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

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

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

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 &

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

Competing Viruses - Attacks Beutel et. al

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

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

Competing Viruses Beutel et. al εβ 2 &

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

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]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 )?

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

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

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

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

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

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

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

Any Questions? Alex Beutel 37

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

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

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

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