Paper Search

Complex Systems and Complexity Science >

2017 , Vol. 14 >Issue 4: 14 - 31

DOI: https://doi.org/10.13306/j.1672-3813.2017.04.002

The Thresholds of Some Epidemic Models

  • CUI Yumei ,
  • CHEN Shanshan ,
  • FU Xinchu ,
  • CUI Yumei ,
  • CHEN Shanshan ,
  • FU Xinchu
Expand
  • College of Sciences, Shanghai University, Shanghai 200444, China

Received date: 2017-05-08

  Online published: 2019-01-16

Fold

Abstract

This paper introduces the derivation of basic reproduction numbers for several epidemic models. The basic reproduction number plays an important role in describing the dynamic behavior of infectious disease models,which is an important indicator to determine the prevalence of diseases. Therefore, the basic reproductive number is a significant reference for the prevention and control of diseases and the immunization strategy. The basic reproduction numbers can be derived by means of the definition, the monotonicity of the infected individuals at the initial moment, the existence of the positive equilibrium and the local stability of the disease-free equilibrium, the numerical simulation,respectively. This paper introduces many epidemic models especially network model and calculates their basic reproduction numbers. Finally, we analyze the changes in the basic reproduction numbers during different periods.

Key words： transmission dynamics; basic reproduction number; opidemic model

Cite this article

CUI Yumei , CHEN Shanshan , FU Xinchu , CUI Yumei , CHEN Shanshan , FU Xinchu . The Thresholds of Some Epidemic Models[J]. Complex Systems and Complexity Science, 2017 , 14(4) : 14 -31 . DOI: 10.13306/j.1672-3813.2017.04.002

