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Interacting Viruses: Interacting Viruses:

Interacting Viruses: - PowerPoint Presentation

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Interacting Viruses: - PPT Presentation

Can Both Survive Alex Beutel B Aditya Prakash Roni Rosenfeld Christos Faloutsos Carnegie Mellon University USA KDD 2012 Beijing Competing Contagions Beutel et al 2012 2 ID: 198004

2012 beutel viruses virus beutel 2012 virus viruses chrome real sketch proof firefox competing sex rate fixed modelproblem cooperation

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Slide1

Interacting Viruses:Can Both Survive?

Alex Beutel, B. Aditya Prakash, Roni Rosenfeld, Christos Faloutsos

Carnegie Mellon University, USAKDD 2012, BeijingSlide2

Competing ContagionsBeutel et. al. 2012

2Firefox v Chrome

Blockbuster v Hulu

Biological

common flu/avian fluSlide3

OutlineIntroduction

Propagation ModelProblem and ResultProof SketchSimulations and Real ExamplesImplications and Subtleties ConclusionsBeutel et. al. 2012

3Slide4

A simple model: SI1|2S

Modified flu-like (SIS) Susceptible-Infected1 or 2-SusceptibleInteraction Factor εFull Mutual Immunity:

ε = 0Partial Mutual Immunity (competition): ε

<

1

Cooperation:

ε

> 1

Beutel et. al. 2012

4

Virus 1

Virus 2

&Slide5

Who-can-Influence-whom GraphBeutel et. al. 2012

5Slide6

Competing Viruses - AttacksBeutel et. al. 2012

6Slide7

Competing Viruses - AttacksBeutel et. al. 2012

7

All attacks are IndependentSlide8

Competing Viruses - CureBeutel et. al. 2012

8

Abandons Chrome

Abandons

FirefoxSlide9

Competing VirusesBeutel et. al. 2012

9

ε

β

2

&Slide10

OutlineIntroduction

Propagation ModelProblem and ResultProof SketchSimulations and Real ExamplesImplications and Subtleties ConclusionsBeutel et. al. 2012

10Slide11

Question: What happens in the end?

Beutel et. al. 201211ASSUME: Virus 1 is stronger than Virus 2

ε = 0Winner takes all

ε

= 1

Co-exist independently

ε

= 2

Viruses cooperate

What about for

0 <

ε

<1

?

Is there a point at which both viruses can

co-exist

?

Clique: [Castillo-Chavez+ 1996]

Arbitrary Graph: [

Prakash

+ 2011]Slide12

Answer: Yes!

There is a phase transitionBeutel et. al. 2012

12ASSUME: Virus 1 is stronger than Virus 2Slide13

Answer: Yes! There is a phase transition

Beutel et. al. 201213ASSUME: Virus 1 is stronger than Virus 2Slide14

Answer: Yes! There is a phase transition

Beutel et. al. 201214ASSUME: Virus 1 is stronger than Virus 2Slide15

Our Result: Viruses can Co-existBeutel et. al. 2012

15Given our SI1|2

S model and a fully connected graph, there exists an εcritical such that for ε ≥ εcritical, there is a fixed point where both viruses survive.

Virus 1 is stronger than Virus 2, if:

strength(Virus 1) > strength(Virus 2)

Strength(Virus)

σ

=

N

β

/

δIn single virus models, threshold is σ

≥ 1

DetailsSlide16

Proof Sketch [Details]View as dynamical system

Define in terms of κ1, κ2, i12κ1 is fraction of population infected with virus 1 (

κ2 for virus 2)i12 is fraction of population infected with bothBeutel et. al. 2012

16

κ

1

κ

2Slide17

Proof Sketch [Details]View as dynamical system

Beutel et. al. 201217

Fixed point when

New Infections

Cured Infections

κ

1

κ

2Slide18

Proof Sketch [Details]

3 previously known fixed points:Beutel et. al. 2012

18

Both viruses die

Virus 2 dies, virus 1 lives on alone

Virus 1 dies, virus 2 lives on alone

I

1

I

2

I

1,2Slide19

Proof Sketch [Details]

