Network Epidemic - cs.rpi.edu

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Network Epidemic Feimi Yu 11/29/2018

Transcript of Network Epidemic - cs.rpi.edu

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Network Epidemic

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

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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: Ξ²

Number of infected individuals: I. Infected ratio i = I/N

Number of susceptible individuals: S Susceptible ratio s = S/N

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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 π‘˜π‘˜ 𝑑𝑑

1βˆ’π‘–π‘–0+𝑖𝑖0𝑒𝑒𝛽𝛽 π‘˜π‘˜ 𝑑𝑑

Final regime: 𝑖𝑖 ∞ = 1

Epidemic threshold: No threshold

Exponential regime: 𝑖𝑖 = 1 βˆ’ πœ‡πœ‡π›½π›½ π‘˜π‘˜

𝐢𝐢𝑒𝑒 𝛽𝛽 π‘˜π‘˜ βˆ’π‘’π‘’ 𝑑𝑑

1βˆ’πΆπΆπ‘’π‘’ 𝛽𝛽 π‘˜π‘˜ βˆ’π‘’π‘’ 𝑑𝑑

Final regime: 𝑖𝑖 ∞ = 1 βˆ’ πœ‡πœ‡π›½π›½ π‘˜π‘˜

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 βˆ’ Individuals can transmit a pathogen only to

those they come into contact with βˆ’ Epidemics spread along links in a network

Real networks are often scale-free βˆ’ Nodes have different dgrees

βˆ’ ⟨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

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Degree block approximation

Degrees is the only variable that matters in network epidemic model

βˆ’ Place all nodes that have the same degree into the same block

βˆ’ No elimination of the compartments based on the state of an individual

βˆ’ Independent of its degree an individual can be susceptible to the disease (empty circles) or infected (full circles).

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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.

π‘–π‘–π‘˜π‘˜ = 𝑖𝑖0(1 + π‘˜π‘˜ π‘˜π‘˜ βˆ’1π‘˜π‘˜2 βˆ’ π‘˜π‘˜

(π‘’π‘’π‘‘π‘‘πœπœπ‘†π‘†π‘†π‘† βˆ’ 1)) πœπœπ‘†π‘†π‘†π‘† =

π‘˜π‘˜π›½π›½( π‘˜π‘˜2 βˆ’ π‘˜π‘˜ )

characteristic time for the spread of the pathogen

𝑖𝑖 = βˆ‘ π‘–π‘–π‘˜π‘˜π‘π‘π‘˜π‘˜ = 𝑖𝑖0(1 + π‘˜π‘˜ 2βˆ’ π‘˜π‘˜π‘˜π‘˜2 βˆ’ π‘˜π‘˜

(π‘’π‘’π‘‘π‘‘πœπœπ‘†π‘†π‘†π‘† βˆ’ 1))

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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

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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.

πœπœπ‘†π‘†π‘†π‘†π‘†π‘† =π‘˜π‘˜

𝛽𝛽 π‘˜π‘˜2 βˆ’ πœ‡πœ‡ π‘˜π‘˜ characteristic time for the spread of the pathogen

Recovery term

πœ†πœ† =π›½π›½πœ‡πœ‡

spreading rate: High Ξ» indicates high spreading likelihood

Threshold Ξ»c

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Network epidemic– SIS model on random network

Epidemic threshold βˆ’ πœ†πœ†π‘π‘ = 1

π‘˜π‘˜ +1

βˆ’ Exceeds threshold: spread until it reaches an endemic state

βˆ’ Less than threshold: dies out

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Network epidemic– SIS model on scale-free network

Epidemic threshold βˆ’ πœ†πœ†π‘π‘ = π‘˜π‘˜

π‘˜π‘˜2

βˆ’ In large networks: πœ†πœ†π‘π‘ β†’ 0, 𝜏𝜏 β†’ 0 βˆ’ This is bad: even viruses that are hard to

pass from individual to individual can spread successfully!

βˆ’ Why? Hubs can broadcast a pathogen to a large number of other nodes!

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Comparison of the network epidemic models

Model Continuum Equation Ο„ Ξ»c

SI π‘‘π‘‘π‘–π‘–π‘˜π‘˜π‘‘π‘‘π‘‘π‘‘

= 𝛽𝛽 1 βˆ’ π‘–π‘–π‘˜π‘˜ π‘˜π‘˜πœƒπœƒπ‘˜π‘˜ π‘˜π‘˜

𝛽𝛽( π‘˜π‘˜2 βˆ’ π‘˜π‘˜ )

0

SIS π‘‘π‘‘π‘–π‘–π‘˜π‘˜π‘‘π‘‘π‘‘π‘‘

= 𝛽𝛽 1 βˆ’ π‘–π‘–π‘˜π‘˜ π‘˜π‘˜πœƒπœƒπ‘˜π‘˜ βˆ’ πœ‡πœ‡π‘–π‘–π‘˜π‘˜ π‘˜π‘˜

𝛽𝛽 π‘˜π‘˜2 βˆ’ πœ‡πœ‡ π‘˜π‘˜

π‘˜π‘˜π‘˜π‘˜2

SIR π‘‘π‘‘π‘–π‘–π‘˜π‘˜

𝑑𝑑𝑑𝑑= 𝛽𝛽 1 βˆ’ π‘–π‘–π‘˜π‘˜ π‘˜π‘˜πœƒπœƒπ‘˜π‘˜ βˆ’ πœ‡πœ‡π‘–π‘–π‘˜π‘˜ π‘ π‘ π‘˜π‘˜ = 1 βˆ’ 𝑖𝑖𝑙𝑙 βˆ’ π‘Ÿπ‘Ÿπ‘˜π‘˜

π‘˜π‘˜ 𝛽𝛽 π‘˜π‘˜2 βˆ’ (πœ‡πœ‡ + 𝛽𝛽) π‘˜π‘˜

1

π‘˜π‘˜2π‘˜π‘˜ βˆ’ 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


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