References

[1]Fe ng Z L, Jorge X, Velasco-Hernández. Competitive exclusion in a vector-host model for the dengue fever[J]. Journal of Mathematical Biology, 1997, 35: 523544.
[2]Chan-Yeung M, Xu R H. SARS: epidemiology[J]. Respirology, 2003, 8: 914.
[3]Small M, Tse C K, Walker D M. Super-spreaders and the rate of transmission of the SARS virus[J]. Physica D, 2006, 215(2): 146158.
[4]Castillo-Chavez C, Feng Z L. To treat or not to treat: the case of tuberculosis[J]. Journal of Mathematical Biology, 1997, 35: 629656.
[5]Castillo-Chavez C, Huang W, Li J. Competitive exclusion in gonorrhea models and other sexually-transmitted diseases[J]. SIAM Journal on Applied Mathematics, 1996, 56: 494508 .
[6]Kermack W O, Mckendrick A G. Contributions to the mathematical theory of epidemics[J]. Proceedings of the Royal Society A, 1927,115: 700721.
[7]Watts D J, Strogatz S H. Collective dynamics of small-world networks[J]. Nature, 1998, 393: 440442.
[8]Barabasi A L, Albert R. Emergence of scaling in random networks[J]. Science, 1999, 286(5439): 509512.
[9]Pastor-Satorras R, Vespignani A. Epidemic spreading in scale-free networks[J]. Physical Review Letters, 2000, 86: 32003203.
[10] Pastor-Satorras R, Vespignani A. Epidemic dynamics in finite size scale-free networks[J]. Physical Review E, 2002, 65(3): 035108.
[11] Pastor-Satorras R, Vespignani A. Epidemic dynamics and endemic states in complex networks[J]. Physical Review E, 2001, 63(6): 066117.
[12] Newman M E J. Spread of epidemic disease on networks[J]. Physical Review E, 2002, 66(1): 016128.
[13] Wang Y, Chakrabarti D, Wang C, et al. Epidemic spreading in real networks: an eigenvalue viewpoint[J]. International Symposium on Reliable Distributed Systems, 2003, 10(13):2543.
[14] Eames K T D. Modelling disease spread through random and regular contacts in clustered populations[J]. Theoretical Population Biology, 2008, 73(1): 104111.
[15] Lindquist J, Ma J, Driessche P V D, et al. Effective degree network disease models[J]. Journal of Mathematical Biology, 2011, 62: 43164.
[16] 李睿琪, 王伟, 舒盼盼, 等. 复杂网络上流行病传播动力学的爆发阈值解析综述[J]. 复杂系统与复杂性科学, 2016, 13(1):139.
Li Ruiqi, Wang Wei, Shu Panpan, et al. Review of threshold theoretical analysis about epidemic spreading dynamics on complex networks[J]. Complex Systems and Complexity Science, 2016, 13(1):1-39.
[17] Van d D P, Watmough J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission[J]. Mathematical Biosciences, 2002, 180(1/2):2948.
[18] Pastor-Satorras R, Vespignani A. Immunization of complex networks[J]. Physical Review E, 2002, 65(3): 036104.
[19] 马知恩, 周义仓. 常微分方程定性与稳定性方法[M]. 北京: 科学出版社, 2001.
[20] Ferreira S C, Castellano C, Pastor-Satorras R. Epidemic thresholds of the susceptible-infected-susceptible model on networks: a comparison of numerical and theoretical results[J]. Physical Review E, 2012, 86(4): 041125.
[21] Shu P, Wang W, Tang M, et al. Simulated identification of epidemic threshold on finite-size networks[J]. Chaos, 2015, 544(10):167.
[22] Liu C, Xie J R, Chen H S, et al. Interplay between the local information based behavioral responses and the epidemic spreading in complex networks[J]. Chaos, 2015, 25(10): 103111.
[23] Newman M E J. Assortative mixing in networks[J]. Physical Review Letters, 2002, 89(20): 208701.
[24] Boguna M, Pastor-Satorras R. Epidemic spreading in correlated complex networks[J]. Physical Review E, 2002, 66(4): 047104.
[25] Boguna M, Pastor-Satorras R, Vespignani A. Absence of epidemic threshold in scale-free networks with degree correlations[J], Physical Review Letters, 2003, 90(2): 028701.
[26] Altmann M. Susceptible-Infected-Removed epidemic models with dynamic partnerships[J]. Journal of Mathematical Biology, 1995, 33: 661675.
[27] Keeling M J, Rand D A, Morris A J. Correlation models for childhood epidemics[J]. Proc Biol Sci, 1997, 264(1385): 11491156.
[28] Filipe J A N, Gibson G J. Comparing approximations to spatio-temporal models for epidemics with local spread[J]. Bulletin of Mathematical Biology, 2001, 63: 603624.
[29] Thomson N A, Ellner S P. Pari-edge approximation for heterogeneous lattice population models[J]. Theoretical Population Biology, 2003, 64: 271280.
[30] Trapman P. Reproduction numbers for epidemics on networks using pair approximation[J]. Mathematical Bioscience, 2007, 210 (2): 464489.
[31] 靳祯, 孙桂全, 刘茂省. 网络传染病动力学建模分析[M]. 北京: 科学出版社, 2014.
[32] Sugimine N, Aihara K. Stability of an equilibrium state in a multi-infectious type SIS model on a truncated network[J]. Artificial Life Robotics, 2007,11: 157161.
[33] Wu Q C, Fu X C, Yang M. Epidemic thresholds in a heterogenous population with competing strains[J]. Chinese Physics B, 2011, 20: 046401.
[34] Read J M, Eames K T D, Edmunds W J. Dynamic social networks and the implicatins for the spread of infectious disease[J]. Journal of the Royal Society Interface, 2008, 5(26): 10011007.
[35] Barrat A, Barthelemy M, Vespignani A. Weighted evolving networks: coupling topology and weight dynamics[J]. Physical Review Letters, 2004,92(22): 228701228704.
[36] Boccaletti S, Latorab V, Morenod Y, et al. Complex networks: structure and dynamics[J]. Physics Reports, 2006, 424: 175308.
[37] Newman M E J. Analysis of weighted networks[J]. Physical Review E, 2004,70(5): 056131
[38] Roshani F, Naimi Y. Effects of degree-biased transmission rate and nonlinear infectivity on rumor spreading in complex social networks[J]. Physical Review E, 2012, 85(3): 036109.
[39] Ma Z E, Li J. Dynamical modeling and analysis of epidemics[J].International Association of Geodesy Symposia, 2009, 106(B11):498.
[40] 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: 15771585.
[41] Liu Q M, Deng C S, Sun M C. The analysis of an epidemic model with time delay on scale-free networks[J]. Physica A, 2014, 410: 7987.
[42] Heesterbeek J A P, Roberts M G. Threshold quantities for helminth infections[J]. Journal of Mathematical Biology, 1995, 33(4): 415434.
[43] Wu Q, Fu X, Jin Z, et al. Influence of dynamic immunization on epidemic spreading in networks[J]. Physica A, 2015, 419:566574.
[44] Chowell G, Nishiura H. Transmission dynamics and control of Ebola virus disease (EVD): a review[J]. BMC Medicine, 2014, 12(1): 196.
Options
Outlines

/

〈 〉

Copyright © 2017 Editorial Office of Complex Systems and Complexity Science
Tel.：0532-85953597 E-mail:fzkxbjb@qdu.edu.cn