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
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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
/δ
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