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**Modeling and analysis of epidemic diffusion within small-world network.**
*(English)*
Zbl 1251.91051

Summary: To depict the rule of epidemic diffusion, two different models, the Susceptible-Exposure-Infected-Recovered-Susceptible (SEIRS) model and the Susceptible-Exposure-Infected-Quarantine-Recovered-Susceptible (SEIQRS) model, are proposed and analyzed within small-world networks. Firstly, the epidemic diffusion models are constructed with mean-filed theory, and condition for the occurrence of disease diffusion is explored. Then, the existence and global stability of the disease-free equilibrium and the endemic equilibrium for these two complex epidemic systems are proved by differential equations knowledge and Routh-Hurwiz theory. At last, a numerical example which includes key parameters analysis and critical topic discussion is presented to test how well the proposed two models may be applied in practice. These works may provide some guidelines for decision makers when coping with epidemic diffusion controlling problems.

### MSC:

91D30 | Social networks; opinion dynamics |

92D30 | Epidemiology |

05C82 | Small world graphs, complex networks (graph-theoretic aspects) |

### Keywords:

Susceptible-Exposure-Infected-Recovered-Susceptible (SEIRS) model; Susceptible-Exposure-Infected-Quarantine-Recovered-Susceptible (SEIQRS) model
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\textit{M. Liu} and \textit{Y. Xiao}, J. Appl. Math. 2012, Article ID 841531, 14 p. (2012; Zbl 1251.91051)

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### References:

[1] | L. M. Wein, D. L. Craft, and E. H. Kaplan, “Emergency response to an anthrax attack,” Proceedings of the National Academy of Sciences of the United States of America, vol. 100, no. 7, pp. 4346-4351, 2003. |

[2] | L. M. Wein, Y. Liu, and T. J. Leighton, “HEPA/vaccine plan for indoor anthrax remediation,” Emerging Infectious Diseases, vol. 11, no. 1, pp. 69-76, 2005. |

[3] | D. L. Craft, L. M. Wein, and A. H. Wilkins, “Analyzing bioterror response logistics: the case of anthrax,” Management Science, vol. 51, no. 5, pp. 679-694, 2005. · Zbl 1232.35172 |

[4] | E. H. Kaplan, D. L. Craft, and L. M. Wein, “Emergency response to a smallpox attack: the case for mass vaccination,” Proceedings of the National Academy of Sciences of the United States of America, vol. 99, no. 16, pp. 10935-10940, 2002. |

[5] | E. H. Kaplan, D. L. Craft, and L. M. Wein, “Analyzing bioterror response logistics: the case of smallpox,” Mathematical Biosciences, vol. 185, no. 1, pp. 33-72, 2003. · Zbl 1035.92039 |

[6] | H. Matsuura, K. Koide, N. Noda, T. Nemoto, M. Nakano, and K. I. Makino, “Stochastic dynamics in biological system and information,” International Journal of Innovative Computing, Information and Control, vol. 4, no. 2, pp. 233-248, 2008. |

[7] | P. L. Shi and L. Z. Dong, “Dynamical models for infectious diseases with varying population size and vaccinations,” Journal of Applied Mathematics, vol. 2012, Article ID 824192, 20 pages, 2012. · Zbl 1235.37036 |

[8] | D. J. Watts and S. H. Strogatz, “Collective dynamics of ’small-world9 networks,” Nature, vol. 393, no. 6684, pp. 440-442, 1998. · Zbl 1368.05139 |

[9] | S. Eubank, H. Guclu, V. S. A. Kumar et al., “Modelling disease outbreaks in realistic urban social networks,” Nature, vol. 429, no. 6988, pp. 180-184, 2004. |

[10] | M. M. Telo Da Gama and A. Nunes, “Epidemics in small world networks,” European Physical Journal B, vol. 50, no. 1-2, pp. 205-208, 2006. |

[11] | J. Saramäki and K. Kaski, “Modelling development of epidemics with dynamic small-world networks,” Journal of Theoretical Biology, vol. 234, no. 3, pp. 413-421, 2005. |

[12] | N. Masuda and N. Konno, “Multi-state epidemic processes on complex networks,” Journal of Theoretical Biology, vol. 243, no. 1, pp. 64-75, 2006. |

[13] | X. J. Xu, H. O. Peng, X. M. Wang, and Y. H. Wang, “Epidemic spreading with time delay in complex networks,” Physica A, vol. 367, pp. 525-530, 2006. |

[14] | X. P. Han, “Disease spreading with epidemic alert on small-world networks,” Physics Letters, Section A, vol. 365, no. 1-2, pp. 1-5, 2007. · Zbl 1203.05143 |

[15] | T. E. Stone, M. M. Jones, and S. R. McKay, “Comparative effects of avoidance and vaccination in disease spread on a dynamic small-world network,” Physica A, vol. 389, no. 23, pp. 5515-5520, 2010. |

[16] | C. I. Hsu and H. H. Shih, “Transmission and control of an emerging influenza pandemic in a small-world airline network,” Accident Analysis and Prevention, vol. 42, no. 1, pp. 93-100, 2010. |

[17] | H. Y. Wang, X. P. Wang, and A. Z. Zeng, “Optimal material distribution decisions based on epidemic diffusion rule and stochastic latent period for emergency rescue,” International Journal of Mathematics in Operational Research, vol. 1, no. 1-2, pp. 76-96, 2009. · Zbl 1176.90060 |

[18] | K. Y. Tham, “An emergency department response to severe acute respiratory syndrome: a prototype response to bioterrorism,” Annals of Emergency Medicine, vol. 43, no. 1, pp. 6-14, 2004. |

[19] | J. Marro and R. Dickman, Nonequilibrium Phase Transitions in Lattice Models, Collection Aléa-Saclay: Monographs and Texts in Statistical Physics, Cambridge University Press, Cambridge, UK, 1999. · Zbl 0931.70002 |

[20] | M. Liu and L. Zhao, “Analysis for epidemic diffusion and emergency demand in an anti-bioterrorism system,” International Journal of Mathematical Modelling and Numerical Optimisation, vol. 2, no. 1, pp. 51-68, 2011. · Zbl 1205.92067 |

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