Abstract：We consider an SIRS epidemic model with a general direct immunization rate on networks. By constructing suitable Lyapunov functions, we find that the dynamical behvaior of the model is completely determined by the epidemic threshold λc. When λ≤λc, the disease-free equilibrium is globally asymptotically stable; when λ>λc, the endemic equilibrium is globally asymptotically stable. In addition, we propose a uniform direct immunization and a targeted direct immunization. The results show that under the same average immunization rate s there exists a critical immunization-lost rate δc so that the epidemic threshold of the targeted direct immunization is smaller (larger) than that of the uniform direct immunization if δ<δc(δ>δc).
张斐, 吴庆初, 曾广洪. 网络上具有一般直接免疫的SIRS传染病模型分析[J]. 复杂系统与复杂性科学, 2017, 14(1): 81-87.
ZHANG Fei, WU Qingchu, ZENG Guanghong. An Analysis of an SIRS Epidemic Model with General Direct Immunization in Networks[J]. Complex Systems and Complexity Science, 2017, 14(1): 81-87.
[1] Wang X F,Chen G R. Complex networks: small-world, scale-free and beyond[J].Circuits and Systems Magazine,IEEE,2003,3(1): 6-20. [2] Newman M E J. The structure and function of complex networks[J].SIAM Rev,2003,45(2):167-256. [3] Pastor-Satorras R,Vespignani A. Epidemic spreading in scale-free networks[J].Phys Rev Lett,2001,86(14): 3200-3203. [4] Newman M E J.The spread of epidemic disease on networks[J].Phys Rev E,2002,66(1):016128. [5] Pastor-Satorras R, Castellano C, Van Mieghem P, et al. Epidemic processes in complex networks[J].Rev Mod Phys, 2015, 87(3): 925. [6] Xia C Y,Wang Z, Sanz J,et al. Effects of delayed recovery and nonuniform transmission on the spreading of diseases in complex networks[J].Physica A,2013,392:1577-1585. [7] Fu X C, Small M, Walker D M, et al. Epidemic dynamics on scale-free networks with piecewise linear infectivity and immunization[J].Phys Rev E, 2008, 77(3): 036113. [8] 李婵婵, 蒋国平, 宋玉蓉. 动态小世界社团网络上的病毒传播研究[J].复杂系统与复杂性科学, 2014, 11(3): 33-39. Li Chanchan, Jiang Guoping, Song Yurong. Epidemic spreading in dynamic small-world networks with commuity structure[J].Complex Systems and Complexity Sicence, 2014, 11(3): 33-39. [9] Zhang H F, Xie J R,Tang M, et al.Suppression of epidemic spreading in complex networks by local information based behavioral response[J].Chaos,2014,24(4):043106. [10] Grassly N C, Fraser C, Garnett G P. Host immunity and synchronized epidemics of syphilis across the United States[J].Nature, 2005, 433(7024): 417-421. [11] Xu R,Ma Z E,Wang Z P.Global stability of a delayed SIRS epidemic model with saturation incidence and temporary immunity[J].Comput Math Appl,2010, 59:3211-3221. [12] Nagy V. Mean-field theory of a recurrent epidemiological model[J].Phys Rev E, 2009,79(6):066105. [13] 夏承遗,刘忠信,陈增强,等.复杂网络上带有直接免疫的SIRS类传染模型研究[J].控制与决策,2008, 23(4): 468-472. Xia Chengyi, Liu Zhongxin, Chen Zengqiang, et al. SIRS epidemic model with direct immunization on complex networks[J].Control and Decision, 2008, 23(4): 468-472. [14] 石定琴,柯林,周金贵,等. 带直接免疫的SIRS模型中疾病以2种途径传播的动力学分析[J].江西师范大学学报:自然科学版,2013,37(6):637-640. Shi Dingqin, Ke Lin, Zhou Jingui, et al. The dynamical analysis for epidemic transmission on an SIRS model with direct immunization via two distinct routes[J].J Jiangxi Norm Univ (Nat Sci Ed), 2013,37(6):637-640. [15] Yu R Z, Li K Z, Chen B D, et al. Dynamical analysis of an SIRS network model with direct immunization and infective vector[J].Adv Differ Equ-NY, 2015, 2015(1): 1-14. [16] Chen L J, Sun J T.Global stability and optimal control of an SIRS epidemic model on heterogeneous networks[J].Physica A,2014, 410:196-204. [17] Peng X L, Xu X J, Fu X C, et al. Vaccination intervention on epidemic dynamics in networks[J].Phys Rev E, 2013, 87(2): 022813. [18] Michalik D E, Steinberg S P, LaRussa P S, et al. Primary vaccine failure after 1 dose of varicella vaccine in healthy children[J].J Infect Dis,2008,197: 944-949. [19] 王伟, 杨慧, 龚凯, 等. 复杂网络上的局域免疫研究[J].电子科技大学学报, 2013, 42(6): 817-830. Wang Wei, Yang Hui, Gong Kai, et al. Local immunization algorithm on complex networks[J].Journal of University of Electronic Science and Technology of China, 2013, 42(6): 817-830. [20] Li K Z,Xu Z P, Zhu G H, et al. Global stability of a susceptible-infected-susceptible epidemic model on networks with individual awareness[J].Chin Phys B,2014,23(11):118904. [21] Li C H, Tsai C C, Yang S Y. Analysis of epidemic spreading of an SIRS model in complex heterogeneous networks[J].Commun Nonlinear Sci Numer Simulat, 2014, 19(4):1042-1054. [22] 张海峰,王阳阳,汪秉宏.行为反应对复杂网络上传染病动力学的影响[J].复杂系统与复杂性科学,2012,9(3):13-21. Zhang Haifeng, Wang Yangyang, Wang Binghong. The impacts of behavioral responses on the spread of infectious diseases on complex networks[J].Complex Systems and Complexity Sicence, 2012,9(3):13-21. [23] Wu Q C, Zhang H F,Zeng G H. Responsive immunization and intervention for infectious diseases in social networks[J].Chaos,2014,24(2):023108. [24] Pastor-Satorras R, Vespignani A. Immunization of complex networks[J].Phys Rev E, 2002, 65(3): 036104. [25] Lou J, Ruggeri T. The dynamics of spreading and immune strategies of sexually transmitted diseases on scale-free network[J].J Math Anal Appl, 2010, 365(1): 210-219. [26] 王琴, 祝光湖, 傅新楚. 有向网络上流行病阈值比较和免疫分析[J].复杂系统与复杂性科学, 2012, 9(4): 26-33. Wang Qin, Zhu Guanghu, Fu Xinchu. Comparison of epidemic thresholds on directednetworks and immunizatio nanalysis[J].Complex Systems and Complexity Sicence, 2012, 9(4): 26-33. [27] Barabási A L, Albert R. Emergence of scaling in random networks[J].Science, 1999,286(5439):509-512. [28] 陈淑芳,吴庆初.网络上局部行为反应对爆发阈值的影响[J].江西师范大学学报:自然科学版,2015,39(5):531-535. Chen Shufang, Wu qingchu. The impact of local behavioral response on epidemic spreading in networks[J].J Jiangxi Norm Univ Nat Sci Ed, 2015,39(5):531-535.
