Two Species Model of Epidemic Spreading

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Two Species Model of Epidemic Spreading Ahn, Yong-Yeol. 2005.4.14

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Two Species Model of Epidemic Spreading. Ahn, Yong-Yeol. 2005.4.14. Epidemics and Contact Process. Epidemics is about self-replicators which can die . Contact process : A simple model consists of branching and spontaneous annihilation. Contact Process. λ. λ = ν / δ. 1. v. - PowerPoint PPT Presentation

Transcript of Two Species Model of Epidemic Spreading

Page 1: Two Species Model of  Epidemic Spreading

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 .