For co-existing fixed point, κ1, κ2, i12 must be:RealPositive

Less than 1Beutel et. al. 201219Slide20

Result

Enforcing system constraints, we get:Beutel et. al. 2012

20

Again, there exists a valid fixed point

for all

ε

ε

critical

Slide21

OutlineIntroduction

Propagation ModelProblem and ResultProof SketchSimulations and Real ExamplesImplications and Subtleties Conclusions

Beutel et. al. 201221Slide22

Simulation: σ1

= 6, σ2 = 4Beutel et. al. 2012

22Slide23

Beutel et. al. 2012

23Simulation: σ

1 = 6, σ2 = 4, ε

= 0.4Slide24

Beutel et. al. 2012

24Simulation: σ1

= 6, σ2 = 4, ε

= 0.4Slide25

Real Examples

Beutel et. al. 2012

25Hulu v

Blockbuster

[Google Search Trends data]Slide26

Real Examples

Beutel et. al. 2012

26Chrome v

Firefox

[Google Search Trends data]Slide27

Real Examples with Prediction

Beutel et. al. 2012

27Chrome v

Firefox

[Google Search Trends data]Slide28

Outline

IntroductionPropagation ModelProblem and ResultProof SketchSimulations and Real ExamplesImplications and Subtleties Arbitrary

GraphsCooperationConclusionsBeutel et. al. 2012

28Slide29

Arbitrary graphs?Beutel et. al. 2012

29

Equivalent to single-virus SIS model

with strength

ε

β

2

2

Therefore,

What if virus 1 is infinitely strong (

δ

1

=0

)?Slide30

Cooperation: ε

> 1Two Cases:Piggyback σ1 ≥ 1 > σ2 : Strong virus helps weak virus surviveTeamwork 1 > σ1

≥ σ2 : Two weak viruses help each other surviveBeutel et. al. 2012

30Slide31

Cooperation: Piggyback Setting

Beutel et. al. 201231

σ

1

= 3

,

σ

2

= 0.5

ε

= 1 : IndependentSlide32

Cooperation: Piggyback Setting

Beutel et. al. 201232

σ

1

= 3

,

σ

2

= 0.5

ε

= 3.5Slide33

Cooperation: Teamwork Setting

Beutel et. al. 201233

σ

1

= 0.8, σ

2

= 0.6

ε

= 1 : IndependentSlide34

Cooperation: Teamwork Setting

Beutel et. al. 201234

σ

1

= 0.8, σ

2

= 0.6

ε

= 8Slide35

OutlineIntroduction

Propagation ModelProblem and ResultSimulationsReal ExamplesProof SketchConclusions

Beutel et. al. 201235Slide36

Conclusions

Interacting Contagions (Chrome vs Firefox)Flu-like modelIncludes partial or full mutual immunity (competition) as well as cooperationQ: Can competing viruses co-exist? A: YesSimulations and Case Studies on real data

Beutel et. al. 201236Slide37

Any Questions?

Alex Beutelabeutel@cs.cmu.eduhttp://alexbeutel.com37Slide38

Proof Sketch

View as dynamical system (Chrome vs Firefox)Beutel et. al. 2012

38rate of change in κ1 =

rate of new additions

rate of people leaving

rate of new additions

=

current Chrome users

x available susceptibles

x

transmissibility

+

current

Chrome

users

x

current Firefox users

x

ε

x transmissibility

rate people leaving = current Chrome users x curing rateSlide39

Proof SketchView as dynamical system

3 previously known fixed points:Beutel et. al. 201239

Both viruses die

Virus 2 dies, virus 1 lives on alone

Virus 1 dies, virus 2 lives on aloneSlide40

Proof Sketch

Just enforcing the constraint that the terms be positive, we get:Beutel et. al. 201240Slide41

A Qualitative Case Study: Sex Ed.Beutel et. al. 2012

41

Abstinence-Only Education

Comprehensive Sex Education

Virus 1: Sexual Activity

Virus 2: Abstinence Pledge

Virus 1: Sexual Activity

Virus 2: Safe Sex Practices

Sexually Inactive and Uneducated

Sexually Active

Abstinent

Practices Unsafe Sex

Believes in Safe Sex

Practices Safe Sex

Sexually Inactive and Uneducated