April, Beijing 全局接种与个体保护对流行病传播的影响 许新建 上海大学数学系 上海大学系统科学研究所

1 流行病传播的数学建模 2 SIV接种模型 3 自适应网络模型 4 静态自适应网络上的SIV模型 5 动态自适应网络上的SIV模型 6 结论

流行病传播的数学建模 1760年Bernoulli研究天花的预防接种 Daniel Bernoulli (1700–1782)

1926年Kermack和McKendrick提出SIR仓室模型

leaky/imperfect vaccine 接种模型 S: Susceptible I: Infected V: Vaccinated temporary immunity leaky/imperfect vaccine C. M. Kribs-Zaleta and J. X. Velasco-Hernandez, A simple vaccination model with multiple endemic states, Math. Biosci. 164, 183 (2000).

density of infected /susceptible individuals at time t 静态均匀网络上SIV模型动力学 density of infected /susceptible individuals at time t X.-L. Peng, X.-J. Xu, X. Fu and T. Zhou, Vaccination intervention on epidemic dynamics in networks, Phys. Rev. E 87 (2013) 022813.

静态非均匀网络上SIV模型动力学 X.-L. Peng, X.-J. Xu, X. Fu and T. Zhou, Vaccination intervention on epidemic dynamics in networks, Phys. Rev. E 87 (2013) 022813.

Theorem 2 Consider system (2) ,

静态自适应网络上SIS模型动力学 S I T. Gross, C. J. D. D’Lima and B. Blasius, Epidemic Dynamics on an Adaptive Network, Phys. Rev. Lett. 96 (2006) 208701.

静态自适应网络上SIV模型动力学 S I (V) ω

Equations for the evolution of node states: δα α β φ ϕ S V I

Equations for the evolution of links:

Triples S I V I S S I V I V I S I V

Bifurcation Structure

Dependence on infection probability and vaccination rate for ω=0 Dependence on infection probability and vaccination rate for ω=0.04 (a) and ω=0.2

Dependence on infection probability and rewiring rate

Network Topology Degree distributions of susceptible(black), infected(red) and vaccinated(blue) nodes for different density of the infected. (a) i=0.99; (b) i=0.53.

Mean nearest-neighbor degree of susceptible(black), infected(red) and vaccinated(blue) nodes as a function of degree for different density of infected. (a) i=0.99; (b) i=0.53.

动态自适应网络上SIV模型动力学 Individuals enter with rate η1 ; Newcomers connect to the existing nodes randomly ; Every node die with rate η2 ; Infected nodes die from disease with rate μ ; Assume that all nodes with the same degree are statistically equivalent in heterogeneous networks.

Susceptible nodes in degree class k : assuming

Infected and vaccinated nodes in degree class k :

Summing up above equations yields

Total degree of the susceptible, infected and vaccinated nodes

Evolutionary equations for links

in which

Triples as I-S-I can be estimated by

η1=η2= μ=0.01 Evolution of the numbers of infected individuals. Degree distribution of new nodes : Poisson. (a) Initial network: Poisson; (b) Initial network: Scale-free.

η1=0.013, η2= μ=0.01

(a) (b) Degree distribution of susceptible (black), infected (red) and vaccinated (blue) individuals at time t=2000.

结论 Larger threshold Bistable and Oscillation Dual degree correlation Profound of birth and death Characteristic degree

谢谢大家！