Download presentation

Presentation is loading. Please wait.

Published byMadeleine Everest Modified over 2 years ago

1
Interacting Viruses: Can Both Survive? Alex Beutel, B. Aditya Prakash, Roni Rosenfeld, Christos Faloutsos Carnegie Mellon University, USA KDD 2012, Beijing

2
Competing Contagions Beutel et. al. 2012 2 Firefox v ChromeBlockbuster v Hulu Biological common flu/avian flu

3
Outline Introduction Propagation Model Problem and Result Proof Sketch Simulations and Real Examples Implications and Subtleties Conclusions Beutel et. al. 2012 3

4
A simple model: SI 1|2 S Modified flu-like (SIS) Susceptible-Infected 1 or 2 -Susceptible Interaction Factor ε ▫Full Mutual Immunity: ε = 0 ▫Partial Mutual Immunity (competition): ε < 1 ▫Cooperation: ε > 1 Beutel et. al. 2012 4 Virus 1 Virus 2 &

5
Who-can-Influence-whom Graph Beutel et. al. 2012 5

6
Competing Viruses - Attacks Beutel et. al. 2012 6

7
Competing Viruses - Attacks Beutel et. al. 2012 7 All attacks are Independent

8
Competing Viruses - Cure Beutel et. al. 2012 8 Abandons Chrome Abandons Firefox

9
Competing Viruses Beutel et. al. 2012 9 εβ 2 &

10
Outline Introduction Propagation Model Problem and Result Proof Sketch Simulations and Real Examples Implications and Subtleties Conclusions Beutel et. al. 2012 10

11
Question: What happens in the end? Beutel et. al. 2012 11 ASSUME: Virus 1 is stronger than Virus 2 ε = 0 Winner 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? 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]

12
Answer: Yes! There is a phase transition Beutel et. al. 2012 12 ASSUME: Virus 1 is stronger than Virus 2

13
Answer: Yes! There is a phase transition Beutel et. al. 2012 13 ASSUME: Virus 1 is stronger than Virus 2

14
Answer: Yes! There is a phase transition Beutel et. al. 2012 14 ASSUME: Virus 1 is stronger than Virus 2

15
Our Result: Viruses can Co-exist Beutel et. al. 2012 15 Given our SI 1|2 S model and a fully connected graph, there exists an ε critical such that for ε ≥ ε critical, there is a fixed point where both viruses survive. 1.Virus 1 is stronger than Virus 2, if: strength(Virus 1) > strength(Virus 2) 2.Strength(Virus) σ = N β / δ In single virus models, threshold is σ ≥ 1

16
Proof Sketch [Details] View as dynamical system Define in terms of κ 1, κ 2, i 12 ▫κ 1 is fraction of population infected with virus 1 (κ 2 for virus 2) ▫i 12 is fraction of population infected with both Beutel et. al. 2012 16 κ1κ1 κ2κ2

17
Proof Sketch [Details] View as dynamical system Beutel et. al. 2012 17 Fixed point when New Infections Cured Infections κ1κ1 κ2κ2

18
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 I1I1 I2I2 I 1,2

19
Proof Sketch [Details] For co-existing fixed point, κ 1, κ 2, i 12 must be: 1.Real 2.Positive 3.Less than 1 Beutel et. al. 2012 19

20
Result Enforcing system constraints, we get: Beutel et. al. 2012 20 Again, there exists a valid fixed point for all ε ≥ ε critical

21
Outline Introduction Propagation Model Problem and Result Proof Sketch Simulations and Real Examples Implications and Subtleties Conclusions Beutel et. al. 2012 21

22
Simulation: σ 1 = 6, σ 2 = 4 Beutel et. al. 2012 22

23
Beutel et. al. 2012 23 Simulation: σ 1 = 6, σ 2 = 4, ε = 0.4

24
Beutel et. al. 2012 24 Simulation: σ 1 = 6, σ 2 = 4, ε = 0.4

25
Real Examples Beutel et. al. 2012 25 Hulu v Blockbuster [Google Search Trends data]

26
Real Examples Beutel et. al. 2012 26 Chrome v Firefox [Google Search Trends data]

27
Real Examples with Prediction Beutel et. al. 2012 27 Chrome v Firefox [Google Search Trends data]

28
Outline Introduction Propagation Model Problem and Result Proof Sketch Simulations and Real Examples Implications and Subtleties ▫Arbitrary Graphs ▫Cooperation Conclusions Beutel et. al. 2012 28

29
Arbitrary graphs? Beutel et. al. 2012 29 Equivalent to single-virus SIS model with strength εβ 2 /δ 2 Therefore, What if virus 1 is infinitely strong ( δ 1 =0 )?

30
Cooperation: ε > 1 Two Cases: ▫Piggyback σ 1 ≥ 1 > σ 2 : Strong virus helps weak virus survive ▫Teamwork 1 > σ 1 ≥ σ 2 : Two weak viruses help each other survive Beutel et. al. 2012 30

31
Cooperation: Piggyback Setting Beutel et. al. 2012 31 σ 1 = 3, σ 2 = 0.5 ε = 1 : Independent

32
Cooperation: Piggyback Setting Beutel et. al. 2012 32 σ 1 = 3, σ 2 = 0.5 ε = 3.5

33
Cooperation: Teamwork Setting Beutel et. al. 2012 33 σ 1 = 0.8, σ 2 = 0.6 ε = 1 : Independent

34
Cooperation: Teamwork Setting Beutel et. al. 2012 34 σ 1 = 0.8, σ 2 = 0.6 ε = 8

35
Outline Introduction Propagation Model Problem and Result Simulations Real Examples Proof Sketch Conclusions Beutel et. al. 2012 35

36
Conclusions Interacting Contagions (Chrome vs Firefox) ▫Flu-like model ▫Includes partial or full mutual immunity (competition) as well as cooperation Q: Can competing viruses co-exist? A: Yes Simulations and Case Studies on real data Beutel et. al. 2012 36

37
Any Questions? Alex Beutel abeutel@cs.cmu.edu http://alexbeutel.com 37

38
Proof Sketch View as dynamical system (Chrome vs Firefox) Beutel et. al. 2012 38 rate of change in κ 1 = 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 rate

39
Proof Sketch View as dynamical system 3 previously known fixed points: Beutel et. al. 2012 39 Both viruses die Virus 2 dies, virus 1 lives on alone Virus 1 dies, virus 2 lives on alone

40
Proof Sketch Just enforcing the constraint that the terms be positive, we get: Beutel et. al. 2012 40

41
A Qualitative Case Study: Sex Ed. Beutel et. al. 2012 41 Abstinence-Only EducationComprehensive Sex Education Virus 1: Sexual Activity Virus 2: Abstinence Pledge Virus 1: Sexual Activity Virus 2: Safe Sex Practices Sexually Inactive and Uneducated Sexually ActiveAbstinent Practices Unsafe Sex Believes in Safe Sex Practices Safe Sex Sexually Inactive and Uneducated

Similar presentations

OK

On the Spread of Viruses on the Internet Noam Berger Joint work with C. Borgs, J.T. Chayes and A. Saberi.

On the Spread of Viruses on the Internet Noam Berger Joint work with C. Borgs, J.T. Chayes and A. Saberi.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google