SIS病毒传播模型在单向网络中的动力学研究
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摘要
SIS(Susceptible-Infected-Susceptible)病毒传播模型属于病毒传播中较为经典的一类传播模型。研究病毒在单向网络中传播的临界现象,以及传播概率、恢复概率和出度平均度对临界值的影响。理论分析表明,当恢复概率为定值时,网络传播临界值与网络的出度平均度呈反比例关系。恢复概率为1时,传播临界值与初度平均度同样呈反比例关系。传播概率为定值时,网络的恢复临界值与网络的出度平均度呈正比例关系,发生临界现象时,传播概率和恢复概率呈线性关系。实验结果表明,随着初始感染密度的不同,临界值的仿真值也存在差异,当恢复概率和出度平均度为定值时,随着初始感染密度的增大,临界值的仿真值也逐渐增大。同时,当传播概率和出度平均度为定值时,临界值的仿真值也随着初始感染密度的增大而增大。随着初始感染密度的增大,临界值的仿真值与理论值之间的误差逐渐变小。当初始感染密度接近于1时,临界值的仿真值大于理论值。
SIS epidemic spreading model is a classical propagation model. This paper investigated the critical phenomena of virus propagating in unidirectional networks and the factors affecting the threshold by propagation probability,recovery probability and average degree of output. Theoretical analysis shows that the threshold decreases with the increase of one-way network average degree when the recovery probability is constant. The propagation threshold is also inversely proportional to the initial average when the recovery probability is one. When the propagation probability is constant,the threshold increases with the increase of the average degree. When a critical phenomenon occurs,the probabilities of propagation and recovery are linear. The simulation results also show that the thresholds vary with the initial density of infection. When the recovery probability and average degree of output are constant,the simulation value of threshold increases with the increase of the density of the initial infection. When the propagation probability and the average degree of output are fixed,the simulation value of threshold increases with the increase of the initial infection density. With the initial infection density increasing,the error between theoretical and simulation values reduces gradually. When the initial infection density closes to one,the simulation value will exceed the theoretical value.
SIS epidemic spreading model is a classical propagation model. This paper investigated the critical phenomena of virus propagating in unidirectional networks and the factors affecting the threshold by propagation probability,recovery probability and average degree of output. Theoretical analysis shows that the threshold decreases with the increase of one-way network average degree when the recovery probability is constant. The propagation threshold is also inversely proportional to the initial average when the recovery probability is one. When the propagation probability is constant,the threshold increases with the increase of the average degree. When a critical phenomenon occurs,the probabilities of propagation and recovery are linear. The simulation results also show that the thresholds vary with the initial density of infection. When the recovery probability and average degree of output are constant,the simulation value of threshold increases with the increase of the density of the initial infection. When the propagation probability and the average degree of output are fixed,the simulation value of threshold increases with the increase of the initial infection density. With the initial infection density increasing,the error between theoretical and simulation values reduces gradually. When the initial infection density closes to one,the simulation value will exceed the theoretical value.
引文
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[15]李婵婵,蒋国平.社团结构网络环境下SIS病毒传播建模与分析[J].复杂系统与复杂性科学,2016,13(2):67-73.
[16]杨康,张仲义.基于复杂网络理论的供应链网络风险传播机理研究[J].系统科学与数学,2013,33(10):1224-1232.
[17]Kang H,Fu X.Epidemic spreading and global stability of an SIS model with an infective vector on complex networks[J].Communications in Nonlinear Science&Numerical Simulation,2015,27(1-3):30-39.
[18]Xu C.Global Threshold Dynamics of a Stochastic Differential Equation SIS Model[J].Journal of Mathematical Analysis&Applications,2017,447(2):736-757.
[19]Teng Z,Wang L.Persistence and extinction for a class of stochastic SIS epidemic models with nonlinear incidence rate[J].Physica A Statistical Mechanics&Its Applications,2016,451:507-518.
[20]Rifhat R,Wang L,Teng Z.Dynamics for a class of stochastic SIS epidemic models with nonlinear incidence and periodic coefficients[J].Physica A Statistical Mechanics&Its Applications,2017,481:176-190.
[2]Morita S,Jin Y.Disadvantages of Preferential Dispersals in Fluctuating Environments[J].Journal of the Physical Society of Japan,2015,84(3):034801.
[3]鲁延玲,蒋国平,宋玉蓉.自适应网络中病毒传播的稳定性和分岔行为研究[J].物理学报,2013,62(13):130202.
[4]Hung W,Jiang R,Hu M B,et al.Effect of incubation period on epidemic spreading in complex networks[J].Chinese Physics,2009,18(4):1306-1311.
[5]Zhang H,Small M,Chu A F X.Different Epidemic Models on Complex Networks[J].Communications in Theoretical Physics,2009,52(7):180-184.
[6]王延,郑志刚.无标度网络上的传播动力学[J].物理学报,2009,58(7):4421-4425.
[7]李翔.复杂动态网络传播动力学[J].力学进展,2008,38(6):723-732.
[8]汪小帆.复杂网络理论及其应用[M].北京:清华大学出版社,2006.
[9]周涛,傅忠谦,牛永伟,等.复杂网络上传播动力学研究综述[J].自然科学进展,2005,15(5):513-518.
[10]Pastor-Satorras R,Vespignani A.Epidemic dynamics and endemic states in complex networks[J].Physical Review E Statistical Nonlinear&Soft Matter Physics,2001,63(2):066117.
[11]Shi H,Duan Z,Chen G.An SIS model with infective medium on complex networks[J].Physica A Statistical Mechanics&Its Applications,2008,387(8-9):2133-2144.
[12]Erdor P,Renyi A.On the evolution of random graphs[J].Publ Math Inst Hung Acad Sci,1960,5:17-61.
[13]Watts D J,Strogatz S H.Collective dynamic of small-world networks[J].Nature,1998,393:440-442.
[14]王亚奇,蒋国平.一种新的具有传播媒介的SIS病毒传播模型[C]//全国复杂网络学术会议论文.2009.
[15]李婵婵,蒋国平.社团结构网络环境下SIS病毒传播建模与分析[J].复杂系统与复杂性科学,2016,13(2):67-73.
[16]杨康,张仲义.基于复杂网络理论的供应链网络风险传播机理研究[J].系统科学与数学,2013,33(10):1224-1232.
[17]Kang H,Fu X.Epidemic spreading and global stability of an SIS model with an infective vector on complex networks[J].Communications in Nonlinear Science&Numerical Simulation,2015,27(1-3):30-39.
[18]Xu C.Global Threshold Dynamics of a Stochastic Differential Equation SIS Model[J].Journal of Mathematical Analysis&Applications,2017,447(2):736-757.
[19]Teng Z,Wang L.Persistence and extinction for a class of stochastic SIS epidemic models with nonlinear incidence rate[J].Physica A Statistical Mechanics&Its Applications,2016,451:507-518.
[20]Rifhat R,Wang L,Teng Z.Dynamics for a class of stochastic SIS epidemic models with nonlinear incidence and periodic coefficients[J].Physica A Statistical Mechanics&Its Applications,2017,481:176-190.