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 network– 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 network– 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: β
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 – 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: 𝑖𝑖 = 𝑖𝑖0𝑒𝑒𝛽𝛽 𝑘𝑘 𝑡𝑡
1−𝑖𝑖0+𝑖𝑖0𝑒𝑒𝛽𝛽 𝑘𝑘 𝑡𝑡
Final regime: 𝑖𝑖 ∞ = 1
Epidemic threshold: No threshold
Exponential regime: 𝑖𝑖 = 1 − 𝜇𝜇𝛽𝛽 𝑘𝑘
𝐶𝐶𝑒𝑒 𝛽𝛽 𝑘𝑘 −𝑢𝑢 𝑡𝑡
1−𝐶𝐶𝑒𝑒 𝛽𝛽 𝑘𝑘 −𝑢𝑢 𝑡𝑡
Final regime: 𝑖𝑖 ∞ = 1 − 𝜇𝜇𝛽𝛽 𝑘𝑘
Epidemic threshold: 𝑅𝑅0 = 1
Exponential regime: No closed solution
Final regime: 𝑖𝑖 ∞ = 0
Epidemic threshold: 𝑅𝑅0 = 1
11/29/2018 Network Epidemic 5
Real networks are not homogeneous
Real contagions are regional − Individuals can transmit a pathogen only to
those they come into contact with − Epidemics spread along links in a network
Real networks are often scale-free − Nodes have different dgrees
− ⟨k⟩ 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
− Place all nodes that have the same degree into the same block
− No elimination of the compartments based on the state of an individual
− 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 – 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 Θk(t)
of my neighbors are infected.
𝑖𝑖𝑘𝑘 = 𝑖𝑖0(1 + 𝑘𝑘 𝑘𝑘 −1𝑘𝑘2 − 𝑘𝑘
(𝑒𝑒𝑡𝑡𝜏𝜏𝑆𝑆𝑆𝑆 − 1)) 𝜏𝜏𝑆𝑆𝑆𝑆 =
𝑘𝑘𝛽𝛽( 𝑘𝑘2 − 𝑘𝑘 )
characteristic time for the spread of the pathogen
𝑖𝑖 = ∑ 𝑖𝑖𝑘𝑘𝑝𝑝𝑘𝑘 = 𝑖𝑖0(1 + 𝑘𝑘 2− 𝑘𝑘𝑘𝑘2 − 𝑘𝑘
(𝑒𝑒𝑡𝑡𝜏𝜏𝑆𝑆𝑆𝑆 − 1))
11/29/2018 Network Epidemic 9
Network epidemic– 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 – 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 Θk(t)
of my neighbors are infected.
𝜏𝜏𝑆𝑆𝑆𝑆𝑆𝑆 =𝑘𝑘
𝛽𝛽 𝑘𝑘2 − 𝜇𝜇 𝑘𝑘 characteristic time for the spread of the pathogen
Recovery term
𝜆𝜆 =𝛽𝛽𝜇𝜇
spreading rate: High λ indicates high spreading likelihood
Threshold λc
11/29/2018 Network Epidemic 11
Network epidemic– SIS model on random network
Epidemic threshold − 𝜆𝜆𝑐𝑐 = 1
𝑘𝑘 +1
− Exceeds threshold: spread until it reaches an endemic state
− Less than threshold: dies out
11/29/2018 Network Epidemic 12
Network epidemic– SIS model on scale-free network
Epidemic threshold − 𝜆𝜆𝑐𝑐 = 𝑘𝑘
𝑘𝑘2
− In large networks: 𝜆𝜆𝑐𝑐 → 0, 𝜏𝜏 → 0 − This is bad: even viruses that are hard to
pass from individual to individual can spread successfully!
− 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 τ λc
SI 𝑑𝑑𝑖𝑖𝑘𝑘𝑑𝑑𝑑𝑑
= 𝛽𝛽 1 − 𝑖𝑖𝑘𝑘 𝑘𝑘𝜃𝜃𝑘𝑘 𝑘𝑘
𝛽𝛽( 𝑘𝑘2 − 𝑘𝑘 )
0
SIS 𝑑𝑑𝑖𝑖𝑘𝑘𝑑𝑑𝑑𝑑
= 𝛽𝛽 1 − 𝑖𝑖𝑘𝑘 𝑘𝑘𝜃𝜃𝑘𝑘 − 𝜇𝜇𝑖𝑖𝑘𝑘 𝑘𝑘
𝛽𝛽 𝑘𝑘2 − 𝜇𝜇 𝑘𝑘
𝑘𝑘𝑘𝑘2
SIR 𝑑𝑑𝑖𝑖𝑘𝑘
𝑑𝑑𝑑𝑑= 𝛽𝛽 1 − 𝑖𝑖𝑘𝑘 𝑘𝑘𝜃𝜃𝑘𝑘 − 𝜇𝜇𝑖𝑖𝑘𝑘 𝑠𝑠𝑘𝑘 = 1 − 𝑖𝑖𝑙𝑙 − 𝑟𝑟𝑘𝑘
𝑘𝑘 𝛽𝛽 𝑘𝑘2 − (𝜇𝜇 + 𝛽𝛽) 𝑘𝑘
1
𝑘𝑘2𝑘𝑘 − 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
Top Related