Post on 21-Jan-2016
description
Two Species Model of Epidemic Spreading
Ahn, Yong-Yeol. 2005.4.14
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Epidemics and Contact Process
• Epidemics is about self-replicators which can die.
• Contact process : A simple model consists of branching and spontaneous annihilation.
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Contact Process
1
λ
v
λ =ν/ δ
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Transition of C. P. (1d)
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Contact Process
iiidt
di)1(
Approx. Malthus–Verhulst eq.
For λ < 0 infection vanishes.For λ > 0 steady state ( i = 1 – 1/ λ)
This equation is “wrong”, but correctly describes the qualitative dynamics of contact process, which is that…
Contact process exhibits an absorbing phase transition.
We can NOT solve exactly this terribly simple process!
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Absorbing Phase Transition
• Order parameter
• Correlation length
• Correlation time
ρ
p
activeinactive
cpp :
x
cx pp :
t
ct pp :
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Directed Percolation
• Contact process is same as the directed percolation process.
• In the lattice with randomly occupied edges,
• Follow only occupied edges in only designated direction.
• This process can continue ad infinitum?
Directed Percolation problem
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Surface Roughening Model and DP
XNo spontaneous infection no digging condition
- Let’s see only one layer
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S. R. and Coupled DP
A A A A A
B
D D
C
B B B B B B B B
C C C C C C C CCC
D D D D D D D D DDD
C
(RSOS : Restricted solid-on-solid interface)
active
inactive
B
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Asymmetrically Coupled DP
A B C D ..
Positive coupling
Simulation results in 1D
A : DP critical behavior B, C, … : dressed negative coupling is irrelevant
Negative coupling
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ACDP : Diagram
BA
AntigenContagion
AntibodyWorm killing worm
Detection and initiation
Healing
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ACDP on Networks
• Antigen – antibody interaction• Worm – worm killing worm
• Motto:
“Never flood the network”
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Phase Diagram
• If there is no virus, vaccine spreading has a epidemic threshold.
• If there is no vaccine, virus has no epidemic threshold.
Ap
0
0A
B
Bp
0
0A
B
0
0A
B
Ma
c
b
There is the happy region where there is no virus and no vaccine.
Vaccine Annihilation probability
Virus Annihilation probability
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Mean Field Theory
ABBBB
BAAAAA
a
a
~
,~
2
2
Aa
Ba
0A B
0
/A
B B Ba b
0
0A
B
M
DP
DP
DP
2, 1/ 2, 1
1, DP1/ 2, 1A xA tA
B xB tB
At ‘M’
Mean field eq.
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On a Scale-Free Network
• We should consider each degree seperately.
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SIS Model on a Scale-free Network
))(())(1()()(
titiktit
tikk
k
Because a scale-free network is very inhomogenous, we must consider the density of infected nodes seperately for each degree.
))(( ti is the probability that any given link points to an infected node.
(We don’t need to consider the excess degree because we’re dealing with SIS model. )
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SIS Model on a Scale-free Network
))(())(1()()(
titiktit
tikk
k
))(( ti
The probability that the selected node is not infected
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SIS Model on a Scale-free Network
At Steady state,
Where Average degree
Edge selection
With these two equations and continum approximation,
We get
ρk = ik
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SIS Model on a Scale-free Network
• By solving previous equation, we get, “no epidemic threshold
on the scale-free network”
Homogeneous network
Scale-freeVanishing
Epidemic threshold
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CDP on a Scale-free Network
spontaneous annihilationwith prob. Ap
A
(1 )
infecting or activating vaccine at “all neighbors” with prob. 1 Ap
B
branching “one offspring”with prob. 1 Bp
spontaneous annihilationwith prob. Bp
From J.D. Noh’s ppt
predation
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Different Branching Rule
• Because we do not want the vaccine occupying the whole network and vaccine is more active in the condition that virus is abundant, we assign a different rule for the vaccine.
• Both rules are possible in our CDP framework
Previous SIS rule Modified rule
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On a Scale-Free Network
kk
Bk
kk
Ak
BkkB
AkkAkBk
AkA
BkkBkAk
k
kkPykkPx
kypkypypy
kpxpxpx
)|( ,)|(
where,
)1)(1()1()1(
)1)(1()1(
Mean field equation
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When There is no Virus
kk
Bk
kk
Ak
BkkB
AkkAkBk
AkA
BkkBkAk
k
kkPykkPx
kypkypypy
kpxpxpx
)|( ,)|(
where,
)1)(1()1()1(
)1)(1()1(
' '''
)'(
'
)|'(
k kkk
Bk k
kPy
k
kkPy
Without degree-degree correlation,
Mean field eq.
