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Mean-field theories and quasi-stationarysimulations of epidemic models on complex
networks
Angelica S. Mata, Silvio C. FerreiraDepartamento de Fısica - Universidade Federal de Vicosa
[email protected] Systems Foundations and Applications
Rio de Janeiro, October 29th - November 1st
Outline
IntroductionPair quenched mean-field theoryQuasi-stationary simulationsNature of epidemics on complex networksProspects
Vicosa - Minas Gerais
Universidade Federal de Vicosa
How do we get sick?
Network
Ravasz et al., Science 297 1551 (2002).
Basic network properties
32
1
4
k
log
P(k
)
log k
ij
lij = 3
Small-world (SW): <l>~log N
Scale-free (SF): P(k)~k-γ
〈k2〉 → ∞ for 2 < γ < 3 - Scale-free
Epidemic dynamicsStates: Susceptible (S), Infected (I), Removed (R) and so forth . . .
S
R
I
The susceptible-infected-susceptible (SIS) model
n
INFECTION
1
CURE
Absorbing-state and phase transitions
λ
ρ
ActiveAbsorbing
λc
Competing mean-field theories
Heterogeneous mean-fieldtheory (HMF)[Pastor-Satorras and Vepignani, PRL 863200 (2001)]
k=5 k=3
dρk
dt= −ρk + λk(1− ρk )
∑k ′
P(k ′|k)ρ′k ⇒ λc =〈k〉〈k2〉
Quenched mean-field theory (QMF)[Chakrabarti at al., ACM Trans. Inf. Syst. Secur. 10 1 (2008)]
dρi
dt= −ρi + λ(1− ρi )
∑j
Aijρj ⇒ λc =1
Λm,
where Λm is the largest eigenvalue of the adjacency matrix Ai,j
Thresholds for P(k) ∼ k−γ
HMFλc =
〈k〉〈k2〉
→{
const . > 0 γ > 30 γ ≤ 3
Conclusion: Scale-free networks (γ < 3) do not have a threshold inthe thermodynamical limit (N →∞) while networks with finite 〈k2〉 do.
QMF [Castallano and Pastor-Satorrras, PRL 105, 218701 (2010)]
λc =1
Λm'{
1/√
kmax γ > 2.5〈k〉/〈k2〉 2 < γ ≤ 2.5
Conclusion: does not exist a finite threshold for SIS in the limitN →∞ for any network with diverging cutoff.
Quenched pair-approximation
Mata and Ferreira, EPL 103 48003 (2013))
Simple QMF theory assumes that the probability that a vertex isoccupied or empty does not depend on the states of itsneighbors.
φi,j ≈ (1− ρi )ρj
Pair approximation is the simplest MF theory that includesdynamical correlations.
Dynamical equation for a pair of vertices are investigated.
Notation and normalization conditions
Infected vertex→ 1Heath vertex→ 0
ρi = [1i ]
[0i ] = 1− ρi
ψij = [1i ,1j ]
ωij = [0i ,0j ]
φij = [0i ,1j ]
φij = [1i ,0j ]
ψij = ψji
ωij = ωji
φij = φji
ψij + φij = ρj
ψij + φij = ρi
ωij + φij = 1− ρi
ωij + φij = 1− ρj
Dynamical equations
dρi
dt= −ρi + λ
∑j
φijAij
dφij
dt= −φij − λφij + ψij + λ
∑l∈N (j)
l 6=i
[0i0j1l ]− λ∑
l∈N (i)l 6=j
[1l0i1j ]
Pair approximation [ben Avraham and Kohler, PRA 45, 8358 (1992)]
[Ai ,Bj ,Cl ] ≈[Ai ,Bj ][Bj ,Cl ]
[Bj ]
dφij
dt= −(1 + λ)φij +ψij + λ
∑l
ωijφjl
1− ρj(Ajl − δil )− λ
∑l
φij φli
1− ρi(Ail − δlj )
Thresholds
Performing a linear stability analysis around the fixed pointρi = φij = ψij = 0 and using a quasi-static approximation forlong times we find
dρi
dt=∑
j
Lijρj
with the Jacobian
Lij = −(
1 +λ2ki
2λ+ 2
)δij +
λ(2 + λ)
2λ+ 2Aij .
