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## Two Species Model of Epidemic Spreading

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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

We can NOT solve exactly this terribly simple process!

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.

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

Surface Roughening Model and DP

- Let’s see only one layer

No spontaneous infection no digging condition

X

S. R. and Coupled DP

(RSOS : Restricted solid-on-solid interface)

A

A

A

A

A

B

B

B

B

B

B

B

B

B

B

C

C

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C

C

D

D

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D

D

D

active

inactive

Asymmetrically Coupled DP

Positive coupling

A

B

C

D

..

Negative coupling

- Simulation results in 1D
- A : DP critical behavior
- B, C, … : dressed
- negative coupling is irrelevant

ACDP on Networks

- Antigen – antibody interaction
- Worm – worm killing worm
- Motto:

“Never flood the network”

Phase Diagram

- If there is no virus, vaccine spreading has a epidemic threshold.
- If there is no vaccine, virus has no epidemic threshold.

Vaccine Annihilation probability

There is the happy region where there is no virus and no vaccine.

Virus Annihilation probability

On a Scale-Free Network

- We should consider each degree seperately.

SIS Model on a Scale-free Network

Because a scale-free network is very inhomogenous, we must consider the density of infected nodes seperately for each degree.

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. )

SIS Model on a Scale-free Network

The probability that the selected node is not infected

SIS Model on a Scale-free Network

ρk = ik

At Steady state,

Edge selection

Average degree

Where

With these two equations and continum approximation,

We get

SIS Model on a Scale-free Network

- By solving previous equation, we get,

“no epidemic threshold

on the scale-free network”

Scale-free

Vanishing

Epidemic threshold

Homogeneous network

infecting or activating vaccine

at “all neighbors” with prob.

spontaneous annihilation

with prob.

predation

spontaneous annihilation

with prob.

branching“one offspring”

with prob.

CDP on a Scale-free NetworkFrom

J.D. Noh’s ppt

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

Modified rule

Previous SIS rule

On a Scale-Free Network

Mean field equation

Mean Field eq.

By assuming a steady state, multiplying p(k), and summing over k, we get a equation

Note that the critical point pB=½.

MF eq.

- Calculating with γ=3 β = 1 with logarithmic correction
- Calculating with γ>3, β = 1

β = 1 w/ log. correction

β = 1/(γ-2)

β = 1

γ

3

2

Diffusion – Annihilation Process

Meanwhile, diffusion-annihilation process can be described as,

Same eq.

And by summing over P(k),

3

2

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.

Simulation Result

γ =3

Critical point = 0.425 (mean field : 0.5)

δ = 1.3 (mf: 1 with log. correction )

δ=1.3

If we draw with logarithmic term, δ ~ 1.1

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.

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 .

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