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Mean Field eq.
'
' )ky/)'(
(where )1)(1(k
kBkkBkBk k
kPyy
k
kypypy
By assuming a steady state, multiplying p(k), and summing over k, we get a equation
kp
pky
B
Bk 1
21
Note that the critical point pB=½.
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MF eq.
• Calculating with 2<γ<3,
) ,1
(where
,1
)( 1
2
kuk
y
p
pu
uuPduky
B
B
k
kk
m
And by direct calculation,
1
)21(~ Bpy pc
As γ decreases, β increases.
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MF eq.
• Calculating with γ=3 β = 1 with logarithmic correction
• Calculating with γ>3, β = 1
γ2 3
β = 1/(γ-2) β = 1
β = 1 w/ log. correction
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Diffusion – Annihilation Process
'' )
)'( (where )1)(1(
kk
Bk
BkkBkBk k
kPykypypy
Out CDP eq.
by multiflying P(k) and summing over k,
2/1Bp
k
kyypypypy kBBB )1()1(
When ,
k
kyyy k
2
1
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Diffusion – Annihilation Process
),)](21[)()(
ttk
kt
dt
tdkk
k
))(2)(
ttdt
td
And by summing over P(k),
Meanwhile, diffusion-annihilation process can be described as,
k
kyyy k
2
1
Same eq.
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DA Process and Time Evolution
• At critical point, DA process and our model is described by the same mf eq. And we apply the DA result to our model.
γ2 3
tt
1~)(
ttt
ln
1~)(
1
~)( tt
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Simulation Result
Critical point = 0.425 (mean field : 0.5)
δ = 1.3 (mf: 1 with log. correction )
γ =3
δ=1.3
If we draw with logarithmic term, δ ~ 1.1
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Finite Size Scaling
p
p
pppp
c
c
~
)( ~
)(
)( ~),(
LL
LpLt
Assuming a scaling form,
LL
LpLpfLLp
:
: )(),(
( θ=β/ ν )
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Finite Size Scaling
β =1.3
γ =3
θ~β/ ν
Mf: β = 1 with log. correction
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Another γ s
• γ =2.5 Critical point = 0.425 (mean field : 0.5)
δ = 2.0 (mf: 1/(γ-2) = 2 )
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Another γ s
• γ =2.5
β = 1.75 (mf: 2)
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Another γ s
• γ =2.75, 4
δ ~ 1.5
pc = 0.425 pc = 0.426
δ ~ 1.3
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Summary : δ
γ2 3 1
cor.) log.(w/ 1
1
~1.3 (mf: 2)
~1.3 (1.1 w l.c.) (mf:1 w lc)
2 (mf: 2)
2.5 2.75 4
~1.5(mf:1.33)
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Summary : β
γ2 3β = 1/(γ-2) β = 1
β = 1 w/ log. correction
~1.3
1.75(mf:2)
2.5 2.75
Not yet Not yet
4
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Summary
• We put the CDP model on a scale-free network for simulating the behavior of an epidemic and reacting anti-pathogen.
• The simulation results are consistent to the mean field calculation.
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References
• U. C. Täuber et al., Phys. Rev. Lett. 80, 2165 (1998); Y. Y. Goldschmidt, ibid. 81, 2178 (1998).
• J. D. Noh and H. Park, CondMat/051542 (to be published in Phys. Rev. Lett.).
• J. Marro and R. Dickman, “Nonequilibrium Phase Transition in Lattice Models”.
• Papers in http://janice.kaist.ac.kr/bbs/viewtopic.php?t=30 and related papers.
• Haye Hinrichsen, “Nonequilibrium Critical Phenomena and Phase Transitions into Absorbing States”, arXiv:cond-mat/0001070.
• Arno Karlen, “Man and Microbes”.• J. D. Murray, “Mathematical Biology”.• M. E. J. Newman, SIAM Review, 45, 167 (2003).• Malcolm Gladwell, “The Tipping Point”.• About percolation : 손승우 , http://stat.kaist.ac.kr/~sonswoo/prl.html .