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

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

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

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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 Virus 1 Virus 2 &

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

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Competing Viruses - Attacks Beutel et. al

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

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

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Competing Viruses Beutel et. al εβ 2 &

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

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Question: What happens in the end? Beutel et. al 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]

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

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

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

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Our Result: Viruses can Co-exist Beutel et. al 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

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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 κ1κ1 κ2κ2

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

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Proof Sketch [Details] 3 previously known fixed points: Beutel et. al Both viruses die Virus 2 dies, virus 1 lives on alone Virus 1 dies, virus 2 lives on alone I1I1 I2I2 I 1,2

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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

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

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

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

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

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

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

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

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

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

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

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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

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

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

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

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

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

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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

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Any Questions? Alex Beutel 37

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

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

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

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

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