Epidemic Dynamics on an Adaptive Network Gross et al. Published - - PowerPoint PPT Presentation

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epidemic dynamics on an adaptive network
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Epidemic Dynamics on an Adaptive Network Gross et al. Published - - PowerPoint PPT Presentation

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Epidemic Dynamics on an Adaptive Network Gross et al. Published 2006 in Physical Review Letters Presented by Bernhard Liepe 6/8/2018 1 Network topology has a strong impact on node dynamics

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Epidemic Dynamics on an Adaptive Network

Gross et al.

Published 2006 in Physical Review Letters Presented by Bernhard Liepe

6/8/2018 1

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Network topology has a strong impact on node dynamics

https://de.wikipedia.org/wiki/Topologie_(Rechnernetz)

6/8/2018 2

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Simple, but dynamic rules can generate a complex network topology

  • eg. Preferential attachment or

selective rewiring

  • Shape real world networks like

the internet

http://guruprasad.net/posts/world-maps-data-patterns/

6/8/2018 3

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Susceptible-Infected-Susceptible (SIS) model

  • n adaptive an network
  • Nodes can only be
  • Suceptible
  • Infected
  • Parameters
  • Number of nodes

N

  • Bidirectional links

K

  • Infection probability

p

  • Recovery probability

r

  • Rewiring

w

  • Link to infected is cut with probability w

a new link to other suceptible is formed

Epidemic Rewiring

6/8/2018 4

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High rewiring rate increases epidemic threshold

  • Epidemic threshold p* =>

Infection probability necessary to maintain stable infection level

<k> = 2𝐿

𝑂 mean degree of nodes

w - rewiring probability r – recovery probability

6/8/2018 5

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Rewiring leads to seperated sup-populations and a dynamical equilibrium

Random rewiring Rewiring but no local dynamics (p = r = 0) Adaptive network Mean degree of nearest-nieghbor Distribution of node degree

6/8/2018 6

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Adaptive SIS network leads to temporal fluctuations of degree of nodes

  • Susceptible degree shows linear

increase in time

  • Infected degree decays exponentially
  • > dynamic equilibrium
  • Local effect of rewiring suppresses

epidemic

  • Topology effect of rewiring promotes

epidemic

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Dynamic model as a system of coupled differential equations

Infection of susceptibles Recovery of infected SI to II link due to infection II to SI link due to recovery

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Hysteresis loops and discontinous transitions are generic features of the adaptive model

  • adaptive network with rewiring -
  • >Second threshold

(saddle-node bifurcation)

  • > formation of hysteresis loop

6/8/2018 9

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Two parameter bifurcation diagram summarizing model states

  • White and gray -> single attractor
  • Medium gray -> both stable
  • Dark grey -> stable healthy state

coexists with epidemic cycle

  • Dash-dot -> transcritical

bifurcation

  • Dashed -> saddle-node
  • Dotted -> cycle fold

6/8/2018 10

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Thank you for you attention! Feel free to ask questions.

6/8/2018 11

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Additional resources

  • https://de.wikipedia.org/wiki/Topologie_(Rechnernetz); 06.06.18
  • http://guruprasad.net/posts/world-maps-data-patterns/; 06.06.18

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