Leonid E. Zhukov...Leonid E. Zhukov (HSE) Lecture 13 22.04.2016 1 / 38 Lecture outline 1...
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Epidemics on networks
Leonid E. Zhukov
School of Data Analysis and Artificial IntelligenceDepartment of Computer Science
National Research University Higher School of Economics
Network Science
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Lecture outline
1 Probabilistic modelSI modelSIS modelSIR model
2 Simulations
3 Experimental results
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Probabilistic model
network of potential contacts (adjacency matrix A)
probabilistic model (state of a node):si (t) - probability that at t node i is susceptiblexi (t) - probability that at t node i is infectedri (t) - probability that at t node i is recovered
β - infection rate (probably to get infected on a contact in time δt)γ - recovery rate (probability to recover in a unit time δt)
from deterministic to probabilistic description
connected component - all nodes reachable
network is undirected (matrix A is symmetric)
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Probabilistic model
Two processes:
Node infection:
Pinf = si (t)
1−∏
j∈N (i)
(1− βxj(t)δt)
≈ βsi (t)∑
j∈N (i)
xj(t)δt
Node recovery:
Prec = γxi (t)δt
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SI model
SI ModelS −→ I
Probabilities that node i : si (t) - susceptible, xi (t) -infected at t
xi (t) + si (t) = 1
β - infection rate, probability to get infected in a unit time
xi (t + δt) = xi (t) + βsi∑j
Aijxjδt
infection equations
dxi (t)
dt= βsi (t)
∑j
Aijxj(t)
xi (t) + si (t) = 1
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SI model
Differential equation
dxi (t)
dt= β(1− xi (t))
∑j
Aijxj
early time approximation, t → 0, xi (t)� 1
dxi (t)
dt= β
∑j
Aijxj
dx(t)
dt= βAx(t)
Solution in the basisAvk = λkvk
x(t) =∑k
ak(t)vk
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SI model
∑k
dakdt
vk = β∑k
Aak(t)vk = β∑k
ak(t)λkvk
dak(t)
dt= βλkak(t)
ak(t) = ak(0)eβλk t , ak(0) = vTk x(0)
Solutionx(t) =
∑k
ak(0)eλkβtvk
t → 0, λmax = λ1 > λk
x(t) = v1eλ1βt
1 growth rate of infections depends on λ1
2 probability of infection of nodes depends on v1 , i.e v1i
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SI model
late- time approximation, t →∞, xi (t)→ const
dxi (t)
dt= β(1− xi (t))
∑j
Aijxj = 0
Ax 6= 0 since λmin 6= 0, 1− xi (t) ≈ 0
All nodes in connected component get infected t →∞ xi (t)→ 1
Connected component structure and distribution. Does initiallyinfected node belong to GCC?
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SI model
image from M. Newman, 2010
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SI simulation
1 Every node at any time step is in one state {S , I}2 Initialize c nodes in state I
3 On each time step each I node has a probability β to infect itsnearest neighbors (NN), S → I
Model dynamics:
I + Sβ−→ 2I
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SI model simulation
β = 0.5
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SI model simulation
β = 0.5
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SI model simulation
β = 0.5
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SI model simulation
β = 0.5
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SI model simulation
β = 0.5
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SI model simulation
β = 0.5
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SI model
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SIS model
SIS ModelS −→ I −→ S
Probabilites that node i : si (t) - susceptable, xi (t) -infected at t
xi (t) + si (t) = 1
β - infection rate, γ - recovery rate
infection equations:
dxi (t)
dt= βsi (t)
∑j
Aijxj(t)− γxi
xi (t) + si (t) = 1
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SIS model
Differential equation
dxi (t)
dt= β(1− xi (t))
∑j
Aijxj − γxi
early time approximation, xi (t)� 1
dxi (t)
dt= β
∑j
Aijxj − γxi
dxi (t)
dt= β
∑j
(Aij −γ
βδij)xj
dx(t)
dt= β(A− (
γ
β)I)x(t)
dx(t)
dt= βMx(t), M = A− (
γ
β)I
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SIS model
Eigenvector basis
Mv′k = λ′kv′k , M = A− (
γ
β)I, Avk = λkvk
v′k = vk , λ′k = λk −γ
β
Solution
x(t) =∑k
ak(t)v′k =∑k
ak(0)v′keλ′kβt =
∑k
ak(0)vke(βλk−γ)t
λ1 ≥ λk , critical: βλ1 = γ-if βλ1 > γ, x(t)→ v1e
(βλ1−γ)t - growth-if βλ1 < γ, x(t)→ 0 - decay
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SIS model
Epidemic threshold R0:
if βγ < R0 - infection dies over time
if βγ > R0 - infection survives and becomes epidemic
In SIS model:
R0 =1
λ1, Av1 = λ1v1
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SIS model
long time t →∞:
xi (t)→ const
dxi (t)
dt= β(1− xi )
∑j
Aijxj − γxi = 0
xi =
∑j Aijxj
γβ +
∑j Aijxj
Above the epidemic threshold (β/γ > R0)- if β � γ, xi (t)→ 1- if β ∼ γ, xi
γβ =
∑j Aijxj , then λ1 = γ
β , xi (t)→ (v1)i
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SIS simulation
1 Every node at any time step is in one state {S , I}2 Initialize c nodes in state I
3 Each node stays infected τγ =∫∞
0 τe−τγdτ = 1/γ time steps
4 On each time step each I node has a prabability β to infect itsnearest neighbours (NN), S → I
5 After τγ time steps node recovers, I → S
Model dynamics: {I + S
β−→ 2I
Iγ−→ S
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SIS model simulation
β = 0.5, τ = 2
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SIS model simulation
