network epidemic - cs.rpi.edu

Network Epidemic

F e i m i Yu 11 / 2 9 / 2 0 1 8

11/29/2018 Network Epidemic 2

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Recall of epidemic models without network– basic states

Susceptible (healthy)

Infected (sick)

Removed (immune / dead)

S I R Infection

Recovery

Recovery

Removal

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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

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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!

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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

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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

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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