgotten exponentially fast, so that all trajectories of this system converge to a unique trajectory. In addition, synchronization of star-shaped complex networks is investigated via the partial contraction principle. This example further verifies the interrelation between contraction and transverse stability.

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http://dx.doi.org/10.1103/PhysRevE.76.056213 | DOI Listing |

November 2007