network epidemic - cs.rpi.edu
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Network Epidemic
F e i m i Yu 11 / 2 9 / 2 0 1 8
11/29/2018 Network Epidemic 2
11/29/2018
Recall of epidemic models without networkโ basic states
Susceptible (healthy)
Infected (sick)
Removed (immune / dead)
S I R Infection
Recovery
Recovery
Removal
11/29/2018 Network Epidemic 3
11/29/2018
Recall of epidemic models without networkโ definitions
Each individual has <k> contacts (degrees)
Likelihood that the disease will be transmitted from an infected to a healthy individual in a unit time: ฮฒ
Number of infected individuals: I. Infected ratio i = I/N
Number of susceptible individuals: S Susceptible ratio s = S/N
11/29/2018 Network Epidemic 4
11/29/2018
Recall of epidemic models without network โ comparisons
Susceptible (healthy)
Infected (sick)
Removed (immune / dead)
S I R Infection
Recovery
Recovery
Removal
Susceptible (healthy)
Infected (sick)
Removed (immune / dead)
S I R Infection
Recovery
Recovery
Removal
Susceptible (healthy)
Infected (sick)
Removed (immune / dead)
S I R Infection
Recovery
Recovery
Removal
Time (t)
Frac
tion
Infe
cted
i(t
) Fr
actio
n In
fect
ed i(
t)
Time (t)
Exponential regime: ๐๐ = ๐๐0๐๐๐ฝ๐ฝ ๐๐ ๐ก๐ก
1โ๐๐0+๐๐0๐๐๐ฝ๐ฝ ๐๐ ๐ก๐ก
Final regime: ๐๐ โ = 1
Epidemic threshold: No threshold
Exponential regime: ๐๐ = 1 โ ๐๐๐ฝ๐ฝ ๐๐
๐ถ๐ถ๐๐ ๐ฝ๐ฝ ๐๐ โ๐ข๐ข ๐ก๐ก
1โ๐ถ๐ถ๐๐ ๐ฝ๐ฝ ๐๐ โ๐ข๐ข ๐ก๐ก
Final regime: ๐๐ โ = 1 โ ๐๐๐ฝ๐ฝ ๐๐
Epidemic threshold: ๐ ๐ 0 = 1
Exponential regime: No closed solution
Final regime: ๐๐ โ = 0
Epidemic threshold: ๐ ๐ 0 = 1
11/29/2018 Network Epidemic 5
Real networks are not homogeneous
Real contagions are regional โ Individuals can transmit a pathogen only to
those they come into contact with โ Epidemics spread along links in a network
Real networks are often scale-free โ Nodes have different dgrees
โ โจkโฉ is not sufficient to characterize the topology
The black death pandemic
Homogeneous mixing model is not realistic!
11/29/2018 Network Epidemic 6
11/29/2018
Comparison between homogeneous mixing and real network
11/29/2018 Network Epidemic 7
Degree block approximation
Degrees is the only variable that matters in network epidemic model
โ Place all nodes that have the same degree into the same block
โ No elimination of the compartments based on the state of an individual
โ Independent of its degree an individual can be susceptible to the disease (empty circles) or infected (full circles).
11/29/2018 Network Epidemic 8
Network epidemic โ SI model
SI model:
Density of infected neighbors of nodes
with degree k
Average degree of the
block k
Split nodes by their degrees
I am susceptible with k neighbors, and ฮk(t)
of my neighbors are infected.
๐๐๐๐ = ๐๐0(1 + ๐๐ ๐๐ โ1๐๐2 โ ๐๐
(๐๐๐ก๐ก๐๐๐๐๐๐ โ 1)) ๐๐๐๐๐๐ =
๐๐๐ฝ๐ฝ( ๐๐2 โ ๐๐ )
characteristic time for the spread of the pathogen
๐๐ = โ ๐๐๐๐๐๐๐๐ = ๐๐0(1 + ๐๐ 2โ ๐๐๐๐2 โ ๐๐
(๐๐๐ก๐ก๐๐๐๐๐๐ โ 1))
11/29/2018 Network Epidemic 9
Network epidemicโ SI model
An ER random network with <k> = 2
At any time, the fraction of high degree nodes that are infected is higher than the fraction of low degree nodes.
All hubs are infected at any time
11/29/2018 Network Epidemic 10
Network epidemic โ SIS model
SI model:
Density of infected neighbors of nodes
with degree k
Average degree of the
block k
Split nodes by their degrees
I am susceptible with k neighbors, and ฮk(t)
of my neighbors are infected.
๐๐๐๐๐๐๐๐ =๐๐
๐ฝ๐ฝ ๐๐2 โ ๐๐ ๐๐ characteristic time for the spread of the pathogen
Recovery term
๐๐ =๐ฝ๐ฝ๐๐
spreading rate: High ฮป indicates high spreading likelihood
Threshold ฮปc
11/29/2018 Network Epidemic 11
Network epidemicโ SIS model on random network
Epidemic threshold โ ๐๐๐๐ = 1
๐๐ +1
โ Exceeds threshold: spread until it reaches an endemic state
โ Less than threshold: dies out
11/29/2018 Network Epidemic 12
Network epidemicโ SIS model on scale-free network
Epidemic threshold โ ๐๐๐๐ = ๐๐
๐๐2
โ In large networks: ๐๐๐๐ โ 0, ๐๐ โ 0 โ This is bad: even viruses that are hard to
pass from individual to individual can spread successfully!
โ Why? Hubs can broadcast a pathogen to a large number of other nodes!
11/29/2018 Network Epidemic 13
11/29/2018
Comparison of the network epidemic models
Model Continuum Equation ฯ ฮปc
SI ๐๐๐๐๐๐๐๐๐๐
= ๐ฝ๐ฝ 1 โ ๐๐๐๐ ๐๐๐๐๐๐ ๐๐
๐ฝ๐ฝ( ๐๐2 โ ๐๐ )
0
SIS ๐๐๐๐๐๐๐๐๐๐
= ๐ฝ๐ฝ 1 โ ๐๐๐๐ ๐๐๐๐๐๐ โ ๐๐๐๐๐๐ ๐๐
๐ฝ๐ฝ ๐๐2 โ ๐๐ ๐๐
๐๐๐๐2
SIR ๐๐๐๐๐๐
๐๐๐๐= ๐ฝ๐ฝ 1 โ ๐๐๐๐ ๐๐๐๐๐๐ โ ๐๐๐๐๐๐ ๐ ๐ ๐๐ = 1 โ ๐๐๐๐ โ ๐๐๐๐
๐๐ ๐ฝ๐ฝ ๐๐2 โ (๐๐ + ๐ฝ๐ฝ) ๐๐
1
๐๐2๐๐ โ 1
11/29/2018 Network Epidemic 14
11/29/2018
Summary
Network epidemics behave very differently than simple epidemic models
Nodes with higher degree (hubs) are more vulnerable than those with lower degree
In SIS and SIR models, even pathogen with low spreading rate persists iin scale-free networks because of the broadcasting ability of the hubs