The critical point is obtained when absorbing state ρi = 0 losesstability or, equivalently, when the largest eigenvalue of Lijvanishes.
Analytical results for simple networks
(c)(b)(a)
Absorbing states and finite sizes
The absorbing configuration is the unique actual stationary statefor finite size systems.
Quasi-stationary approach: the averages are restricted to anensemble of active configurations.
P(σ) =P(σ, t)Ps(t)
, t →∞
Quasi-stationary simulations
Oliveira and Dickman, PRE 71 016129 (2005)
QS simulations in complex networks
Finite-size effect are strongly enhanced by the small-worldproperty.QS state is a suitable framework to handle absorbing states oncomplex networks:Contact processFerreira, Ferreira, Pastor-Satorras PRE 83, 066113 (2011)Ferreira, Ferreira, Castellano, Pastor-Satorras PRE 84, 066102 (2011)Ferreira and Ferreira EPJB (2013) [arXiv:1307.6186 (2013)]
Epidemic spreadingFerreira, Castellano, Pastor-Satorras et al. PRE 86 041125 (2012)Mata and Ferreira EPL 103 48003 (2013)
Infinitely many absorbing statesSander, Ferreira, Pastor-Satorras et al. PRE 87, 022820 (2013)
Metapopulation dynamicsMata,Ferreira, Pastor-Satorras PRE 88 042820 (2013).
Numerical determination of the thresholds
Ferreira, Castellano, Pastor-Satorras, PRE 86 041125 (2012).
Modified susceptibility:
χ = N〈ρ2〉 − 〈ρ〉2
〈ρ〉∼ Nϑ, ϑ > 0
Validating the method: annealed networks
101
102
103
χ
104
4 x 104
16 x 104
64 x 104
256 x 104
1024 x 104
a)
Thresholds for annealed networks
103
104
105
106
107
N
10-3
10-2
10-1
λp(N) [γ=2.25]
λc
HMF [γ=2.25]
λp(N) [γ=3.50]
λc
HMF [γ=3.50]
b)
Simple homogeneous case: RRN
0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28
λ
100
101
102
103
104
χ
N=103
N=104
N=105
N=106
N=107
103
104
105
106
107
N
0.19
0.20
0.21
0.22
λp
λc
qmfλc
pqmf
Star/Wheel graphs
10-4
10-3
10-2
10-1
λ
101
102
103
104
χ
N = 103
N = 104
N = 105
N = 106
N = 107
103
104
105
106
107
N
10-4
10-3
10-2
10-1
thre
shold
λp(N)
λc
qmf
λc
pqmf
Quenched networks: γ < 2.5
103
104
105
106
107
N
10-3
10-2
10-1
100
threshold
HMFQMFPQMF
λp
10-2
10-110
0
101
102
103
χ
103
104
105
106
107
N
-0.03
-0.02
-0.01
0
λ−
λp
λ
Quenched networks: 2.5 < γ < 3
103
104
105
106
107
N
10-2
10-1
threshold
HMFQMFPQMF
λp
(2/kmax
)0.5
10-110
0
101
102
χ
104
105
106
107
108
N
-0.02
-0.01
0
0.01
0.02
λ−
λp
λ
Quenched networks: γ > 3 and kmax = 〈kmax〉
103
104
105
106
107
108
N
10-2
10-1
100
threshold
HMFQMFPQMF
λp
(2/kmax
)0.5
10-2
10-1
λ
101
102
χ
103
104
105
10610
7
108
3x107
The nature of the epidemic thresholds
The peak at low infection rate reproduced by the PQMF theory isassociated to a localized epidemics.
Goltsev et al., PRL 109 128702 (2012).
The second peak if associated to an endemic phase where allvertices are infected at some time.
Boguna, Castellano, Pastor-Satorras PRL 111, 068701 (2013).
Lifespan method [B,C,P-S, op cit.]
0 0.05 0.1 0.15 0.2 0.25
λ
100
101
102
103
104
105
106
107
τ
103
3x103
104
3x104
105
3x105
106
3x106
107
3x107
108
103
104
105
106
107
108
N
10-2
10-1
100
λc
χτ
Prospects
Acknowledgments
CollaboratorsStudentsApplied statistical physics groupGISCSupporting agencies