β = 0.5, τ = 2
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SIS model simulation
β = 0.5, τ = 2
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SIS model simulation
β = 0.5, τ = 2
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SIS model simulation
β = 0.5, τ = 2
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SIS model simulation
β = 0.5, τ = 2
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SIS model simulation
β = 0.5, τ = 2
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SIS model
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SIS model simulation
β = 0.2, τ = 2
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SIS model simulation
β = 0.2, τ = 2
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SIS model simulation
β = 0.2, τ = 2
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SIS model simulation
β = 0.2, τ = 2
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SIS model simulation
β = 0.2, τ = 2
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SIS model simulation
β = 0.2, τ = 2
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SIS model simulation
β = 0.2, τ = 2
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SIS model
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SIR model
SIR ModelS −→ I −→ R
probabilities si (t) -susceptable , xi (t) - infected, ri (t) - recovered
si (t) + xi (t) + ri (t) = 1
β - infection rate, γ - recovery rate
Infection equation:
dxidt
= βsi∑j
Aijxj − γxi
dridt
= γxi
xi (t) + si (t) + ri (t) = 1
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SIR model
Differential equation
dxi (t)
dt= β(1− ri − xi )
∑j
Aijxj − γxi
early time, t → 0, ri ∼ 0, SIS = SIR
dxi (t)
dt= β(1− xi )
∑j
Aijxj − γxi
Solutionx(t) ∼ v1e
(βλ1−γ)t
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SIR model
image from M. Newman, 2010
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SIR simulation
1 Every node at any time step is in one state {S , I ,R}2 Initialize c nodes in state I
3 Each node stays infected τγ = 1/γ time steps
4 On each time step each I node has a prabability β to infect itsnearest neighbours (NN), S → I
5 After τγ time steps node recovers, I → R
6 Nodes R do not participate in further infection propagation
Model dynamics: {I + S
β−→ 2I
Iγ−→ R
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SIR model
β = 0.5, τ = 2
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SIR model
β = 0.5, τ = 2
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SIR model
β = 0.5, τ = 2
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SIR model
β = 0.5, τ = 2
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SIR model
β = 0.5, τ = 2
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SIR model
β = 0.5, τ = 2
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SIR model
β = 0.5, τ = 2
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SIR model
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SIR model
β = 0.2, τ = 2
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SIR model
β = 0.2, τ = 2
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SIR model
β = 0.2, τ = 2
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SIR model
β = 0.2, τ = 2
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SIR model
β = 0.2, τ = 2
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SIR model
β = 0.2, τ = 2
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SIR model
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SIR epidemics as percolation
image from D. Easley and J. Kleinberg, 2010
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SIR epidemics as percolation
Bond percolation: only those nodes that are on a percolation path withinfected node will get infected
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5 Networks, SIR
Networks: 1) random, 2) lattice, 3) small world, 4) spatial, 5) scale-free
image from Keeling et al, 2005
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5 Networks, SIR
Networks: 1) random, 2) lattice, 3) small world, 4) spatial, 5) scale-free
Keeling et al, 2005
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Effective distance
J. Manitz, et.al. 2014, D. Brockman, D. Helbing, 2013
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Network synchronization, SIRS
Small-world network at different values of disorder parameter p
Kuperman et al, 2001
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Flu contagion
Infected - red, friends of infected - yellowN. Christakis, J. Fowler, 2010
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References
Epidemic outbreaks in complex heterogeneous networks. Y. Moreno,R. Pastor-Satorras, and A. Vespignani., Eur. Phys. J. B 26, 521?529,2002.
Networks and Epidemics Models. Matt. J. Keeling and Ken.T.D.Eames, J. R. Soc. Interfac, 2, 295-307, 2005
Simulations of infections diseases on networks. G. Witten and G.Poulter. Computers in Biology and Medicine, Vol 37, No. 2, pp195-205, 2007
Small World Effect in an Epidemiological Model. M. Kuperman andG. Abramson, Phys. Rev. Lett., Vol 86, No 13, pp 2909-2912, 2001
Manitz J, Kneib T, Schlather M, Helbing D, Brockmann D. OriginDetection During Food-borne Disease Outbreaks - A Case Study ofthe 2011 EHEC/HUS Outbreak in Germany. PLoS Currents, 2014
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